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Radial Turbine Thermo-Mechanical Stress Optimization by Multidisciplinary Discrete Adjoint Method

Radial Turbine Thermo-Mechanical Stress Optimization by Multidisciplinary Discrete Adjoint Method International Journal of Turbomachinery Propulsion and Power Article Radial Turbine Thermo-Mechanical Stress Optimization by Multidisciplinary Discrete Adjoint Method 1 , 2 , 3 , 1 2 Alberto Racca * , Tom Verstraete and Lorenzo Casalino Turbomachinery and Propulsion Department, von Karman Institute for Fluid Dynamics, Chaussée de Waterloo 72, 1640 Rhode-St.-Genèse, Belgium; tom.verstraete@vki.ac.be Mechanical and Aerospace Engineering Department, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy; lorenzo.casalino@polito.it Advanced Engineering Department, PUNCH Torino S.p.A., Corso Castelfidardo 36, 10129 Torino, Italy * Correspondence: alberto.racca@polito.it Received: 31 August 2020; Accepted: 24 November 2020; Published: 25 November 2020 Abstract: This paper addresses the problem of the design optimization of turbomachinery components under thermo-mechanical constraints, with focus on a radial turbine impeller for turbocharger applications. Typically, turbine components operate at high temperatures and are exposed to important thermal gradients, leading to thermal stresses. Dealing with such structural requirements necessitates the optimization algorithms to operate a coupling between fluid and structural solvers that is computationally intensive. To reduce the cost during the optimization, a novel multiphysics gradient-based approach is developed in this work, integrating a Conjugate Heat Transfer procedure by means of a partitioned coupling technique. The discrete adjoint framework allows for the ecient computation of the gradients of the thermo-mechanical constraint with respect to a large number of design variables. The contribution of the thermal strains to the sensitivities of the cost function extends the multidisciplinary outlook of the optimization and the accuracy of its predictions, with the aim of reducing the empirical safety factors applied to the design process. Finally, a turbine impeller is analyzed in a demanding operative condition and the gradient information results in a perturbation of the grid coordinates, reducing the stresses at the rotor back-plate, as a demonstration of the suitability of the presented method. Keywords: radial turbine; fluid–structure interaction; adjoint method; conjugate heat transfer; thermo-mechanical stress; partitioned coupling algorithm 1. Introduction Multidisciplinary optimization methods are widely adopted in the development cycle of turbomachinery-related technologies, such as in the energy and mobility businesses [1–3]. The increasing complexity of the products in terms of geometry and flow characteristics in design and o -design conditions, along with the fulfillment of requirements of lifetime, cost, and packaging, is reflected in the growth of concurrent design techniques that provide the experts with a comprehensive view on the problem [4,5]. The exploration of the design space is enriched by multiple physical disciplines whose interactions are captured along the optimization, resulting in accurate predictions of the component performance and any possible violation of the structural constraints. This holistic approach to the design problem di ers from classical staggered methods, presenting a series of mono-discipline optimizations performed in cascade, leading typically to significantly longer development times. Moreover, such staggered methods may not converge to the same optimum as the holistic one and thus lead to suboptimal designs. Int. J. Turbomach. Propuls. Power 2020, 5, 30; doi:10.3390/ijtpp5040030 www.mdpi.com/journal/ijtpp Int. J. Turbomach. Propuls. Power 2020, 5, 30 2 of 31 1.1. Background The recent evolution of the market request towards aggressive reductions in emissions and fuel consumption o ers new challenges to the research community. The need for further advances in multidisciplinary design optimization techniques is presented in the NASA 2040 roadmap [6] as one of nine core development areas in the context of integrated multiscale modelling frameworks contributing to the success in ecient design, manufacturing, and certification of future aerospace systems. Similarly, the engineering of advanced turbocharging solutions in the automotive sector demands further increases in eciency to support novel combustion strategies, with no compromise on the durability targets [7,8]. Enhancements in turbomachinery components performance can be achieved by increasing the detail in the geometrical representation of the domain and reflecting it in a higher number of parameters engaged during the optimization process. This higher resolution allows the exploration of more complex shapes and discloses hidden interactions among features now present in the design space. Additionally, an increased fidelity in the physical description of the multidisciplinary problem is highly regarded and the aim is twofold. First, the optimizer is able to make more reliable predictions of the responses to the geometrical modifications, having access to lumped information accounting for the interactions of more disciplines involved in the computation of the objective functions and constraints. Therefore, the evolution of the optimization benefits from adhering more to the real behavior of the component, finally resulting in a reduced amount of experimental validation testing. Second, a more sophisticated physical description opens the path to adopting a richer set of constraints, and the safety margins usually applied to the design process can be relaxed in favor of more performance-oriented layout choices. Therefore, the optimizer can target more diversified paths towards the achievement of the desired solution, thus reducing the sti ness of the algorithm in performing the shape modifications. The increased complexity in the models is reflected in the choice of a suitable optimization method between two major classes, notably denoted as gradient-free and gradient-based techniques. Gradient-free methods [9], also known as “zero-order” methods, solely make use of the function values for the search of the global optimum, neglecting any knowledge of its gradients. The solution is identified after the evaluation of the output values of a large population of candidates with characteristics scattered within the boundaries of the selected design space. The larger the population, the higher the likelihood of identifying the global optimum. In the case of a multidisciplinary problem, each sample in the population must be processed through all the disciplines (FEM and FVM analyses, among others) [10]. Therefore, the number of computations necessary to identify a solution scales up with the number of involved disciplines and the number of design variables necessary to investigate the full domain, resulting quickly in a computationally expensive approach [11,12]. Even if, nowadays, gradient-free methods still represent the state-of-the-art approach to optimizations in industry because of their robustness and the high quality of the solutions, the advent of more stringent requirements and the need of evaluating larger domains are exposing an intrinsic limitation, notably referred to as the “curse of dimensionality” [13]. Gradient-based methods [14,15], with specific reference to “first-order” methods, make use of the gradient information of the function of interest in the search of the optimal solution, and therefore are more ecient than zero-order methods. The evaluation of the gradients may be computationally intensive if performed by Finite Di erences or complex step techniques [16], especially in the presence of a large number of design variables and in the case of disciplines involving expensive computations. However, if the number of objective functions is lower than the number of optimization parameters, an ecient calculation of the gradients can be obtained by the adjoint method [17]. This technique, first introduced in [18] for fluid problems, is characterized by a cost for the computation of the sensitivity derivatives of the objective function that is essentially independent of the number of design variables. Therefore, it is particularly suited for problems requiring detailed geometrical descriptions with many degrees of freedom. Moreover, the exploration of the design space through its gradients decreases the number of iterations necessary to achieve the local optimal solution. Therefore, the reduced overhead Int. J. Turbomach. Propuls. Power 2020, 5, 30 3 of 31 can be traded for the introduction of more computationally demanding disciplines, returning a higher accuracy in the numerical solutions. A performance comparison between these two techniques is o ered in [19], whose application to the multi-point optimization of a radial turbine impeller shows that the adjoint method is capable of extending the Pareto front identified by a di erential evolution (DE) algorithm. In particular, the gradient-based approach delivers a solution with a higher total-to-static eciency, improving also the moment of inertia figure at the cost of a fraction of the computational time required by the gradient-free method to achieve its global optimum (roughly 6–10%). Other examples of applications of the adjoint methods to the optimization of turbines and compressors are reported in [20–25], with focus on the opportunity of increasing the number of variables in the design space and the multidisciplinary aspect of the physical evaluations (including aerodynamics, stresses, and modal analysis). The higher maturity recently gained by gradient-based approaches and their improved robustness in the analysis of complex flow conditions [26,27] provides evidence of their application to problems of industrial relevance. Therefore, adjoint methods are considered a suitable choice to address the previously mentioned design advancements in turbomachinery-related applications. 1.2. Motivation The present document focuses on the development of a radial turbine impeller for an automotive turbocharger. In contrast to aeronautical applications, the typical duty cycles for passenger cars span the entire turbine map, from very low mass flow rates and pressure ratios in urban driving situations up to peak power, resulting in requirements stretching the machine design over a wide range of operative conditions. Moreover, these turbines present tight clearances between rotor and housing in the order of few hundreds of microns, whilst the typical thermal operative range spans from ambient temperature up to 1050 C, without the possibility of any blade-dedicated cooling circuit. Such scenario clearly demonstrates the challenges in performance requirements and durability the designer is confronted with during the product development process. The structural integrity of the rotor experiencing centrifugal and fluid-induced stresses is a critical aspect in the search for design solutions fulfilling the eciency and permeability targets. The problem of the Fluid–Structure Interaction (FSI) in radial turbines in relation to the thermal stresses induced by the fluid in contact with the blades is discussed in [28], in an e ort to accurately predict the heat exchange between the two media for a reliable estimation of the rotor lifetime in steady state and dynamic operative conditions. The study shows how the incorrect estimation of the thermal stresses could impair the robustness of the component, justifying the inclusion of this additional constraint on top of the computation of the centrifugal stresses. The introduction of thermal analyses in the framework of an optimization process is a topic of recent interest [29–31]. The physical phenomenon is described through the application of a Conjugate Heat Transfer (CHT) procedure, as originally proposed in [32,33], which involves the thermal interaction between a solid body and its surrounding fluid by a coupled solution of the two domains. The challenge in the implementation of this analysis within the landscape of a multidisciplinary optimization of a complex geometry consists of the trade-o between accuracy and significant computational overhead, so far limiting thermal predictions only to the final validation phase in industrial design procedures. In fact, it is common practice during an optimization to replace such intensive computations with reduced-order models or safety margins based on empirical experience. This approach to the evaluation of the thermal stresses would su er in terms of accuracy or, in the case of large safety margins, would over-constrain the optimization problem. 1.3. Goal of the Paper This paper addresses the problem of robust optimizations w.r.t. thermo-mechanical stresses by developing a discrete adjoint framework tailored to the implementation of the CHT analysis within the in-house multidisciplinary design and optimization platform “CADO” developed at the von Karman Int. J. Turbomach. Propuls. Power 2020, 5, 30 4 of 31 Institute for Fluid Dynamics [34]. Indeed, the selection of a gradient-based method is justified by its Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 4 of 31 computational eciency in identifying a local minimum, although the computation of the gradients is more convoluted due to the multiple disciplines involved in the performance analysis. The calculation the in-house multidisciplinary design and optimization platform “CADO” developed at the von of the thermal stresses in the solid domain and their sensitivities with respect to the grid coordinates Karman Institute for Fluid Dynamics [34]. Indeed, the selection of a gradient-based method is is therefore the subject of the present paper, with the aim of including this analysis in a Sequential justified by its computational efficiency in identifying a local minimum, although the computation of Quadratic Programming-based optimizer [35]. the gradients is more convoluted due to the multiple disciplines involved in the performance analysis. The calculation of the thermal stresses in the solid domain and their sensitivities with respect To the best of our knowledge, only a few studies have considered the CHT analysis in adjoint-based to the grid coordinates is therefore the subject of the present paper, with the aim of including this optimizations [36,37], with only a limited number adopting a discrete adjoint formulation [38–40] and analysis in a Sequential Quadratic Programming-based optimizer [35]. none treating a test case with a complex three-dimensional flow field or extending the problem to To the best of our knowledge, only a few studies have considered the CHT analysis in adjoint- thermo-mechanical evaluations. Therefore, this work is aimed at covering critical gaps in the design of based optimizations [36,37], with only a limited number adopting a discrete adjoint formulation [38– radial turbines with the aim of creating a framework suitable for industrial applications. 40] and none treating a test case with a complex three-dimensional flow field or extending the In light of this objective, the remaining portion of the document is structured as follows. problem to thermo-mechanical evaluations. Therefore, this work is aimed at covering critical gaps in First, the framework of the primal solver for the execution of the CHT analysis is presented, the design of radial turbines with the aim of creating a framework suitable for industrial applications. with extension to the computation of the thermo-mechanical stresses. Then, the implementation of the In light of this objective, the remaining portion of the document is structured as follows. First, adjoint thworkflow e frameworfor k ofthe the p calculation rimal solverof fothe r thesensitivities execution of th ofethe CHT constraint analysis isfunction presentedwith , withr e espect xtensioto n the solid and to th fluid e com grid puta coor tion dinates of the th isediscussed, rmo-mechan outl icalining stressthe es. steps Then, towar the im ds ple the men coupli tationng of of ththe e addomains joint workflow for the calculation of the sensitivities of the constraint function with respect to the solid in reverse mode. Finally, the validation of the model and the results of the application of the procedure and fluid grid coordinates is discussed, outlining the steps towards the coupling of the domains in to a radial turbine rotor test case are presented and the conclusions are drawn from this study. reverse mode. Finally, the validation of the model and the results of the application of the procedure to a radial turbine rotor test case are presented and the conclusions are drawn from this study. 2. Primal Solver The CHT analysis implemented in the current work follows the partitioned coupling approach 2. Primal Solver proposed by [41], in which a fluid and a solid solver are sequentially called in an iterative process The CHT analysis implemented in the current work follows the partitioned coupling approach with the mutual exchange of boundary conditions till reaching the convergence of the temperature proposed by [41], in which a fluid and a solid solver are sequentially called in an iterative process at their interface. The advantage of this strategy is related to the possibility of adopting dedicated with the mutual exchange of boundary conditions till reaching the convergence of the temperature meshes and specialized numerical methods for the solution of each domain, increasing the robustness at their interface. The advantage of this strategy is related to the possibility of adopting dedicated of the m conv eshe er s gence and spe of cia the lizetwo d nuanalyses. merical meMor thodeover s for th , e in so the luticase on of of eac an h dunsteady omain, inccomputation, reasing the robus the tne partial ss of the convergence of the two analyses. Moreover, in the case of an unsteady computation, the partial decoupling of the two fields, which show very di erent characteristic time scales, would lead to a decoupling of the two fields, which show very different characteristic time scales, would lead to a faster convergence of the assembly [42–45]. faster convergence of the assembly [42–45]. 2.1. Mapping Procedure 2.1. Mapping Procedure The problem under investigation is represented in Figure 1, in which a turbine rotor is analyzed The problem under investigation is represented in Figure 1, in which a turbine rotor is analyzed considering a periodic sector. A multi-block structured fluid grid with boundary layer refinement is considering a periodic sector. A multi-block structured fluid grid with boundary layer refinement is interfaced to an unstructured solid mesh of second-order tetrahedral elements for the computation of interfaced to an unstructured solid mesh of second-order tetrahedral elements for the computation the heat flux between the two domains. of the heat flux between the two domains. Figure 1. Radial turbine rotor: fluid and solid sector meshes. Figure 1. Radial turbine rotor: fluid and solid sector meshes. Int. J. Turbomach. Propuls. Power 2020, 5, 30 5 of 31 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 5 of 31 Because the meshes are non-matching at the common interface, the exchange of information Because the meshes are non-matching at the common interface, the exchange of information is is performed through the distance-weighted interpolation (DWI) technique presented in [46]. performed through the distance-weighted interpolation (DWI) technique presented in [46]. The The procedure implies the mapping of the two regions in order to record the correspondences procedure implies the mapping of the two regions in order to record the correspondences of each of each fluid cell center with the closest surrounding solid nodes and vice versa. For each fluid cell, the fluid cell center with the closest surrounding solid nodes and vice versa. For each fluid cell, the information of heat flux or temperature detected in the mating solid nodes is processed by a weighting information of heat flux or temperature detected in the mating solid nodes is processed by a function based on their mutual distances, resulting in the coupling of the two regions. The same weighting function based on their mutual distances, resulting in the coupling of the two regions. The routine is applied to the solid grid with respect to its fluid counterpart. same routine is applied to the solid grid with respect to its fluid counterpart. This method exhibits a good level of generality in treating di erent coupling problems, irrespective This method exhibits a good level of generality in treating different coupling problems, of their grids characteristics. For instance, in the turbine test case the exchange of information is irrespective of their grids characteristics. For instance, in the turbine test case the exchange of e ectively performed also at the blade hub fillet, a feature which is present in the solid domain but not information is effectively performed also at the blade hub fillet, a feature which is present in the solid detailed in the corresponding fluid region. domain but not detailed in the corresponding fluid region. An improvement in the accuracy of this coupling procedure derives from the local labelling of An improvement in the accuracy of this coupling procedure derives from the local labelling of the interface. With reference to Figure 2, the surface between the two media is split in sub-regions the interface. With reference to Figure 2, the surface between the two media is split in sub-regions (e.g., hub, blade pressure side, blade suction side, tip, etc.) and the corresponding solid faces and fluid (e.g., hub, blade pressure side, blade suction side, tip, etc.) and the corresponding solid faces and blocks boundaries are named accordingly. Therefore, the algorithm searching for the fluid cells–solid fluid blocks boundaries are named accordingly. Therefore, the algorithm searching for the fluid cells– nodes correspondences is separately executed for each sub-region, instead of attempting the coupling solid nodes correspondences is separately executed for each sub-region, instead of attempting the of the two fields through a global search within the entire domain. This arrangement is particularly coupling of the two fields through a global search within the entire domain. This arrangement is e ective in locations presenting thin walls, such as in proximity of the blade tip, as it avoids any particularly effective in locations presenting thin walls, such as in proximity of the blade tip, as it erroneous fluid-solid matching between cells and nodes actually sitting on opposite sides of the blade avoids any erroneous fluid-solid matching between cells and nodes actually sitting on opposite sides (i.e., suction side versus pressure side). of the blade (i.e., suction side versus pressure side). Figure 2. Radial turbine rotor surface split in sub-regions (left), example of FVM-FEM meshes Figure 2. Radial turbine rotor surface split in sub-regions (left), example of FVM-FEM meshes correspondence identification (right). correspondence identification (right). Additionally, the original algorithm seeks the nodes–cells correspondences through a Additionally, the original algorithm seeks the nodes–cells correspondences through a “search “search radius” delimiting for each grid node a spherical volume within which the neighbors are radius” delimiting for each grid node a spherical volume within which the neighbors are identified. identified. This technique may induce interpolation errors in proximity of the walls, where temperature This technique may induce interpolation errors in proximity of the walls, where temperature gradients and fluid grid stretching are significant. In fact, the temperature assigned to a solid node gradients and fluid grid stretching are significant. In fact, the temperature assigned to a solid node may result from the weighting of temperature values derived from fluid cells located within the may result from the weighting of temperature values derived from fluid cells located within the boundary layer but not in direct contact with the interface. Moreover, as demonstrated in [46], distinct boundary layer but not in direct contact with the interface. Moreover, as demonstrated in [46], distinct grid refinements at the solid and fluid side may influence the local accuracy of the wall temperature grid refinements at the solid and fluid side may influence the local accuracy of the wall temperature information passed to the FEM. In order to cope with these issues, the searching criterion is limited to information passed to the FEM. In order to cope with these issues, the searching criterion is limited the sole layer of nodes and cells in direct contact with the walls, and the concept of “search radius” to the sole layer of nodes and cells in direct contact with the walls, and the concept of “search radius” is dismissed in favor of a ranking assigned to each node or cell according to their mutual distance. Finally, consistently with the solution proposed in [46] addressing the problem of the aspect ratio of Int. J. Turbomach. Propuls. Power 2020, 5, 30 6 of 31 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 6 of 31 is dismissed in favor of a ranking assigned to each node or cell according to their mutual distance. Finally, consistently with the solution proposed in [46] addressing the problem of the aspect ratio cells on a wall boundary, additional virtual grid points are introduced on the FVM side between two of cells on a wall boundary, additional virtual grid points are introduced on the FVM side between neighboring cell centers. Such virtual points are aimed at improving the uniformity of the local two neighboring cell centers. Such virtual points are aimed at improving the uniformity of the local distribution of grid points in regions of significant mesh stretching, and their temperature is distribution of grid points in regions of significant mesh stretching, and their temperature is interpolated interpolated between the closest “master” cells of the original mesh. Indeed, the goal is to increase between the closest “master” cells of the original mesh. Indeed, the goal is to increase the density the density of the attractors in proximity of each solid node, avoiding possible inconsistencies of gethe nerattractors ated from in capr ptur oximity ing thof e ieach nform solid ation node, fromavoiding too far flpossible uid cellsinconsistencies . Thus, the dom generated ain couplfr inom g is capturing the information from too far fluid cells. Thus, the domain coupling is executed through the executed through the distance-weighting interpolation of information belonging to cells and nodes distance-weighting in actual close prox interpolation imity. of information belonging to cells and nodes in actual close proximity. The quality of the interpolation is referenced in Figure 3. The fluid temperature at the walls The quality of the interpolation is referenced in Figure 3. The fluid temperature at the walls (Figur (Figur ee3 3 a) a)is isinterpolated interpolatedby bythe theDWI DWI pr pocedur rocedur eeand andpassed passedto tothe thesolid solidnodes nodeslaying layingon onthe thesurface surface (Figure 3b). During a CHT iteration, the solid returns the heat flux to the fluid, as discussed in the (Figure 3b). During a CHT iteration, the solid returns the heat flux to the fluid, as discussed in the next next section. section. Figure 3. Distance-weighted interpolation method applied to the exchange of information between Figure 3. Distance-weighted interpolation method applied to the exchange of information between solid and fluid domains: (a) fluid temperature at walls, (b) interpolated fluid temperature assigned to solid and fluid domains: (a) fluid temperature at walls, (b) interpolated fluid temperature assigned to the solid surface. the solid surface. 2.2. Partitioned Coupling 2.2. Partitioned Coupling The data exchange between the fluid and solid fields is performed through the “heat transfer The data exchange between the fluid and solid fields is performed through the “heat transfer forward flux back” method (hFFB) [47], whose stability properties are discussed in [48]. In summary, forward flux back” method (hFFB) [47], whose stability properties are discussed in [48]. In summary, the convergence rate of the conjugate problem depends on the local Biot number, which expresses the the convergence rate of the conjugate problem depends on the local Biot number, which expresses ratio between the conductive over the convective thermal resistances. In complex geometries, as in the ratio between the conductive over the convective thermal resistances. In complex geometries, as the case of a radial turbine impeller, the Biot number may locally change to values greater or lower in the case of a radial turbine impeller, the Biot number may locally change to values greater or lower than unity, according to the variations in the blade thickness. Hence, [46] demonstrates that methods than unity, according to the variations in the blade thickness. Hence, [46] demonstrates that methods such as the hFFB or the “heat transfer forward temperature back” (hFTB) method would promote the such as the hFFB or the “heat transfer forward temperature back” (hFTB) method would promote the stabilization of the fluid-solid coupling w.r.t. other techniques, such as the “flux forward temperature stabilization of the fluid-solid coupling w.r.t. other techniques, such as the “flux forward temperature back” (FFTB) or the “temperature forward flux back” (TFFB) methods, which strictly require a modulus back” (FFTB) or the “temperature forward flux back” (TFFB) methods, which strictly require a of the Biot number respectively lower or greater than unity. Instead, the hFTB and hFFB methods modulus of the Biot number respectively lower or greater than unity. Instead, the hFTB and hFFB exhibit a wider range of convergence. Finally, the hFFB method was selected for the present study since methods exhibit a wider range of convergence. Finally, the hFFB method was selected for the present study since a heat flux boundary condition imposed to the fluid domain would generally improve the convergence stability of the CFD computation more than an imposed wall temperature boundary condition. Figure 4 reports the partitioned coupling workflow, described as follows. Int. J. Turbomach. Propuls. Power 2020, 5, 30 7 of 31 a heat flux boundary condition imposed to the fluid domain would generally improve the convergence stability of the CFD computation more than an imposed wall temperature boundary condition. Figure 4 reports the partitioned coupling workflow, described as follows. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 7 of 31 Figure 4. hFFB coupling method for the CHT analysis. Figure 4. hFFB coupling method for the CHT analysis. The CHT analysis is addressed with an initial CFD computation considering adiabatic walls. The CHT analysis is addressed with an initial CFD computation considering adiabatic walls. The The fluid temperature T and the heat flux normal to the walls q are extracted at the boundaries in FW FW fluid temperature and the heat flux normal to the walls are extracted at the boundaries in contact with the solid, and a virtual fluid bulk temperature T is calculated as: f l contact with the solid, and a virtual fluid bulk temperature is calculated as: FW T = =T − ,, (1) FW f l (1) with ℎ indicating a constant user-defined virtual heat transfer coefficient. with h indicating a constant user-defined virtual heat transfer coecient. Since the initial fluid simulation is adiabatic, Equation (1) results in a virtual fluid bulk Since the initial fluid simulation is adiabatic, Equation (1) results in a virtual fluid bulk temperature temperature equal to the fluid temperature at the walls. The DWI procedure returns the field equal to the fluid temperature at the walls. The DWI procedure returns the T field associated with f l associated with the solid grid nodes. the solid grid nodes. Consequently, ℎ and are assigned to the FEM heat transfer model, such that the heat flux Consequently, h and T are assigned to the FEM heat transfer model, such that the heat flux to the f l to the solid results from the following boundary condition: solid results from the following boundary condition: = ℎ − , (2) q = h T T , (2) SW f l with T as the unknown solid temperature. After solving the FEM heat transfer problem, the temperature and heat flux at this boundary are known. with T as the unknown solid temperature. After solving the FEM heat transfer problem, the temperature The heat flux at the interface is calculated from the temperature field T by means of the Fourier’s and heat flux at this boundary are known. law applied to all the elements exposing at least one face to the fluid: The heat flux at the interface is calculated from the temperature field T by means of the Fourier ’s law applied to all the elements exposing at least one = −face ,to the fluid: (3) dT with indicating the temperature gradient normal to the wall and k the thermal conductivity q = k , (3) SW dn coefficient. The heat flux is processed by the DWI procedure, returning the heat flux assigned to the fluid cells at the interface. The fluid simulation, now accounting for an external heat flux at the viscous walls, is recomputed and the entire process is re-iterated for several loops till the achievement of the continuity of temperatures at the interface between the two fields. The user-imposed virtual heat transfer coefficient ℎ influences the predicted wall heat fluxes at any intermediate cycle of the CHT procedure, thus determining the path to convergence of the whole Int. J. Turbomach. Propuls. Power 2020, 5, 30 8 of 31 with dT/dn indicating the temperature gradient normal to the wall and k the thermal conductivity coecient. The heat flux q is processed by the DWI procedure, returning the heat flux q assigned to the SW FW fluid cells at the interface. The fluid simulation, now accounting for an external heat flux at the viscous walls, is recomputed and the entire process is re-iterated for several loops till the achievement of the continuity of temperatures at the interface between the two fields. The user-imposed virtual heat transfer coecient h influences the predicted wall heat fluxes at any intermediate cycle of the CHT procedure, thus determining the path to convergence of the whole coupling. In the case of the hFFB method, [46] shows that a suitable choice of the virtual coecient h < 2h, with h as the physical value of the heat transfer coecient, would guarantee the convergence of the conjugate problem for any local value of the Biot number. The higher the value of h within the stable region, the faster the convergence of the partitioned coupling method. A discussion about the determination of the h value for a problem of industrial relevance is reported in Section 4.2. 2.3. Solid Heat Transfer Solver The steady-state energy balance within the solid domain is described by Equation (4), with q as the rate of heat transfer into the system, and is computed through a FEM steady linear solver according to [49]. rq = 0, (4) The boundary conditions applied to the problem are classified in three families: Tj = const @W kr Tj = q @W 2 , (5) kr Tj = h Tj T @W @W 3 3 f l with the surface of the solid domain defined as @W [ @W [ @W = @W. 1 2 3 The first equation in (5) implies a fixed temperature imposed at the surface @W . The second type of boundary conditions specifies a heat flux at the surface @W , while Fourier ’s law at LHS describes the heat flux through the medium. If q = 0, the adiabatic wall boundary condition is defined. The third equation represents the convection condition, whose heat flux through the surface @W is proportional to the temperature di erence w.r.t. the surrounding fluid. The virtual heat transfer coecient h and the interpolated thermal field T are reported for the sake of consistency with Equation (2). f l Once integrated over the entire domain, Equation (4) is discretized, resulting in the linear system: A T = b, (6) with the semi-positive definite sti ness matrix A comprising the conductive and convective terms, n,n n A = [A + A ]2 R , and the load vector b2 R accounting for the contribution of the boundary conv cond conditions. The linear system in (6) is solved by an iterative conjugate gradient method. 2.4. Fluid Solver The compressible Reynolds-Averaged Navier–Stokes solver with cell-centered spatial discretization over a structured multi-block grid, developed in [19], is invoked in the CHT framework. The solver computes the convective fluxes by Roe’s upwind scheme with MUSCL extrapolation, while a central discretization is applied to the calculation of viscous fluxes. The time marching technique is implemented according to the JT-KIRK scheme proposed in [26] for the improved stabilization of the discrete adjoint solver. The Spalart–Allmaras turbulence model is adopted in this study. In order to establish the coupling with the solid domain, heat fluxes are imposed at the viscous wall boundaries adopting a thin shear layer approximation. The fluid mesh presents two layers of ghost cells in order to facilitate the computation of the fluxes at the interfaces [50]. The heat fluxes calculated Int. J. Turbomach. Propuls. Power 2020, 5, 30 9 of 31 in Equation (3) and processed through the DWI technique are associated with the corresponding fluid block boundaries in contact with the solid walls. With reference to Figure 5, D1 is the first layer of the inner fluid domain; G1 and G2 are the corresponding ghost layers. The fluid temperature in D1 is detected from the previous solver iteration, and the ghost cell temperature in the first layer is updated according to: Dn T = T + q , (7) G1 D1 FW Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 9 of 31 f l with Dn as the distance between the cell centers and k as the locally computed fluid conductivity. with ∆ as the distance between the cell centers and kfl as the locally computed fluid conductivity. Similarly, the temperature in the G2 layer is updated in cascade, with reference to the newly computed Similarly, the temperature in the G2 layer is updated in cascade, with reference to the newly temperature in G1. The temperature values are used to calculate the corresponding cell densities computed temperature in G1. The temperature values are used to calculate the corresponding cell considering a zero-order pressure extrapolation in wall normal direction, while the no-slip condition densities considering a zero-order pressure extrapolation in wall normal direction, while the no-slip is assured by reversing the fluid tangential velocity components in the ghost layers. Equation (7) is condition is assured by reversing the fluid tangential velocity components in the ghost layers. linearized and accounted in the implicit scheme. Equation (7) is linearized and accounted in the implicit scheme. Figure 5. Fluid domain discretization at block boundaries: D1 as the inner domain layer, G1-G2 as the Figure 5. Fluid domain discretization at block boundaries: D1 as the inner domain layer, G1-G2 as the ghost layers. ghost layers. The rate of convergence induced by the Neumann boundary condition is influenced by the actual The rate of convergence induced by the Neumann boundary condition is influenced by the magnitude of the imposed heat flux in Equation (7). Since the hFFB method is employed in the present actual magnitude of the imposed heat flux in Equation (7). Since the hFFB method is employed in the development, no under-relaxation factors are invoked for the update of the ghost cells temperatures, present development, no under-relaxation factors are invoked for the update of the ghost cells as they are replaced by the selection of a suitable virtual heat transfer coecient h, determinant for the temperatures, as they are replaced by the selection of a suitable virtual heat transfer coefficient ℎ, stability of the coupling process [46]. determinant for the stability of the coupling process [46]. 2.5. Solid Mechanical Solver 2.5. Solid Mechanical Solver Once the continuity of temperatures and heat flux is achieved at the interface of the two domains, Once the continuity of temperatures and heat flux is achieved at the interface of the two domains, an FEM linear elastic solver with quadratic elements presented in [51] is invoked for the solution of the an FEM linear elastic solver with quadratic elements presented in [51] is invoked for the solution of von Mises stresses. the von Mises stresses. S u = f , (8) = , (8) n,n n n with S 2 R as the sti ness matrix, u 2 R referencing the vector of nodal displacements, and f 2 R with ∈ as the stiffness matrix, ∈ referencing the vector of nodal displacements, and ∈ indicating the load. indicating the load. Within the framework of the present analysis, the solid mechanical solver shares the same mesh Within the framework of the present analysis, the solid mechanical solver shares the same mesh as the heat transfer solver, as remarked in Section 3. Therefore, the previously computed nodal as the heat transfer solver, as remarked in Section 3. Therefore, the previously computed nodal temperatures are directly assigned to the new linear system. temperatures are directly assigned to the new linear system. The right-hand side vector accounts for the centrifugal loading and the thermal strains, considering The right-hand side vector accounts for the centrifugal loading and the thermal strains, the e ect of the material contractions and expansions induced by the temperature variations, which are considering the effect of the material contractions and expansions induced by the temperature computed as: variations, which are computed as: " = (T ) T T , (9) thermal i i re f = ( ) − , (9) with α(Ti) indicating the temperature-dependent thermal expansion coefficient; Ti pointing to the node temperatures resulting from the CHT analysis; and Tref as the reference temperature at the initial state, at which no thermal stresses are present within the material [52]. In the current study, we refer to Tref as the ambient temperature. Through the thermal strains, the stresses in the material are dependent on the outcome of the CHT computation, which increases the complexity of the gradient calculation, as presented later on. The maximum von Mises stress is computed using a p-norm function according to Equation (10), with p = 75. Int. J. Turbomach. Propuls. Power 2020, 5, 30 10 of 31 with (T ) indicating the temperature-dependent thermal expansion coecient; T pointing to the node i i temperatures resulting from the CHT analysis; and T as the reference temperature at the initial state, ref at which no thermal stresses are present within the material [52]. In the current study, we refer to T ref as the ambient temperature. Through the thermal strains, the stresses in the material are dependent on the outcome of the CHT computation, which increases the complexity of the gradient calculation, as presented later on. The maximum von Mises stress is computed using a p-norm function according to Equation (10), with p = 75. t R dV VM R . (10) max VM dV Equation (10) fulfills the requirement of global di erentiability, as it is a continuous function, and is invoked as a constraint within the optimization problem with the aim of keeping the thermo-mechanical stresses, induced through the Conjugate Heat Transfer, bounded to a prescribed value. The gradient of this constraint is computed by an adjoint methodology, as described in the next section. 3. Adjoint Solver The structure of the primal solver evaluating the thermo-mechanical stresses is computationally expensive because the CHT procedure requires several loops of CFD-FEM analysis to reach the equilibrium of the temperatures and heat flux at the fluid–solid interface. For this reason, such a framework is less suited for gradient-free optimization methods, as a prohibitive large population of candidates would need to be evaluated using this expensive tool. A gradient-based optimization technique, on the other hand, may o er a faster convergence through a more guided search in the design space. Especially when combined with the adjoint method, the computation of the derivatives is very ecient and almost insensitive to the number of design parameters. The evaluation of the cost function J in primal mode is a two-step process. The first one considers the values of the design variables 2 R and generates the correspondent geometry and finally the grid, whose points coordinates are expressed by the vector X 2 R . The second step involves the flow solver and some post-processing routines, returning the value of J. Following the same approach in reverse mode, also the adjoint evaluation procedure is split in two parts: the first one computing the sensitivities of the cost function J w.r.t. the perturbations of the grid coordinates X, and the second one calculating the sensitivities of the grid coordinates X w.r.t the changes in the design variable . Therefore, the CAD-based parametrization described in [21] allows transferring the grid sensitivities of the cost function to the design variables, controlling the component shape modifications by the application of the chain rule of di erentiation expressed in Equation (11): dJ @J dX = . (11) d @X d Within the thermo-mechanical analysis process developed in the present work, Equation (11) is invoked twice in order to account for the cost function sensitivities w.r.t. the coordinates of the fluid grid points and the solid grid nodes, respectively. Therefore, the global sensitivity expression can be decomposed as follows: dX dJ @J f l @J dX sl = + . (12) d @X d @X d f l sl TM The remaining portion of this section is devoted to the calculation of the cost function sensitivities with respect to the fluid and solid grid coordinates, respectively @J/@X and @J/@X , operated through f l sl the application of a discrete adjoint method. Such sensitivities provide the descent direction for constrained or unconstrained multidisciplinary optimization problems with the aim of minimizing the thermally related response functions. Int. J. Turbomach. Propuls. Power 2020, 5, 30 11 of 31 Consistently with the structure of the primal solver, the adjoint framework treating this interdisciplinary problem is formulated according to a “loose-coupling” approach, di erent from the “strong-coupling” techniques as described in [53]. In fact, the latter accounts for cross-discipline Jacobian terms implicitly exchanging boundary conditions between the fluid and solid domains, and therefore directly solving the global adjoint system at once. Instead, the selected partitioned-coupling method is less intrusive in the structure of the existing adjoint CFD solver from [19] and therefore more suited to a continuously growing modular multidisciplinary platform. Figure 6 reports the framework of the thermo-mechanical evaluation, with the adjoint workflow Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 11 of 31 on the RHS. The iterative structure characterizing the solution of the primal problem is maintained in its adjoint counterpart, which walks through the entire chain in reverse mode by the manual in its adjoint counterpart, which walks through the entire chain in reverse mode by the manual di erentiation of the full process. Therefore, the adjoint branch is discussed starting from the bottom, differentiation of the full process. Therefore, the adjoint branch is discussed starting from the bottom, till achieving the final grid sensitivities on the top. till achieving the final grid sensitivities on the top. Figure Figure 6. 6.W Wo orkrfk lo fl w ow fo rfo thre p th re im p ar lia m nd ala d an jod in t ac d o jo m in pt u tc ao tim on psu o tf as tt ie oa nd sy o -sf tas tt eea thd ey rm -so ta -m te et ch he ar nm ico al -m coec nsh tr a an in ic ta s.l constraints. 3.1. Adjoint Variables The most demanding terms in the computation of Equation (12) are the sensitivities of the response function w.r.t. the grid coordinates. Since: ( ) , , (13) with u denoting the vector of the state variables, the application of the chain rule of differentiation results in the following decomposition: (14) = + . Equation (14) holds for both the fluid and the solid grids. The second term at RHS presents the partial derivative of the cost function w.r.t. the state variables ⁄ . Since the current problem exhibits a multidisciplinary footprint, the hierarchical structure of the workflow in Figure 6 implies that the output states contributing to the p-norm calculation in Equation (10) depend on the intermediate ones evaluated by the preceding CFD and FEM solvers. Therefore, according to the Int. J. Turbomach. Propuls. Power 2020, 5, 30 12 of 31 3.1. Adjoint Variables The most demanding terms in the computation of Equation (12) are the sensitivities of the response function w.r.t. the grid coordinates. Since: J(X, u(X)), (13) with u denoting the vector of the state variables, the application of the chain rule of di erentiation results in the following decomposition: dJ @J @J @u = + . (14) dX @X @u @X Equation (14) holds for both the fluid and the solid grids. The second term at RHS presents the partial derivative of the cost function w.r.t. the state variables @J/@u. Since the current problem exhibits a multidisciplinary footprint, the hierarchical structure of the workflow in Figure 6 implies that the output states contributing to the p-norm calculation in Equation (10) depend on the intermediate ones evaluated by the preceding CFD and FEM solvers. Therefore, according to the principle of the “reverse di erentiation” [54] we choose for instance an output variable u , whose sensitivity dJ/du is known, 3 3 and calculate its sensitivities w.r.t. each intermediate state till the initial one, such that: dJ dJ du du du du 2 3 2 J(u (u (u ))) ! . (15) 3 2 dJ dJ du du du du 1 2 1 Since the work focuses on the single thermo-mechanical output J expressed by Equation (10), TM herein named y, we can associate with any intermediate state variable u a new variable u , called the i i adjoint variable, defined as: @y u = . (16) @u The upper bar notation follows the convention for adjoint variables reported in [54]. Hence, we apply the chain rule walking throughout the original trace in Figure 6 in backward mode, paying attention to the opposite propagation of the adjoint variables with respect to the physical ones. This technique finally returns the sensitivities of the response function w.r.t. the initial state variables. The following sections describe the backward propagation process in detail, with the aim of o ering a guidance to the development of the adjoint method. 3.2. Adjoint Response Function The adjoint framework is initiated by seeding the maximum von Mises constraint. Since y = J =  , the input is: TM maxVM @y = J =  = 1. (17) TM maxVM @J TM The maximum von Mises stress is function of the components of the Cauchy stress tensor. Therefore, in reverse mode the adjoint stress components are calculated as: t R dV dy dJ d p TM V VM =  = J = R . (18) i, j TM d d d i, j dV On the other hand, Equation (10) shows an explicit dependence over the solid grid coordinates through the two volume integrals appearing at the numerator and denominator. Int. J. Turbomach. Propuls. Power 2020, 5, 30 13 of 31 Therefore, the constraint function sensitivities with respect to the solid grid coordinates are accumulated as follows: t R dV dy dJ d TM V VM = X + = J = . (19) TM sl dX dX dX dV sl sl sl In Equation (19), the operator “+=” is reported to emphasize the concept of accumulation of the sensitivities w.r.t. the solid grid coordinates that is propagated backwards throughout the adjoint workflow till the grid generation procedure. Moreover, the X notation is adopted for the sake of simplicity, while in reality the sensitivities over the three dimensions (X,Y,Z) are accounted in the development. Finally, in the remaining portion of the document the notations dy/du and dy/dX will be skipped in favor of the more convenient u and X. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 13 of 31 3.3. Adjoint Mechanical Solver 3.3. Adjoint Mechanical Solver The evaluations through the mechanical solver involve the multi-step approach outlined in The evaluations through the mechanical solver involve the multi-step approach outlined in Figure 7, where the primal mode is represented along with its adjoint counterpart. Notably, the reverse Figure 7, where the primal mode is represented along with its adjoint counterpart. Notably, the computation follows the backward propagation of the adjoint variables, closely resembling the primal reverse computation follows the backward propagation of the adjoint variables, closely resembling scheme. The related procedure is discussed in detail in this section. the primal scheme. The related procedure is discussed in detail in this section. Figure 7. Evaluation by mechanical solver: primal and adjoint modes. Figure 7. Evaluation by mechanical solver: primal and adjoint modes. In primal mode, the stress components  are computed from the elasticity matrix of the material i, j In primal mode, the stress components are computed from the elasticity matrix of the under consideration and the strain components. Thus, the di erentiation in reverse mode returns the material under consideration and the strain components. Thus, the differentiation in reverse mode adjoint strains " . i, j returns the adjoint strains ̅ . ̅ = ( ) _ , " =  E(T) m_i j i, j ( ) ̅ = − ( ) = ( ) − ( ) → . ( _) ( ,) (20) , = E(T) " _ " _ (T) ! " T =  E T . (20) > i, j i, j m_i j th_i j th_i j : ( ) = − ( ) E(T) =  " , _" (T ) _ i, j m_i j th_i j Here, the adjoint stresses are the once computed by the reverse differentiation discussed in Here, the adjoint stresses  are the once computed by the reverse di erentiation discussed in i, j Section 3.2. Section 3.2. The adjoint mechanical strains ̅ and the adjoint thermal strains ̅ appear in Equation (20). The adjoint mechanical strains " and the adjoint thermal strains " appear in Equation (20). th Since the thermal strains exhibit direct dependence from the nodal temperatures through the Since the thermal strains exhibit direct dependence from the nodal temperatures through the T T i re f − term and indirect dependence through the thermal expansion coefficient α(T), the relative term and indirect dependence through the thermal expansion coecient (T), the relative contributions contributions to the adjoint solid node temperatures are accumulated and propagated to the next step to the adjoint solid node temperatures are accumulated and propagated to the next step in the in the reverse workflow. reverse workflow. 8 += ̅ ( ) ( ) > ( ) ( ) = − → T+ = " T . (21) ( ) _ < th_i j += ̅ − " (T) = (T) T T ! > . (21) th_i j re f d (T) T+ = " T T th_i j re f dT Additionally, also the elasticity matrix E depends on the nodal temperature, which leads Additionally, also the elasticity matrix E depends on the nodal temperature, which leads similarly similarly to contributions for the adjoint solid node temperatures. to contributions for the adjoint solid node temperatures. ( ) ( ) = ( ) → += ( ) . (22) d f (T) E(T) = f (T) ! T+ = E(T) . (22) The process of accumulation of the contributes to the adjoint temperatures field is facilitated by dT the commonality of the solid grid coordinates between the mechanical and heat transfer solvers, as no re-interpolations are necessary for the backward propagation of the sensitivities. Next, recalling that the mechanical strains in primal mode are calculated from the vector of nodal displacements u, the adjoint nodal displacements are derived by the reverse differentiation ( ̅ → ) and provided in input to the adjoint mechanical solver. The iterative linear system solver adopted in primal mode for the solution of Equation (8) is not directly differentiated but, as discussed in [20], a new linear system is formulated with the adjoint displacements vector appearing at RHS as: = . (23) The system (23) is solved for the adjoint load vector , while the transpose of the stiffness matrix S, previously computed in primal mode, appears at LHS. Therefore, the adjoint stiffness matrix is obtained from the system (24). ̅ ̅ = − . (24) Int. J. Turbomach. Propuls. Power 2020, 5, 30 14 of 31 The process of accumulation of the contributes to the adjoint temperatures field is facilitated by the commonality of the solid grid coordinates between the mechanical and heat transfer solvers, as no re-interpolations are necessary for the backward propagation of the sensitivities. Next, recalling that the mechanical strains " in primal mode are calculated from the vector of nodal displacements u, the adjoint nodal displacements are derived by the reverse di erentiation (" ! u ) and provided in input to the adjoint mechanical solver. The iterative linear system solver adopted in primal mode for the solution of Equation (8) is not directly di erentiated but, as discussed in [20], a new linear system is formulated with the adjoint displacements vector appearing at RHS as: S f = u. (23) The system (23) is solved for the adjoint load vector f , while the transpose of the sti ness matrix S, previously computed in primal mode, appears at LHS. Therefore, the adjoint sti ness matrix is obtained from the system (24). S = u f . (24) i j j Finally, the system assembly process is algorithmically di erentiated, returning the grid sensitivity accumulation: d f dS X + = S + f . (25) sl dX dX sl sl The system (25) accumulates the grid sensitivities attributed to the mechanical solver. The adjoint temperatures deriving from Equations (21) and (22) are provided as input to the adjoint heat transfer solver at the next section, thus linking the structural solver to the iterative adjoint CHT process. 3.4. Adjoint Heat Transfer Solver Consistently with the previous section, the iterative linear system solver adopted in primal mode for the solution of Equation (6) is not directly di erentiated, whilst the adjoint load vector and the adjoint sti ness matrix are computed through the new linear systems in Equation (26). A b = T. (26a) A = T b . (26b) i j j i Equation (26a) shows the transposed sti ness matrix A accounting for the conductive and convective terms previously calculated in primal mode, while the adjoint temperature vector T from Section 3.3 is imposed at the RHS. The resulting adjoint load vector b is assigned to Equation (26b), together with the temperature solution stored at the last CHT loop in primal mode, contributing to the calculation of the adjoint sti ness matrix A. Similarly to Equation (25), the system assembly process is algorithmically di erentiated and the adjoint contributes to the solid grid coordinates are accumulated. Since the thermal solver and the mechanical solver share the same grid, a unique vector of solid grid sensitivities is accumulated and propagated backwards throughout the CHT workflow. The convective loading at the RHS of the system (6) accounts for the virtual bulk fluid temperature. Its adjoint counterpart T is computed from b (returned by Equation (26a)) and from the user-defined f l virtual heat transfer coecient h. e 0 e b = h T ! T = b h. (27) i f l i f l i T is not accumulated along the entire adjoint workflow but recomputed at every CHT loop in reverse f l mode and then passed to the adjoint hFFB routine. Int. J. Turbomach. Propuls. Power 2020, 5, 30 15 of 31 3.5. Adjoint hFFB Procedure (Solid! Fluid) The adjoint virtual bulk fluid temperature field T associated with the solid nodes is processed f l by algorithmic di erentiation, returning the adjoint T assigned to the neighboring fluid cells centers. f l Equation (28) reports the DWI procedure in primal mode on the left, with focus on the calculation of the solid temperature at node j, leveraging the information of the cluster of i = (1, ..., n) neighboring fluid cells. On the right, the backward di erentiation of the interpolation procedure results in the adjoint temperature field associated to the fluid domain. The operator “+=” stresses the fact each fluid cell may belong to the “neighboring cluster” of di erent solid nodes j, and therefore would accumulate their adjoint values. P T (i) n f l 2 2 i=1 (dist(i)) (dist(i)) ( ) ( ) ( ) T j = ! T i + = T j . (28) P P f l f l f l n 1 n 1 2 2 i=1 i=1 (dist(i)) (dist(i)) Equation (28) suggests also that additional contributions to the solid grid sensitivities X and to sl the fluid grid sensitivities X can be accumulated through the adjoint vector of mapped distances f l between each solid node and the corresponding fluid neighbors, dist(i), as detailed in Section 2.1. Thus, after deriving the adjoint distances dist(i) as T ( j)dT ( j)/d(dist(i)), the cells-nodes mapping f l f l routine is di erentiated and the contributions to the two grids are obtained. Finally, the algorithmic di erentiation of Equation (1) returns the adjoint temperature at the fluid walls T , which is used as input to the adjoint CFD solver at the next step, and the adjoint wall heat FW flux q , stored for a later accumulation. FW T = T > FW q f l FW < T = T ! > . (29) FW f l > 1 e q = T f l h FW 3.6. Adjoint Fluid Solver The partitioned coupling technique adopted in this work allows for non-intrusive calls to the in-house adjoint CFD solver. The flow field is initialized with the converged flow solution U from the corresponding CHT loop in primal mode, and the viscous wall temperatures in the ghost cells layers are updated again imposing the Neumann boundary condition, as reported in Equation (7). The adjoint fluid wall temperature T , resulting from the reverse hFFB procedure at the previous FW step, contributes to the linearization of the constraint function with respect to the conservative flow variables U and the fluid grid coordinates X . The adjoint equation discussed in [19] is repeated here f l for convenience: " # " # T T @R @J = , (30) @U @U with R as the non-linear residuals of the primal flow solver, indicating the unknown adjoint variables, and J the objective or constraint function of interest. Therefore, the RHS term linearized at the viscous walls exposed to the heat flux from the solid develops by the chain rule as follows: @J @J @V @V @J @T @V @V bnd domain FW bnd domain = = = @U @V @V @U @T @V @V @U bnd domain domain FW bnd domain domain (31) @T @V @V FW bnd domain T , FW @V @V @U bnd domain domain with V as the primitive variables computed at the boundary of interest, V as the corresponding bnd domain variables from the interior of the fluid domain, and the last term @V/@U indicating the transformation matrix from primitive to conservative variables. Int. J. Turbomach. Propuls. Power 2020, 5, 30 16 of 31 Similarly, the linearization of the cost function w.r.t. the fluid grid coordinates is the following: dJ @J @R = . (32) dX @X @X f l f l f l Therefore, the first term at RHS linearized at the viscous walls with heat flux results in: @J @J @V @J @T @V @T @V FW FW bnd bnd bnd = = = T . (33) FW @X @V @X @T @V @X @V @X f l bnd f l FW bnd f l bnd f l Equations (31) and (33) show the contributions of the adjoint fluid wall temperatures T FW calculated in Section 3.5 to the fluid grid sensitivities at the wall boundaries exposed to the heat flux. For the principle of the reverse accumulation, the wall heat flux term, entering in primal mode as the update of the boundary conditions in Equation (7), results in an adjoint fluid wall heat flux q that FW is computed once the adjoint CFD solver approaches full convergence. This contribution is summed up with the q returned in Section 3.5 by the reversed hFFB procedure. FW Dn Dn T = T + q ! q = q + T . (34) G1 D1 FW G1 FW FW k hFFB k f l f l A further contribution derives also from the second layer of ghost cells, not reported in Equation (34) for the sake of simplicity. 