"Advanced Modeling and Simulation in Engineering Sciences"
, Volume 8 (1) – Apr 15, 2021

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- Springer Journals
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- Copyright Â© The Author(s) 2021
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- 2213-7467
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- 10.1186/s40323-021-00194-5
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tristan.maquart@hotmail.fr; thomas.elguedj@insa-lyon.fr This paper presents an eﬀective framework to automatically construct 3D quadrilateral Univ Lyon, INSA-Lyon, CNRS meshes of complicated geometry and arbitrary topology adapted for parametric UMR5259, LaMCoS, 69621 Lyon, France studies. The input is a triangulation of the solid 3D model’s boundary provided from Full list of author information is B-Rep CAD models or scanned geometry. The triangulated mesh is decomposed into a available at the end of the article set of cuboids in two steps: pants decomposition and cuboid decomposition. This workﬂow includes an integration of a geometry-feature-aware pants-to-cuboids decomposition algorithm. This set of cuboids perfectly replicates the input surface topology. Using aligned global parameterization, patches are re-positioned on the surface in a way to achieve low overall distortion, and alignment to principal curvature directions and sharp features. Based on the cuboid decomposition and global parameterization, a 3D quadrilateral mesh is extracted. For diﬀerent parametric instances with the same topology but diﬀerent geometries, the MEG-IsoQuad method allows to have the same representation: isotopological meshes holding the same connectivity where each point on a mesh has an analogous one into all other meshes. Faithful 3D numerical charts of parametric geometries are then built using standard data-based techniques. Geometries are then evaluated in real-time. The eﬃciency and the robustness of the proposed approach are illustrated through a few parametric examples. Keywords: 3D quadrilateral meshes, Parametric geometry, Global parameterization, Data-based models Introduction Since few years, data-based models and reduced order modeling [10,11,47] have been particularly studied in many physics applications. Such approaches allows to determine complex solutions in real-time. By real-time we mean a computational cost drastically reduced in comparison with a classical approach. In most reduced order approaches, an oﬄine step is ﬁrst needed in order to build the database model. For parametric studies, this means constructing and running the model for given sets of parameters wisely cho- sen. In the online phase, executed in near real-time, this database is used to construct the solution of a new set of parameters. Common reduced order modeling techniques classi- cally used in computational mechanics often require structured meshes to be eﬃcient by avoiding inaccurate projection steps between all meshes. This is obviously encountered © 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 Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 2 of 28 when geometric parameters are considered. One way to circumvent this issue is to build isotopological meshes of all the geometric instances. Previous work has been done to build the foundations of our approach [2,3]. Moreover, research on the same problematic has been recently published [40–42]. For diﬀerent geometric instances with the same topo- logical properties (e.g. the same shape), being able to construct isotopological meshes of all these instances makes it very easy to compare these meshes. Consequently data- based approaches with geometric parameters are mainly guided by topological informa- tion throughout the oﬄine step. Generating isotopological quad meshes therefore seems as the right way to solve this problem. In numerous applications in domains like computational mechanics, aerodynamics, crash simulations and ﬂuid-structure interactions, 3D quadrilateral meshes are recom- mended due to their highly regular structure. Generation of high quality quadrilateral meshes from B-Rep triangulated meshes has been recently extensively studied. All devel- oped methods seek to construct such meshes with local and global speciﬁed properties. Most common desired properties are element quality, alignment with principal curva- ture directions or boundaries, global topology structure and respect of sharp features. Most recent techniques for quadrangulation use a global parameterization of the input triangulated surface [6,27,45,48]. Global parameterization is of particular interest for parameterizing the whole surface and handling discontinuities along seams. Constructing such global parameterization is usually done using a reliable ﬁeld that serves as a guidance to drive the parameterization gradient. Cross ﬁelds can be used for that purpose, and in addition allow to capture principal curvature directions, boundaries and sharp features while understanding the topology. Diﬀerent techniques exist for 2D or 3D quadrilateral meshes purposes [5,15,30,49]. Quadrilateral layouting (i.e., partitioning the triangulated mesh into generalized squares) is particularly eﬃcient to build structured quadrilateral meshes [8,9]. In the present work, we seek to decompose objects deﬁned by their triangulated bound- ary into geometry-aware quadrilateral layouts understanding geometric features as well as possible. This tedious task is ﬁrst done using pants decomposition which was widely stud- ied [12,21,34,53]. Secondly, we perform cuboid decomposition [32,35,36] on each pants patch that allow us to obtain a decomposition in quadrilateral patches. We present a novel decomposition method that splits automatically the surface into geometry-feature-aware patches. We use an optimization process of quadrilateral patches embedded on surfaces. An optimal global parameterization gives the ﬁnal surface parameterization. Topological properties of the designed cross ﬁeld is entirely provided from the layout. Our main con- tribution is a new way to design such quadrilateral layouts to ensure an optimized resulting quadrilateral patch decomposition of the surface suitable for isotopological meshing pur- poses. With this generic setup, our method can be applied in automatic scenarios involving speciﬁc cuboid sketches per pant patch. These kind of automatic processes are really useful to compute the oﬄine phase needed for parametric studies using reduced order modeling and data-based models. The paper is organized as follows. We ﬁrst extend the decomposition pipeline thought to understand the geometry in “Smart model decomposition” section. Then using global parameterization aligned with a cross ﬁeld capturing geometric information, we extract feature-aligned quadrilateral meshes in “Direction ﬁeld generation” and “ Aligned global Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 3 of 28 Fig. 1 Parametric analysis problematic of diﬀerent meshes with the same topology parameterization computation from geometric direction ﬁeld” sections. Handling iso- topological datas throughout the 3D quadrilateral meshing process, geometric snapshots are used to build a reduced order model in a full autonomous integrated algorithm as pre- sented in the various examples of “Application of the MEG-IsoQuad method togeometric parametric analysis” section. Problem statement In this article, we want to compute 3D quadrilateral meshes whose properties belong to a very restricted class. The goal is to be able to compare diﬀerent shapes, i.e., diﬀerent geometries of the same topology class. This kind of meshes are required in most parametric applications such as statistical shape analysis and reduced order modeling [19,24]. Figure 1 shows a graphical illustration of the problem. The most desirable properties of the targeted meshes are: • Minimize the number of singularities, deﬁned to be the