cghnatios@ndu.edu.lb Department of mechanical A nonparametric method assessing the error and variability margins in solutions engineering, Notre Dame depicted in a separated form using experimental results is illustrated in this work. The University-Louaize, PO Box 72, Zouk Mosbeh, Lebanon method assess the total variability of the solution including the modeling error and the Full list of author information is truncation error when experimental results are available. The illustrated method is available at the end of the article based on the use of the PGD separated form solutions, enriched by transforming a part of the PGD basis vectors into probabilistic one. The constructed probabilistic vectors are restricted to the physical solution’s Stiefel manifold. The result is a real-time parametric PGD solution enhanced with the solution variability and the conﬁdence intervals. Keywords: Proper Generalized Decomposition, Model reduction, Automated tape placement, hybrid modeling, Nonparametric probabilistic method, Machine learning Introduction Nowadays, simulation-based decision making in engineering and sciences is widely accepted and used in predicting material and parts behavior. Many partial diﬀerential equation (PDE) models are widely accepted and used in the industry, while constantly being improved to increase their ﬁdelity. On the other hand, many advanced methods for solving PDEs are being designed to improve solution speed, aiming for real-time appli- cations [4,11,13,30,37]. These techniques are commonly known by model order reduc- tion techniques (or simply model reduction techniques), mainly aiming at constructing a reduced basis for the solution of PDEs [22,40]. Regarding model order reduction techniques, the methods are divided into two large categories: (i) “a priori” model reduction techniques, which aims at constructing a solution in a reduced basis separated form, before having any knowledge of the full order model or also the high deﬁnition model (HDM) solution [21,27]. (ii) “a posteriori” model reduction techniques, where the reduced order basis of projection is constructed from previous HDM solutions computed using eﬃcient greedy algorithms to construct as accurate as © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 0123456789().,–: volV Ghnatios and Barasinski Adv. Model. and Simul. in Eng. Sci. (2021) 8:20 Page 2 of 13 possible reduced order basis [5,6,34,39]. The PGD method appears to be the only “a priori” model reduction technique which is widely used nowadays to compute solutions in a fairly separable domain [10,25,26]. Using model reduction techniques or reduced order basis methods lead in general to a good approximation, which tends to the ﬁnite element approximation when increasing the size of the projection basis [2,3,41]. However, it is known that model reduction techniques in general involve a truncation error (either being an “a priori” or “a posteriori” techniques) as well as modeling errors, inherited from their full-order models counterparts, also known as high deﬁnition models (HDM) [18]. Therefore, a margin of improvement of the solution do exist. On the other hand, with the current development of machine learning and data-driven techniques, ﬁtting data using surrogate models based on regressions, neural network is becoming widely available and increasingly popular [14,24,32,47]. However, these surro- gate models do not impose naturally physical conditions like the conservation of mass or energy. Therefore, some other PDE ﬁtting techniques are being developed, based on satis- fying thermodynamic constraints, using methods like the GENERIC formalism [20,29]or the General Equilibrium for Non-Equilibrium Reversible-Irreversible Coupling [31,38]. However, often engineering applications data is scarce and expensive to generate in suﬃ- cient quantities to solely rely on data models. Therefore, to leverage recent developments of artiﬁcial intelligence and machine learning algorithms, combinations of model reduc- tion techniques with machine learning are built to address the modeling and truncation errors, either by error correction [1] or by improvement of modal selection using the quantities of interest [33]. Another appealing approach for creating data-driven surrogate models while leveraging previously well-established models is the “Digital Twins” approach [1,9]. Digital