3.7. Adjoint hFFB Procedure (Fluid! Solid) The partitioned coupling scheme in reverse mode completes by transferring the adjoint terms from the fluid to the solid domain. q is processed by the di erentiated DWI procedure, similarly to FW Equation (28), and returns the adjoint heat flux q associated with the solid grid nodes. Consistently SW with the previous call, additional contributions are accumulated to the solid grid sensitivities X and sl to the fluid grid sensitivities X through the adjoint vector of mapped distances between each fluid f l cell center and the neighboring solid nodes. Finally, the routine adopted to calculate the wall heat flux within the solid domain in Equation (3) is algorithmically di erentiated, accounting only for the nodes laying on the interface. This process results in the new field of adjoint solid temperatures T , passed to the adjoint heat transfer solver SW described in Section 3.4. The whole adjoint CHT process is iteratively repeated from Sections 3.4–3.7 for as many loops as the ones performed during the primal computation till convergence. Once walked through the entire workflow in reverse mode, X and X , enclosing the contributions f l sl from the entire multidisciplinary chain, are introduced in Equation (12), and the constraint function sensitivities w.r.t. the design variables are finally computed. Hence, dJ/d is evaluated by the SQP-based optimizer and the geometry is updated, opening the path to a new thermo-mechanical evaluation within the history of the shape optimization problem. 4. Validation 4.1. Flat Plate (Primal Mode) The validation of the CHT process in primal mode is performed with the consideration of the analytic solution o ered by [33] for the conjugate problem applied to a flat plate subjected to an incompressible flow. The mesh characteristics and boundary conditions are reported in Table 1, while the fluid and solid meshes are qualitatively shown in Figure 8. The thermal conductivity adopted for the solid phase is aimed at obtaining an average Biot number around unity. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 16 of 31 3.7. Adjoint hFFB Procedure (Fluid → Solid) The partitioned coupling scheme in reverse mode completes by transferring the adjoint terms from the fluid to the solid domain. is processed by the differentiated DWI procedure, similarly to Equation (28), and returns the adjoint heat flux associated with the solid grid nodes. Consistently with the previous call, additional contributions are accumulated to the solid grid sensitivities and to the fluid grid sensitivities through the adjoint vector of mapped distances between each fluid cell center and the neighboring solid nodes. Finally, the routine adopted to calculate the wall heat flux within the solid domain in Equation (3) is algorithmically differentiated, accounting only for the nodes laying on the interface. This process results in the new field of adjoint solid temperatures , passed to the adjoint heat transfer solver described in Section 3.4. The whole adjoint CHT process is iteratively repeated from Section 3.4 to Section 3.7 for as many loops as the ones performed during the primal computation till convergence. Int. J. Turbomach. Propuls. Power 2020, 5, 30 17 of 31 Once walked through the entire workflow in reverse mode, and , enclosing the contributions from the entire multidisciplinary chain, are introduced in Equation (12), and the Table 1. Flat plate domain characteristics and boundary conditions. constraint function sensitivities w.r.t. the design variables are finally computed. Hence, ⁄ is evaluated by the SQP-based optimizer and the geometry is updated, opening the path to a new Domain Settings Value thermo-mechanical evaluation within the history of the shape optimization problem. Fluid domain length/height 0.25 m/0.1 m Fluid mesh cells count 365,000 4. Validation Plate thickness/length 0.01 m/0.2 m Solid mesh nodes count/elements count 35,000 4.1. Flat Plate (Primal Mode) Fluid type Air Inlet flow total pressure 1.03 10 Pa The validation of the CHT process in primal mode is performed with the consideration of the Inlet flow temperature 1000 K analytic solution offered by [33] for the conjugate problem applied to a flat plate subjected to an Outlet flow static pressure 1.029 10 Pa incompressible flow. The mesh characteristics and boundary conditions are reported in Table 1, while Plate temperature at bottom face 600 K Plate thermal conductivity 0.29 W/m K the fluid and solid meshes are qualitatively shown in Figure 8. The thermal conductivity adopted for 100 W/m K Virtual heat transfer coecient h the solid phase is aimed at obtaining an average Biot number around unity. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 17 of 31 Figure 8. Fluid and solid meshes for the conjugate problem applied to a flat plate. Figure 8. Fluid and solid meshes for the conjugate problem applied to a flat plate. T The he cconver onverggence ence hihistory story of of thethe max maximum imum temp temperatur erature dife fer di en c er e ence betwe between en two suc two cess successive ive fluid– Table 1. Flat plate domain characteristics and boundary conditions. s fluid–solid olid iteratiiterations ons isL rep is or reported ted in Fi in gur Figur e 9, ew 9h , wher ere a eth arthr esh eshold old of of 1K 1K isis se selected lected aas s th the e sst ta abilization bilization c criterion riterion f for or th the e m model. odel. Domain Settings Value s s L = T T , (35) Fluid domain length/height 0.25 m/0.1 m n+1 max = | − | , (35) Fluid mesh cells count 365,000 Plate thickness/length 0.01 m/0.2 m Solid mesh nodes count/elements count 35,000 Fluid type Air Inlet flow total pressure 1.03 × 10 Pa Inlet flow temperature 1000 K Outlet flow static pressure 1.029 × 10 Pa Plate temperature at bottom face 600 K Plate thermal conductivity 0.29 W/m K Virtual heat transfer coefficient ℎ 100 W/m K Figure 9. Convergence history of L . Figure 9. Convergence history of L1 ∞. The distribution of temperature at the interface between the two domains is presented in Figure 10a, The distribution of temperature at the interface between the two domains is presented in Figure comparing Luikov’s di erential heat transfer (DHT) solution with the numerical one. Figure 10b 10a, comparing Luikov’s differential heat transfer (DHT) solution with the numerical one. Figure 10b shows the temperature variation in a vertical section located at x = 0.05m. In both cases, a sucient shows the temperature variation in a vertical section located at x = 0.05m. In both cases, a sufficient agreement between the numerical results and the analytic solution is obtained within the limitations agreement between the numerical results and the analytic solution is obtained within the limitations of the DHT approach, which does not capture the e ect of the solid conductivity in the streamwise of the DHT approach, which does not capture the effect of the solid conductivity in the streamwise direction, as explained in [55]. direction, as explained in [55]. Figure 10. CHT problem for flat plate: (a) temperature at the domain interface, (b) temperature profile at x = 0.05 m. Numerical vs. analytic solution. 4.2. Rotor Mesh Sensitivity Analysis A mesh sensitivity study for the CHT analysis is performed on a three-dimensional test case presented in [19], considering a radial turbine impeller with 10 blades and an inlet diameter of 50 mm. A two-dimensional sketch of the domain on the meridional plane of the rotor is reported in Figure 11, along with the locations of the imposed boundary conditions. The method of the characteristics [56] is adopted to establish the number of physical conditions to be assigned at the inflow and outflow boundaries. The total upstream temperature and pressure are specified at the inlet of the domain, along with flow velocity components in the corresponding coordinate system. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 17 of 31 The convergence history of the maximum temperature difference between two successive fluid– solid iterations is reported in Figure 9, where a threshold of 1K is selected as the stabilization criterion for the model. | | = − , (35) Figure 9. Convergence history of L∞. The distribution of temperature at the interface between the two domains is presented in Figure 10a, comparing Luikov’s differential heat transfer (DHT) solution with the numerical one. Figure 10b shows the temperature variation in a vertical section located at x = 0.05m. In both cases, a sufficient agreement between the numerical results and the analytic solution is obtained within the limitations of the DHT approach, which does not capture the effect of the solid conductivity in the streamwise Int. J. Turbomach. Propuls. Power 2020, 5, 30 18 of 31 direction, as explained in [55]. Figure 10. CHT problem for flat plate: (a) temperature at the domain interface, (b) temperature profile Figure 10. CHT problem for flat plate: (a) temperature at the domain interface, (b) temperature profile at x = 0.05 m. Numerical vs. analytic solution. at x = 0.05 m. Numerical vs. analytic solution. 4.2. Rotor Mesh Sensitivity Analysis 4.2. Rotor Mesh Sensitivity Analysis A mesh sensitivity study for the CHT analysis is performed on a three-dimensional test case A mesh sensitivity study for the CHT analysis is performed on a three-dimensional test case pr p esented resented in in [19 [1], 9]considering , considering a radial a radia turbine l turbinimpeller e impelle wi r w th ith 10 1 blades 0 blade and s anan d a inlet n inldiameter et diamete of r 50 of mm. 50 Am two-dimensional m. A two-dimensketch sional s of ke the tch domain of the don omthe ain meridional on the meri plane dionaof l ptlhe ane r otor of th is e rreported otor is rin epFigur orted ein 11 , Figure 11, along with the locations of the imposed boundary conditions. The method of the along with the locations of the imposed boundary conditions. The method of the characteristics [56] iscadopted haracteristo ticestablish s [56] is athe dopnumber ted to esof tabphysical lish the n conditions umber of pto hybe sica assigned l conditio at nsthe to inflow be assig and ned outflow at the inflow and outflow boundaries. The total upstream temperature and pressure are specified at the boundaries. The total upstream temperature and pressure are specified at the inlet of the domain, Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 18 of 31 inlet of the domain, along with flow velocity components in the corresponding coordinate system. along with flow velocity components in the corresponding coordinate system. At the subsonic outflow boundary At the su,ba so single nic outf flow low variable boundis ary imposed, , a single specifically flow variathe ble downstr is impoeam sed, static specifpr ica essur lly th e.e In do conclusion, wnstream the boundary conditions applied to the problem are summarized in Table 2, in which the impeller static pressure. In conclusion, the boundary conditions applied to the problem are summarized in r T otational able 2, inspeed whichis th also e im included. peller rotational speed is also included. Figure 11. Turbine rotor meridional view and boundary condition locations. Figure 11. Turbine rotor meridional view and boundary condition locations. Table 2. Turbine rotor boundary conditions. Boundary Conditions Value Inlet total pressure p0 173 kPa Inlet total temperature T0 1080 K Inlet flow angle α from radial direction 62 deg Outlet static pressure ps 101 kPa Blade rotational speed ω 140,000 RPM The interactions of the rotor fluid and solid meshes are explored over three levels of refinement, as summarized in Table 3. Additionally, three values of the virtual heat transfer coefficient ℎ are investigated in order to determine a suitable trade-off between stability and computational time. These factors are explored following the combinations expressed by the L9 Orthogonal Array method [57], returning nine test cases (Table 4). Table 3. Fluid and solid mesh refinements, ℎ levels. Factors Levels Value Fluid domain “coarse”—cells count 0.8 M Fluid domain “mid”—cells count 1.3 M Fluid domain “fine”—cells count 2.1 M Solid domain “coarse”—nodes count 105 k Solid domain “mid”—nodes count 295 k Solid domain “fine”—nodes count 1.1 M ℎ low–mid–high (W/m K) 800–1000–1300 Int. J. Turbomach. Propuls. Power 2020, 5, 30 19 of 31 Table 2. Turbine rotor boundary conditions. Boundary Conditions Value Inlet total pressure p 173 kPa Inlet total temperature T 1080 K Inlet flow angle from radial direction 62 deg Outlet static pressure p 101 kPa Blade rotational speed ! 140,000 RPM The interactions of the rotor fluid and solid meshes are explored over three levels of refinement, as summarized in Table 3. Additionally, three values of the virtual heat transfer coecient h are investigated in order to determine a suitable trade-o between stability and computational time. These factors are explored following the combinations expressed by the L9 Orthogonal Array method [57], returning nine test cases (Table 4). Table 3. Fluid and solid mesh refinements, h levels. Factors Levels Value Fluid domain “coarse”—cells count 0.8 M Fluid domain “mid”—cells count 1.3 M Fluid domain “fine”—cells count 2.1 M Solid domain “coarse”—nodes count 105 k Solid domain “mid”—nodes count 295 k Solid domain “fine”—nodes count 1.1 M e 2 800–1000–1300 h low–mid–high (W/m K) Table 4. L9 Orthogonal Array applied to the turbine rotor CHT analysis. Test Case Number CFD Mesh FEM Mesh h [W/m K] 1 coarse coarse 800 2 coarse mid 1000 3 coarse fine 1300 4 mid coarse 1000 5 mid mid 1300 6 mid fine 800 7 fine coarse 1300 8 fine mid 800 9 fine fine 1000 All the simulations are run till convergence at a relative residual drop of 10 for the computational fluid dynamics (CFD) solver and 10 for the FEM solver, while the CHT workflow is stopped for a maximum deviation in wall temperature between two successive fluid-solid iterations L below 1 K. y+ values below unity are obtained for the “mid” and “fine” levels of the fluid grid refinement, while the coarsest one exhibits a peak in y+ around 2.5 at the blade leading edge. Concerning the maximum number of nodes in the solid mesh, the code was tested up to the finest possible grid before incurring in memory leakage issues with the adopted Conjugate Gradient iterative solver. The results in Table 5 demonstrate a low sensitivity of the maximum temperature in the solid domain due to the settings applied to the coupling process, except for the first test case with coarse meshes at both the fluid and solid side. A critical aspect is represented by the quality of the interpolation of the information between the two domains realized by the fine-tuned distribution of virtual grid points on the interface at fluid side, as discussed in Section 2.1. Moreover, the finest solid mesh increases the resolution of the convective loading, resulting in more accurate temperature predictions in the material and, therefore, also more reliable heat fluxes returned to the fluid. Figure 14 shows two cases of solid mesh size, the coarse one (Figure 14a) and the intermediate one (Figure 14b), with the addition Int. J. Turbomach. Propuls. Power 2020, 5, 30 20 of 31 of the trailing edge refinement described hereafter. In the case of finer solid grids, the interpolation procedure returns a richer temperature pattern at the interface, closely mirroring the fluid conditions at the walls, whilst the coarsest one approximates the thermal loading distribution with the highest deviations localized in the blade tip region, where large secondary flows and leakages are present, and at the trailing edge. In this respect, a surface temperature analysis at a cross section located at about 25% of the chord was performed in order to evaluate the integral of the temperature deviations between each investigated test case w.r.t. test case 9. The axial position of the planar section was chosen in a region of high convective perturbations due to the presence of a significant flow detachment at the blade leading edge. The results, summarized in Table 5, are supported by the example in Figure 12, showing the temperature contours for case 1 and case 9. It is interesting to note that, in both set-ups, the blade tip returns significant thermal gradients because of the thinner geometry exposed to the flow vortices. However, the coarsest mesh overestimates the temperature drop, resulting in the integral reported in Table 5. Table 5. Fluid–solid grid sensitivity: summary of CHT computations. #CHT Loops to Norm. Computational Computational Time: Maximum Solid Delta Temperature Case# Convergence [-] Time for CHT Iteration CFD—FEM w.r.t. Total Temperature [K] Integral 1 11 1.0 97.0–0.5% 1019 0.121 2 10 1.02 95.2–1.9% 1025 0.053 3 9 1.36 71.3–25.0% 1027 0.045 4 8 1.76 98.3–0.3% 1026 0.092 5 10 1.78 97.2–1.1% 1028 0.038 6 12 2.12 81.6–16.1% 1028 0.023 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 20 of 31 7 11 2.4 98.6–0.2% 1026 0.074 8 11 2.44 97.2–0.8% 1028 0.019 99 8 8 2.76 2 .76× 86.5–12.4% 86.5–12.4% 1028 1028 0.000 0.000 Figure 12. Temperature contours for test case 1 and test case 9 at the planar cross section on the blade. Figure 12. Temperature contours for test case 1 and test case 9 at the planar cross section on the blade. Finally, considerations about the computational time are synthesized in Table 5, with a dominant Finally, considerations about the computational time are synthesized in Table 5, with a dominant portion covered by the CFD solver and a significant rise in overhead emerging from the FEM heat portion covered by the CFD solver and a significant rise in overhead emerging from the FEM heat transfer computations at the finest grid level. The normalization refers to the total duration of a single transfer computations at the finest grid level. The normalization refers to the total duration of a single CHT iteration, including also the hFFB procedure. The comparison is performed w.r.t. the reference CHT iteration, including also the hFFB procedure. The comparison is performed w.r.t. the reference duration of test case 1, presenting the coarsest meshes at both fluid and solid side. duration of test case 1, presenting the coarsest meshes at both fluid and solid side. The interactions among the three factors in the Orthogonal Array from Table 4 are studied through the compounded signal S, defined as: = + (1 − ) , (36) with the term A enclosing the normalized difference between the maximum temperature predicted by the test case of interest and test case 9 (showing the finest grids), B representing the normalized computational time, and ω as the weighting coefficient. In the current study, ω = 0.7 in order to bias the objective function towards the accuracy of the coupling process. Indeed, the factors levels finally expected from this study are the ones minimizing the signal S. The analysis of the Orthogonal Array [57], resulting in Figure 13, reveals the dependence of the signal S from the three variables and their correspondent levels. The chart illustrates the virtual heat transfer coefficient ℎ is selected at its highest value within the stability region of the Biot number discussed in [46], as it influences the B term in Equation (36), determining the rate of convergence of the CHT coupling process. Indeed, the current problem exhibits an optimal ℎ = 1000 W/m K, since the highest value approaches the limit of the stability region, with incipient oscillations in the value of heat flux exchanged at the interface, before reaching convergence. The solid grid size, instead, characterizes the accuracy of the interpolation of the quantities passed between the two domains through the DWI process and affects the A term in Equation (36). In general, finer solid grids are favored by the current analysis. Remarkably, the fluid mesh size follows an opposite trend because its influence on the A term is less pronounced, as the coarsest level already provides good-quality solutions. On the other hand, the signal S is driven by the increase in computational overhead experienced with the fluid mesh refinement. Int. J. Turbomach. Propuls. Power 2020, 5, 30 21 of 31 The interactions among the three factors in the Orthogonal Array from Table 4 are studied through the compounded signal S, defined as: S = ! A + (1 !) B, (36) with the term A enclosing the normalized di erence between the maximum temperature predicted by the test case of interest and test case 9 (showing the finest grids), B representing the normalized computational time, and ! as the weighting coecient. In the current study, ! = 0.7 in order to bias the objective function towards the accuracy of the coupling process. Indeed, the factors levels finally expected from this study are the ones minimizing the signal S. The analysis of the Orthogonal Array [57], resulting in Figure 13, reveals the dependence of the signal S from the three variables and their correspondent levels. The chart illustrates the virtual heat transfer coecient h is selected at its highest value within the stability region of the Biot number discussed in [46], as it influences the B term in Equation (36), determining the rate of convergence of the CHT coupling process. Indeed, the current problem exhibits an optimal h = 1000 W/m K, since the highest value approaches the limit of the stability region, with incipient oscillations in the value of heat flux exchanged at the interface, before reaching convergence. The solid grid size, instead, characterizes the accuracy of the interpolation of the quantities passed between the two domains through the DWI process and a ects the A term in Equation (36). In general, finer solid grids are favored by the current analysis. Remarkably, the fluid mesh size follows an opposite trend because its influence on the A term is less pronounced, as the coarsest level already provides good-quality solutions. On the other hand, the signal S is driven by the increase in computational overhead experienced with the fluid mesh refinement. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 21 of 31 Figure 13. Normalized signal S dependence from factor levels in the Orthogonal Array. Figure 13. Normalized signal S dependence from factor levels in the Orthogonal Array. Since the weighting coecient ! privileges the accuracy of the CHT coupling process, the final Since the weighting coefficient ω privileges the accuracy of the CHT coupling process, the final selection of the factors levels results in the intermediate values for the refinements of both the selection of the factors levels results in the intermediate values for the refinements of both the grids grids and for the virtual heat transfer coecient. However, it is recognized that smaller solid and for the virtual heat transfer coefficient. However, it is recognized that smaller solid element sizes element sizes improve the stability of the CHT coupling process because it avoids local poor improve the stability of the CHT coupling process because it avoids local poor quality in the quality in the discretization of the blade surface in correspondence with thin regions (Figure 14a), discretization of the blade surface in correspondence with thin regions (Figure 14a), potentially potentially inducing drops in the local Biot number and inconsistencies with the selected h value from inducing drops in the local Biot number and inconsistencies with the selected ℎ value from a stability a stability standpoint. Therefore, the issue is addressed by the generation of a “hybrid” configuration standpoint. Therefore, the issue is addressed by the generation of a “hybrid” configuration (Figure (Figure 14b), envisaging a local solid mesh refinement in correspondence with the blade tip surface 14b), envisaging a local solid mesh refinement in correspondence with the blade tip surface and and trailing edge, whilst maintaining the intermediate element size in the rest of the domain. This last trailing edge, whilst maintaining the intermediate element size in the rest of the domain. This last setup increases the mesh density to a total node count of about 420 k. setup increases the mesh density to a total node count of about 420 k. Figure 14. Comparison of solid mesh refinements: (a) coarse grid, (b) hybrid-mid grid. Finally, the Orthogonal Array analysis is repeated, assigning the integral of temperature deviations to the A term in Equation (36). The outcome, shown in Figure 15, confirms the previous considerations, except for the fluid mesh size, which promotes the trend towards the coarsest mesh because it still provides sufficiently accurate results. However, since the difference between the coarse and intermediate meshes is moderate, the previous selection of the medium size grid will be pursued for the sake of improved prediction accuracy. Based on such considerations, the settings returned by the analysis of the Orthogonal Array are considered for the further assessments of the coupling problem. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 21 of 31 Figure 13. Normalized signal S dependence from factor levels in the Orthogonal Array. Since the weighting coefficient ω privileges the accuracy of the CHT coupling process, the final selection of the factors levels results in the intermediate values for the refinements of both the grids and for the virtual heat transfer coefficient. However, it is recognized that smaller solid element sizes improve the stability of the CHT coupling process because it avoids local poor quality in the discretization of the blade surface in correspondence with thin regions (Figure 14a), potentially inducing drops in the local Biot number and inconsistencies with the selected ℎ value from a stability standpoint. Therefore, the issue is addressed by the generation of a “hybrid” configuration (Figure 14b), envisaging a local solid mesh refinement in correspondence with the blade tip surface and Int. J. Turbomach. Propuls. Power 2020, 5, 30 22 of 31 trailing edge, whilst maintaining the intermediate element size in the rest of the domain. This last setup increases the mesh density to a total node count of about 420 k. Figure 14. Comparison of solid mesh refinements: (a) coarse grid, (b) hybrid-mid grid. Figure 14. Comparison of solid mesh refinements: (a) coarse grid, (b) hybrid-mid grid. Finally, the Orthogonal Array analysis is repeated, assigning the integral of temperature Finally, the Orthogonal Array analysis is repeated, assigning the integral of temperature deviations deviations to the A term in Equation (36). The outcome, shown in Figure 15, confirms the previous to the A term in Equation (36). The outcome, shown in Figure 15, confirms the previous considerations, considerations, except for the fluid mesh size, which promotes the trend towards the coarsest mesh except for the fluid mesh size, which promotes the trend towards the coarsest mesh because it because it still provides sufficiently accurate results. However, since the difference between the still provides suciently accurate results. However, since the di erence between the coarse and coarse and intermediate meshes is moderate, the previous selection of the medium size grid will be intermediate meshes is moderate, the previous selection of the medium size grid will be pursued for pursued for the sake of improved prediction accuracy. the sake of improved prediction accuracy. Based on such considerations, the settings returned by the analysis of the Orthogonal Array are Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 22 of 31 considered for the further assessments of the coupling problem. Figure 15. Normalized signal S dependence from factor levels in Orthogonal Array—integral of Figure 15. Normalized signal S dependence from factor levels in Orthogonal Array—integral of temperate deviation assigned to term A. temperate deviation assigned to term A. 4.3. Gradients Accuracy Evaluation Based on such considerations, the settings returned by the analysis of the Orthogonal Array are considered for the further assessments of the coupling problem. The present work accounts for the manual differentiation of the multidisciplinary process shown in Figure 6. The sensitivities are accurately computed by the reverse Algorithmic Differentiation 4.3. Gradients Accuracy Evaluation technique [54] for what concerns the FEM solvers and the hFFB process, while the partitioned coupling approach opens the path to the direct integration of the in-house adjoint CFD solver. The present work accounts for the manual di erentiation of the multidisciplinary process shown The validation of the differentiated FEM solvers is presented herein with reference to the rotor in Figure 6. The sensitivities are accurately computed by the reverse Algorithmic Di erentiation test case from Section 4.2. The adjoint sensitivities ⁄ of the maximum temperature and of the technique [54] for what concerns the FEM solvers and the hFFB process, while the partitioned coupling maximum von Mises stress including the thermal strain w.r.t. selected design parameters (referenced approach opens the path to the direct integration of the in-house adjoint CFD solver. in Table 6 and Figure 16) are compared to the correspondent gradients computed by the Finite The validation of the di erentiated FEM solvers is presented herein with reference to the rotor Differences (FD) technique. The latter accounts for evaluations by a central differencing scheme, test case whofr seom optiSection mal step4.2 siz.e The is seaadjoint rched wisensitivities th the aim of a@cJc/ ur@ ate of grthe adiemaximum nts computatemperatur tions. An exe em and plary of the −6 maximum case isvon repo Mises rted in str Fess igurincluding e 17, showthe ing th thermal e identistrain ficatiow n .r o.t. f a selec suitab ted le sdesign tep sizeparameters at the value(r oefer f 10 enced m in for the seventh design variable in the framework of thermal evaluations by the heat transfer solver. Table 6 and Figure 16) are compared to the correspondent gradients computed by the Finite Di erences The magnitude of the perturbation must avoid too-small values, resulting in round-off errors, and (FD) technique. The latter accounts for evaluations by a central di erencing scheme, whose optimal too large values as well, because they introduce significant truncation errors. This outcome results step size is searched with the aim of accurate gradients computations. An exemplary case is reported from the evaluation of the cost function at each step size by solving the linear system obtained after the perturbation of the mesh through the morphing technique described in [20]. Table 6. Design variables adopted in the dJ/dα gradient validation. Design Variables α 1 Rotor hub meridional contour: Y-coordinate at 20% chord; 2 Rake angle; 3 Back-plate thickness; 4 Turbine shaft diameter; 5 Turbine shaft length; 6 Rotor maximum diameter; 7 Blade height at leading edge; 8 Back-plate/shaft connection axial position; 9 Blade hub thickness ; 10 Blade hub fillet radius. Int. J. Turbomach. Propuls. Power 2020, 5, 30 23 of 31 in Figure 17, showing the identification of a suitable step size at the value of 10 m for the seventh design variable in the framework of thermal evaluations by the heat transfer solver. The magnitude of the perturbation must avoid too-small values, resulting in round-o errors, and too large values as well, because they introduce significant truncation errors. This outcome results from the evaluation of the cost function at each step size by solving the linear system obtained after the perturbation of the mesh through the morphing technique described in [20]. Table 6. Design variables adopted in the dJ/d gradient validation. Design Variables 1 Rotor hub meridional contour: Y-coordinate at 20% chord; 2 Rake angle; 3 Back-plate thickness; 4 Turbine shaft diameter; 5 Turbine shaft length; 6 Rotor maximum diameter; 7 Blade height at leading edge; 8 Back-plate/shaft connection axial position; 9 Blade hub thickness; 10 Blade hub fillet radius. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 23 of 31 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 23 of 31 Figure 16. Design variables from Table 2. Figure 16. Design variables from Table 2. Figure 16. Design variables from Table 2. Figure Fig 17. ure Step 17. size Step evaluation size evaluat for ionthe for computation the computatiof ondJ o/f dd J/d by α Finite by Fin Di ite Di erences: fferences Heat : Hea Transfer t Transfer solver , Figure 17. Step size evaluation for the computation of dJ/dα by Finite Differences: Heat Transfer solver, variable 7. variable 7. solver, variable 7. The sensitivities of the maximum solid temperature to perturbations of the ten design variables The sensitivities of the maximum solid temperature to perturbations of the ten design variables are summarized in Figure 18a, which shows a close agreement between the two methods, both in are summarized in Figure 18a, which shows a close agreement between the two methods, both in sign and magnitude. To understand the sensitivity analysis in detail, let us consider the thermal paths sign and magnitude. To understand the sensitivity analysis in detail, let us consider the thermal paths in the turbine rotor experiencing the convective loading resulting from the operative condition in the turbine rotor experiencing the convective loading resulting from the operative condition reported in Table 2, along with a convective condition on the back-plate surface with a uniform fluid reported in Table 2, along with a convective condition on the back-plate surface with a uniform fluid temperature of 950 K and a Dirichlet boundary condition of 500 K assigned to the extreme section of temperature of 950 K and a Dirichlet boundary condition of 500 K assigned to the extreme section of the shaft. the shaft. Int. J. Turbomach. Propuls. Power 2020, 5, 30 24 of 31 The sensitivities of the maximum solid temperature to perturbations of the ten design variables are summarized in Figure 18a, which shows a close agreement between the two methods, both in sign and magnitude. To understand the sensitivity analysis in detail, let us consider the thermal paths in the turbine rotor experiencing the convective loading resulting from the operative condition reported in Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 24 of 31 Table 2, along with a convective condition on the back-plate surface with a uniform fluid temperature of Int950 . J. Tu K rbo and macha . Pr Dirichlet opuls. Powboundary er 2020, 5, x Fcondition OR PEER REV ofIEW 500 K assigned to the extreme section of the shaft. 24 of 31 Figure 18. Comparison of the dJ/dα sensitivities of the solid maximum temperature (a) and the Figure 18. Comparison of the dJ/d sensitivities of the solid maximum temperature (a) and the maximum von Mises stress (b): gradients computed by the adjoint method vs. Finite Differences. Figure 18. Comparison of the dJ/dα sensitivities of the solid maximum temperature (a) and the maximum von Mises stress (b): gradients computed by the adjoint method vs. Finite Di erences. maximum von Mises stress (b): gradients computed by the adjoint method vs. Finite Differences. Figure 19 reveals that the maximum temperature is detected at the nodes in the region of the Figure 19 reveals that the maximum temperature is detected at the nodes in the region of the hub, hub, in proximity with the leading edge. The computation of such constraint through a p-norm Figure 19 reveals that the maximum temperature is detected at the nodes in the region of the in proximity with the leading edge. The computation of such constraint through a p-norm function function returns a marked sensitivity from the rotor back-plate thickness, whose enlargement would hub, in proximity with the leading edge. The computation of such constraint through a p-norm returns a marked sensitivity from the rotor back-plate thickness, whose enlargement would reduce reduce the influence of the heat sink located at the back-plate outer surface. Similarly, an increase in function returns a marked sensitivity from the rotor back-plate thickness, whose enlargement would the influence of the heat sink located at the back-plate outer surface. Similarly, an increase in blade blade height at the leading edge would extend the thermal path to the colder areas highlighted at the reduce the influence of the heat sink located at the back-plate outer surface. Similarly, an increase in height at the leading edge would extend the thermal path to the colder areas highlighted at the tip, tip, while an elongation of the shaft itself would reduce the gradient field. On the contrary, an increase blade height at the leading edge would extend the thermal path to the colder areas highlighted at the while an elongation of the shaft itself would reduce the gradient field. On the contrary, an increase in in the turbine shaft diameter, an advancement of the axial position of the intersection between the tip, while an elongation of the shaft itself would reduce the gradient field. On the contrary, an increase the turbine shaft diameter, an advancement of the axial position of the intersection between the rotor rotor back-plate and the shaft, and an enlargement of the blade hub thickness would favor the cooling in the turbine shaft diameter, an advancement of the axial position of the intersection between the back-plate and the shaft, and an enlargement of the blade hub thickness would favor the cooling of of the hub upper region. Such sensitivities provide the descent direction for a constrained rotor back-plate and the shaft, and an enlargement of the blade hub thickness would favor the cooling the hub upper region. Such sensitivities provide the descent direction for a constrained optimization optimization problem aimed at limiting the maximum rotor temperature for the sake of extended of the hub upper region. Such sensitivities provide the descent direction for a constrained problem aimed at limiting the maximum rotor temperature for the sake of extended lifetime. lifetime. optimization problem aimed at limiting the maximum rotor temperature for the sake of extended lifetime. Figure 19. Solid temperature on turbine rotor (operative condition from Table 2). Figure 19. Solid temperature on turbine rotor (operative condition from Table 2). Figure 19. Solid temperature on turbine rotor (operative condition from Table 2). Similarly, the accuracy of the adjoint structural solver sensitivities is assessed by comparison with the same gradients computed by Finite Differences. The results reported in Figure 18b confirm Similarly, the accuracy of the adjoint structural solver sensitivities is assessed by comparison the suitability of the manually differentiated framework. with the same gradients computed by Finite Differences. The results reported in Figure 18b confirm Moreover, the time for the computation of the adjoint-based gradients for the heat transfer solver the suitability of the manually differentiated framework. and the structural solver was measured. If X is the time required for the computation of the Moreover, the time for the computation of the adjoint-based gradients for the heat transfer solver multidisciplinary workflow in primal mode and the problem accounts for n design variables, the and the structural solver was measured. If X is the time required for the computation of the calculation of the gradients by Finite Differences would approximately cost n ∙ X. On the contrary, multidisciplinary workflow in primal mode and the problem accounts for n design variables, the the application of the adjoint method herein described costs about 8.6X for the heat transfer module calculation of the gradients by Finite Differences would approximately cost n ∙ X. On the contrary, the application of the adjoint method herein described costs about 8.6X for the heat transfer module Int. J. Turbomach. Propuls. Power 2020, 5, 30 25 of 31 Similarly, the accuracy of the adjoint structural solver sensitivities is assessed by comparison with the same gradients computed by Finite Di erences. The results reported in Figure 18b confirm the suitability of the manually di erentiated framework. Moreover, the time for the computation of the adjoint-based gradients for the heat transfer Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 25 of 31 solver and the structural solver was measured. If X is the time required for the computation of and 2.3X for the structural analysis, in fair agreement with the original expectations. Such costs the multidisciplinary workflow in primal mode and the problem accounts for n design variables, mostly depend on the assembly process of the differentiated system (Equation (25)) rather than the the calculation of the gradients by Finite Di erences would approximately cost n X. On the contrary, solver itself. the application of the adjoint method herein described costs about 8.6X for the heat transfer module Finally, the loose-coupling approach adopted for the development of the CHT workflow allows and 2.3X for the structural analysis, in fair agreement with the original expectations. Such costs mostly the direct integration of the in-house adjoint CFD solver, and the accuracy of the computed gradients depend on the assembly process of the di erentiated system (Equation (25)) rather than the solver itself. was already demonstrated by the same author. Finally, the loose-coupling approach adopted for the development of the CHT workflow allows the direct integration of the in-house adjoint CFD solver, and the accuracy of the computed gradients 5. Results and Discussion was already demonstrated by the same author. The design work aims at reducing the thermo-mechanical stresses in the turbine impeller 5. Results and Discussion presented in Section 4.2. The operative condition in Table 2 is considered, along with a Dirichlet The design work aims at reducing the thermo-mechanical stresses in the turbine impeller presented boundary condition with a temperature of 500 K imposed at the extreme cross section of the shaft, in Section 4.2. The operative condition in Table 2 is considered, along with a Dirichlet boundary emulating the heat sink provided by the oil cooling circuit in the turbocharger bearing housing. The condition with a temperature of 500 K imposed at the extreme cross section of the shaft, emulating the contribution of the thermal strains introduced by the present work improves the accuracy of the heat sink provided by the oil cooling circuit in the turbocharger bearing housing. The contribution of computation of the von Mises stresses, whose gradients provide information for the geometry the thermal strains introduced by the present work improves the accuracy of the computation of the modifications. von Mises stresses, whose gradients provide information for the geometry modifications. According to the grids selection anticipated in Section 4.2, the fluid domain is discretized with a According to the grids selection anticipated in Section 4.2, the fluid domain is discretized with a multi-block structured mesh of about 1.3M cells with boundary layer refinement. The solid domain multi-block structured mesh of about 1.3 M cells with boundary layer refinement. The solid domain accounts for an unstructured grid of approximately 420 k nodes and second-order tetrahedral accounts for an unstructured grid of approximately 420 k nodes and second-order tetrahedral elements. elements. Figure 20 presents the temperature distribution in the baseline geometry under the selected Figure 20 presents the temperature distribution in the baseline geometry under the selected steady-state operative condition. A significant thermal gradient is identified at the connection between steady-state operative condition. A significant thermal gradient is identified at the connection the shaft and the rotor back-plate, with a correspondent local variation of thermal strains according to between the shaft and the rotor back-plate, with a correspondent local variation of thermal strains Equation (9). This is reflected in increased localized stresses, as demonstrated in Figure 21. In fact, according to Equation (9). This is reflected in increased localized stresses, as demonstrated in Figure Figure 21a shows the mechanical pattern of the baseline geometry, whose structural computation is 21. In fact, Figure 21a shows the mechanical pattern of the baseline geometry, whose structural performed considering only the centrifugal loading. On the other hand, Figure 21b reveals the new computation is performed considering only the centrifugal loading. On the other hand, Figure 21b stress distribution accounting also for the thermal field, resulting from the convective input and the reveals the new stress distribution accounting also for the thermal field, resulting from the convective heat sink at the shaft. Indeed, a local deviation in von Mises stresses of about 20% is identified in the input and the heat sink at the shaft. Indeed, a local deviation in von Mises stresses of about 20% is region around the connection with the back-plate. identified in the region around the connection with the back-plate. Figure 20. Solid temperature distribution with rotor experiencing convective loading from the exhaust Figure 20. Solid temperature distribution with rotor experiencing convective loading from the gases and shaft cooling (baseline geometry). exhaust gases and shaft cooling (baseline geometry). Int. J. Turbomach. Propuls. Power 2020, 5, 30 26 of 31 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 26 of 31 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 26 of 31 Figure 21. Normalized von Mises stresses pattern comparison in baseline layout: (a) mechanical Figure Figure21. 21Normalized . Normalizevon d vo Mises n Misstr esesses strespattern ses pattcomparison ern comparin iso baseline n in bas layout: eline la (a y)omechanical ut: (a) mecversus hanical versus (b) thermo-mechanical. (b v) er thermo-mechanical. sus (b) thermo-mechanical. The suitability of the process described in Section 3 is tested with the computation of the The suitability of the process described in Section 3 is tested with the computation of the The suitability of the process described in Section 3 is tested with the computation of the sensitivities of the thermo-mechanical constraint w.r.t. the grid coordinates, as reported in Figure 22, sensitivities of the thermo-mechanical constraint w.r.t. the grid coordinates, as reported in Figure 22, sensitivities of the thermo-mechanical constraint w.r.t. the grid coordinates, as reported in Figure 22, providing the information for a grid perturbation. Therefore, Figure 23 compares the distribution of providing the information for a grid perturbation. Therefore, Figure 23 compares the distribution of providing the information for a grid perturbation. Therefore, Figure 23 compares the distribution of von Mises stresses in the baseline geometry (left) and in the updated layout (right). A local reduction von Mises stresses in the baseline geometry (left) and in the updated layout (right). A local reduction von Mises stresses in the baseline geometry (left) and in the updated layout (right). A local reduction in the maximum von Mises stress in excess of 35% is achieved with the new configuration. Most of in the maximum von Mises stress in excess of 35% is achieved with the new configuration. Most of the in the maximum von Mises stress in excess of 35% is achieved with the new configuration. Most of the advancements can be attributed to the increased curvature of the back-plate contour, in particular advancements can be attributed to the increased curvature of the back-plate contour, in particular in the advancements can be attributed to the increased curvature of the back-plate contour, in particular in the region of the connection with the shaft, originally experiencing the highest concentrated the region of the connection with the shaft, originally experiencing the highest concentrated stresses. in the region of the connection with the shaft, originally experiencing the highest concentrated stresses. The new shape reveals a more gradual mechanical pattern at such intersection, supported The new shape reveals a more gradual mechanical pattern at such intersection, supported also by a stresses. The new shape reveals a more gradual mechanical pattern at such intersection, supported also by a smoother temperature evolution lowering the influence of the local thermal strains. The smoother temperature evolution lowering the influence of the local thermal strains. The new layout also by a smoother temperature evolution lowering the influence of the local thermal strains. The new layout returns lower sensitivity values, in line with the goal of satisfying the thermo-structural returns lower sensitivity values, in line with the goal of satisfying the thermo-structural constraint new layout returns lower sensitivity values, in line with the goal of satisfying the thermo-structural constraint within an optimization problem. within an optimization problem. constraint within an optimization problem. Figure 22. Sensitivities of the thermo-mechanical constraint w.r.t. grid coordinates. Figure 22. Sensitivities of the thermo-mechanical constraint w.r.t. grid coordinates. Figure 22. Sensitivities of the thermo-mechanical constraint w.r.t. grid coordinates. Other areas of the blade are highlighted in Figure 22, in particular at high blade span and in the upper portion of the back-plate. The activation of the sensitivities in such regions is favored by the contribution of the thermal input. However, these portions of the rotor present a pattern of smaller stresses, not violating the original constraint, and therefore their sensitivity fields are neglected. Int. J. Turbomach. Propuls. Power 2020, 5, 30 27 of 31 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 27 of 31 Figure Figure23. 23N . N oro m rm alia zle iz ded vo n vM oni se Mi s ss tes res s se tr ses dis ses tri bd uitsit o rn ib iu nttiu or n b ii n n e itm ur p b eiln le e r e im xp p eel rie le nrc iex ngp th er eiren mo ci-n m ge cth ha er nm ica o l- loading: (a) baseline configuration, (b) updated geometry. mechanical loading: (a) baseline configuration, (b) updated geometry. The second positive e ect of the implementation of the CHT analysis within the design loop Other areas of the blade are highlighted in Figure 22, in particular at high blade span and in the is about the more accurate prediction of the actual turbine eciency, as no more adiabatic walls upper portion of the back-plate. The activation of the sensitivities in such regions is favored by the assumptions are in place. The deviation in the total-to-static eciency w.r.t. an adiabatic simulation is contribution of the thermal input. However, these portions of the rotor present a pattern of smaller about 0.2% in the selected operative condition while considering the sole impeller. However, more stresses, not violating the original constraint, and therefore their sensitivity fields are neglected. marked di erences are expected in case the CHT analysis is extended to the whole turbine stage, The second positive effect of the implementation of the CHT analysis within the design loop is including the volute and the nozzle guide vanes. about the more accurate prediction of the actual turbine efficiency, as no more adiabatic walls assumptions are in place. The deviation in the total-to-static efficiency w.r.t. an adiabatic simulation 6. Conclusions is about 0.2% in the selected operative condition while considering the sole impeller. However, more mark This ed d paper ifferen addr ces esses are ex the pec pr teoblem d in ca of sethe the Fluid–Str CHT an uctu alysr ie s Interaction is extended in to the thdesign e wholoptimization e turbine stag of e, turbomachinery components under thermo-mechanical constraints. The activity focuses on a radial including the volute and the nozzle guide vanes. turbine rotor for turbocharger applications, with the aim of reducing the maximum von Mises stress 6. Conclusions located in a critical area of the impeller experiencing the influence of the thermal loading. The problem of the integration of high-fidelity thermal predictions within the framework of an This paper addresses the problem of the Fluid–Structure Interaction in the design optimization optimization is addressed by the development of a novel multidisciplinary gradient-based approach, of turbomachinery components under thermo-mechanical constraints. The activity focuses on a suitable for the ecient computation of the gradients of the cost function with respect to a large radial turbine rotor for turbocharger applications, with the aim of reducing the maximum von Mises number of design variables. The introduction of a Conjugate Heat Transfer evaluation procedure, stress located in a critical area of the impeller experiencing the influence of the thermal loading. supported by the heat transfer forward flux back method, is undertaken in order to establish the energy The problem of the integration of high-fidelity thermal predictions within the framework of an coupling of the solid domain with its surrounding fluid, according to a partitioned coupling approach. optimization is addressed by the development of a novel multidisciplinary gradient-based approach, The gradient calculation is addressed by a discrete adjoint technique, and the resulting sensitivities of suitable for the efficient computation of the gradients of the cost function with respect to a large the thermo-mechanical constraint w.r.t. the grid coordinates allow for the mechanical optimization of number of design variables. The introduction of a Conjugate Heat Transfer evaluation procedure, the component, including considerations related to the impact of its thermal operative environment. supported by the heat transfer forward flux back method, is undertaken in order to establish the The solid–fluid grid coupling process, the analysis procedure, and the multidisciplinary energy coupling of the solid domain with its surrounding fluid, according to a partitioned coupling di erentiation technique are described. Moreover, the validation of the conjugate problem is discussed approach. The gradient calculation is addressed by a discrete adjoint technique, and the resulting with reference to the comparison with an analytic solution and a mesh sensitivity study. The accuracy sensitivities of the thermo-mechanical constraint w.r.t. the grid coordinates allow for the mechanical of the computed gradients is also presented. optimization of the component, including considerations related to the impact of its thermal The application of the multidisciplinary workflow demonstrates the possibility of controlling the operative environment. local von Mises stresses within admissible ranges, extending the outreach of the original mechanical The solid–fluid grid coupling process, the analysis procedure, and the multidisciplinary evaluations with the inclusion of the thermal stresses. New regions of the impeller, before not a ected differentiation technique are described. Moreover, the validation of the conjugate problem is by such considerations, are now involved, as their relative sensitivities are enhanced by the local thermal discussed with reference to the comparison with an analytic solution and a mesh sensitivity study. field. In particular, in the tested sample an improvement exceeding 35% in local von Mises stresses The accuracy of the computed gradients is also presented. is achieved by the application of the presented method. Additionally, more accurate performance The application of the multidisciplinary workflow demonstrates the possibility of controlling predictions result from the introduction of the CHT analysis within the design optimization process, the local von Mises stresses within admissible ranges, extending the outreach of the original no more relying on the adiabatic walls assumption. mechanical evaluations with the inclusion of the thermal stresses. New regions of the impeller, before not affected by such considerations, are now involved, as their relative sensitivities are enhanced by the local thermal field. In particular, in the tested sample an improvement exceeding 35% in local von Int. J. Turbomach. Propuls. Power 2020, 5, 30 28 of 31 This contribution is a new step towards the large-scale optimization of turbomachinery components by gradient-based methods, with the goal of the future introduction of fatigue life evaluations in transient conditions. Author Contributions: Conceptualization, A.R. and T.V.; methodology, A.R. and T.V.; software, A.R.; validation, A.R. and T.V.; formal analysis, A.R.; investigation, A.R.; resources, T.V. and L.C.; writing—original draft preparation, A.R.; writing—review and editing, A.R., T.V., and L.C.; visualization, A.R.; supervision, T.V. and L.C.; project administration, T.V. and L.C.; funding acquisition, T.V. and L.C. All authors have read and agreed to the published version of the manuscript. Funding: The research leading to these results is co-funded by General Motors Global Propulsion Systems and PUNCH Torino S.p.A. under the grant of the Industrial PhD Program. Acknowledgments: We gratefully thank General Motors Global Propulsion Systems for the permission to publish this paper. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the collection, analysis, or interpretation of data or in the writing of the manuscript. Nomenclature CFD Computational Fluid Dynamics FEM Finite Element Method DWI Distance-Weighted Interpolation CHT Conjugate Heat Transfer FSI Fluid Structure Interaction JT-KIRK Jacobian Trained Krylov Implicit Runge Kutta MUSCL Monotonic Upstream-centered Scheme for Conservation Laws design variable ( ) T thermal expansion coecient [K ] " thermal strain [-] von Mises stresses [Pa] VM adjoint variable dist(i) distance between i-th fluid cell-solid node coupling in DWI procedure [m] dT/dn temperature gradient normal to wall [K/m] E(T) elasticity-strain matrix h virtual heat transfer coecient [W/m K] J cost function, objective function k thermal conductivity coecient [W/mK] q heat flux from fluid domain normal to the wall [W/m ] FW q heat flux in solid domain normal to the wall [W/m ] SW R non-linear residuals in fluid analysis T fluid temperature in first layer of inner domain cells [K] D1 T virtual fluid bulk temperature [K] f l T0 virtual fluid bulk temperature interpolated by DWI procedure [K] f l T fluid temperature at the wall [K] FW T fluid temperature in first layer of ghost cells [K] G1 T reference temperature for thermal strains calculation [K] re f U conservative flow variables in fluid analysis u adjoint variable V primitive flow variables at domain boundaries in fluid analysis bnd X fluid/solid grid coordinate (x,y,z) [m] Int. 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Introduction to Quality Engineering: Designing Quality into Products and Processes; Asian Productivity Organization/UNIPUB: White Plain, NY, USA; Unipub/Kraus: New York, NY, USA, 1986. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional aliations. © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY-NC-ND) license (http://creativecommons.org/licenses/by-nc-nd/4.0/). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Journal of Turbomachinery, Propulsion and Power Multidisciplinary Digital Publishing Institute