nodes with a valency diﬀerent from 4. • Alignment with features, curvature directions and boundaries. • Determine high quality elements, as close to a square as possible. • Constrain the number of elements, connectivity and features locations to enable relevant comparison of all studied meshes. Proposed approach We give a workﬂow to compute high quality 3D quadrilateral meshes with speciﬁc prop- erties needed in parametric analysis. Our goal is to generate such meshes for objects with complex geometry and arbitrary topology provided from boundary representation or scanned techniques. We want our framework to be as robust as possible. We present an integrated pipeline partitioning the input triangulated surface to relevant domains until having comparable meshes for parametric studies. The ﬁrst contribution consists in understanding the geometry while decomposing the mesh. The second contri- bution consists in computing an optimized aligned global parameterization using a very coarse quadrilateral layout helping us to deﬁne the required quadrilateral mesh. Reduced order modeling is a tedious task that requires isotopological meshes in most cases. All geometric instances of the population must satisfy such mesh properties to be compared and studied. The issue of having isotopological meshes is addressed by Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 4 of 28 Fig. 2 The MEG-IsoQuad method. Building isotopological analogous feature-aligned quadrilateral meshes from triangulated meshes with the same topology. Resulting isotopological meshes have non-uniform isotropy generating the same parameterization for all members of the given population. Figure 2 provides a simple understanding of our approach: the MEG-IsoQuad method. Building isotopological analogous quadrilateral meshes: the MEG-IsoQuad method In this section we present a method thought to build 3D quadrilateral instances from trian- gulated meshes suitable for data-based models with geometric parameters. This method allows us to identify the main shape variations. Quadrilateral mesh topology needs to be the same among all snapshots. We also strive to place vertices in a geometry-aware manner, i.e., all isotopological meshes have analogous points in the whole mesh. These constraints can be satisﬁed using our framework based on topology and geometry aspects. Topology and parameterization prerequisites Following prerequisites introduce brieﬂy basic notions of geometry and topology that are needed to present our work. The interested reader unfamiliar with those concepts can ﬁnd relevant detailed information in the cited references and in the corresponding literature. Topology context and related work Topology is the study of properties like continuity, connectedness and boundaries of a space that are preserved under continuous deforma- tions, such as bending and stretching, but not tearing and gluing. A homeomorphism is an isomorphism that admits a continuous function between two topological equivalent spaces that has a continuous inverse function. We are interested on transformations that Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 5 of 28 preserve all the topological properties of a given space. Homeomorphic spaces admit a homeomorphism between them, thus topological spaces are equivalent. See Hatcher [22] for more details. A surface M is a 2-manifold, i.e., a topological space in which each point has a neigh- 2 2 borhood homeomorphic to either the plane R or the closed half plane R . Points with closed half-plane neighborhood are deﬁned as the boundary ∂M of the surface M.Inthe following we investigate only connected, orientable, compact surfaces with boundaries. Genus of a connected and orientable surface M (i.e., a 2-manifold embedded in R )isthe maximum number of non-intersecting closed curves which can be drawn on it without disconnecting the surface. With these deﬁnitions, taking into account a genus-g surface M possibly with b boundary components, we can now deﬁne the topological invariant. This invariant used to classify surfaces is called Euler characteristic: χ(M) = 2 − 2g − b. If M is a triangulated surface with vertices V,edges E and faces F,wecan deﬁnethe characteristic as χ(M) = dim(V ) − dim(E) + dim(F). An example of Euler characteristic is provided in Fig. 3a. We note that surfaces with diﬀerent Euler characteristic cannot be homeomorphic. We deﬁne a homology basis for M to be any set of 2gcycles whose homology classes generate the ﬁrst homology group H [17]. Cutting along these cycles yields a genus-0 surface. The set of all handle and tunnel loops form a homology basis. Suppose a closed 3 3 surface M ⊂ R separates R into a bounded space I and an unbounded space O. Handle and tunnel loops on M can be deﬁned as follows. A loop a is a tunnel if it spans a disk in the unbounded space O. A loop b is a handle if it spans a disk in the bounded space I. Pants decomposition was studied in Hatcher et al.[23], subsequent work has been done to ﬁnd the optimal segmentation of a given surface into relevant pants patches [12] using the shortest homology basis [26]. Geometry-aware pants decomposition has been also investigated by Zhang and Li [53]. Li et al. [34] developed a pants decomposition framework for computing maps between surfaces with arbitrary topologies. An example of pants decomposition using a homology basis can be seen in Fig. 3b. More recently, Hajij et al. [21] tried to segment surfaces into pants using a morse function. Let M be a surface of genus g with b boundary components. A pants decomposition g,b of M is a collection of pairwise disjoint simple cycles that splits the surface into pants g,b patches. Each pants patch is a genus-0 surface (topological sphere) with 3 boundaries. We assume that M is a surface with negative Euler characteristic, i.e., M is none of the surfaces M (topological sphere), M (topological disk), M (topological cylinder) and 0,0 0,1 0,2 M (topological torus). In this case pants decompositions of M do exist, and each pants 1,0 decomposition consists of 3g + b − 3 curves and divides M into 2g + b − 2 pants patches. A direction ﬁeld deﬁned on a surface M is a tangent unit vector ﬁeld: at each point of the surface, there exists a direction u such that u= 1and u.n = 0, where n is the normal of M.A N-symmetry direction ﬁeld U is a multivalued direction ﬁeld:ateachpoint of the surface M, there exists a N-symmetry direction u which is a set of N directions {u , u , ..., u , u } preserved by rotations of 2π/N around the normal n of M.These 1 2 N −1 N ﬁelds are subject to the well known Poincaré-Hopf theorem for a 2-dimensional manifold M.If M has boundaries, the vector ﬁeld must be pointing in the outward normal direction along them (for higher symmetries, see the boundary number theorem [49]). Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 6 of 28 Fig. 3 Topological properties and pants decomposition of a double torus. a Triangulated genus-2 surface without boundaries has Euler characteristic χ = 2 − 2g − b =−2. b Pants decomposition provides two pants (gray and blue meshes) by cutting along handle loops and using symmetry. Homology basis is composed by two handle loops and two tunnel loops respectively depicted in red and green Used mapping techniques Diﬀerent approaches of mapping exist. In our case, we are interested in mappings or parameterizations which map a surface M embedded in R to a canonical domain D in R . Throughout the paper, triangulated disk-like surfaces M with vertices V,edges E and faces F are considered for parameterization. For instance, techniques to map a multiply connected surface already exist [52]. We use discrete harmonic mapping to solve parameterizations on disks. Harmonic mappings have attributes derived from conformal parameterization, but there is no guarantee on angles. To proceed we construct a harmonic function f : M −→ R such that f = 0. Harmonic maps minimize Dirichlet energy: E (f ) = ∇f dM. (1) The surface boundary ∂M is ﬁrst mapped to the boundary of the parametric domain and then the parameterization for the interior vertices is obtained by solving the linear system: f (v ) = w (f (v ) − f (v )). (2) w i ij j i j∈N where v , v ∈ V , N is the neighborhood of v ,and w is the scalar weight assigned to i j i i ij the oriented edge e (v ,v ). Diﬀerent parameterization methods assign diﬀerent weights ij i j w . The ﬁrst deﬁnition of weight was introduced by Tutte [51]. In the parameter space, ij each vertex is placed at the barycenter of its neighbors. Recently, Saboret et al. [50]have implemented a CGAL package handling some of the state-of-the-art surface parameter- ization methods such as least squares conformal maps, discrete conformal map, discrete authalic parameterization, Floater mean value coordinates or Tutte barycentric mapping. In this context, we use mean value coordinates introduced by Floater [18]. According to the Rado-Kneser-Choquet theorem, if weights are positive and the boundary ∂M is mapped homeomorphically in a convex square parametric space, the mapping driven by the harmonic function f has to be bijective. This allows us to determine bijective mappings in order to avoid fold-overs in the smart model decomposition part, see e.g. “Smart model decomposition” section. Smart model decomposition In order to build valid quadrilateral layout to determine the surface parameteriza- tion, we proceed in two steps: geometry-aware pants decomposition (“Geometry-aware Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 7 of 28 pants decomposition” section) and feature-aware cuboid decomposition (“Feature-aware cuboid decomposition” section). Pants decomposition and cuboid decomposition are con- structed to respect the topology of the input triangulated surface and consider features. Cuboid decomposition approximates very roughly the geometry while faithfully replicat- ing its topology taking into account sharp edges and vertices. Due to its patch regular structure, it can serve as parametric domain needed for pure 3D quadrilateral mesh com- putation. Geometry-aware pants decomposition Closely related literature approaches and algorithmic prerequisites We focus on the decomposition of a triangulated surface into a set of pants patches {T }. Pants decom- position provides a canonical decomposition scheme for common surfaces. χ =−1 for a pants patch, therefore a pant is the simplest topology after the sphere, disk, cylinder and torus. Many algorithms take as input handle or tunnel loops for genus-g surfaces to segment into pants patches [34]. The handle loops required for automatic pants decompo- sition are computed using the technique presented by K. Dey et al. [25]. An improvement was also developed by the same group [16]. In the following, we reuse the pants decom- position algorithm of Al-Akhras et al. [4] in order to integrate a most recent method. New extensions According to previous work presented by Al-Akhras [3], we introduce an extension to enumerate the entire space of topological pants decomposition possibilities at each step. In other words, instead of picking 2 boundaries in an arbitrary manner, we suggest to take all couples of 2 boundaries among all non-repeating and commutative combinations when determining a pants patch relative to the considered step. Besides, giving some feature points, our new algorithm is able to determine a pant by slicing the mesh along a cycle passing across these locations. User inputs User manual inputs are reduced to feature points selection in order to guide the decomposition passing through these points of interest. For other used geometric criteria, they are suggested to the user. Feature-aware cuboid decomposition algorithm Given the homology basis formed by handle and tunnel loops, we can take a subset H composed of g simple pairwise disjoint handle loops {h ,h , ..., h ,h }. Slicing surface M with b boundary components along its 1 2 g −1 g g handle loops will lead to a genus-0 surface with 2g + b boundary components denoted as W ={w ,w , ..., w ,w }. We iteratively pick two boundaries w and w from 1 2 i j 2g +b−1 2g +b W (see algorithm 1) taking into account combinations and compute a new simple cycle w to bound them, i.e., w is homotopic to w ◦ w . The three cycles w , w and w bound ij ij i j i j ij a pants patch T . We remove this pants patch T from M. The remaining patch is still k k genus-0 but its boundary number reduces by 1: the two cycles w and w are removed, i j and one new cycle w is inserted. This is iteratively performed until |W|= 3. This idea is ij formulated in Algorithm 1, and the operation that traces a cycle w homotopic to cycle ij w ◦ w is formulated in Algorithms 2 and 3. i j Choice of the geometric criterion Shortest length of cycles on weighted triangulated meshes are computed using a Dijkstra’s algorithm based on weighted tree graphs. A CGAL package exists to compute these paths [29]. Diﬀerent geometric criteria can be used to guide the pants decomposition. Geometric criterion can be adapted to the mesh M. In algorithm 2 and also in Algorithm 1 to sort loops in L, this can be changed for areas Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 8 of 28 Algorithm 1 Geometry-aware pants decomposition Input 1: Triangulated genus-g surface M with b boundary components. Input 2: g geometrically relevant handle loops of M. Input 3: Global geometric criterion for L. Output: Set of −χ(M) pants patches T ={T , ..., T }, with M =∪T . 1 i −χ(M) 01: k = 1. 02: Slice M along all its handle loops and get a surface M with 2g + b boundaries. 03: Put all boundaries of M in a set W ={w , ..., w }. k 1 2g +b 04: While |W | > 3 do 05: Build or reset an empty set of loops L ={0}. Size(W )! 06: Compute N combinations: dim(N ) = . c c 2!(Size(W )−2)! 07: For all couples [w ,w ]in N : i j c 08: Compute a cycle w homotopic to w ◦ w . ij i j 09: Add loop to L. 10: End For 11: Sort relevant loops in L ={l , ..., l } using a global geometric criterion. 1 dim(N ) 12: The optimal w cycle is classiﬁed in L. ij 13: {w ,w ,w } bound a pants patch T . Remove T from M : M ← M \ T . 1 j ij k k k k k k 14: Remove w and w from W,and add w into W . i j ij 15: k ← k + 1. 16: End While Algorithm 2 Homotopic cycle computation Input 1: Genus-0 surface M with b boundary components {w , ..., w }. Input 2: Geometric criterion. Output: Acycle w homotopic to cycle w ◦ w . ij i j 01: Compute the shortest path connecting w to w . i j 02: Slice M along this path to get one new large boundary c . ij 03: Connect all other boundaries together using shortest paths. 04: Slice M along these paths to get one new large boundary c . 05: M becomes a topological cylinder. 06: Compute the shortest path γ connecting c and c . ij k 07: Slice M along the path γ . 08: Every point p ∈ γ (n + 1 points) is split into a pair (p , p ). i i i 09: Trace all shortest paths connecting points pairs (p , p ). i i 10: Among these paths, the cycle w satisﬁes the geometric criterion. ij Algorithm 3 Feature homotopic cycle computation Input 1: Genus-0 surface M with b boundary components {w , ..., w }. Input 2: Set of n feature points S ={s , ..., s }. p 0 n Output: Acycle w homotopic to cycle w ◦ w . ij i j 01: Compute the shortest path connecting w to w . i j 02: Slice M along this path to get one new large boundary c . ij 03: Connect all other boundaries together using shortest paths. 04: Slice M along these paths to get one new large boundary c . 05: M becomes a topological cylinder. 06: Compute the shortest path γ connecting c and c . ij k 07: Slice M along the path γ . 08: Every point p ∈ γ (n + 1 points) is split into a pair (p , p ). i i i 09: {p , p ,p , p } are set to be the corners of a square harmonic mapping. 0 0 n n 10: Partial line inverse mapping passing through n points in S is performed. 