twins are built based on the correction of previously established PDEs using surrogate models, to learn new physics or establish new models quantifying the error [19,28], but also to incorporate the variability of the experimental results in the HDM or reduced order models solutions [15,45,46]. Multiple other works aimed to quantify the errors induced by modeling, either for the HDM solutions [16,17,36,42], or for using model reduction techniques relying on reduced order basis (ROB) [8,44,45]. Moreover, a non-parametric probabilistic method (NPM) for error quantiﬁcation was introduced in [43] and extended to μ-parametric ROB methods in [45]. The NPM method has the advantage of quantifying uncertainties independently of their sources, either modeling errors, nonlinear modeling, truncation errors... This is performed by replacing the deterministic reduced basis by a stochastic counterpart [45]. However, to the best knowledge of the authors, non of these techniques where adapted to be used for the Proper Generalized Decomposition (PGD) framework, but are rather used for the Finite elements HDM approach, or for a μ-parametric ROB solution [45]. In this work, we aim to extend the NPM method presented in [45] to the PGD framework. The extension results in a stochastic solution based on a stochastic reduced basis, who’s variations are restricted to the Stiefel manifold of the original PGD solution. The variations are controlled by several hyper parameters identiﬁed through solving an optimization problem on the ﬂy, as the method do not require any direct solution of the PDE modeling the problem during optimization. First of all, we will revisit NPM approach in “NPM method for PGD solutions” section of this work. Later on, we introduce the PGD simulation and experimental measurements of Ghnatios and Barasinski Adv. Model. and Simul. in Eng. Sci. (2021) 8:20 Page 3 of 13 the automated tape placement, the application selected for this work, in “The automated tape placement process, simulation and measurements” section. In “NPM method applied for the PGD simulation of the automated tape placement process” section we illustrate the NPM results applied on the PGD solution obtained in “The automated tape placement process, simulation and measurements” section. Finally the article is wrapped up with discussions and conclusions in “Conclusion” section. NPM method for PGD solutions Short review of NPM for reduced order basis Let’s consider for illustration purpose a PDE having the following semi-discrete formula- tion: [M][y¨(t)] + [C][y˙(t)] + [K ][y(t)] + [F(t)] = 0, (1) with t being the time domain, and the solution [y] can be projected on a reduced basis [V ] such as: [y(t)] = [V ][q(t)] (2) The variable [q(t)] is the solution of the PDE equation depicted in Eq. (1) in the reduced coordinate system, after projection on the reduced order basis, and solving as a function of the time. The NPM method consists of replacing a reduced order basis [V ] ∈ R (N,n) by a stochastic counterpart [V] ∈ R , where N is the number of degrees of freedom (N,n) in the HDM model, and n is the dimension of the built reduced basis. Considering a set of constraints or Dirichlet boundary conditions deﬁned by a matrix [B] ∈ R ,the (N,N ) BC reduced basis [V ] should satisfy the following orthogonality and boundary conditions constraints: [V ] [M][V ] = [I ] (3) [B] [V ] = [0] where [I ] being the identity matrix in a domain having the dimension of the reduced basis n,and [M] ∈ [N, N] the mass matrix of the problem in question. Therefore, the stochastic reduced basis should satisfy the same constraints as the initial reduced basis: [V] [M][V] = [I ] (4) [B] [V] = [0] Using NPM, the stochastic basis [V] is constructed using the maximum entropy princi- ple, while variation of [V ] to construct [V] should be performed on the tangent vector T space deﬁned by [45]: T T T = [Z] ∈ R :[V] [M][Z] + [Z] [M][V] = [0] (5) v (N,n) with [Z] being therefore the ﬁrst derivative of [V ] using the inner product << V ,V >>= 1 2 tr [V ] [M][V ] . Therefore, [Z] can be built in the following form: 1 2 [Z] = [V][a] + [V ][b](6) ⊥ Ghnatios and Barasinski Adv. Model. and Simul. in Eng. Sci. (2021) 8:20 Page 4 of 13 Deﬁning [a], [b]and [V ] leads to the following construction algorithm [45]: [U] = [G(β)][σ ] [A] = [U] − [B]([B] [U]) T T [D] = [V ] [M][A] + [A] [M][V ] /2 (7) [Z] = [A] − [V ][D] −1/2 ⎪ 2 T ⎪ [H] = [I ] + s [Z] [M][Z] [V] = [V ] + s[Z] [H] ( ) with [σ]being an n × n upper diagonal matrix of hyper-parameters ∈ R to be identiﬁed, [G(β)] is a second order centered random matrix ∈ R built from two uniform random (N,n) distributions as well as the hyper-parameter β ∈ [0; 1], as described in the appendix D in [45]. s ∈ [0; 1] is also a hyper-parameter to deﬁne. The problem hyper-parameters are found using the following minimization problem: [σ ], β,s = argmin w E O([V],t) − E d+ ( ) ( ) exp (8) (1 − w) Var (O([V],t)) − Var d exp We deﬁne w ∈ [0; 1] as a weight to insure as much as possible a convexity in the cost function. E(·) deﬁnes the mean of a distribution, Var(·) deﬁne the standard deviation of a given distribution. E and Var are the experimental values of the target values of exp exp the quantities of interest O([V],t), t being the time domain. The quantities of interest are deﬁned as a function of the solution of the PDE equations of the modeled problem: O([V]) = H([V][p(t)],t), (9) with [p(t)] ∈ R[n, m] the solution of the PDE in the reduced order basis coordinate system, m being the length of the mesh in the time domain. Computing [p(t)] would require a direct solution of the PDE (1) in the reduced coordinate system, for every variation of the reduced basis [V]. This gets even more time consuming when the derivatives are to be computed numerically during the optimization process. Interested readers are referred to the [15,45,46] and their references therein. To evaluate stochastic quantities of interests of the generated stochastic solution y, derived from the expression of NPM, one may need to repeat the algorithm (7) m times, to construct several solutions for diﬀerent selection values of [G(β)]. In fact, [G(β)] is the only random distribution, generated out of two uniform random distributions as explained in the appendix of [45]. Therefore, one evaluation of the cost function depicted in (8) requires m solutions of the problem depicted in the reduced basis [V]. Such approach can lead to prohibitive calculation times when solving complex PDEs in a large computation domain expressing multiple number of reduced coordinates. NPM method for PGD solutions The PGD solutions are computed in a separated form such as [11]: j=D i=N y(q ; q ; ··· ; q ) = Q (q ), (10) 1 2 D ij j i=1 j=1 Ghnatios and Barasinski Adv. Model. and Simul. in Eng. Sci. (2021) 8:20 Page 5 of 13 with q the separated coordinates of the problem, and Q (q ) the recurrent solutions in j ij j the domain q . In a discrete form, Q becomes a matrix [Q] ∈ R[N , N ], with N the j ij j j j number of degrees of freedom in q domain, and N the number of products of functions required to converge the PGD solution of the problem. Using NPM, we choose to enrich some (or all) of the PGD reduced coordinates solution using the same algorithm depicted in Eq. (7). The results for a reduced basis of a domain [q ] results in the following: ⎪ [U] = [G(β)][σ ] [A] = [U] − [B]([B] [U]) ⎨ T T [D] = [Q] [M][A] + [A] [M][Q] /2 (11) [Z] = [A] − [Q] [D] −1/2 2 T [H] = [I ] + s [Z] [M][Z] [Q ] = [Q] + s[Z] [H] j j The algorithm illustrated in Eq. (11) represents an enrichment of one dimension in the PGD solution domain. The same algorithm can be designed to enrich as much dimensions as required, by generating the random basis [G(β)] as a higher dimensionality array. The number of hyper-parameters of the problem with increase accordingly. In the optimization process, evaluating the quantities of interests O([Q],t)donot require any solution of the PDE problem, as the PGD deﬁnes a surrogate model replac- ing the PDE with a product of functions deﬁned in every dimension of the domain. For instance, the evaluation of the quantities of interest for any solution requires only the knowledge of the solution y deﬁned by: i=N y(q ; q ; ··· ; q ) = Q (q ) Q (q ), (12) 1 2 D ij j ij j i=1 j∈D j∈D where D are the deterministic, non-enriched basis, while D are the probabilistic basis, d p enriched using NPM. Therefore, one solution and thus one evaluation of the quantities of interests, is computed only by using the product and sum of matrices. The automated tape placement process, simulation and measurements The automated tape placement process (ATP) is a one step, out-of-autoclave, composite manufacturing process [12,35]. It aims a continuous