Radial Turbine Thermo-Mechanical Stress Optimization by Multidisciplinary Discrete Adjoint Method

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International Journal of Turbomachinery Propulsion and Power Article Radial Turbine Thermo-Mechanical Stress Optimization by Multidisciplinary Discrete Adjoint Method 1 , 2 , 3 , 1 2 Alberto Racca * , Tom Verstraete and Lorenzo Casalino Turbomachinery and Propulsion Department, von Karman Institute for Fluid Dynamics, Chaussée de Waterloo 72, 1640 Rhode-St.-Genèse, Belgium; tom.verstraete@vki.ac.be Mechanical and Aerospace Engineering Department, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy; lorenzo.casalino@polito.it Advanced Engineering Department, PUNCH Torino S.p.A., Corso Castelfidardo 36, 10129 Torino, Italy * Correspondence: alberto.racca@polito.it Received: 31 August 2020; Accepted: 24 November 2020; Published: 25 November 2020 Abstract: This paper addresses the problem of the design optimization of turbomachinery components under thermo-mechanical constraints, with focus on a radial turbine impeller for turbocharger applications. Typically, turbine components operate at high temperatures and are exposed to important thermal gradients, leading to thermal stresses. Dealing with such structural requirements necessitates the optimization algorithms to operate a coupling between fluid and structural solvers that is computationally intensive. To reduce the cost during the optimization, a novel multiphysics gradient-based approach is developed in this work, integrating a Conjugate Heat Transfer procedure by means of a partitioned coupling technique. The discrete adjoint framework allows for the ecient computation of the gradients of the thermo-mechanical constraint with respect to a large number of design variables. The contribution of the thermal strains to the sensitivities of the cost function extends the multidisciplinary outlook of the optimization and the accuracy of its predictions, with the aim of reducing the empirical safety factors applied to the design process. Finally, a turbine impeller is analyzed in a demanding operative condition and the gradient information results in a perturbation of the grid coordinates, reducing the stresses at the rotor back-plate, as a demonstration of the suitability of the presented method. Keywords: radial turbine; fluid–structure interaction; adjoint method; conjugate heat transfer; thermo-mechanical stress; partitioned coupling algorithm 1. Introduction Multidisciplinary optimization methods are widely adopted in the development cycle of turbomachinery-related technologies, such as in the energy and mobility businesses [1–3]. The increasing complexity of the products in terms of geometry and flow characteristics in design and o -design conditions, along with the fulfillment of requirements of lifetime, cost, and packaging, is reflected in the growth of concurrent design techniques that provide the experts with a comprehensive view on the problem [4,5]. The exploration of the design space is enriched by multiple physical disciplines whose interactions are captured along the optimization, resulting in accurate predictions of the component performance and any possible violation of the structural constraints. This holistic approach to the design problem di ers from classical staggered methods, presenting a series of mono-discipline optimizations performed in cascade, leading typically to significantly longer development times. Moreover, such staggered methods may not converge to the same optimum as the holistic one and thus lead to suboptimal designs. Int. J. Turbomach. Propuls. Power 2020, 5, 30; doi:10.3390/ijtpp5040030 www.mdpi.com/journal/ijtpp Int. J. Turbomach. Propuls. Power 2020, 5, 30 2 of 31 1.1. Background The recent evolution of the market request towards aggressive reductions in emissions and fuel consumption o ers new challenges to the research community. The need for further advances in multidisciplinary design optimization techniques is presented in the NASA 2040 roadmap [6] as one of nine core development areas in the context of integrated multiscale modelling frameworks contributing to the success in ecient design, manufacturing, and certification of future aerospace systems. Similarly, the engineering of advanced turbocharging solutions in the automotive sector demands further increases in eciency to support novel combustion strategies, with no compromise on the durability targets [7,8]. Enhancements in turbomachinery components performance can be achieved by increasing the detail in the geometrical representation of the domain and reflecting it in a higher number of parameters engaged during the optimization process. This higher resolution allows the exploration of more complex shapes and discloses hidden interactions among features now present in the design space. Additionally, an increased fidelity in the physical description of the multidisciplinary problem is highly regarded and the aim is twofold. First, the optimizer is able to make more reliable predictions of the responses to the geometrical modifications, having access to lumped information accounting for the interactions of more disciplines involved in the computation of the objective functions and constraints. Therefore, the evolution of the optimization benefits from adhering more to the real behavior of the component, finally resulting in a reduced amount of experimental validation testing. Second, a more sophisticated physical description opens the path to adopting a richer set of constraints, and the safety margins usually applied to the design process can be relaxed in favor of more performance-oriented layout choices. Therefore, the optimizer can target more diversified paths towards the achievement of the desired solution, thus reducing the sti ness of the algorithm in performing the shape modifications. The increased complexity in the models is reflected in the choice of a suitable optimization method between two major classes, notably denoted as gradient-free and gradient-based techniques. Gradient-free methods [9], also known as “zero-order” methods, solely make use of the function values for the search of the global optimum, neglecting any knowledge of its gradients. The solution is identified after the evaluation of the output values of a large population of candidates with characteristics scattered within the boundaries of the selected design space. The larger the population, the higher the likelihood of identifying the global optimum. In the case of a multidisciplinary problem, each sample in the population must be processed through all the disciplines (FEM and FVM analyses, among others) [10]. Therefore, the number of computations necessary to identify a solution scales up with the number of involved disciplines and the number of design variables necessary to investigate the full domain, resulting quickly in a computationally expensive approach [11,12]. Even if, nowadays, gradient-free methods still represent the state-of-the-art approach to optimizations in industry because of their robustness and the high quality of the solutions, the advent of more stringent requirements and the need of evaluating larger domains are exposing an intrinsic limitation, notably referred to as the “curse of dimensionality” [13]. Gradient-based methods [14,15], with specific reference to “first-order” methods, make use of the gradient information of the function of interest in the search of the optimal solution, and therefore are more ecient than zero-order methods. The evaluation of the gradients may be computationally intensive if performed by Finite Di erences or complex step techniques [16], especially in the presence of a large number of design variables and in the case of disciplines involving expensive computations. However, if the number of objective functions is lower than the number of optimization parameters, an ecient calculation of the gradients can be obtained by the adjoint method [17]. This technique, first introduced in [18] for fluid problems, is characterized by a cost for the computation of the sensitivity derivatives of the objective function that is essentially independent of the number of design variables. Therefore, it is particularly suited for problems requiring detailed geometrical descriptions with many degrees of freedom. Moreover, the exploration of the design space through its gradients decreases the number of iterations necessary to achieve the local optimal solution. Therefore, the reduced overhead Int. J. Turbomach. Propuls. Power 2020, 5, 30 3 of 31 can be traded for the introduction of more computationally demanding disciplines, returning a higher accuracy in the numerical solutions. A performance comparison between these two techniques is o ered in [19], whose application to the multi-point optimization of a radial turbine impeller shows that the adjoint method is capable of extending the Pareto front identified by a di erential evolution (DE) algorithm. In particular, the gradient-based approach delivers a solution with a higher total-to-static eciency, improving also the moment of inertia figure at the cost of a fraction of the computational time required by the gradient-free method to achieve its global optimum (roughly 6–10%). Other examples of applications of the adjoint methods to the optimization of turbines and compressors are reported in [20–25], with focus on the opportunity of increasing the number of variables in the design space and the multidisciplinary aspect of the physical evaluations (including aerodynamics, stresses, and modal analysis). The higher maturity recently gained by gradient-based approaches and their improved robustness in the analysis of complex flow conditions [26,27] provides evidence of their application to problems of industrial relevance. Therefore, adjoint methods are considered a suitable choice to address the previously mentioned design advancements in turbomachinery-related applications. 1.2. Motivation The present document focuses on the development of a radial turbine impeller for an automotive turbocharger. In contrast to aeronautical applications, the typical duty cycles for passenger cars span the entire turbine map, from very low mass flow rates and pressure ratios in urban driving situations up to peak power, resulting in requirements stretching the machine design over a wide range of operative conditions. Moreover, these turbines present tight clearances between rotor and housing in the order of few hundreds of microns, whilst the typical thermal operative range spans from ambient temperature up to 1050 C, without the possibility of any blade-dedicated cooling circuit. Such scenario clearly demonstrates the challenges in performance requirements and durability the designer is confronted with during the product development process. The structural integrity of the rotor experiencing centrifugal and fluid-induced stresses is a critical aspect in the search for design solutions fulfilling the eciency and permeability targets. The problem of the Fluid–Structure Interaction (FSI) in radial turbines in relation to the thermal stresses induced by the fluid in contact with the blades is discussed in [28], in an e ort to accurately predict the heat exchange between the two media for a reliable estimation of the rotor lifetime in steady state and dynamic operative conditions. The study shows how the incorrect estimation of the thermal stresses could impair the robustness of the component, justifying the inclusion of this additional constraint on top of the computation of the centrifugal stresses. The introduction of thermal analyses in the framework of an optimization process is a topic of recent interest [29–31]. The physical phenomenon is described through the application of a Conjugate Heat Transfer (CHT) procedure, as originally proposed in [32,33], which involves the thermal interaction between a solid body and its surrounding fluid by a coupled solution of the two domains. The challenge in the implementation of this analysis within the landscape of a multidisciplinary optimization of a complex geometry consists of the trade-o between accuracy and significant computational overhead, so far limiting thermal predictions only to the final validation phase in industrial design procedures. In fact, it is common practice during an optimization to replace such intensive computations with reduced-order models or safety margins based on empirical experience. This approach to the evaluation of the thermal stresses would su er in terms of accuracy or, in the case of large safety margins, would over-constrain the optimization problem. 1.3. Goal of the Paper This paper addresses the problem of robust optimizations w.r.t. thermo-mechanical stresses by developing a discrete adjoint framework tailored to the implementation of the CHT analysis within the in-house multidisciplinary design and optimization platform “CADO” developed at the von Karman Int. J. Turbomach. Propuls. Power 2020, 5, 30 4 of 31 Institute for Fluid Dynamics [34]. Indeed, the selection of a gradient-based method is justified by its Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 4 of 31 computational eciency in identifying a local minimum, although the computation of the gradients is more convoluted due to the multiple disciplines involved in the performance analysis. The calculation the in-house multidisciplinary design and optimization platform “CADO” developed at the von of the thermal stresses in the solid domain and their sensitivities with respect to the grid coordinates Karman Institute for Fluid Dynamics [34]. Indeed, the selection of a gradient-based method is is therefore the subject of the present paper, with the aim of including this analysis in a Sequential justified by its computational efficiency in identifying a local minimum, although the computation of Quadratic Programming-based optimizer [35]. the gradients is more convoluted due to the multiple disciplines involved in the performance analysis. The calculation of the thermal stresses in the solid domain and their sensitivities with respect To the best of our knowledge, only a few studies have considered the CHT analysis in adjoint-based to the grid coordinates is therefore the subject of the present paper, with the aim of including this optimizations [36,37], with only a limited number adopting a discrete adjoint formulation [38–40] and analysis in a Sequential Quadratic Programming-based optimizer [35]. none treating a test case with a complex three-dimensional flow field or extending the problem to To the best of our knowledge, only a few studies have considered the CHT analysis in adjoint- thermo-mechanical evaluations. Therefore, this work is aimed at covering critical gaps in the design of based optimizations [36,37], with only a limited number adopting a discrete adjoint formulation [38– radial turbines with the aim of creating a framework suitable for industrial applications. 40] and none treating a test case with a complex three-dimensional flow field or extending the In light of this objective, the remaining portion of the document is structured as follows. problem to thermo-mechanical evaluations. Therefore, this work is aimed at covering critical gaps in First, the framework of the primal solver for the execution of the CHT analysis is presented, the design of radial turbines with the aim of creating a framework suitable for industrial applications. with extension to the computation of the thermo-mechanical stresses. Then, the implementation of the In light of this objective, the remaining portion of the document is structured as follows. First, adjoint thworkflow e frameworfor k ofthe the p calculation rimal solverof fothe r thesensitivities execution of th ofethe CHT constraint analysis isfunction presentedwith , withr e espect xtensioto n the solid and to th fluid e com grid puta coor tion dinates of the th isediscussed, rmo-mechan outl icalining stressthe es. steps Then, towar the im ds ple the men coupli tationng of of ththe e addomains joint workflow for the calculation of the sensitivities of the constraint function with respect to the solid in reverse mode. Finally, the validation of the model and the results of the application of the procedure and fluid grid coordinates is discussed, outlining the steps towards the coupling of the domains in to a radial turbine rotor test case are presented and the conclusions are drawn from this study. reverse mode. Finally, the validation of the model and the results of the application of the procedure to a radial turbine rotor test case are presented and the conclusions are drawn from this study. 2. Primal Solver The CHT analysis implemented in the current work follows the partitioned coupling approach 2. Primal Solver proposed by [41], in which a fluid and a solid solver are sequentially called in an iterative process The CHT analysis implemented in the current work follows the partitioned coupling approach with the mutual exchange of boundary conditions till reaching the convergence of the temperature proposed by [41], in which a fluid and a solid solver are sequentially called in an iterative process at their interface. The advantage of this strategy is related to the possibility of adopting dedicated with the mutual exchange of boundary conditions till reaching the convergence of the temperature meshes and specialized numerical methods for the solution of each domain, increasing the robustness at their interface. The advantage of this strategy is related to the possibility of adopting dedicated of the m conv eshe er s gence and spe of cia the lizetwo d nuanalyses. merical meMor thodeover s for th , e in so the luticase on of of eac an h dunsteady omain, inccomputation, reasing the robus the tne partial ss of the convergence of the two analyses. Moreover, in the case of an unsteady computation, the partial decoupling of the two fields, which show very di erent characteristic time scales, would lead to a decoupling of the two fields, which show very different characteristic time scales, would lead to a faster convergence of the assembly [42–45]. faster convergence of the assembly [42–45]. 2.1. Mapping Procedure 2.1. Mapping Procedure The problem under investigation is represented in Figure 1, in which a turbine rotor is analyzed The problem under investigation is represented in Figure 1, in which a turbine rotor is analyzed considering a periodic sector. A multi-block structured fluid grid with boundary layer refinement is considering a periodic sector. A multi-block structured fluid grid with boundary layer refinement is interfaced to an unstructured solid mesh of second-order tetrahedral elements for the computation of interfaced to an unstructured solid mesh of second-order tetrahedral elements for the computation the heat flux between the two domains. of the heat flux between the two domains. Figure 1. Radial turbine rotor: fluid and solid sector meshes. Figure 1. Radial turbine rotor: fluid and solid sector meshes. Int. J. Turbomach. Propuls. Power 2020, 5, 30 5 of 31 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 5 of 31 Because the meshes are non-matching at the common interface, the exchange of information Because the meshes are non-matching at the common interface, the exchange of information is is performed through the distance-weighted interpolation (DWI) technique presented in [46]. performed through the distance-weighted interpolation (DWI) technique presented in [46]. The The procedure implies the mapping of the two regions in order to record the correspondences procedure implies the mapping of the two regions in order to record the correspondences of each of each fluid cell center with the closest surrounding solid nodes and vice versa. For each fluid cell, the fluid cell center with the closest surrounding solid nodes and vice versa. For each fluid cell, the information of heat flux or temperature detected in the mating solid nodes is processed by a weighting information of heat flux or temperature detected in the mating solid nodes is processed by a function based on their mutual distances, resulting in the coupling of the two regions. The same weighting function based on their mutual distances, resulting in the coupling of the two regions. The routine is applied to the solid grid with respect to its fluid counterpart. same routine is applied to the solid grid with respect to its fluid counterpart. This method exhibits a good level of generality in treating di erent coupling problems, irrespective This method exhibits a good level of generality in treating different coupling problems, of their grids characteristics. For instance, in the turbine test case the exchange of information is irrespective of their grids characteristics. For instance, in the turbine test case the exchange of e ectively performed also at the blade hub fillet, a feature which is present in the solid domain but not information is effectively performed also at the blade hub fillet, a feature which is present in the solid detailed in the corresponding fluid region. domain but not detailed in the corresponding fluid region. An improvement in the accuracy of this coupling procedure derives from the local labelling of An improvement in the accuracy of this coupling procedure derives from the local labelling of the interface. With reference to Figure 2, the surface between the two media is split in sub-regions the interface. With reference to Figure 2, the surface between the two media is split in sub-regions (e.g., hub, blade pressure side, blade suction side, tip, etc.) and the corresponding solid faces and fluid (e.g., hub, blade pressure side, blade suction side, tip, etc.) and the corresponding solid faces and blocks boundaries are named accordingly. Therefore, the algorithm searching for the fluid cells–solid fluid blocks boundaries are named accordingly. Therefore, the algorithm searching for the fluid cells– nodes correspondences is separately executed for each sub-region, instead of attempting the coupling solid nodes correspondences is separately executed for each sub-region, instead of attempting the of the two fields through a global search within the entire domain. This arrangement is particularly coupling of the two fields through a global search within the entire domain. This arrangement is e ective in locations presenting thin walls, such as in proximity of the blade tip, as it avoids any particularly effective in locations presenting thin walls, such as in proximity of the blade tip, as it erroneous fluid-solid matching between cells and nodes actually sitting on opposite sides of the blade avoids any erroneous fluid-solid matching between cells and nodes actually sitting on opposite sides (i.e., suction side versus pressure side). of the blade (i.e., suction side versus pressure side). Figure 2. Radial turbine rotor surface split in sub-regions (left), example of FVM-FEM meshes Figure 2. Radial turbine rotor surface split in sub-regions (left), example of FVM-FEM meshes correspondence identification (right). correspondence identification (right). Additionally, the original algorithm seeks the nodes–cells correspondences through a Additionally, the original algorithm seeks the nodes–cells correspondences through a “search “search radius” delimiting for each grid node a spherical volume within which the neighbors are radius” delimiting for each grid node a spherical volume within which the neighbors are identified. identified. This technique may induce interpolation errors in proximity of the walls, where temperature This technique may induce interpolation errors in proximity of the walls, where temperature gradients and fluid grid stretching are significant. In fact, the temperature assigned to a solid node gradients and fluid grid stretching are significant. In fact, the temperature assigned to a solid node may result from the weighting of temperature values derived from fluid cells located within the may result from the weighting of temperature values derived from fluid cells located within the boundary layer but not in direct contact with the interface. Moreover, as demonstrated in [46], distinct boundary layer but not in direct contact with the interface. Moreover, as demonstrated in [46], distinct grid refinements at the solid and fluid side may influence the local accuracy of the wall temperature grid refinements at the solid and fluid side may influence the local accuracy of the wall temperature information passed to the FEM. In order to cope with these issues, the searching criterion is limited to information passed to the FEM. In order to cope with these issues, the searching criterion is limited the sole layer of nodes and cells in direct contact with the walls, and the concept of “search radius” to the sole layer of nodes and cells in direct contact with the walls, and the concept of “search radius” is dismissed in favor of a ranking assigned to each node or cell according to their mutual distance. Finally, consistently with the solution proposed in [46] addressing the problem of the aspect ratio of Int. J. Turbomach. Propuls. Power 2020, 5, 30 6 of 31 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 6 of 31 is dismissed in favor of a ranking assigned to each node or cell according to their mutual distance. Finally, consistently with the solution proposed in [46] addressing the problem of the aspect ratio cells on a wall boundary, additional virtual grid points are introduced on the FVM side between two of cells on a wall boundary, additional virtual grid points are introduced on the FVM side between neighboring cell centers. Such virtual points are aimed at improving the uniformity of the local two neighboring cell centers. Such virtual points are aimed at improving the uniformity of the local distribution of grid points in regions of significant mesh stretching, and their temperature is distribution of grid points in regions of significant mesh stretching, and their temperature is interpolated interpolated between the closest “master” cells of the original mesh. Indeed, the goal is to increase between the closest “master” cells of the original mesh. Indeed, the goal is to increase the density the density of the attractors in proximity of each solid node, avoiding possible inconsistencies of gethe nerattractors ated from in capr ptur oximity ing thof e ieach nform solid ation node, fromavoiding too far flpossible uid cellsinconsistencies . Thus, the dom generated ain couplfr inom g is capturing the information from too far fluid cells. Thus, the domain coupling is executed through the executed through the distance-weighting interpolation of information belonging to cells and nodes distance-weighting in actual close prox interpolation imity. of information belonging to cells and nodes in actual close proximity. The quality of the interpolation is referenced in Figure 3. The fluid temperature at the walls The quality of the interpolation is referenced in Figure 3. The fluid temperature at the walls (Figur (Figur ee3 3 a) a)is isinterpolated interpolatedby bythe theDWI DWI pr pocedur rocedur eeand andpassed passedto tothe thesolid solidnodes nodeslaying layingon onthe thesurface surface (Figure 3b). During a CHT iteration, the solid returns the heat flux to the fluid, as discussed in the (Figure 3b). During a CHT iteration, the solid returns the heat flux to the fluid, as discussed in the next next section. section. Figure 3. Distance-weighted interpolation method applied to the exchange of information between Figure 3. Distance-weighted interpolation method applied to the exchange of information between solid and fluid domains: (a) fluid temperature at walls, (b) interpolated fluid temperature assigned to solid and fluid domains: (a) fluid temperature at walls, (b) interpolated fluid temperature assigned to the solid surface. the solid surface. 2.2. Partitioned Coupling 2.2. Partitioned Coupling The data exchange between the fluid and solid fields is performed through the “heat transfer The data exchange between the fluid and solid fields is performed through the “heat transfer forward flux back” method (hFFB) [47], whose stability properties are discussed in [48]. In summary, forward flux back” method (hFFB) [47], whose stability properties are discussed in [48]. In summary, the convergence rate of the conjugate problem depends on the local Biot number, which expresses the the convergence rate of the conjugate problem depends on the local Biot number, which expresses ratio between the conductive over the convective thermal resistances. In complex geometries, as in the ratio between the conductive over the convective thermal resistances. In complex geometries, as the case of a radial turbine impeller, the Biot number may locally change to values greater or lower in the case of a radial turbine impeller, the Biot number may locally change to values greater or lower than unity, according to the variations in the blade thickness. Hence, [46] demonstrates that methods than unity, according to the variations in the blade thickness. Hence, [46] demonstrates that methods such as the hFFB or the “heat transfer forward temperature back” (hFTB) method would promote the such as the hFFB or the “heat transfer forward temperature back” (hFTB) method would promote the stabilization of the fluid-solid coupling w.r.t. other techniques, such as the “flux forward temperature stabilization of the fluid-solid coupling w.r.t. other techniques, such as the “flux forward temperature back” (FFTB) or the “temperature forward flux back” (TFFB) methods, which strictly require a modulus back” (FFTB) or the “temperature forward flux back” (TFFB) methods, which strictly require a of the Biot number respectively lower or greater than unity. Instead, the hFTB and hFFB methods modulus of the Biot number respectively lower or greater than unity. Instead, the hFTB and hFFB exhibit a wider range of convergence. Finally, the hFFB method was selected for the present study since methods exhibit a wider range of convergence. Finally, the hFFB method was selected for the present study since a heat flux boundary condition imposed to the fluid domain would generally improve the convergence stability of the CFD computation more than an imposed wall temperature boundary condition. Figure 4 reports the partitioned coupling workflow, described as follows. Int. J. Turbomach. Propuls. Power 2020, 5, 30 7 of 31 a heat flux boundary condition imposed to the fluid domain would generally improve the convergence stability of the CFD computation more than an imposed wall temperature boundary condition. Figure 4 reports the partitioned coupling workflow, described as follows. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 7 of 31 Figure 4. hFFB coupling method for the CHT analysis. Figure 4. hFFB coupling method for the CHT analysis. The CHT analysis is addressed with an initial CFD computation considering adiabatic walls. The CHT analysis is addressed with an initial CFD computation considering adiabatic walls. The The fluid temperature T and the heat flux normal to the walls q are extracted at the boundaries in FW FW fluid temperature and the heat flux normal to the walls are extracted at the boundaries in contact with the solid, and a virtual fluid bulk temperature T is calculated as: f l contact with the solid, and a virtual fluid bulk temperature is calculated as: FW T = =T − ,, (1) FW f l (1) with ℎ indicating a constant user-defined virtual heat transfer coefficient. with h indicating a constant user-defined virtual heat transfer coecient. Since the initial fluid simulation is adiabatic, Equation (1) results in a virtual fluid bulk Since the initial fluid simulation is adiabatic, Equation (1) results in a virtual fluid bulk temperature temperature equal to the fluid temperature at the walls. The DWI procedure returns the field equal to the fluid temperature at the walls. The DWI procedure returns the T field associated with f l associated with the solid grid nodes. the solid grid nodes. Consequently, ℎ and are assigned to the FEM heat transfer model, such that the heat flux Consequently, h and T are assigned to the FEM heat transfer model, such that the heat flux to the f l to the solid results from the following boundary condition: solid results from the following boundary condition: = ℎ − , (2) q = h T T , (2) SW f l with T as the unknown solid temperature. After solving the FEM heat transfer problem, the temperature and heat flux at this boundary are known. with T as the unknown solid temperature. After solving the FEM heat transfer problem, the temperature The heat flux at the interface is calculated from the temperature field T by means of the Fourier’s and heat flux at this boundary are known. law applied to all the elements exposing at least one face to the fluid: The heat flux at the interface is calculated from the temperature field T by means of the Fourier ’s law applied to all the elements exposing at least one = −face ,to the fluid: (3) dT with indicating the temperature gradient normal to the wall and k the thermal conductivity q = k , (3) SW dn coefficient. The heat flux is processed by the DWI procedure, returning the heat flux assigned to the fluid cells at the interface. The fluid simulation, now accounting for an external heat flux at the viscous walls, is recomputed and the entire process is re-iterated for several loops till the achievement of the continuity of temperatures at the interface between the two fields. The user-imposed virtual heat transfer coefficient ℎ influences the predicted wall heat fluxes at any intermediate cycle of the CHT procedure, thus determining the path to convergence of the whole Int. J. Turbomach. Propuls. Power 2020, 5, 30 8 of 31 with dT/dn indicating the temperature gradient normal to the wall and k the thermal conductivity coecient. The heat flux q is processed by the DWI procedure, returning the heat flux q assigned to the SW FW fluid cells at the interface. The fluid simulation, now accounting for an external heat flux at the viscous walls, is recomputed and the entire process is re-iterated for several loops till the achievement of the continuity of temperatures at the interface between the two fields. The user-imposed virtual heat transfer coecient h influences the predicted wall heat fluxes at any intermediate cycle of the CHT procedure, thus determining the path to convergence of the whole coupling. In the case of the hFFB method, [46] shows that a suitable choice of the virtual coecient h < 2h, with h as the physical value of the heat transfer coecient, would guarantee the convergence of the conjugate problem for any local value of the Biot number. The higher the value of h within the stable region, the faster the convergence of the partitioned coupling method. A discussion about the determination of the h value for a problem of industrial relevance is reported in Section 4.2. 2.3. Solid Heat Transfer Solver The steady-state energy balance within the solid domain is described by Equation (4), with q as the rate of heat transfer into the system, and is computed through a FEM steady linear solver according to [49]. rq = 0, (4) The boundary conditions applied to the problem are classified in three families: Tj = const @W kr Tj = q @W 2 , (5) kr Tj = h Tj T @W @W 3 3 f l with the surface of the solid domain defined as @W [ @W [ @W = @W. 1 2 3 The first equation in (5) implies a fixed temperature imposed at the surface @W . The second type of boundary conditions specifies a heat flux at the surface @W , while Fourier ’s law at LHS describes the heat flux through the medium. If q = 0, the adiabatic wall boundary condition is defined. The third equation represents the convection condition, whose heat flux through the surface @W is proportional to the temperature di erence w.r.t. the surrounding fluid. The virtual heat transfer coecient h and the interpolated thermal field T are reported for the sake of consistency with Equation (2). f l Once integrated over the entire domain, Equation (4) is discretized, resulting in the linear system: A T = b, (6) with the semi-positive definite sti ness matrix A comprising the conductive and convective terms, n,n n A = [A + A ]2 R , and the load vector b2 R accounting for the contribution of the boundary conv cond conditions. The linear system in (6) is solved by an iterative conjugate gradient method. 2.4. Fluid Solver The compressible Reynolds-Averaged Navier–Stokes solver with cell-centered spatial discretization over a structured multi-block grid, developed in [19], is invoked in the CHT framework. The solver computes the convective fluxes by Roe’s upwind scheme with MUSCL extrapolation, while a central discretization is applied to the calculation of viscous fluxes. The time marching technique is implemented according to the JT-KIRK scheme proposed in [26] for the improved stabilization of the discrete adjoint solver. The Spalart–Allmaras turbulence model is adopted in this study. In order to establish the coupling with the solid domain, heat fluxes are imposed at the viscous wall boundaries adopting a thin shear layer approximation. The fluid mesh presents two layers of ghost cells in order to facilitate the computation of the fluxes at the interfaces [50]. The heat fluxes calculated Int. J. Turbomach. Propuls. Power 2020, 5, 30 9 of 31 in Equation (3) and processed through the DWI technique are associated with the corresponding fluid block boundaries in contact with the solid walls. With reference to Figure 5, D1 is the first layer of the inner fluid domain; G1 and G2 are the corresponding ghost layers. The fluid temperature in D1 is detected from the previous solver iteration, and the ghost cell temperature in the first layer is updated according to: Dn T = T + q , (7) G1 D1 FW Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 9 of 31 f l with Dn as the distance between the cell centers and k as the locally computed fluid conductivity. with ∆ as the distance between the cell centers and kfl as the locally computed fluid conductivity. Similarly, the temperature in the G2 layer is updated in cascade, with reference to the newly computed Similarly, the temperature in the G2 layer is updated in cascade, with reference to the newly temperature in G1. The temperature values are used to calculate the corresponding cell densities computed temperature in G1. The temperature values are used to calculate the corresponding cell considering a zero-order pressure extrapolation in wall normal direction, while the no-slip condition densities considering a zero-order pressure extrapolation in wall normal direction, while the no-slip is assured by reversing the fluid tangential velocity components in the ghost layers. Equation (7) is condition is assured by reversing the fluid tangential velocity components in the ghost layers. linearized and accounted in the implicit scheme. Equation (7) is linearized and accounted in the implicit scheme. Figure 5. Fluid domain discretization at block boundaries: D1 as the inner domain layer, G1-G2 as the Figure 5. Fluid domain discretization at block boundaries: D1 as the inner domain layer, G1-G2 as the ghost layers. ghost layers. The rate of convergence induced by the Neumann boundary condition is influenced by the actual The rate of convergence induced by the Neumann boundary condition is influenced by the magnitude of the imposed heat flux in Equation (7). Since the hFFB method is employed in the present actual magnitude of the imposed heat flux in Equation (7). Since the hFFB method is employed in the development, no under-relaxation factors are invoked for the update of the ghost cells temperatures, present development, no under-relaxation factors are invoked for the update of the ghost cells as they are replaced by the selection of a suitable virtual heat transfer coecient h, determinant for the temperatures, as they are replaced by the selection of a suitable virtual heat transfer coefficient ℎ, stability of the coupling process [46]. determinant for the stability of the coupling process [46]. 2.5. Solid Mechanical Solver 2.5. Solid Mechanical Solver Once the continuity of temperatures and heat flux is achieved at the interface of the two domains, Once the continuity of temperatures and heat flux is achieved at the interface of the two domains, an FEM linear elastic solver with quadratic elements presented in [51] is invoked for the solution of the an FEM linear elastic solver with quadratic elements presented in [51] is invoked for the solution of von Mises stresses. the von Mises stresses. S u = f , (8) = , (8) n,n n n with S 2 R as the sti ness matrix, u 2 R referencing the vector of nodal displacements, and f 2 R with ∈ as the stiffness matrix, ∈ referencing the vector of nodal displacements, and ∈ indicating the load. indicating the load. Within the framework of the present analysis, the solid mechanical solver shares the same mesh Within the framework of the present analysis, the solid mechanical solver shares the same mesh as the heat transfer solver, as remarked in Section 3. Therefore, the previously computed nodal as the heat transfer solver, as remarked in Section 3. Therefore, the previously computed nodal temperatures are directly assigned to the new linear system. temperatures are directly assigned to the new linear system. The right-hand side vector accounts for the centrifugal loading and the thermal strains, considering The right-hand side vector accounts for the centrifugal loading and the thermal strains, the e ect of the material contractions and expansions induced by the temperature variations, which are considering the effect of the material contractions and expansions induced by the temperature computed as: variations, which are computed as: " = (T ) T T , (9) thermal i i re f = ( ) − , (9) with α(Ti) indicating the temperature-dependent thermal expansion coefficient; Ti pointing to the node temperatures resulting from the CHT analysis; and Tref as the reference temperature at the initial state, at which no thermal stresses are present within the material [52]. In the current study, we refer to Tref as the ambient temperature. Through the thermal strains, the stresses in the material are dependent on the outcome of the CHT computation, which increases the complexity of the gradient calculation, as presented later on. The maximum von Mises stress is computed using a p-norm function according to Equation (10), with p = 75. Int. J. Turbomach. Propuls. Power 2020, 5, 30 10 of 31 with (T ) indicating the temperature-dependent thermal expansion coecient; T pointing to the node i i temperatures resulting from the CHT analysis; and T as the reference temperature at the initial state, ref at which no thermal stresses are present within the material [52]. In the current study, we refer to T ref as the ambient temperature. Through the thermal strains, the stresses in the material are dependent on the outcome of the CHT computation, which increases the complexity of the gradient calculation, as presented later on. The maximum von Mises stress is computed using a p-norm function according to Equation (10), with p = 75. t R dV VM R . (10) max VM dV Equation (10) fulfills the requirement of global di erentiability, as it is a continuous function, and is invoked as a constraint within the optimization problem with the aim of keeping the thermo-mechanical stresses, induced through the Conjugate Heat Transfer, bounded to a prescribed value. The gradient of this constraint is computed by an adjoint methodology, as described in the next section. 3. Adjoint Solver The structure of the primal solver evaluating the thermo-mechanical stresses is computationally expensive because the CHT procedure requires several loops of CFD-FEM analysis to reach the equilibrium of the temperatures and heat flux at the fluid–solid interface. For this reason, such a framework is less suited for gradient-free optimization methods, as a prohibitive large population of candidates would need to be evaluated using this expensive tool. A gradient-based optimization technique, on the other hand, may o er a faster convergence through a more guided search in the design space. Especially when combined with the adjoint method, the computation of the derivatives is very ecient and almost insensitive to the number of design parameters. The evaluation of the cost function J in primal mode is a two-step process. The first one considers the values of the design variables 2 R and generates the correspondent geometry and finally the grid, whose points coordinates are expressed by the vector X 2 R . The second step involves the flow solver and some post-processing routines, returning the value of J. Following the same approach in reverse mode, also the adjoint evaluation procedure is split in two parts: the first one computing the sensitivities of the cost function J w.r.t. the perturbations of the grid coordinates X, and the second one calculating the sensitivities of the grid coordinates X w.r.t the changes in the design variable . Therefore, the CAD-based parametrization described in [21] allows transferring the grid sensitivities of the cost function to the design variables, controlling the component shape modifications by the application of the chain rule of di erentiation expressed in Equation (11): dJ @J dX = . (11) d @X d Within the thermo-mechanical analysis process developed in the present work, Equation (11) is invoked twice in order to account for the cost function sensitivities w.r.t. the coordinates of the fluid grid points and the solid grid nodes, respectively. Therefore, the global sensitivity expression can be decomposed as follows: dX dJ @J f l @J dX sl = + . (12) d @X d @X d f l sl TM The remaining portion of this section is devoted to the calculation of the cost function sensitivities with respect to the fluid and solid grid coordinates, respectively @J/@X and @J/@X , operated through f l sl the application of a discrete adjoint method. Such sensitivities provide the descent direction for constrained or unconstrained multidisciplinary optimization problems with the aim of minimizing the thermally related response functions. Int. J. Turbomach. Propuls. Power 2020, 5, 30 11 of 31 Consistently with the structure of the primal solver, the adjoint framework treating this interdisciplinary problem is formulated according to a “loose-coupling” approach, di erent from the “strong-coupling” techniques as described in [53]. In fact, the latter accounts for cross-discipline Jacobian terms implicitly exchanging boundary conditions between the fluid and solid domains, and therefore directly solving the global adjoint system at once. Instead, the selected partitioned-coupling method is less intrusive in the structure of the existing adjoint CFD solver from [19] and therefore more suited to a continuously growing modular multidisciplinary platform. Figure 6 reports the framework of the thermo-mechanical evaluation, with the adjoint workflow Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 11 of 31 on the RHS. The iterative structure characterizing the solution of the primal problem is maintained in its adjoint counterpart, which walks through the entire chain in reverse mode by the manual in its adjoint counterpart, which walks through the entire chain in reverse mode by the manual di erentiation of the full process. Therefore, the adjoint branch is discussed starting from the bottom, differentiation of the full process. Therefore, the adjoint branch is discussed starting from the bottom, till achieving the final grid sensitivities on the top. till achieving the final grid sensitivities on the top. Figure Figure 6. 6.W Wo orkrfk lo fl w ow fo rfo thre p th re im p ar lia m nd ala d an jod in t ac d o jo m in pt u tc ao tim on psu o tf as tt ie oa nd sy o -sf tas tt eea thd ey rm -so ta -m te et ch he ar nm ico al -m coec nsh tr a an in ic ta s.l constraints. 3.1. Adjoint Variables The most demanding terms in the computation of Equation (12) are the sensitivities of the response function w.r.t. the grid coordinates. Since: ( ) , , (13) with u denoting the vector of the state variables, the application of the chain rule of differentiation results in the following decomposition: (14) = + . Equation (14) holds for both the fluid and the solid grids. The second term at RHS presents the partial derivative of the cost function w.r.t. the state variables ⁄ . Since the current problem exhibits a multidisciplinary footprint, the hierarchical structure of the workflow in Figure 6 implies that the output states contributing to the p-norm calculation in Equation (10) depend on the intermediate ones evaluated by the preceding CFD and FEM solvers. Therefore, according to the Int. J. Turbomach. Propuls. Power 2020, 5, 30 12 of 31 3.1. Adjoint Variables The most demanding terms in the computation of Equation (12) are the sensitivities of the response function w.r.t. the grid coordinates. Since: J(X, u(X)), (13) with u denoting the vector of the state variables, the application of the chain rule of di erentiation results in the following decomposition: dJ @J @J @u = + . (14) dX @X @u @X Equation (14) holds for both the fluid and the solid grids. The second term at RHS presents the partial derivative of the cost function w.r.t. the state variables @J/@u. Since the current problem exhibits a multidisciplinary footprint, the hierarchical structure of the workflow in Figure 6 implies that the output states contributing to the p-norm calculation in Equation (10) depend on the intermediate ones evaluated by the preceding CFD and FEM solvers. Therefore, according to the principle of the “reverse di erentiation” [54] we choose for instance an output variable u , whose sensitivity dJ/du is known, 3 3 and calculate its sensitivities w.r.t. each intermediate state till the initial one, such that: dJ dJ du du du du 2 3 2 J(u (u (u ))) ! . (15) 3 2 dJ dJ du du du du 1 2 1 Since the work focuses on the single thermo-mechanical output J expressed by Equation (10), TM herein named y, we can associate with any intermediate state variable u a new variable u , called the i i adjoint variable, defined as: @y u = . (16) @u The upper bar notation follows the convention for adjoint variables reported in [54]. Hence, we apply the chain rule walking throughout the original trace in Figure 6 in backward mode, paying attention to the opposite propagation of the adjoint variables with respect to the physical ones. This technique finally returns the sensitivities of the response function w.r.t. the initial state variables. The following sections describe the backward propagation process in detail, with the aim of o ering a guidance to the development of the adjoint method. 3.2. Adjoint Response Function The adjoint framework is initiated by seeding the maximum von Mises constraint. Since y = J =  , the input is: TM maxVM @y = J =  = 1. (17) TM maxVM @J TM The maximum von Mises stress is function of the components of the Cauchy stress tensor. Therefore, in reverse mode the adjoint stress components are calculated as: t R dV dy dJ d p TM V VM =  = J = R . (18) i, j TM d d d i, j dV On the other hand, Equation (10) shows an explicit dependence over the solid grid coordinates through the two volume integrals appearing at the numerator and denominator. Int. J. Turbomach. Propuls. Power 2020, 5, 30 13 of 31 Therefore, the constraint function sensitivities with respect to the solid grid coordinates are accumulated as follows: t R dV dy dJ d TM V VM = X + = J = . (19) TM sl dX dX dX dV sl sl sl In Equation (19), the operator “+=” is reported to emphasize the concept of accumulation of the sensitivities w.r.t. the solid grid coordinates that is propagated backwards throughout the adjoint workflow till the grid generation procedure. Moreover, the X notation is adopted for the sake of simplicity, while in reality the sensitivities over the three dimensions (X,Y,Z) are accounted in the development. Finally, in the remaining portion of the document the notations dy/du and dy/dX will be skipped in favor of the more convenient u and X. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 13 of 31 3.3. Adjoint Mechanical Solver 3.3. Adjoint Mechanical Solver The evaluations through the mechanical solver involve the multi-step approach outlined in The evaluations through the mechanical solver involve the multi-step approach outlined in Figure 7, where the primal mode is represented along with its adjoint counterpart. Notably, the reverse Figure 7, where the primal mode is represented along with its adjoint counterpart. Notably, the computation follows the backward propagation of the adjoint variables, closely resembling the primal reverse computation follows the backward propagation of the adjoint variables, closely resembling scheme. The related procedure is discussed in detail in this section. the primal scheme. The related procedure is discussed in detail in this section. Figure 7. Evaluation by mechanical solver: primal and adjoint modes. Figure 7. Evaluation by mechanical solver: primal and adjoint modes. In primal mode, the stress components  are computed from the elasticity matrix of the material i, j In primal mode, the stress components are computed from the elasticity matrix of the under consideration and the strain components. Thus, the di erentiation in reverse mode returns the material under consideration and the strain components. Thus, the differentiation in reverse mode adjoint strains " . i, j returns the adjoint strains ̅ . ̅ = ( ) _ , " =  E(T) m_i j i, j ( ) ̅ = − ( ) = ( ) − ( ) → . ( _) ( ,) (20) , = E(T) " _ " _ (T) ! " T =  E T . (20) > i, j i, j m_i j th_i j th_i j : ( ) = − ( ) E(T) =  " , _" (T ) _ i, j m_i j th_i j Here, the adjoint stresses are the once computed by the reverse differentiation discussed in Here, the adjoint stresses  are the once computed by the reverse di erentiation discussed in i, j Section 3.2. Section 3.2. The adjoint mechanical strains ̅ and the adjoint thermal strains ̅ appear in Equation (20). The adjoint mechanical strains " and the adjoint thermal strains " appear in Equation (20). th Since the thermal strains exhibit direct dependence from the nodal temperatures through the Since the thermal strains exhibit direct dependence from the nodal temperatures through the T T i re f − term and indirect dependence through the thermal expansion coefficient α(T), the relative term and indirect dependence through the thermal expansion coecient (T), the relative contributions contributions to the adjoint solid node temperatures are accumulated and propagated to the next step to the adjoint solid node temperatures are accumulated and propagated to the next step in the in the reverse workflow. reverse workflow. 8 += ̅ ( ) ( ) > ( ) ( ) = − → T+ = " T . (21) ( ) _ < th_i j += ̅ − " (T) = (T) T T ! > . (21) th_i j re f d (T) T+ = " T T th_i j re f dT Additionally, also the elasticity matrix E depends on the nodal temperature, which leads Additionally, also the elasticity matrix E depends on the nodal temperature, which leads similarly similarly to contributions for the adjoint solid node temperatures. to contributions for the adjoint solid node temperatures. ( ) ( ) = ( ) → += ( ) . (22) d f (T) E(T) = f (T) ! T+ = E(T) . (22) The process of accumulation of the contributes to the adjoint temperatures field is facilitated by dT the commonality of the solid grid coordinates between the mechanical and heat transfer solvers, as no re-interpolations are necessary for the backward propagation of the sensitivities. Next, recalling that the mechanical strains in primal mode are calculated from the vector of nodal displacements u, the adjoint nodal displacements are derived by the reverse differentiation ( ̅ → ) and provided in input to the adjoint mechanical solver. The iterative linear system solver adopted in primal mode for the solution of Equation (8) is not directly differentiated but, as discussed in [20], a new linear system is formulated with the adjoint displacements vector appearing at RHS as: = . (23) The system (23) is solved for the adjoint load vector , while the transpose of the stiffness matrix S, previously computed in primal mode, appears at LHS. Therefore, the adjoint stiffness matrix is obtained from the system (24). ̅ ̅ = − . (24) Int. J. Turbomach. Propuls. Power 2020, 5, 30 14 of 31 The process of accumulation of the contributes to the adjoint temperatures field is facilitated by the commonality of the solid grid coordinates between the mechanical and heat transfer solvers, as no re-interpolations are necessary for the backward propagation of the sensitivities. Next, recalling that the mechanical strains " in primal mode are calculated from the vector of nodal displacements u, the adjoint nodal displacements are derived by the reverse di erentiation (" ! u ) and provided in input to the adjoint mechanical solver. The iterative linear system solver adopted in primal mode for the solution of Equation (8) is not directly di erentiated but, as discussed in [20], a new linear system is formulated with the adjoint displacements vector appearing at RHS as: S f = u. (23) The system (23) is solved for the adjoint load vector f , while the transpose of the sti ness matrix S, previously computed in primal mode, appears at LHS. Therefore, the adjoint sti ness matrix is obtained from the system (24). S = u f . (24) i j j Finally, the system assembly process is algorithmically di erentiated, returning the grid sensitivity accumulation: d f dS X + = S + f . (25) sl dX dX sl sl The system (25) accumulates the grid sensitivities attributed to the mechanical solver. The adjoint temperatures deriving from Equations (21) and (22) are provided as input to the adjoint heat transfer solver at the next section, thus linking the structural solver to the iterative adjoint CHT process. 3.4. Adjoint Heat Transfer Solver Consistently with the previous section, the iterative linear system solver adopted in primal mode for the solution of Equation (6) is not directly di erentiated, whilst the adjoint load vector and the adjoint sti ness matrix are computed through the new linear systems in Equation (26). A b = T. (26a) A = T b . (26b) i j j i Equation (26a) shows the transposed sti ness matrix A accounting for the conductive and convective terms previously calculated in primal mode, while the adjoint temperature vector T from Section 3.3 is imposed at the RHS. The resulting adjoint load vector b is assigned to Equation (26b), together with the temperature solution stored at the last CHT loop in primal mode, contributing to the calculation of the adjoint sti ness matrix A. Similarly to Equation (25), the system assembly process is algorithmically di erentiated and the adjoint contributes to the solid grid coordinates are accumulated. Since the thermal solver and the mechanical solver share the same grid, a unique vector of solid grid sensitivities is accumulated and propagated backwards throughout the CHT workflow. The convective loading at the RHS of the system (6) accounts for the virtual bulk fluid temperature. Its adjoint counterpart T is computed from b (returned by Equation (26a)) and from the user-defined f l virtual heat transfer coecient h. e 0 e b = h T ! T = b h. (27) i f l i f l i T is not accumulated along the entire adjoint workflow but recomputed at every CHT loop in reverse f l mode and then passed to the adjoint hFFB routine. Int. J. Turbomach. Propuls. Power 2020, 5, 30 15 of 31 3.5. Adjoint hFFB Procedure (Solid! Fluid) The adjoint virtual bulk fluid temperature field T associated with the solid nodes is processed f l by algorithmic di erentiation, returning the adjoint T assigned to the neighboring fluid cells centers. f l Equation (28) reports the DWI procedure in primal mode on the left, with focus on the calculation of the solid temperature at node j, leveraging the information of the cluster of i = (1, ..., n) neighboring fluid cells. On the right, the backward di erentiation of the interpolation procedure results in the adjoint temperature field associated to the fluid domain. The operator “+=” stresses the fact each fluid cell may belong to the “neighboring cluster” of di erent solid nodes j, and therefore would accumulate their adjoint values. P T (i) n f l 2 2 i=1 (dist(i)) (dist(i)) ( ) ( ) ( ) T j = ! T i + = T j . (28) P P f l f l f l n 1 n 1 2 2 i=1 i=1 (dist(i)) (dist(i)) Equation (28) suggests also that additional contributions to the solid grid sensitivities X and to sl the fluid grid sensitivities X can be accumulated through the adjoint vector of mapped distances f l between each solid node and the corresponding fluid neighbors, dist(i), as detailed in Section 2.1. Thus, after deriving the adjoint distances dist(i) as T ( j)dT ( j)/d(dist(i)), the cells-nodes mapping f l f l routine is di erentiated and the contributions to the two grids are obtained. Finally, the algorithmic di erentiation of Equation (1) returns the adjoint temperature at the fluid walls T , which is used as input to the adjoint CFD solver at the next step, and the adjoint wall heat FW flux q , stored for a later accumulation. FW T = T > FW q f l FW < T = T ! > . (29) FW f l > 1 e q = T f l h FW 3.6. Adjoint Fluid Solver The partitioned coupling technique adopted in this work allows for non-intrusive calls to the in-house adjoint CFD solver. The flow field is initialized with the converged flow solution U from the corresponding CHT loop in primal mode, and the viscous wall temperatures in the ghost cells layers are updated again imposing the Neumann boundary condition, as reported in Equation (7). The adjoint fluid wall temperature T , resulting from the reverse hFFB procedure at the previous FW step, contributes to the linearization of the constraint function with respect to the conservative flow variables U and the fluid grid coordinates X . The adjoint equation discussed in [19] is repeated here f l for convenience: " # " # T T @R @J = , (30) @U @U with R as the non-linear residuals of the primal flow solver, indicating the unknown adjoint variables, and J the objective or constraint function of interest. Therefore, the RHS term linearized at the viscous walls exposed to the heat flux from the solid develops by the chain rule as follows: @J @J @V @V @J @T @V @V bnd domain FW bnd domain = = = @U @V @V @U @T @V @V @U bnd domain domain FW bnd domain domain (31) @T @V @V FW bnd domain T , FW @V @V @U bnd domain domain with V as the primitive variables computed at the boundary of interest, V as the corresponding bnd domain variables from the interior of the fluid domain, and the last term @V/@U indicating the transformation matrix from primitive to conservative variables. Int. J. Turbomach. Propuls. Power 2020, 5, 30 16 of 31 Similarly, the linearization of the cost function w.r.t. the fluid grid coordinates is the following: dJ @J @R = . (32) dX @X @X f l f l f l Therefore, the first term at RHS linearized at the viscous walls with heat flux results in: @J @J @V @J @T @V @T @V FW FW bnd bnd bnd = = = T . (33) FW @X @V @X @T @V @X @V @X f l bnd f l FW bnd f l bnd f l Equations (31) and (33) show the contributions of the adjoint fluid wall temperatures T FW calculated in Section 3.5 to the fluid grid sensitivities at the wall boundaries exposed to the heat flux. For the principle of the reverse accumulation, the wall heat flux term, entering in primal mode as the update of the boundary conditions in Equation (7), results in an adjoint fluid wall heat flux q that FW is computed once the adjoint CFD solver approaches full convergence. This contribution is summed up with the q returned in Section 3.5 by the reversed hFFB procedure. FW Dn Dn T = T + q ! q = q + T . (34) G1 D1 FW G1 FW FW k hFFB k f l f l A further contribution derives also from the second layer of ghost cells, not reported in Equation (34) for the sake of simplicity. 