11: The cycle w is reconstructed using line segments. ij Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 9 of 28 Fig. 4 Genus-2 plate pants decomposition. Triangulated surface (a) with its handle and tunnel loops in red and green, respectively (b). Pants decomposition using loops with shortest distance (c) and with symmetry (d) Fig. 5 Geometry-aware pants decomposition with feature points. Double T (a)isdecomposedintopants using 4 feature points given by the user or determined automatically. A 2-torus pants decomposition passing through feature points (b). Notice that using shortest length loops, the result will be the same for (b) of minimum curvature [31], minimum length or symmetry. For instance in Fig. 4c pants decomposition is performed using loops with minimum length, whereas in Fig. 4d decom- position is made by symmetry. In Fig. 5, user selected feature points are given to guide the pants decomposition. All of these guidings yield a geometry-aware decomposition. Discussion on robustness A common surface admits inﬁnitely many pants decomposi- tions. In general, not all pants decomposition results are suitable for the next step of our algorithm. By correctly deﬁning feature points (in general a very small set of points) to lead the segmentation, the algorithm succeeds. If the pants decomposition is guided by the diﬀerent geometric criteria presented above, we obtain satisfying results for all our test cases. Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 10 of 28 Feature-aware cuboid decomposition Closely related literature approaches Pants patches provide a very simple topology and hence each pants patch can be treated separately. Li et al. [32] presented a method that generates cuboids per pants patch with a given user data input. Al-Akhras [3] improved it in order to perform the decomposition in a way that it does not require any input from the user. Nevertheless, developed techniques do not consider sharp features and deal with a simple cuboid decomposition scheme. They both try to generate corners and polyedges on each pants patch T and decompose it into a set of cuboids, each having 8 corners and 12 polyedges in order to construct a volumetric parameterization. In addition, these two previous approaches do not handle complicated cuboid conﬁgurations and generated arcs and nodes are not necessarily positioned onto features of the input surface mesh. Finally, prior automatic processes restrict the placement of the quadrilateral layout nodes in speciﬁc areas, e.g. using harmonic functions. New extensions Our work presents an extension to handle more complicated geometry especially with sharp features. First, on each pants patch the number of quadrilateral patches n is set to respect sharp vertices and edges: we aim to perform a feature-aware cuboid decomposition. Indeed, optimal number of quadrilateral patches depends on the features embedded into the considered pants patch. Secondly, irregular vertices of the layout with a valence v = 4 are placed onto relevant areas with high concentrated Gaussian curvature and/or with signiﬁcant features. Algorithmic prerequisites The space of valid quadrilateral layouts Q is deﬁned in equa- tion (3). A quadrilateral layout is composed by arcs and n nodes of valence v .Cuboid n i conﬁgurations C decompose a surface into a set of quadrilateral patches which are ready for hexahedral meshing due their cubic structure. In this paper, the cuboid decomposition is dedicated only to surface decomposition. Indeed, this work only focuses on structured quadrilateral meshes. ∀v ∈ Q, 1 − = χ(M) = 2 − 2g. (3) i Closed i=1 User inputs To reach the feature-aware cuboid decomposition goal, manual user inputs are necessary. Depending on the pant geometry and features to replicate in the cuboid decomposition, 2 inputs are mandatory. Using a cuboid conﬁguration templates library, a segmentation layout is choosen by the user; thus involving the wish to suit the desired features of the pant into the decomposition. Then, the structure of the requested template requires the manual selection of its asscociated feature points. Nature of these features is automatically suggested and the user has just to specify their location. Feature-aware cuboid decomposition algorithm The 3 boundaries of a given pants patch will be arbitrarily denoted by B , B and B . We process these pants patches one by one in 1 2 3 an arbitrary order. To guarantee cuboid corner alignment, when we determine one pants patch’s result, we transfer its corners on the boundaries of the adjacent pants patches if they are not processed yet. Notice that just a general algorithm with the most relevant speciﬁcations is provided using a simple decomposition scheme of 4 cuboids per pant patch. Step 1. We generate 3 feature cutting curves W , W and W : S1 S2 S3 [A] We compute 3 discrete harmonic functions: f , f and f . To compute f ,weset 1 2 3 i f = 0 for vertices on the boundary B and f = 1 for vertices on the remaining two i i i Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 11 of 28 Fig. 6 Temporary cutting curves generation. a Discrete harmonic function f . Blue color represents parametric values close to 0, red ones are close to 1. Using a set C of n feature cutting points, f and f ,the c 2 1 three temporary cutting curves are computed in (b) Fig. 7 Boundary seed points computation. Temporary patches P , P and P respectively in (a–c). Using T3 T1 T2 an approach close to circular conformal mapping, temporary seed points s are determined into each Tk patches P illustrated in pictures (d–f) Tk boundaries B and B .Thenwesolve f = 0 using mean value coordinates. Figure j i 6a. [B] Each harmonic function f has one minimum component and two maximum com- ponents. Among a set C of n feature cutting points C ={c ,c , ..., c ,c },let c 0 1 n −1 n c c f = min[f (c )] , ∀c ∈ C. Then a temporary non-feature cutting curve W is deﬁned i i i i Ti as the isoparametric curve of the function f for the value f .Figure 6b. i i [C] We remove a long branch, i.e., by cutting along its temporary cutting curve W . Tk After ﬁlling the cutting hole, the resulting patch is a topological cylinder with 2 boundaries B and B . We denote this temporary patch by P .Figure 7a–c. i j Tk [D] Using previously computed temporary patches P ,weﬁrstset u = 0 for vertices on Tk B and u = 1 for vertices on B ,thensolve u = 0. Using an approach close to Zeng i j et al. [52], among all iso-v curves along ∇u starting from B and B ,weautomatically i j select two relevant curves who are intersecting the ﬁlled boundary B , one starting k Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 12 of 28 Fig. 8 Feature cutting curves generation. a The three temporary unit square maps U , U and U (left to T3 T1 T2 right). b Partial line inverse mapping using feature cutting points in C performed in U , U and U (left to T3 T1 T2 right). c Computed feature cutting curves W , W and W S1 S2 S3 Fig. 9 Quadrilateral layout arcs generation. a We keep W passing through all cutting feature points. Into S3 each relevant U partial line inverse mapping is performed using points in C, D and possibly in O. (b) Curves in the physical space with supplementary boundary curves. Other arcs are determined to have a valid quadrilateral layout or cuboid conﬁguration (c) from temporary seed point s on B and one starting from temporary seed point Tik i s on B .Figure 7d–f. Tjk j [E] By performing this previous task on each temporary patch P , we obtain 6 temporary Tk seed points s , s , s , s , s and s . Working on all patches P we plot an Tij Tik Tji Tjk Tki Tkj Tk iso-v curve along ∇u from the seed vertex s on the boundary B to the seed Tik i vertex s on boundary B .Weslice P along this iso-curve and get two duplicated Tjk j Tk boundary paths. We ﬁnally set v = 0and v = 1 on them respectively and solve v = 0 to obtain 3 oriented temporary unit-square maps denoted U with corners Tk {s ,s , s , s }. s and s are duplicated points of s and s respectively. Tik Tjk Tik Tjk