deposit of prepreg tapes, while heat- ing the deposit region and compressing the incoming tape to achieve a certain degree of consolidation, from low cohesion of the tapes to consolidated panels. The process has the potential to achieve in a unique step the manufacturing of composite parts in the ﬁnal shape. Experimental process The studied process is the deposition of prepregs on a heated cylindrical shape substrate, while heating with three external radiative heat source, the RHS controls the temperature Tin and will not be modeled in the rest of this work. The process is illustrated in Fig. 1. The system is also equipped with two infrared camera to measure the temperature ﬁelds during deposition. The measurement is performed on 6 selected points as marked in Fig. 2. Figure 2 also illustrates the position of selected probing points. Figure 2a illustrates the top Ghnatios and Barasinski Adv. Model. and Simul. in Eng. Sci. (2021) 8:20 Page 6 of 13 Fig. 1 The ATP winding process with the three radiative heat sources marked as RHS and RHS ,the 1 2 incoming tape temperature is T , the cylindrical deposition base has a radius R = 54 mm and a controlled in c temperature T (a) Thermal camera placed above the in- (b) Thermal camera placed below the in- coming tape, point 3 is an input for the coming tape, point 5 is an input for the simulation simulation Fig. 2 The thermal camera with the control points 1 and 2 from a and 1 to 4 from b view of the currently deposited prepreg, referred as layer l later on, with 3 selected points 1, 2, and 3. Point 3 is considered as an input to the simulation, the temperature of the incoming prepreg T .Figure 2b shows a bottom view of the deposited prepreg with point in 5 considered as the temperature of the drum unrolling the prepreg, not represented in the simulated region. Points 1, 2, 3 and 4 in Fig. 2b illustrate the top region of the previously deposited layer, referred as layer l −1 later in this manuscript. The used material is prepreg tapes made of unidirectional carbon ﬁbers impregnated with a thermoplastic matrix. The measurements are performed at a constant time step t = 0.5 s. After measure- ments, the data is formatted and ﬁltered using a moving least squares approach [7,23]. The experiment will deposit 4 times each of the ﬁrst ply, then 4 times the second ply and so on, until depositing 6 plies in each experiment. We dispose of experiments with 2 sets of parameters used. The parameters are illustrated in Table 1. The ﬁnal ﬁltered results from camera 1, for test one, one of the four measurements, are illustrated in Fig. 3.InFig. 3, we use the deposition velocity and the perimeter to ﬁnd the deposition time of each of the 6 overlapping layers. The means and standard deviations of the selected points for the parameters set 1 is illustrated in Figs. 6 and 7, while the ones for the set 2 appears in Figs. 8 and 9. Ghnatios and Barasinski Adv. Model. and Simul. in Eng. Sci. (2021) 8:20 Page 7 of 13 Table 1 Used experimental parameters Parameter Values for set 1 (SI units) Values for set 2 (SI units) w 0.1852 rad/s 0.5556 rad/s 2 2 Q 9250 W/m 9250 W/m 2 2 Q 4500 W/m 4500 W/m Selected point 1 Selected point 2 Time (min) Fig. 3 The measurement data performed by thermal camera 1 at control points 1 and 2 PGD simulation In this part we explain the modeling and simulation performed in the context of the ATP process. The modeling is performed in the cylindrical (r, θ,z) coordinate system. When considering the reference as the RHS for example, one can work in a steady state formulation considering the convection-diﬀusion equation. In cylindrical coordinates, the governing equation becomes: 2 2 ρC ∂U 1 ∂ ∂U 1 ∂ U ∂ U − w + K r + K + K = 0 (13) r θ z 2 2 2 r ∂θ r ∂r ∂r r ∂θ ∂z with U being the temperature, ρ being the density, C the thermal capacity at constant pressure, w the angular velocity, (K ,K ,K ) the diagonal components of the conductivity r θ z tensor. One may note that the deposition in the studied process orients the ﬁbers in the tangential direction only, therefore the out-of-diagonal components of the conductivity tensor are all set to zero. Multiplying Eq. (13)by r ﬁrst and by a test function U , then integrating by part, Eq. (13) will lead to the weak form of the problem: ∂U ∂U ∂U 1 ∂U ∗ ∂U ∂U ∗ ∂U ρC U w + K r + K + K r rdrdθdz P r θ z ∂θ ∂r ∂r r ∂θ ∂θ ∂z ∂z = U n · K∇U K being the conductivity tensor, n the outward normal and the boundary of the domain . Equation (14) is therefore the governing equation to be solved, coupled with the bound- ary conditions depicted in Fig. 4. A boundary condition of symmetry impose the continuity Temperature (C) Ghnatios and Barasinski Adv. Model. and Simul. in Eng. Sci. (2021) 8:20 Page 8 of 13 Fig. 4 The modeling of the ATP process of the temperature between ply l at θ = 2π and ply l − 1at θ = 0. The tip of the very ﬁrst layer is considered adiabatic. The boundary conditions are reviewed in Eq. 14. U(r = R , θ) = T ⎪ c b U(r ∈]R + t × (l − 1); R + t × l], 0) = T ⎪ c l c l in ⎪ U(r ∈]R ; R + t × (l − 1)], 0) = U(r ∈]R + t ; R + t × l], 2π) c c l c l c l ∇U | · n = 0 (14) r∈]R ;R +t ],θ =2π c c l ∇U | − · n = h U − U ⎪ ( ) air ∞ r= R +t ∗(l−1) ,θ ∈[0;l ] ⎪ ( ) c l θ ∇U | + · n = h (U − U ) ⎪ air ∞ r=(R +t ∗(l−1)) ,θ ∈[0;l ] ⎪ c l θ ∇U | · n = h (U − U ) r=R +t ∗(l),θ air ∞ with t = 0.2 mm the thickness of a prepreg layer, and l ≈ 3.55 rad the angular length between the probe 3 in Fig. 2a and the point of contact between layers l and l − 1, h = 15 W/m k the coeﬃcient of convection with the air at ambient temperature T = air ∞ 25 C. In the experimental results, T and T are constantly changing, therefore they in are considered as extra parameters of the problem, varying inside a predeﬁned interval: ◦ ◦ T ∈ [25; 350] Cand T ∈ [25; 250] C. Moreover, the deposit angular velocity w as in b well as the thermal heat ﬂux generated by the radiatives sources RHS and RHS , Q and 1 2 1 Q respectively, are also considered as extra parameters of the problem. The solution is therefore obtained in a separated form as: i=N U(r, θ,z,T ,T , w,Q ,Q ) = R (r)Z (θ,z)T (T )U (T )Q (Q )P (Q ) (15) in b 1 2 i i i in i b i 1 i 2 i=1 Readers unfamiliar with the PGD solution in separated form are referred to the following references and their references therein [10,11]. Moreover, a thermal contact resistance is introduced between deposited ply, as described in [12,18], with a thermal conductance h = 6000 W/m K. The involved prob- lem parameters are depicted in Table 2. The simulation results are depicted in Fig. 5 for the deposition of layers 3 and 5. The results show a discontinuity between deposited layers, as induced by the modeled thermal contact resistance. The simulated results are compared to the experimental ones in Figs. 6 Ghnatios and Barasinski Adv. Model. and Simul. in Eng. Sci. (2021) 8:20 Page 9 of 13 Table 2 Process and material parameters Parameter Values (SI units) rho 1350 kg/m C 1700 J/kg K T [25;350] C in T [25;250] C K = K = K 0.6 w/mK r z ⊥ K = K 6W/mK θ // w [0.18;2.06] rad/s Q [0;20000] W/m Q [0;20000] W/m Fig. 5 Vertical section in the deposited layers during the deposition of the 3rd and 5th layers. Temperatures in C 400 400 100 Simulated temperature at the deposited layer 50 Simulated temperature at the deposited layer Experimental temperature at the deposit layer Experimental temperature at the deposit layer 0 50 01234567 01234567 (rad) (rad) (a) Deposition of the third layer along with (b) Deposition of the ﬁfth layer along with the temperature measured by probes 1 and the temperature measured by probes 1 and 2 illustrated in ﬁgure 2(a) for set 1 of exper- 2 illustrated in ﬁgure 2(a) for set 1 of exper- imental parameters imental parameters Fig. 6 Comparison of the simulated temperature in the middle of the top layer, with experimental measurements, for the 3rd and 5th layer deposition. Temperatures in C and 7. The experimental results show large variability. Only 4 experiments were available to compute the mean and the standard deviation. The results are shown for w = 0.0533 2 2 rad/s, Q = 9250 W/m and Q = 4500 W/m . T and T are experimental inputs. 