3.7. Adjoint hFFB Procedure (Fluid! Solid) The partitioned coupling scheme in reverse mode completes by transferring the adjoint terms from the fluid to the solid domain. q is processed by the di erentiated DWI procedure, similarly to FW Equation (28), and returns the adjoint heat flux q associated with the solid grid nodes. Consistently SW with the previous call, additional contributions are accumulated to the solid grid sensitivities X and sl to the fluid grid sensitivities X through the adjoint vector of mapped distances between each fluid f l cell center and the neighboring solid nodes. Finally, the routine adopted to calculate the wall heat flux within the solid domain in Equation (3) is algorithmically di erentiated, accounting only for the nodes laying on the interface. This process results in the new field of adjoint solid temperatures T , passed to the adjoint heat transfer solver SW described in Section 3.4. The whole adjoint CHT process is iteratively repeated from Sections 3.4–3.7 for as many loops as the ones performed during the primal computation till convergence. Once walked through the entire workflow in reverse mode, X and X , enclosing the contributions f l sl from the entire multidisciplinary chain, are introduced in Equation (12), and the constraint function sensitivities w.r.t. the design variables are finally computed. Hence, dJ/d is evaluated by the SQP-based optimizer and the geometry is updated, opening the path to a new thermo-mechanical evaluation within the history of the shape optimization problem. 4. Validation 4.1. Flat Plate (Primal Mode) The validation of the CHT process in primal mode is performed with the consideration of the analytic solution o ered by [33] for the conjugate problem applied to a flat plate subjected to an incompressible flow. The mesh characteristics and boundary conditions are reported in Table 1, while the fluid and solid meshes are qualitatively shown in Figure 8. The thermal conductivity adopted for the solid phase is aimed at obtaining an average Biot number around unity. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 16 of 31 3.7. Adjoint hFFB Procedure (Fluid → Solid) The partitioned coupling scheme in reverse mode completes by transferring the adjoint terms from the fluid to the solid domain. is processed by the differentiated DWI procedure, similarly to Equation (28), and returns the adjoint heat flux associated with the solid grid nodes. Consistently with the previous call, additional contributions are accumulated to the solid grid sensitivities and to the fluid grid sensitivities through the adjoint vector of mapped distances between each fluid cell center and the neighboring solid nodes. Finally, the routine adopted to calculate the wall heat flux within the solid domain in Equation (3) is algorithmically differentiated, accounting only for the nodes laying on the interface. This process results in the new field of adjoint solid temperatures , passed to the adjoint heat transfer solver described in Section 3.4. The whole adjoint CHT process is iteratively repeated from Section 3.4 to Section 3.7 for as many loops as the ones performed during the primal computation till convergence. Int. J. Turbomach. Propuls. Power 2020, 5, 30 17 of 31 Once walked through the entire workflow in reverse mode, and , enclosing the contributions from the entire multidisciplinary chain, are introduced in Equation (12), and the Table 1. Flat plate domain characteristics and boundary conditions. constraint function sensitivities w.r.t. the design variables are finally computed. Hence, ⁄ is evaluated by the SQP-based optimizer and the geometry is updated, opening the path to a new Domain Settings Value thermo-mechanical evaluation within the history of the shape optimization problem. Fluid domain length/height 0.25 m/0.1 m Fluid mesh cells count 365,000 4. Validation Plate thickness/length 0.01 m/0.2 m Solid mesh nodes count/elements count 35,000 4.1. Flat Plate (Primal Mode) Fluid type Air Inlet flow total pressure 1.03 10 Pa The validation of the CHT process in primal mode is performed with the consideration of the Inlet flow temperature 1000 K analytic solution offered by [33] for the conjugate problem applied to a flat plate subjected to an Outlet flow static pressure 1.029 10 Pa incompressible flow. The mesh characteristics and boundary conditions are reported in Table 1, while Plate temperature at bottom face 600 K Plate thermal conductivity 0.29 W/m K the fluid and solid meshes are qualitatively shown in Figure 8. The thermal conductivity adopted for 100 W/m K Virtual heat transfer coecient h the solid phase is aimed at obtaining an average Biot number around unity. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 17 of 31 Figure 8. Fluid and solid meshes for the conjugate problem applied to a flat plate. Figure 8. Fluid and solid meshes for the conjugate problem applied to a flat plate. T The he cconver onverggence ence hihistory story of of thethe max maximum imum temp temperatur erature dife fer di en c er e ence betwe between en two suc two cess successive ive fluid– Table 1. Flat plate domain characteristics and boundary conditions. s fluid–solid olid iteratiiterations ons isL rep is or reported ted in Fi in gur Figur e 9, ew 9h , wher ere a eth arthr esh eshold old of of 1K 1K isis se selected lected aas s th the e sst ta abilization bilization c criterion riterion f for or th the e m model. odel. Domain Settings Value s s L = T T , (35) Fluid domain length/height 0.25 m/0.1 m n+1 max = | − | , (35) Fluid mesh cells count 365,000 Plate thickness/length 0.01 m/0.2 m Solid mesh nodes count/elements count 35,000 Fluid type Air Inlet flow total pressure 1.03 × 10 Pa Inlet flow temperature 1000 K Outlet flow static pressure 1.029 × 10 Pa Plate temperature at bottom face 600 K Plate thermal conductivity 0.29 W/m K Virtual heat transfer coefficient ℎ 100 W/m K Figure 9. Convergence history of L . Figure 9. Convergence history of L1 ∞. The distribution of temperature at the interface between the two domains is presented in Figure 10a, The distribution of temperature at the interface between the two domains is presented in Figure comparing Luikov’s di erential heat transfer (DHT) solution with the numerical one. Figure 10b 10a, comparing Luikov’s differential heat transfer (DHT) solution with the numerical one. Figure 10b shows the temperature variation in a vertical section located at x = 0.05m. In both cases, a sucient shows the temperature variation in a vertical section located at x = 0.05m. In both cases, a sufficient agreement between the numerical results and the analytic solution is obtained within the limitations agreement between the numerical results and the analytic solution is obtained within the limitations of the DHT approach, which does not capture the e ect of the solid conductivity in the streamwise of the DHT approach, which does not capture the effect of the solid conductivity in the streamwise direction, as explained in [55]. direction, as explained in [55]. Figure 10. CHT problem for flat plate: (a) temperature at the domain interface, (b) temperature profile at x = 0.05 m. Numerical vs. analytic solution. 4.2. Rotor Mesh Sensitivity Analysis A mesh sensitivity study for the CHT analysis is performed on a three-dimensional test case presented in [19], considering a radial turbine impeller with 10 blades and an inlet diameter of 50 mm. A two-dimensional sketch of the domain on the meridional plane of the rotor is reported in Figure 11, along with the locations of the imposed boundary conditions. The method of the characteristics [56] is adopted to establish the number of physical conditions to be assigned at the inflow and outflow boundaries. The total upstream temperature and pressure are specified at the inlet of the domain, along with flow velocity components in the corresponding coordinate system. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 17 of 31 The convergence history of the maximum temperature difference between two successive fluid– solid iterations is reported in Figure 9, where a threshold of 1K is selected as the stabilization criterion for the model. | | = − , (35) Figure 9. Convergence history of L∞. The distribution of temperature at the interface between the two domains is presented in Figure 10a, comparing Luikov’s differential heat transfer (DHT) solution with the numerical one. Figure 10b shows the temperature variation in a vertical section located at x = 0.05m. In both cases, a sufficient agreement between the numerical results and the analytic solution is obtained within the limitations of the DHT approach, which does not capture the effect of the solid conductivity in the streamwise Int. J. Turbomach. Propuls. Power 2020, 5, 30 18 of 31 direction, as explained in [55]. Figure 10. CHT problem for flat plate: (a) temperature at the domain interface, (b) temperature profile Figure 10. CHT problem for flat plate: (a) temperature at the domain interface, (b) temperature profile at x = 0.05 m. Numerical vs. analytic solution. at x = 0.05 m. Numerical vs. analytic solution. 4.2. Rotor Mesh Sensitivity Analysis 4.2. Rotor Mesh Sensitivity Analysis A mesh sensitivity study for the CHT analysis is performed on a three-dimensional test case A mesh sensitivity study for the CHT analysis is performed on a three-dimensional test case pr p esented resented in in [19 [1], 9]considering , considering a radial a radia turbine l turbinimpeller e impelle wi r w th ith 10 1 blades 0 blade and s anan d a inlet n inldiameter et diamete of r 50 of mm. 50 Am two-dimensional m. A two-dimensketch sional s of ke the tch domain of the don omthe ain meridional on the meri plane dionaof l ptlhe ane r otor of th is e rreported otor is rin epFigur orted ein 11 , Figure 11, along with the locations of the imposed boundary conditions. The method of the along with the locations of the imposed boundary conditions. The method of the characteristics [56] iscadopted haracteristo ticestablish s [56] is athe dopnumber ted to esof tabphysical lish the n conditions umber of pto hybe sica assigned l conditio at nsthe to inflow be assig and ned outflow at the inflow and outflow boundaries. The total upstream temperature and pressure are specified at the boundaries. The total upstream temperature and pressure are specified at the inlet of the domain, Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 18 of 31 inlet of the domain, along with flow velocity components in the corresponding coordinate system. along with flow velocity components in the corresponding coordinate system. At the subsonic outflow boundary At the su,ba so single nic outf flow low variable boundis ary imposed, , a single specifically flow variathe ble downstr is impoeam sed, static specifpr ica essur lly th e.e In do conclusion, wnstream the boundary conditions applied to the problem are summarized in Table 2, in which the impeller static pressure. In conclusion, the boundary conditions applied to the problem are summarized in r T otational able 2, inspeed whichis th also e im included. peller rotational speed is also included. Figure 11. Turbine rotor meridional view and boundary condition locations. Figure 11. Turbine rotor meridional view and boundary condition locations. Table 2. Turbine rotor boundary conditions. Boundary Conditions Value Inlet total pressure p0 173 kPa Inlet total temperature T0 1080 K Inlet flow angle α from radial direction 62 deg Outlet static pressure ps 101 kPa Blade rotational speed ω 140,000 RPM The interactions of the rotor fluid and solid meshes are explored over three levels of refinement, as summarized in Table 3. Additionally, three values of the virtual heat transfer coefficient ℎ are investigated in order to determine a suitable trade-off between stability and computational time. These factors are explored following the combinations expressed by the L9 Orthogonal Array method [57], returning nine test cases (Table 4). Table 3. Fluid and solid mesh refinements, ℎ levels. Factors Levels Value Fluid domain “coarse”—cells count 0.8 M Fluid domain “mid”—cells count 1.3 M Fluid domain “fine”—cells count 2.1 M Solid domain “coarse”—nodes count 105 k Solid domain “mid”—nodes count 295 k Solid domain “fine”—nodes count 1.1 M ℎ low–mid–high (W/m K) 800–1000–1300 Int. J. Turbomach. Propuls. Power 2020, 5, 30 19 of 31 Table 2. Turbine rotor boundary conditions. Boundary Conditions Value Inlet total pressure p 173 kPa Inlet total temperature T 1080 K Inlet flow angle from radial direction 62 deg Outlet static pressure p 101 kPa Blade rotational speed ! 140,000 RPM The interactions of the rotor fluid and solid meshes are explored over three levels of refinement, as summarized in Table 3. Additionally, three values of the virtual heat transfer coecient h are investigated in order to determine a suitable trade-o between stability and computational time. These factors are explored following the combinations expressed by the L9 Orthogonal Array method [57], returning nine test cases (Table 4). Table 3. Fluid and solid mesh refinements, h levels. Factors Levels Value Fluid domain “coarse”—cells count 0.8 M Fluid domain “mid”—cells count 1.3 M Fluid domain “fine”—cells count 2.1 M Solid domain “coarse”—nodes count 105 k Solid domain “mid”—nodes count 295 k Solid domain “fine”—nodes count 1.1 M e 2 800–1000–1300 h low–mid–high (W/m K) Table 4. L9 Orthogonal Array applied to the turbine rotor CHT analysis. Test Case Number CFD Mesh FEM Mesh h [W/m K] 1 coarse coarse 800 2 coarse mid 1000 3 coarse fine 1300 4 mid coarse 1000 5 mid mid 1300 6 mid fine 800 7 fine coarse 1300 8 fine mid 800 9 fine fine 1000 All the simulations are run till convergence at a relative residual drop of 10 for the computational fluid dynamics (CFD) solver and 10 for the FEM solver, while the CHT workflow is stopped for a maximum deviation in wall temperature between two successive fluid-solid iterations L below 1 K. y+ values below unity are obtained for the “mid” and “fine” levels of the fluid grid refinement, while the coarsest one exhibits a peak in y+ around 2.5 at the blade leading edge. Concerning the maximum number of nodes in the solid mesh, the code was tested up to the finest possible grid before incurring in memory leakage issues with the adopted Conjugate Gradient iterative solver. The results in Table 5 demonstrate a low sensitivity of the maximum temperature in the solid domain due to the settings applied to the coupling process, except for the first test case with coarse meshes at both the fluid and solid side. A critical aspect is represented by the quality of the interpolation of the information between the two domains realized by the fine-tuned distribution of virtual grid points on the interface at fluid side, as discussed in Section 2.1. Moreover, the finest solid mesh increases the resolution of the convective loading, resulting in more accurate temperature predictions in the material and, therefore, also more reliable heat fluxes returned to the fluid. Figure 14 shows two cases of solid mesh size, the coarse one (Figure 14a) and the intermediate one (Figure 14b), with the addition Int. J. Turbomach. Propuls. Power 2020, 5, 30 20 of 31 of the trailing edge refinement described hereafter. In the case of finer solid grids, the interpolation procedure returns a richer temperature pattern at the interface, closely mirroring the fluid conditions at the walls, whilst the coarsest one approximates the thermal loading distribution with the highest deviations localized in the blade tip region, where large secondary flows and leakages are present, and at the trailing edge. In this respect, a surface temperature analysis at a cross section located at about 25% of the chord was performed in order to evaluate the integral of the temperature deviations between each investigated test case w.r.t. test case 9. The axial position of the planar section was chosen in a region of high convective perturbations due to the presence of a significant flow detachment at the blade leading edge. The results, summarized in Table 5, are supported by the example in Figure 12, showing the temperature contours for case 1 and case 9. It is interesting to note that, in both set-ups, the blade tip returns significant thermal gradients because of the thinner geometry exposed to the flow vortices. However, the coarsest mesh overestimates the temperature drop, resulting in the integral reported in Table 5. Table 5. Fluid–solid grid sensitivity: summary of CHT computations. #CHT Loops to Norm. Computational Computational Time: Maximum Solid Delta Temperature Case# Convergence [-] Time for CHT Iteration CFD—FEM w.r.t. Total Temperature [K] Integral 1 11 1.0 97.0–0.5% 1019 0.121 2 10 1.02 95.2–1.9% 1025 0.053 3 9 1.36 71.3–25.0% 1027 0.045 4 8 1.76 98.3–0.3% 1026 0.092 5 10 1.78 97.2–1.1% 1028 0.038 6 12 2.12 81.6–16.1% 1028 0.023 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 20 of 31 7 11 2.4 98.6–0.2% 1026 0.074 8 11 2.44 97.2–0.8% 1028 0.019 99 8 8 2.76 2 .76× 86.5–12.4% 86.5–12.4% 1028 1028 0.000 0.000 Figure 12. Temperature contours for test case 1 and test case 9 at the planar cross section on the blade. Figure 12. Temperature contours for test case 1 and test case 9 at the planar cross section on the blade. Finally, considerations about the computational time are synthesized in Table 5, with a dominant Finally, considerations about the computational time are synthesized in Table 5, with a dominant portion covered by the CFD solver and a significant rise in overhead emerging from the FEM heat portion covered by the CFD solver and a significant rise in overhead emerging from the FEM heat transfer computations at the finest grid level. The normalization refers to the total duration of a single transfer computations at the finest grid level. The normalization refers to the total duration of a single CHT iteration, including also the hFFB procedure. The comparison is performed w.r.t. the reference CHT iteration, including also the hFFB procedure. The comparison is performed w.r.t. the reference duration of test case 1, presenting the coarsest meshes at both fluid and solid side. duration of test case 1, presenting the coarsest meshes at both fluid and solid side. The interactions among the three factors in the Orthogonal Array from Table 4 are studied through the compounded signal S, defined as: = + (1 − ) , (36) with the term A enclosing the normalized difference between the maximum temperature predicted by the test case of interest and test case 9 (showing the finest grids), B representing the normalized computational time, and ω as the weighting coefficient. In the current study, ω = 0.7 in order to bias the objective function towards the accuracy of the coupling process. Indeed, the factors levels finally expected from this study are the ones minimizing the signal S. The analysis of the Orthogonal Array [57], resulting in Figure 13, reveals the dependence of the signal S from the three variables and their correspondent levels. The chart illustrates the virtual heat transfer coefficient ℎ is selected at its highest value within the stability region of the Biot number discussed in [46], as it influences the B term in Equation (36), determining the rate of convergence of the CHT coupling process. Indeed, the current problem exhibits an optimal ℎ = 1000 W/m K, since the highest value approaches the limit of the stability region, with incipient oscillations in the value of heat flux exchanged at the interface, before reaching convergence. The solid grid size, instead, characterizes the accuracy of the interpolation of the quantities passed between the two domains through the DWI process and affects the A term in Equation (36). In general, finer solid grids are favored by the current analysis. Remarkably, the fluid mesh size follows an opposite trend because its influence on the A term is less pronounced, as the coarsest level already provides good-quality solutions. On the other hand, the signal S is driven by the increase in computational overhead experienced with the fluid mesh refinement. Int. J. Turbomach. Propuls. Power 2020, 5, 30 21 of 31 The interactions among the three factors in the Orthogonal Array from Table 4 are studied through the compounded signal S, defined as: S = ! A + (1 !) B, (36) with the term A enclosing the normalized di erence between the maximum temperature predicted by the test case of interest and test case 9 (showing the finest grids), B representing the normalized computational time, and ! as the weighting coecient. In the current study, ! = 0.7 in order to bias the objective function towards the accuracy of the coupling process. Indeed, the factors levels finally expected from this study are the ones minimizing the signal S. The analysis of the Orthogonal Array [57], resulting in Figure 13, reveals the dependence of the signal S from the three variables and their correspondent levels. The chart illustrates the virtual heat transfer coecient h is selected at its highest value within the stability region of the Biot number discussed in [46], as it influences the B term in Equation (36), determining the rate of convergence of the CHT coupling process. Indeed, the current problem exhibits an optimal h = 1000 W/m K, since the highest value approaches the limit of the stability region, with incipient oscillations in the value of heat flux exchanged at the interface, before reaching convergence. The solid grid size, instead, characterizes the accuracy of the interpolation of the quantities passed between the two domains through the DWI process and a ects the A term in Equation (36). In general, finer solid grids are favored by the current analysis. Remarkably, the fluid mesh size follows an opposite trend because its influence on the A term is less pronounced, as the coarsest level already provides good-quality solutions. On the other hand, the signal S is driven by the increase in computational overhead experienced with the fluid mesh refinement. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 21 of 31 Figure 13. Normalized signal S dependence from factor levels in the Orthogonal Array. Figure 13. Normalized signal S dependence from factor levels in the Orthogonal Array. Since the weighting coecient ! privileges the accuracy of the CHT coupling process, the final Since the weighting coefficient ω privileges the accuracy of the CHT coupling process, the final selection of the factors levels results in the intermediate values for the refinements of both the selection of the factors levels results in the intermediate values for the refinements of both the grids grids and for the virtual heat transfer coecient. However, it is recognized that smaller solid and for the virtual heat transfer coefficient. However, it is recognized that smaller solid element sizes element sizes improve the stability of the CHT coupling process because it avoids local poor improve the stability of the CHT coupling process because it avoids local poor quality in the quality in the discretization of the blade surface in correspondence with thin regions (Figure 14a), discretization of the blade surface in correspondence with thin regions (Figure 14a), potentially potentially inducing drops in the local Biot number and inconsistencies with the selected h value from inducing drops in the local Biot number and inconsistencies with the selected ℎ value from a stability a stability standpoint. Therefore, the issue is addressed by the generation of a “hybrid” configuration standpoint. Therefore, the issue is addressed by the generation of a “hybrid” configuration (Figure (Figure 14b), envisaging a local solid mesh refinement in correspondence with the blade tip surface 14b), envisaging a local solid mesh refinement in correspondence with the blade tip surface and and trailing edge, whilst maintaining the intermediate element size in the rest of the domain. This last trailing edge, whilst maintaining the intermediate element size in the rest of the domain. This last setup increases the mesh density to a total node count of about 420 k. setup increases the mesh density to a total node count of about 420 k. Figure 14. Comparison of solid mesh refinements: (a) coarse grid, (b) hybrid-mid grid. Finally, the Orthogonal Array analysis is repeated, assigning the integral of temperature deviations to the A term in Equation (36). The outcome, shown in Figure 15, confirms the previous considerations, except for the fluid mesh size, which promotes the trend towards the coarsest mesh because it still provides sufficiently accurate results. However, since the difference between the coarse and intermediate meshes is moderate, the previous selection of the medium size grid will be pursued for the sake of improved prediction accuracy. Based on such considerations, the settings returned by the analysis of the Orthogonal Array are considered for the further assessments of the coupling problem. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 21 of 31 Figure 13. Normalized signal S dependence from factor levels in the Orthogonal Array. Since the weighting coefficient ω privileges the accuracy of the CHT coupling process, the final selection of the factors levels results in the intermediate values for the refinements of both the grids and for the virtual heat transfer coefficient. However, it is recognized that smaller solid element sizes improve the stability of the CHT coupling process because it avoids local poor quality in the discretization of the blade surface in correspondence with thin regions (Figure 14a), potentially inducing drops in the local Biot number and inconsistencies with the selected ℎ value from a stability standpoint. Therefore, the issue is addressed by the generation of a “hybrid” configuration (Figure 14b), envisaging a local solid mesh refinement in correspondence with the blade tip surface and Int. J. Turbomach. Propuls. Power 2020, 5, 30 22 of 31 trailing edge, whilst maintaining the intermediate element size in the rest of the domain. This last setup increases the mesh density to a total node count of about 420 k. Figure 14. Comparison of solid mesh refinements: (a) coarse grid, (b) hybrid-mid grid. Figure 14. Comparison of solid mesh refinements: (a) coarse grid, (b) hybrid-mid grid. Finally, the Orthogonal Array analysis is repeated, assigning the integral of temperature Finally, the Orthogonal Array analysis is repeated, assigning the integral of temperature deviations deviations to the A term in Equation (36). The outcome, shown in Figure 15, confirms the previous to the A term in Equation (36). The outcome, shown in Figure 15, confirms the previous considerations, considerations, except for the fluid mesh size, which promotes the trend towards the coarsest mesh except for the fluid mesh size, which promotes the trend towards the coarsest mesh because it because it still provides sufficiently accurate results. However, since the difference between the still provides suciently accurate results. However, since the di erence between the coarse and coarse and intermediate meshes is moderate, the previous selection of the medium size grid will be intermediate meshes is moderate, the previous selection of the medium size grid will be pursued for pursued for the sake of improved prediction accuracy. the sake of improved prediction accuracy. Based on such considerations, the settings returned by the analysis of the Orthogonal Array are Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 22 of 31 considered for the further assessments of the coupling problem. Figure 15. Normalized signal S dependence from factor levels in Orthogonal Array—integral of Figure 15. Normalized signal S dependence from factor levels in Orthogonal Array—integral of temperate deviation assigned to term A. temperate deviation assigned to term A. 4.3. Gradients Accuracy Evaluation Based on such considerations, the settings returned by the analysis of the Orthogonal Array are considered for the further assessments of the coupling problem. The present work accounts for the manual differentiation of the multidisciplinary process shown in Figure 6. The sensitivities are accurately computed by the reverse Algorithmic Differentiation 4.3. Gradients Accuracy Evaluation technique [54] for what concerns the FEM solvers and the hFFB process, while the partitioned coupling approach opens the path to the direct integration of the in-house adjoint CFD solver. The present work accounts for the manual di erentiation of the multidisciplinary process shown The validation of the differentiated FEM solvers is presented herein with reference to the rotor in Figure 6. The sensitivities are accurately computed by the reverse Algorithmic Di erentiation test case from Section 4.2. The adjoint sensitivities ⁄ of the maximum temperature and of the technique [54] for what concerns the FEM solvers and the hFFB process, while the partitioned coupling maximum von Mises stress including the thermal strain w.r.t. selected design parameters (referenced approach opens the path to the direct integration of the in-house adjoint CFD solver. in Table 6 and Figure 16) are compared to the correspondent gradients computed by the Finite The validation of the di erentiated FEM solvers is presented herein with reference to the rotor Differences (FD) technique. The latter accounts for evaluations by a central differencing scheme, test case whofr seom optiSection mal step4.2 siz.e The is seaadjoint rched wisensitivities th the aim of a@cJc/ ur@ ate of grthe adiemaximum nts computatemperatur tions. An exe em and plary of the −6 maximum case isvon repo Mises rted in str Fess igurincluding e 17, showthe ing th thermal e identistrain ficatiow n .r o.t. f a selec suitab ted le sdesign tep sizeparameters at the value(r oefer f 10 enced m in for the seventh design variable in the framework of thermal evaluations by the heat transfer solver. Table 6 and Figure 16) are compared to the correspondent gradients computed by the Finite Di erences The magnitude of the perturbation must avoid too-small values, resulting in round-off errors, and (FD) technique. The latter accounts for evaluations by a central di erencing scheme, whose optimal too large values as well, because they introduce significant truncation errors. This outcome results step size is searched with the aim of accurate gradients computations. An exemplary case is reported from the evaluation of the cost function at each step size by solving the linear system obtained after the perturbation of the mesh through the morphing technique described in [20]. Table 6. Design variables adopted in the dJ/dα gradient validation. Design Variables α 1 Rotor hub meridional contour: Y-coordinate at 20% chord; 2 Rake angle; 3 Back-plate thickness; 4 Turbine shaft diameter; 5 Turbine shaft length; 6 Rotor maximum diameter; 7 Blade height at leading edge; 8 Back-plate/shaft connection axial position; 9 Blade hub thickness ; 10 Blade hub fillet radius. Int. J. Turbomach. Propuls. Power 2020, 5, 30 23 of 31 in Figure 17, showing the identification of a suitable step size at the value of 10 m for the seventh design variable in the framework of thermal evaluations by the heat transfer solver. The magnitude of the perturbation must avoid too-small values, resulting in round-o errors, and too large values as well, because they introduce significant truncation errors. This outcome results from the evaluation of the cost function at each step size by solving the linear system obtained after the perturbation of the mesh through the morphing technique described in [20]. Table 6. Design variables adopted in the dJ/d gradient validation. Design Variables 1 Rotor hub meridional contour: Y-coordinate at 20% chord; 2 Rake angle; 3 Back-plate thickness; 4 Turbine shaft diameter; 5 Turbine shaft length; 6 Rotor maximum diameter; 7 Blade height at leading edge; 8 Back-plate/shaft connection axial position; 9 Blade hub thickness; 10 Blade hub fillet radius. Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 23 of 31 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 23 of 31 Figure 16. Design variables from Table 2. Figure 16. Design variables from Table 2. Figure 16. Design variables from Table 2. Figure Fig 17. ure Step 17. size Step evaluation size evaluat for ionthe for computation the computatiof ondJ o/f dd J/d by α Finite by Fin Di ite Di erences: fferences Heat : Hea Transfer t Transfer solver , Figure 17. Step size evaluation for the computation of dJ/dα by Finite Differences: Heat Transfer solver, variable 7. variable 7. solver, variable 7. The sensitivities of the maximum solid temperature to perturbations of the ten design variables The sensitivities of the maximum solid temperature to perturbations of the ten design variables are summarized in Figure 18a, which shows a close agreement between the two methods, both in are summarized in Figure 18a, which shows a close agreement between the two methods, both in sign and magnitude. To understand the sensitivity analysis in detail, let us consider the thermal paths sign and magnitude. To understand the sensitivity analysis in detail, let us consider the thermal paths in the turbine rotor experiencing the convective loading resulting from the operative condition in the turbine rotor experiencing the convective loading resulting from the operative condition reported in Table 2, along with a convective condition on the back-plate surface with a uniform fluid reported in Table 2, along with a convective condition on the back-plate surface with a uniform fluid temperature of 950 K and a Dirichlet boundary condition of 500 K assigned to the extreme section of temperature of 950 K and a Dirichlet boundary condition of 500 K assigned to the extreme section of the shaft. the shaft. Int. J. Turbomach. Propuls. Power 2020, 5, 30 24 of 31 The sensitivities of the maximum solid temperature to perturbations of the ten design variables are summarized in Figure 18a, which shows a close agreement between the two methods, both in sign and magnitude. To understand the sensitivity analysis in detail, let us consider the thermal paths in the turbine rotor experiencing the convective loading resulting from the operative condition reported in Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 24 of 31 Table 2, along with a convective condition on the back-plate surface with a uniform fluid temperature of Int950 . J. Tu K rbo and macha . Pr Dirichlet opuls. Powboundary er 2020, 5, x Fcondition OR PEER REV ofIEW 500 K assigned to the extreme section of the shaft. 24 of 31 Figure 18. Comparison of the dJ/dα sensitivities of the solid maximum temperature (a) and the Figure 18. Comparison of the dJ/d sensitivities of the solid maximum temperature (a) and the maximum von Mises stress (b): gradients computed by the adjoint method vs. Finite Differences. Figure 18. Comparison of the dJ/dα sensitivities of the solid maximum temperature (a) and the maximum von Mises stress (b): gradients computed by the adjoint method vs. Finite Di erences. maximum von Mises stress (b): gradients computed by the adjoint method vs. Finite Differences. Figure 19 reveals that the maximum temperature is detected at the nodes in the region of the Figure 19 reveals that the maximum temperature is detected at the nodes in the region of the hub, hub, in proximity with the leading edge. The computation of such constraint through a p-norm Figure 19 reveals that the maximum temperature is detected at the nodes in the region of the in proximity with the leading edge. The computation of such constraint through a p-norm function function returns a marked sensitivity from the rotor back-plate thickness, whose enlargement would hub, in proximity with the leading edge. The computation of such constraint through a p-norm returns a marked sensitivity from the rotor back-plate thickness, whose enlargement would reduce reduce the influence of the heat sink located at the back-plate outer surface. Similarly, an increase in function returns a marked sensitivity from the rotor back-plate thickness, whose enlargement would the influence of the heat sink located at the back-plate outer surface. Similarly, an increase in blade blade height at the leading edge would extend the thermal path to the colder areas highlighted at the reduce the influence of the heat sink located at the back-plate outer surface. Similarly, an increase in height at the leading edge would extend the thermal path to the colder areas highlighted at the tip, tip, while an elongation of the shaft itself would reduce the gradient field. On the contrary, an increase blade height at the leading edge would extend the thermal path to the colder areas highlighted at the while an elongation of the shaft itself would reduce the gradient field. On the contrary, an increase in in the turbine shaft diameter, an advancement of the axial position of the intersection between the tip, while an elongation of the shaft itself would reduce the gradient field. On the contrary, an increase the turbine shaft diameter, an advancement of the axial position of the intersection between the rotor rotor back-plate and the shaft, and an enlargement of the blade hub thickness would favor the cooling in the turbine shaft diameter, an advancement of the axial position of the intersection between the back-plate and the shaft, and an enlargement of the blade hub thickness would favor the cooling of of the hub upper region. Such sensitivities provide the descent direction for a constrained rotor back-plate and the shaft, and an enlargement of the blade hub thickness would favor the cooling the hub upper region. Such sensitivities provide the descent direction for a constrained optimization optimization problem aimed at limiting the maximum rotor temperature for the sake of extended of the hub upper region. Such sensitivities provide the descent direction for a constrained problem aimed at limiting the maximum rotor temperature for the sake of extended lifetime. lifetime. optimization problem aimed at limiting the maximum rotor temperature for the sake of extended lifetime. Figure 19. Solid temperature on turbine rotor (operative condition from Table 2). Figure 19. Solid temperature on turbine rotor (operative condition from Table 2). Figure 19. Solid temperature on turbine rotor (operative condition from Table 2). Similarly, the accuracy of the adjoint structural solver sensitivities is assessed by comparison with the same gradients computed by Finite Differences. The results reported in Figure 18b confirm Similarly, the accuracy of the adjoint structural solver sensitivities is assessed by comparison the suitability of the manually differentiated framework. with the same gradients computed by Finite Differences. The results reported in Figure 18b confirm Moreover, the time for the computation of the adjoint-based gradients for the heat transfer solver the suitability of the manually differentiated framework. and the structural solver was measured. If X is the time required for the computation of the Moreover, the time for the computation of the adjoint-based gradients for the heat transfer solver multidisciplinary workflow in primal mode and the problem accounts for n design variables, the and the structural solver was measured. If X is the time required for the computation of the calculation of the gradients by Finite Differences would approximately cost n ∙ X. On the contrary, multidisciplinary workflow in primal mode and the problem accounts for n design variables, the the application of the adjoint method herein described costs about 8.6X for the heat transfer module calculation of the gradients by Finite Differences would approximately cost n ∙ X. On the contrary, the application of the adjoint method herein described costs about 8.6X for the heat transfer module Int. J. Turbomach. Propuls. Power 2020, 5, 30 25 of 31 Similarly, the accuracy of the adjoint structural solver sensitivities is assessed by comparison with the same gradients computed by Finite Di erences. The results reported in Figure 18b confirm the suitability of the manually di erentiated framework. Moreover, the time for the computation of the adjoint-based gradients for the heat transfer Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 25 of 31 solver and the structural solver was measured. If X is the time required for the computation of and 2.3X for the structural analysis, in fair agreement with the original expectations. Such costs the multidisciplinary workflow in primal mode and the problem accounts for n design variables, mostly depend on the assembly process of the differentiated system (Equation (25)) rather than the the calculation of the gradients by Finite Di erences would approximately cost n X. On the contrary, solver itself. the application of the adjoint method herein described costs about 8.6X for the heat transfer module Finally, the loose-coupling approach adopted for the development of the CHT workflow allows and 2.3X for the structural analysis, in fair agreement with the original expectations. Such costs mostly the direct integration of the in-house adjoint CFD solver, and the accuracy of the computed gradients depend on the assembly process of the di erentiated system (Equation (25)) rather than the solver itself. was already demonstrated by the same author. Finally, the loose-coupling approach adopted for the development of the CHT workflow allows the direct integration of the in-house adjoint CFD solver, and the accuracy of the computed gradients 5. Results and Discussion was already demonstrated by the same author. The design work aims at reducing the thermo-mechanical stresses in the turbine impeller 5. Results and Discussion presented in Section 4.2. The operative condition in Table 2 is considered, along with a Dirichlet The design work aims at reducing the thermo-mechanical stresses in the turbine impeller presented boundary condition with a temperature of 500 K imposed at the extreme cross section of the shaft, in Section 4.2. The operative condition in Table 2 is considered, along with a Dirichlet boundary emulating the heat sink provided by the oil cooling circuit in the turbocharger bearing housing. The condition with a temperature of 500 K imposed at the extreme cross section of the shaft, emulating the contribution of the thermal strains introduced by the present work improves the accuracy of the heat sink provided by the oil cooling circuit in the turbocharger bearing housing. The contribution of computation of the von Mises stresses, whose gradients provide information for the geometry the thermal strains introduced by the present work improves the accuracy of the computation of the modifications. von Mises stresses, whose gradients provide information for the geometry modifications. According to the grids selection anticipated in Section 4.2, the fluid domain is discretized with a According to the grids selection anticipated in Section 4.2, the fluid domain is discretized with a multi-block structured mesh of about 1.3M cells with boundary layer refinement. The solid domain multi-block structured mesh of about 1.3 M cells with boundary layer refinement. The solid domain accounts for an unstructured grid of approximately 420 k nodes and second-order tetrahedral accounts for an unstructured grid of approximately 420 k nodes and second-order tetrahedral elements. elements. Figure 20 presents the temperature distribution in the baseline geometry under the selected Figure 20 presents the temperature distribution in the baseline geometry under the selected steady-state operative condition. A significant thermal gradient is identified at the connection between steady-state operative condition. A significant thermal gradient is identified at the connection the shaft and the rotor back-plate, with a correspondent local variation of thermal strains according to between the shaft and the rotor back-plate, with a correspondent local variation of thermal strains Equation (9). This is reflected in increased localized stresses, as demonstrated in Figure 21. In fact, according to Equation (9). This is reflected in increased localized stresses, as demonstrated in Figure Figure 21a shows the mechanical pattern of the baseline geometry, whose structural computation is 21. In fact, Figure 21a shows the mechanical pattern of the baseline geometry, whose structural performed considering only the centrifugal loading. On the other hand, Figure 21b reveals the new computation is performed considering only the centrifugal loading. On the other hand, Figure 21b stress distribution accounting also for the thermal field, resulting from the convective input and the reveals the new stress distribution accounting also for the thermal field, resulting from the convective heat sink at the shaft. Indeed, a local deviation in von Mises stresses of about 20% is identified in the input and the heat sink at the shaft. Indeed, a local deviation in von Mises stresses of about 20% is region around the connection with the back-plate. identified in the region around the connection with the back-plate. Figure 20. Solid temperature distribution with rotor experiencing convective loading from the exhaust Figure 20. Solid temperature distribution with rotor experiencing convective loading from the gases and shaft cooling (baseline geometry). exhaust gases and shaft cooling (baseline geometry). Int. J. Turbomach. Propuls. Power 2020, 5, 30 26 of 31 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 26 of 31 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 26 of 31 Figure 21. Normalized von Mises stresses pattern comparison in baseline layout: (a) mechanical Figure Figure21. 21Normalized . Normalizevon d vo Mises n Misstr esesses strespattern ses pattcomparison ern comparin iso baseline n in bas layout: eline la (a y)omechanical ut: (a) mecversus hanical versus (b) thermo-mechanical. (b v) er thermo-mechanical. sus (b) thermo-mechanical. The suitability of the process described in Section 3 is tested with the computation of the The suitability of the process described in Section 3 is tested with the computation of the The suitability of the process described in Section 3 is tested with the computation of the sensitivities of the thermo-mechanical constraint w.r.t. the grid coordinates, as reported in Figure 22, sensitivities of the thermo-mechanical constraint w.r.t. the grid coordinates, as reported in Figure 22, sensitivities of the thermo-mechanical constraint w.r.t. the grid coordinates, as reported in Figure 22, providing the information for a grid perturbation. Therefore, Figure 23 compares the distribution of providing the information for a grid perturbation. Therefore, Figure 23 compares the distribution of providing the information for a grid perturbation. Therefore, Figure 23 compares the distribution of von Mises stresses in the baseline geometry (left) and in the updated layout (right). A local reduction von Mises stresses in the baseline geometry (left) and in the updated layout (right). A local reduction von Mises stresses in the baseline geometry (left) and in the updated layout (right). A local reduction in the maximum von Mises stress in excess of 35% is achieved with the new configuration. Most of in the maximum von Mises stress in excess of 35% is achieved with the new configuration. Most of the in the maximum von Mises stress in excess of 35% is achieved with the new configuration. Most of the advancements can be attributed to the increased curvature of the back-plate contour, in particular advancements can be attributed to the increased curvature of the back-plate contour, in particular in the advancements can be attributed to the increased curvature of the back-plate contour, in particular in the region of the connection with the shaft, originally experiencing the highest concentrated the region of the connection with the shaft, originally experiencing the highest concentrated stresses. in the region of the connection with the shaft, originally experiencing the highest concentrated stresses. The new shape reveals a more gradual mechanical pattern at such intersection, supported The new shape reveals a more gradual mechanical pattern at such intersection, supported also by a stresses. The new shape reveals a more gradual mechanical pattern at such intersection, supported also by a smoother temperature evolution lowering the influence of the local thermal strains. The smoother temperature evolution lowering the influence of the local thermal strains. The new layout also by a smoother temperature evolution lowering the influence of the local thermal strains. The new layout returns lower sensitivity values, in line with the goal of satisfying the thermo-structural returns lower sensitivity values, in line with the goal of satisfying the thermo-structural constraint new layout returns lower sensitivity values, in line with the goal of satisfying the thermo-structural constraint within an optimization problem. within an optimization problem. constraint within an optimization problem. Figure 22. Sensitivities of the thermo-mechanical constraint w.r.t. grid coordinates. Figure 22. Sensitivities of the thermo-mechanical constraint w.r.t. grid coordinates. Figure 22. Sensitivities of the thermo-mechanical constraint w.r.t. grid coordinates. Other areas of the blade are highlighted in Figure 22, in particular at high blade span and in the upper portion of the back-plate. The activation of the sensitivities in such regions is favored by the contribution of the thermal input. However, these portions of the rotor present a pattern of smaller stresses, not violating the original constraint, and therefore their sensitivity fields are neglected. Int. J. Turbomach. Propuls. Power 2020, 5, 30 27 of 31 Int. J. Turbomach. Propuls. Power 2020, 5, x FOR PEER REVIEW 27 of 31 Figure Figure23. 23N . N oro m rm alia zle iz ded vo n vM oni se Mi s ss tes res s se tr ses dis ses tri bd uitsit o rn ib iu nttiu or n b ii n n e itm ur p b eiln le e r e im xp p eel rie le nrc iex ngp th er eiren mo ci-n m ge cth ha er nm ica o l- loading: (a) baseline configuration, (b) updated geometry. mechanical loading: (a) baseline configuration, (b) updated geometry. The second positive e ect of the implementation of the CHT analysis within the design loop Other areas of the blade are highlighted in Figure 22, in particular at high blade span and in the is about the more accurate prediction of the actual turbine eciency, as no more adiabatic walls upper portion of the back-plate. The activation of the sensitivities in such regions is favored by the assumptions are in place. The deviation in the total-to-static eciency w.r.t. an adiabatic simulation is contribution of the thermal input. However, these portions of the rotor present a pattern of smaller about 0.2% in the selected operative condition while considering the sole impeller. However, more stresses, not violating the original constraint, and therefore their sensitivity fields are neglected. marked di erences are expected in case the CHT analysis is extended to the whole turbine stage, The second positive effect of the implementation of the CHT analysis within the design loop is including the volute and the nozzle guide vanes. about the more accurate prediction of the actual turbine efficiency, as no more adiabatic walls assumptions are in place. The deviation in the total-to-static efficiency w.r.t. an adiabatic simulation 6. Conclusions is about 0.2% in the selected operative condition while considering the sole impeller. However, more mark This ed d paper ifferen addr ces esses are ex the pec pr teoblem d in ca of sethe the Fluid–Str CHT an uctu alysr ie s Interaction is extended in to the thdesign e wholoptimization e turbine stag of e, turbomachinery components under thermo-mechanical constraints. The activity focuses on a radial including the volute and the nozzle guide vanes. turbine rotor for turbocharger applications, with the aim of reducing the maximum von Mises stress 6. Conclusions located in a critical area of the impeller experiencing the influence of the thermal loading. The problem of the integration of high-fidelity thermal predictions within the framework of an This paper addresses the problem of the Fluid–Structure Interaction in the design optimization optimization is addressed by the development of a novel multidisciplinary gradient-based approach, of turbomachinery components under thermo-mechanical constraints. The activity focuses on a suitable for the ecient computation of the gradients of the cost function with respect to a large radial turbine rotor for turbocharger applications, with the aim of reducing the maximum von Mises number of design variables. The introduction of a Conjugate Heat Transfer evaluation procedure, stress located in a critical area of the impeller experiencing the influence of the thermal loading. supported by the heat transfer forward flux back method, is undertaken in order to establish the energy The problem of the integration of high-fidelity thermal predictions within the framework of an coupling of the solid domain with its surrounding fluid, according to a partitioned coupling approach. optimization is addressed by the development of a novel multidisciplinary gradient-based approach, The gradient calculation is addressed by a discrete adjoint technique, and the resulting sensitivities of suitable for the efficient computation of the gradients of the cost function with respect to a large the thermo-mechanical constraint w.r.t. the grid coordinates allow for the mechanical optimization of number of design variables. The introduction of a Conjugate Heat Transfer evaluation procedure, the component, including considerations related to the impact of its thermal operative environment. supported by the heat transfer forward flux back method, is undertaken in order to establish the The solid–fluid grid coupling process, the analysis procedure, and the multidisciplinary energy coupling of the solid domain with its surrounding fluid, according to a partitioned coupling di erentiation technique are described. Moreover, the validation of the conjugate problem is discussed approach. The gradient calculation is addressed by a discrete adjoint technique, and the resulting with reference to the comparison with an analytic solution and a mesh sensitivity study. The accuracy sensitivities of the thermo-mechanical constraint w.r.t. the grid coordinates allow for the mechanical of the computed gradients is also presented. optimization of the component, including considerations related to the impact of its thermal The application of the multidisciplinary workflow demonstrates the possibility of controlling the operative environment. local von Mises stresses within admissible ranges, extending the outreach of the original mechanical The solid–fluid grid coupling process, the analysis procedure, and the multidisciplinary evaluations with the inclusion of the thermal stresses. New regions of the impeller, before not a ected differentiation technique are described. Moreover, the validation of the conjugate problem is by such considerations, are now involved, as their relative sensitivities are enhanced by the local thermal discussed with reference to the comparison with an analytic solution and a mesh sensitivity study. field. In particular, in the tested sample an improvement exceeding 35% in local von Mises stresses The accuracy of the computed gradients is also presented. is achieved by the application of the presented method. Additionally, more accurate performance The application of the multidisciplinary workflow demonstrates the possibility of controlling predictions result from the introduction of the CHT analysis within the design optimization process, the local von Mises stresses within admissible ranges, extending the outreach of the original no more relying on the adiabatic walls assumption. mechanical evaluations with the inclusion of the thermal stresses. New regions of the impeller, before not affected by such considerations, are now involved, as their relative sensitivities are enhanced by the local thermal field. In particular, in the tested sample an improvement exceeding 35% in local von Int. J. Turbomach. Propuls. Power 2020, 5, 30 28 of 31 This contribution is a new step towards the large-scale optimization of turbomachinery components by gradient-based methods, with the goal of the future introduction of fatigue life evaluations in transient conditions. Author Contributions: Conceptualization, A.R. and T.V.; methodology, A.R. and T.V.; software, A.R.; validation, A.R. and T.V.; formal analysis, A.R.; investigation, A.R.; resources, T.V. and L.C.; writing—original draft preparation, A.R.; writing—review and editing, A.R., T.V., and L.C.; visualization, A.R.; supervision, T.V. and L.C.; project administration, T.V. and L.C.; funding acquisition, T.V. and L.C. All authors have read and agreed to the published version of the manuscript. Funding: The research leading to these results is co-funded by General Motors Global Propulsion Systems and PUNCH Torino S.p.A. under the grant of the Industrial PhD Program. Acknowledgments: We gratefully thank General Motors Global Propulsion Systems for the permission to publish this paper. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the collection, analysis, or interpretation of data or in the writing of the manuscript. Nomenclature CFD Computational Fluid Dynamics FEM Finite Element Method DWI Distance-Weighted Interpolation CHT Conjugate Heat Transfer FSI Fluid Structure Interaction JT-KIRK Jacobian Trained Krylov Implicit Runge Kutta MUSCL Monotonic Upstream-centered Scheme for Conservation Laws design variable ( ) T thermal expansion coecient [K ] " thermal strain [-] von Mises stresses [Pa] VM adjoint variable dist(i) distance between i-th fluid cell-solid node coupling in DWI procedure [m] dT/dn temperature gradient normal to wall [K/m] E(T) elasticity-strain matrix h virtual heat transfer coecient [W/m K] J cost function, objective function k thermal conductivity coecient [W/mK] q heat flux from fluid domain normal to the wall [W/m ] FW q heat flux in solid domain normal to the wall [W/m ] SW R non-linear residuals in fluid analysis T fluid temperature in first layer of inner domain cells [K] D1 T virtual fluid bulk temperature [K] f l T0 virtual fluid bulk temperature interpolated by DWI procedure [K] f l T fluid temperature at the wall [K] FW T fluid temperature in first layer of ghost cells [K] G1 T reference temperature for thermal strains calculation [K] re f U conservative flow variables in fluid analysis u adjoint variable V primitive flow variables at domain boundaries in fluid analysis bnd X fluid/solid grid coordinate (x,y,z) [m] Int. 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Introduction to Quality Engineering: Designing Quality into Products and Processes; Asian Productivity Organization/UNIPUB: White Plain, NY, USA; Unipub/Kraus: New York, NY, USA, 1986. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional aliations. © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY-NC-ND) license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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International Journal of Turbomachinery, Propulsion and PowerMultidisciplinary Digital Publishing Institute

Published: Nov 25, 2020

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