Tik Tjk Tik Tjk Notice that triangles from ﬁlling are removed during U computation. Figure 8a. Tk [F] Points in C are mapped into each unit-square map U and automatically classiﬁed Tk using parametric coordinates u and v. Then we perform partial line inverse mapping between each classiﬁed cutting points c related to relevant U and merge computed i Tk lines to obtain valid feature cutting curves W , W and W .Figure 8b, c. S1 S2 S3 Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 13 of 28 Step 2. We generate 3 feature-aware oriented unit-square maps denoted U : [A] We remove a long branch by cutting along its feature cutting curve W . We proceed Sk like Step 1 [C] to obtain a feature-aware cutting patch P . [B] Using the same workﬂow of Step 1 [D] feature seed points s and s are determined. ik jk [C] Using the same workﬂow of Step 1 [E], we obtain 6 feature seed points s , s , s , s , ij ji ik jk s and s . Then 3 feature-aware oriented unit-square maps U are extracted. ki kj k Step 3.Wegenerateall arcs of Q: [A] According to Step 1 [F] feature cutting arcs W have already been computed. We Sk keep W passing through all cutting feature points. Figure 9a. S3 [B] Among a set D of n feature boundary points D ={d ,d , ..., d ,d },wemap d 0 1 n −1 n d d these points into corresponding maps U . Using automatic classiﬁcation with para- metric coordinates u and v, we pair on each relevant U points in D possibly with feature cutting points in C. We perform partial line inverse mapping to obtain arcs of the quadrilateral layout. Boundary arcs are determined directly in the physical space. Figure 9b. [C] Consider a non-empty set O of n common feature points O ={o ,o , ..., o ,o }. o 0 1 n −1 n o o We proceed like Step 3 [B] by pairing on each relevant U points in O possibly with C or D. [D] Remaining arcs of the quadrilateral layout or cuboid conﬁguration (see equation (3)) are computed using partial line inverse mapping into relevant U with points in D, O or in C. Other arcs and points are then obtained by line-line intersection analysis. Figure 9c. With feature points in C (see Step 3 [C]) or other feature points in D or O, a huge variety of shapes can be decomposed into cuboids and hence handling more complex geometry cases is possible. To go further, the pants-to-cuboids decomposition can be divided in two main parts: the biological and mechanical parts, see e.g. Fig. 10. For instance, speciﬁc biological quadrilateral layouts have been studied by Al-Akhras et al. [2]. Details for the mechanical workﬂow are explained in Fig. 11. Given a consistent pants decomposition, our algorithm is able to decompose a triangulated surface into quadrilateral patches suitable for surface parameterization. This previous task is done only by specifying relevant feature points relative to a given quadrilateral layout template. Direction ﬁeld generation Introduced in “Topology and parameterization prerequisites” section, 4-symmetry direc- tions ﬁelds (i.e., cross ﬁelds) are widely used to determine an aligned global parameteri- zation [6–8,45]. In other words, these ﬁelds are useful to guide the parameterization to determine quality quadrilateral surfaces. Designing a smooth cross ﬁeld C is done with a given set of constraints. We categorize these constraints into two groups: topological and geometric constraints. Topological constraints are imposed singularities and numbers induced by the surface topology χ(M). Geometric constraints are intrinsically embedded on the surface. Thus we seek a cross ﬁeld C which is smooth, aligned with the local geom- etry and topologically compatible. The quadrilateral layout Q or cuboid conﬁguration C contains topological properties whereas the surface M hold the geometric information. Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 14 of 28 Fig. 10 Geometry-feature-aware pants-to-cuboids decomposition algorithm overview. Biological (a, b)and mechanical pipeline (c). Resulting quadrilateral layouts of (a–c) respectively in (d–f) We consider a triangulation of the surface M, assumed to be a 2-dimensional manifold of genus-g with b boundary components. Directions will be stored at faces F to avoid deﬁnition of supplementary tangent planes. The ﬁrst step of the discretization consists in ﬁnding a reference direction in each triangle. This is done by choosing a local orthonormal frame (x, y) attached on each face f . The vector x is an unitary vector along one of the oriented edges of face f ,and y = n × x, where n is the normal of f . A direction u on f can be formulated in terms of polar coordinates. Due to the unit norm of such directions, it is completely parameterized by the polar angle α it forms with x.Wedenote α the direction angle. By unfolding adjacent triangles isometrically to a plane along their common edge, th angles can be expressed in a common coordinate frame. To specify the number of N turns the direction u undergoes to match with u when passing from A to B,the period A B jump [33] is adopted. Other works introduce an angle ω named connection angle to solve this ambiguity [14,15]. In the following, the connection angle based discretization is used for topological design and period jump based discretization is used for geometric design. Topological cross ﬁeld generation The ﬁrst discretization computes an angle α from face f expressed in adjacent face f i j which is in general equal to α , see e.g. equation (4). The angle κ ∈ [−π, π] represents ij j Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 15 of 28 Fig. 11 Geometry-feature-aware pants-to-cuboids decomposition algorithm: mechanical pipeline details the diﬀerence between a reference direction x in face f and x in face f .Angle ω is the i i j j ij connection angle when walking on the dual edge linking face f to face f . i j c i α = α + κ + ω . (4) ij ij We seek to ﬁx all topological degrees of freedom to restrict the cross ﬁeld topologically compatible to our quadrilateral layout Q. Technically speaking, we target a smooth cross ﬁeld C that is singular only at speciﬁed vertices: the position of irregular nodes of Q.The approach of Campen and Kobbelt [8] is adopted to determine the cross ﬁeld topological degrees of freedom from the input quadrilateral layout. Figure 12b provides a topological cross ﬁeld calculated using the minimization of energy introduced by Crane et al. [15]. The generated ﬁeld respect the Gauss-Bonnet theorem by distributing the Gaussian curvature in a smart manner [14]. Geometric cross ﬁeld generation The second discretization computes an angle α from face f expressed in adjacent face f i j j j which is in general diﬀerent to α , see e.g. equation (5). Angles α and α are respectively j i j expressed in their native faces. The integer p ∈ Z is the period jump when passing from ij face f to face f . The integer N is the symmetry of the ﬁeld, i.e., 4 for cross ﬁelds. i j 2π p i α = α + κ + p . (5) ij ij i i A smooth cross ﬁeld that interpolates relevant principal curvature directions, sharp features and boundaries restricted to a given topology is targeted. We search the smoothest ﬁeld taking into account constrained directions given by geometric features. The period Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 16 of 28 Fig. 12 Cross ﬁeld generation. a Input triangulated pant with sharp features. b Topological cross ﬁeld that doesn’t respect the sharp features directions. c Geometric cross ﬁeld with boundaries and sharp features interpolation where the ﬁeld total curvature is null jumps of the ﬁeld are ﬁxed. A technique introduced by Bommes et al. [6] and later used by Al-Akhras et al. [2] is adopted to compute such ﬁeld. Figure 12c shows a geometric cross ﬁeld that interpolates sharp features and boundaries. Aligned global parameterization computation from geometric direction ﬁeld We now compute an aligned global parameterization, i.e., a map from the mesh M to a disk-like surface parameter domain ∈ R . We