1 2 in b Temperature (C) Temperature (C) Ghnatios and Barasinski Adv. Model. and Simul. in Eng. Sci. (2021) 8:20 Page 10 of 13 Simulated temperature at layer l-1 Experimental temperature at layer l-1 Simulated temperature at layer l-1 Experimental at layer l-1 (rad) (rad) (a) Deposition of the third layer, l =3, (b) Deposition of the ﬁfth layer, l = 5, along along with the temperature measured by with the temperature measured by probes 1, probes 1, 2, 3 and 4 illustrated in ﬁgure 2(b) 2, 3 and 4 illustrated in ﬁgure 2(b) for set 1 for set 1 of experimental parameters of experimental parameters Fig. 7 Comparison of the simulated temperature in the middle of layer l − 1 during the deposition of layer n, with experimental measurements, for l = 3and l = 5. Temperatures in C NPM method applied for the PGD simulation of the automated tape placement process Once the experimental and simulation results available, the NPM for PGD algorithm can be used to update and correct the simulated results with the experimental data. The experimental data are considered ground truth in this work, despite the large variability shown in some measurements. The cost function to optimize in this problem is given by: i=6 i=6 2 2 J = 0.9 E − E(U ) . + 0.1 σ − σ (U ) . (16) exp sim exp sim i=1 i=1 exp exp exp exp exp exp exp With U = U r , θ ,z = Z /2,T ,T , w ,Q ,Q , the evaluation of sim max 1 2 in b the simulation abacus at the points having the same input parametric values as the per- formed experiments. In Eq. (16), σ refers to the standard deviation. One may note that evaluating the cost function (16) would require multiple evaluation of the PDE solution, however no extra solution is required in the PGD framework. Only multiplications of vectors and matrices are involved. The enriched PGD solution with NPM for PGD algorithm are illustrated in Figs. 8 and 9 for the temperature ﬁelds in the 3rd and 5th deposited layer respectively. The results are computed after performing m = 100 calculation of the enrichment basis. The PGD enrichment is limited to the 5 most energetic PGD product of vectors. Figures 8 and 9 show the results for NPM–PGD enhancement. The results shown an adaptation of the solution to overlap as much as possible with the experimental results, while translating the experimental results’ variability into a simulation conﬁdence interval. Conclusion In this work, we illustrate the possibility of using the Non-parametric Probabilistic Method (NPM) in the Proper Generalized Decomposition (PGD) framework. The formulation for the PGD framework shows the possibility to enrich the solution and correct it using the NPM, without the need to perform any extra PDE solutions. Temperature (C) Temperature (C) Ghnatios and Barasinski Adv. Model. and Simul. in Eng. Sci. (2021) 8:20 Page 11 of 13 Fig. 8 Enrichement of the deposition of the ﬁfth layer using the NPM for PGD algorithm, l = 3. Temperatures in C Fig. 9 Enrichement of the deposition of the ﬁfth layer using the NPM for PGD algorithm, l = 5. Temperatures in C The selected application was the ATP, which deposits unidirectional ﬁber reinforced plastics on a cylindrical shape using a continuous deposition, and radiative heating. The process is modeled using a novel approach and the simulation results are compared to the experimental ones, before and after enrichment. Before enrichment, the results are in good agreement with the experimental ones. The enrichment process induced a large improvement the results and illustrates the experimental variability as reﬂected on the presented results. Acknowledgements The authors acknowledge the contribution of CANOE for data production. Authors’ contributions CG contributed to the theoretical, coding and writing parts. AB contributed to the theoretical, data collection and writing parts. Both authors read and approved the ﬁnal manuscript. Funding This work is made possible by the ﬁnancial support of the Chair partners (E2S UPPA, ARKEMA and CANOE) AWESOME. The authors would like to thank the aforementioned Chair partners (E2S UPPA, ARKEMA and CANOE) for funding their work. Ghnatios and Barasinski Adv. Model. and Simul. in Eng. Sci. (2021) 8:20 Page 12 of 13 Availability of data and materials Raw measurement data are available upon request. Declarations Competing interests Not applicable. Author details 1 2 Department of mechanical engineering, Notre Dame University-Louaize, PO Box 72, Zouk Mosbeh, Lebanon, Universite de Pau et des Pays de l’Adour, E2S UPPA, CNRS, IPREM, Pau, France. Received: 13 June 2021 Accepted: 20 August 2021 References 1. Ahmed Adel and khaled Salah. 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"Advanced Modeling and Simulation in Engineering Sciences" – Springer Journals
Published: Sep 21, 2021
Keywords: Proper Generalized Decomposition; Model reduction; Automated tape placement; hybrid modeling; Nonparametric probabilistic method; Machine learning
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