assign a couple (u, v) of parameter values on each vertex of M. The parameterization should be locally aligned with the features of the mesh, this is done using the guiding geometric cross ﬁeld previously computed. Such parameterization implies that the gradients ∇u and ∇v of the discrete scalar ﬁeld must follow the cross ﬁeld directions on each face. Seamless parameterization A planar parameterization of a mesh M embedded in R into a parametric domain embedded in R is in general done by computing a cut graph. A cut graph G is a connected graph formed by edges of M that splits the mesh into a disk-like surface mesh M .Seams are then deﬁned as duplicated paths of G. Transitions across seams need to belong to a very restricted class. We search for rigid transformations with a rotation angle multiple of . Moreover across each seam edge or vertex, the corresponding transition must be integral, i.e., relative to an integer. Thus we talk about integral seamless parameterization C C [27]. We target the cross ﬁeld ﬁrst and second directions u and v for the gradients of the parametric coordinates ∇u and ∇v. The parameterization is then computed as the solution of a constrained minimization problem: C 2 C 2 min [∇u − u +∇v − v ]A s.t. (7), (6) u,v f ∈F m m ij ij v = R v + t &v = R v + t . (7) π π 1 ij 2 ij 1 2 2 2 Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 17 of 28 Fig. 13 Abdominal aorta global parameterization. a Input triangulated mesh. b Global parameterization and quadrilateral layout after 1 iteration. c Global parameterization and quadrilateral layout after 25 iterations where A is the area of the considered face f . The corresponding ﬁnal positions v and v 1 2 2 2 are related to the cut edge e from face i to face j with endpoints v ∈ R and v ∈ R in ij 1 2 the parametric domain. Given a side of the cut by picking v and v , the parameterization 1 2 on the other side is determined by these previous transitions. The matching m ∈ Z is ij deﬁned between the two local charts which specify the rotation operator R. Rotations are π 2 made by angles multiple of . The integer t ∈ Z is a translation across cut edge e . ij ij Aligned global parameterization can also integrate sharp edges and boundary constraints to ensure that a common relevant edge is mapped to an isoparametric curve. Node connection constraints Once a valid quadrilateral layout and consistent geometric cross ﬁeld are provided, we wish to restrict each arc in a way that two incident nodes lie on a common isoparametric curve. With isoparametric we mean that either the u or v parameter is constant along the curve when taking transitions into account. The minimization problem (6) is then constrained using node connection constraints. These constraints are derived from the quadrilateral layout Q. Typically, each arc must lie on a common isoparametric curve taking seams transitions into account [46]. We are now able to compute a suitable global parameterization of the mesh M, given the associated quadrilateral layout Q. Quadrilateral layout embedding optimization Depending on geometry and position of Q’s nodes, a better parameterization can be found using node relocation. Due to the nature of the quadrilateral layout, ﬁxed nodes rise to large distortions or even local non-injectivities. Mathematically speaking we are going to optimize equation (6). The nodes are re-positioned based on the gradient of the parameter- ization’s objective functional with respect to their positions. This is done iteratively until a global embedding quality is reached. We follow the method developed by Campen and Kobbelt [8] to perform such optimization. Figure 13 shows an abdominal aorta scanned mesh with its global parameterization optimization using an arbitrary valid quadrilateral layout. Observe the ﬁnal global parameterization quality and nodes relocation. The ﬁnal parameterization is then used to construct feature-aligned quadrilateral meshes. Due to patch structure of the quadrilateral layout, such quadrilateral meshes can be computed patch by patch according to connectivity constraints. Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 18 of 28 Isotopological analogous quadrilateral meshes construction from aligned global parameterization Let us reformulate the previous deﬁned problematic in section 1.2 in a more technical manner. Given a set of input triangulated meshes we strive to ﬁnd 3D quadrilateral meshes which respect the four following properties: • Pure quadrilaterals with low distortion. • Feature aligned. • Isotopological with analogous points into other geometric instances. • Non-uniform isotropy. Pure quadrilaterals with low distortion With the deﬁnition of a quadrilateral layout replicating perfectly the topology of the input triangulated surface mesh, it is possible to deﬁne a pure quadrilateral mesh. Low distortion is achievied by minimizing the alignment of the global parameterization with the guiding ﬁeld. Feature aligned Sharp edges, vertices and principal curvature directions are inportant features of meshes. Alignment to these features is required on all instances to fully conserve signiﬁcant features. Thus the input quadrilateral layout must contain all the sharp edges and vertices to ensure right feature alignment into all input meshes. Isotopological with analogous points into other geometric instances A suﬃcient condition to have isotopological meshes is to hold the same connectivity. A representative quadri- lateral mesh of the population with a correct sampling is taken as reference to set the mesh connectivity. It should be noted here that it is not a suﬃcient condition to compare meshes because of orientation. This is why we want each point of the computed mesh to have an analogous point into all other geometric instances of the considered set. It is achieved globally by patch connectivity and locally by classifying sharp features and applying patch parameterization. Non-uniform isotropy Mesh non-uniform isotropy provides the only degrees of freedom to have isotopological meshes from diﬀerent geometric instances. Non-uniform isotropy is intrinsically set because of patch connectivity and discretization. Application of the MEG-IsoQuad method to geometric parametric analysis Preliminaries Reduced order models As mentioned in the introduction, Reduced Order Models (ROMs) enable real-time analy- sis due to their very low computational cost during evaluation. In the context of geometric parametric studies, their construction requires accumulating a certain number of geome- tries depending each on a set of parameters. This set of geometric instances are called snapshots. For ROM construction we need solution vectors with the same dimension associated to all snapshots in order to build matrices required to perform a Singular Value Decomposition (SVD). Isotopological and analogous meshes are thus appropriate. Algorithmic details The software Rhinoceros 5 [43] is used for visualization and RhinoCommon.dll [44] for handling various Non-Uniform Rational B-Spline (NURBS) objects. Our algorithm is entirely incorporated into a Rhinoceros 5 Plug-In implemented in VB.NET. C++ processes Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 19 of 28 Table 1 Range of parameters used for the T-shape part geometric reduced order model Pant part range of parameters Minimum Maximum X1 on boundary 1 − 40 40 Y1 on boundary 1 − 40 40 Z1 on boundary 1 − 35 35 X2 on boundary 2 − 40 40 Y2 on boundary 2 − 40 40 Z2 on boundary 2 − 35 35 X3 on boundary 3 − 40 40 Y3 on boundary 3 − 40 40 Z3 on boundary 3 − 35 35 are called from the Plug-In. Taking a B-Rep object provided from a ﬁle or constructed from the Rhinoceros 5 graphical interface, a geometric reduced order model can be built using our fully integrated workﬂow. We use for that purpose the ROM builder proprietary software developed by ANSYS [1]. For instance, this builder has been used to prevent excessive compression of buttock’s soft tissues by bony structures [39] in real-time for TM paraplegic persons. All computations have been performed on an Intel Core i7- 6820HQ CPU @ 2.70Ghz with 16Go RAM. Mechanical parametric geometry T-shape part We ﬁrst introduce the T-shaped part for a rapid understanding of our approach. Using this very simple shape, we deﬁne the number and the range of geometric parameters for an arbitrary study, see e.g. Table 1. These meshes come from a standard CAD software and each input triangulated mesh has an attached set of 9 geometric parameters based on the zero reference (Fig. 14a). Then a feature-aware cuboid decomposition is performed involving a pant mesh as topological input (Fig. 14b). Thereafter an optimized aligned global parameterization is computed with the quadrilateral layout and hence gives us the desired quadrilateral mesh with required geometric and connectivity properties. We proceed the workﬂow on all 37 input geometric instances to generate all isotopological analogous meshes (Fig. 14c). To avoid large amount of triangulated meshes as input in order to compute an accurate ROM with 9 geometric parameters, we operate a very simple sparse grid sampling of the sets [37,38] (we took 4 snapshots per parameters, involving 37 meshes). ROM is then built using the isotopological snapshots. Once the reduced order model is built, we can generate in real-time all desired geometries handling our ROM evaluation interface (Fig. 15). Casting part This casting part is a classical one that can be found in mechanical systems for coupling shafts together. Starting with a feature-aware pants decomposition, we apply a speciﬁc cuboid conﬁguration on each pant (Fig. 16c, d). Then the worﬂow is very similar to the T-shaped model presented in the previous paragraph. The sampling of the 6 input parameters (Fig. 16a, b) was made involving an optimal space ﬁlling technique so as to generate 65 triangulated meshes covering as best as possible the R space. Parameters Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 20 of 28 Fig. 14 Geometric snapshots generation of the T-shape. a Zero reference of geometric parameters in the global frame. b Feature-aware cuboid decomposition of all geometric instances. c Optimized set of isotopological analogous quadrilateral meshes: snapshots Fig. 15 Real-time data-based model. Geometry is evaluated from the built reduced order model. Row (a)left to right: parameters, right view and perspective. Row (b) left to right: parameters, right view and perspective for diﬀerent parameters ranges have been deﬁned so as to replicate a real industrial case with large and short variations, see e.g. Table 2. We generate in real-time all desired geometries handling the same ROM evaluation interface (Fig. 17) adapted to the casting part study. Biological parametric geometry Heart ventricle Ventricle geometric reduction is dedicated to a speciﬁc biological application. Scanned data from medical imaging related to one patient is provided during one cardiac cycle uniformly sampled by 10 meshes. Meshes are called “phases” and they start from 0 to 90 (Fig. 18). Ventricule volumetric capacity during a cardiac cycle can be seen in Fig. 19. Using a speciﬁc pipeline well-suited for these particular meshes, a quadrilateral layout Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 21 of 28 Fig. 16 Feature-aware cuboid decomposition of the casting part for geometric snapshots generation. a Radius parameters. b Height parameters. c Geometry-aware pants decomposition. d Feature-aware cuboid decomposition Table 2 Range of parameters used for the casting part geometric reduced order model Casting part range of parameters Minimum Maximum Length L 80 81 Radius Rb 120 125 Radius Rh 35 40 Radius Ri 20 25 Height Hb 15 25 Height Hh 140 160 is computed (Fig. 10e). To be more precise, we modify the topology of scanned meshes from disk to cylinder by adding a little hole at the bottom of the ventricule. Afterwards an optimized aligned global parameterization is extracted using leaﬂet (i.e., the leaﬂet of the aortic valve) principal curvature direction interpolation for the geometric cross ﬁeld. A reduced order model is built using the phase (i.e., time) parameter and isotopological analogous snapshots. For instance two geometries have been evaluated at phase 38.7% (systole, i.e., contraction phase) and 81% (diastole, i.e., expansion phase) in Fig. 20.Mesh linear interpolation can be simply performed if quadrilateral meshes have the required properties. An example of consistent interpolation is given in Fig. 21. Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 22 of 28 Fig. 17 Real-time data-based model. Geometry is evaluated from the built reduced order model. Row (a)left to right: parameters and perspective view. Row (b) left to right: parameters and perspective view for diﬀerent parameters Fig. 18 Three scanned ventricle geometries by corresponding phases Fig. 19 Ventricle volumetric capacity during a cycle Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 23 of 28 Fig. 20 Generated ventricle geometric snapshots (left). Real-time data-based model (right). Geometry is evaluated from the built reduced order model. Two geometries have been evaluated at phase 38.7% (systole) and 81% (diastole) Fig. 21 Mesh linear interpolation between phase 0 to phase 20. Phase 0 and phase 20, respectively depicted in black and blue (left). Isotopological non-analogous meshes yield non-consistent interpolated mesh with intersections (right) whereas analogous ones gives the expected result Table 3 Run times and statistics: average number of faces in the input triangulated meshes, patches in the quadrilateral layout, average computation time per snapshot, number of snapshots in the reduced order model, reduced order model building time, number of modes and evaluation time of the geometry ROM avg. Faces Patches avg. Time (min) Snapshots ROM (s) Modes Evaluation (s) T-Shape part 4406 18 1.6 37 11 12 1.5 Casting part 21680 168 15.6 65 19 15 1.8 Heart ventricle 6179 12 2.8 10 7 10 1.1 Results discussion Run times and statistics Any quadrilateral mesh included in the numerical charts range of parameters can be evaluated in seconds. Table 3 shows some statistics related to the whole workﬂow. B- Rep parametric geometries are turned into structured quadrilateral meshes in minutes. Then, the isotopological analogous constraints are applied to the population of meshes in order to obtain the same representation. Thanks to the same data structure and analogous properties, an accurate reduced order model is built with a prescribed number of modes. For each study, evaluation time is constant because equivalent numerical operations are performed. Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 24 of 28 Accuracy of the MEG-IsoQuad Method and geometric parametric analysis Fidelity of the method can be measured in two steps. The ﬁrst focuses on the quality of the generated quadrilateral meshes with respect to the input triangulated B-Rep. We refer the readers to Campen and Kobbelt[8] and Bommes et al.[6] to have more details on quality espects of computed quadrilaterals. In Fig. 22(a, left), we show that the quadrilateral mesh replicates the B-Rep with ﬁdelity. Viewed contrasted shapes are relative to numerical error. Sharp features are conserved during surface conversion. Notice that for a shape with less localized curvature, the structured output mesh is still satisfactory for a correct sampling of quadrilateral elements. A geometry is evaluated in an unfavorable case and compared with a validation snapshot in Fig. 22(a, middle) which was not used for the reduced order model construction. The studied case is located in a corner of the 9-dimensional sparse grid, far from known information. Despite this diﬃcutly, the evaluated geometry looks quite good with only 12 modes and 37 snapshots to cover this high-dimensional space. Unwanted curvature eﬀects are related to the lack of input geometries and dependent geometric parameters. Accuracy can be signiﬁcantly improved by adding snapshots to the reduced order model and modes in the truncated basis. Standard kriging responses surfaces will be thus more faithful. Error is given in Fig. 22(a, right). Secondly, we investigate the truncated basis dimension focalized on a quarter of the casting part. For that purpose, a snapshot is taken and compared with several evaluated geometries approximated by an increased number of modes. In Fig. 22(b, left) 5 modes are used to represent the geometry until 25 in Fig. 22(b, right). Higher is the number of modes, better are the estimated shapes. While global distance error decreases, the error becomes more and more localized. Comparison and contribution for reduced order modeling with geometric parameters Most of current approaches to build reduced order models with geometric parameters are based on the 2 following methods: mesh morphing and immersed methods. Remark that, for elementary geometric parameters manual techniques are legitimate, see e.g. Lu et al.[37]. To our best knowledge, no recent methods using surface segmentation and global parameterization tools exist to construct accurate geometric numerical charts. Our method employs techniques from the graphics community and adapt them to the needs of reduced order modeling applications with geometric parameters. MEG-IsoQuad technique preserves input mesh features and build isotopological mesh structures directly usable in various SVD algorithms. Firstable, mesh morphing is for instance operated to study biological shapes. Femur based Principal Component Analysis (PCA) requires mesh projection and hence morph- ing tools to compare instances with the same data structure, see e.g. Hraiech [24]and Grassi et al. [20]. Morphing is associated with distorted and overlapped elements and as a consequence this method is not robust for large variation of parameters. On the other hand, researchers employ immersed methods to deal with the same mesh connectivity but diﬀerent geometries. However, these approaches [13,28] also suﬀer from some drawbacks such as a clear deﬁnition of the integration rule for the cut ﬁnite elements. Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 25 of 28 Fig. 22 Accuracy of the MEG-IsoQuad Method and geometric parametric analysis. a B-Rep to reduced order acc model evaluation accuracy analysis on the T-shape part. Used set of parameters is P ={X1 = 40, T −shape Y1 = 40,Z1 = 35,X2 = 40,Y2 = 40,Z2 = 35,X3 = 40,Y3 = 40,Z3 = 35}. Triangulated mesh (depicted in blue) and generated 3D quadrilateral mesh (depicted in gray), geometric diﬀerence is almost zero (left). Geometric comparison between the reduced order model evaluation and a validation snapshot for the same acc set of parameters P (middle). Distance error analysis of (a, middle) mapped onto the reduced order T −shape evaluation (right). b Reduced order model truncated basis inﬂuence on the casting part. Used snapshot has acc parameters P ={L = 80,Rb = 120,Rh = 35,Ri = 20,Hb = 15,Hh = 140}. Distance error is analyzed Casting between the reduced order model evaluation and the corresponding snapshot and mapped on the reduced order model evaluation. Error with a basis constructed from the ﬁrst 5 modes (left). Error using 15 modes (middle). Error using 25 modes (right) Limitations and guarantees Limitations Surface segmentation is highly dependent on the mesh discretization and also on the intrinsic geometry. Hence, the smart model decomposition step may have inputs which are problematic. For instance, pants decomposition is known to produce highly deformed segmentations or unwanted geometries [53]. To overcome partially this issue, we propose to use both topological tools and geometric criteria. Cuboid decomposition deals with consistent pants. The quadrilateral layout computed from the cuboid decomposition step is generally coherent but it may fail when exotic layouts are requested: this is the limitation of the constrained border parameterization to determine layout arcs. Global parameterization step, including cross ﬁelds is entirely performed by techniques from the paper of Campen and Kobbelt [8]. We refer the readers to the limitation section of their work. Guarantees In spite of limitations expressed above, once quadrilateral meshes are generated, the reduced order modeling operation can not fail. Maquart et al. Adv. Model. and Simul. in Eng. Sci. (2021) 8:8 Page 26 of 28 Conclusion We have presented in this work a method for designing 3D quadrilateral snapshots adapted for data-based applications. Quadrilateral layouts are computed from a geometry-feature- aware pants-to-cuboids decomposition. These layouts replicate the input mesh topology and contain relevant isotopological information. Depending on the geometry, the pants cuboid conﬁgurations are chosen by the user to lie on the key features. Using aligned global parameterization, arcs and nodes of the suitable quadrilateral layout are optimized to achieve low overall patch distortion and alignment with sharp features. A key contribution of the present work is the automatic construction of a relevant quadrilateral layout per pant, setting automatically well-adapted cross ﬁeld singularities in order to compute a good aligned global parameterization. This allows us to extract a quadrilateral mesh satisfying sharp features, principal curva- ture directions and possibly boundary constraints. With such features, our method allows to produce isotopological meshes of all the geometric instances. They are created dur- ing the oﬄine phase of ROM based parametric studies. Throughout design process of quadrilateral meshes, we have handled data of all geometric instances in a isotopological manner, no matter the input geometry. This helps us to deﬁne a set of isotopological anal- ogous meshes using one mesh as reference. The previous sets were successfully used to construct accurate geometric parametric models. These data-based models are evaluated for any given parameter set in near real-time to give the speciﬁed geometry. Abbreviations CAD: Computer aided design. Acknowledgements T. Maquart was partially supported by a CIFRE fellowship of the french Association Nationale de la Recherche et de la Technologie and ANSYS research. T. Maquart, T. Elguedj and A. Gravouil were also partially supported by a research contract with ANSYS. These supports are gratefully acknowledged. We especially thank Martijn Hoeijmakers for his help in providing the scanned meshes of the ventricle model coming from anonymous data. All the reduced order model computations are performed with the Reduced Order Model builder developed by the ANSYS France research team. Authors’ contributions TM developed the theoretical formalism, performed the geometric calculations and performed the reduced order model construction. TE, AG and MR contributed to the ﬁnal version of the manuscript. TE and AG supervised the project. All authors read and approved the ﬁnal manuscript. Funding Please see “Acknowledgements” for more details. Availability of data and materials All developed numerical tools are not available. They are the property of ANSYS, Inc.. Declarations Competing interests All authors declare that they have no competing interests concerning this presented work. Author details 1 2 Univ Lyon, INSA-Lyon, CNRS UMR5259, LaMCoS, 69621 Lyon, France, Present address: Laboratoire de Mécanique des Contacts et des Structures, 27 bis avenue Jean Capelle, 69621 Villeurbanne Cedex, France, ANSYS, ANSYS France, 11 Avenue Albert Einstein, 69100 Villeurbanne, France. Received: 8 December 2020 Accepted: 19 March 2021 References 1. Reduced order model builder (non-oﬃcial title) by ANSYS, Inc., ﬁled 2019-01-22. U.S. patent application 16/253,635. Maquart et al. Adv. Model. and Simul. in Eng. Sci. 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