A distortion measure to validate and generate curved high-order meshes on CAD surfaces with independence of parameterization

This is the accepted version of the following article: [Gargallo-Peiró, A., Roca, X., Peraire, J., and Sarrate, J. (2016) A distortion measure to validate and generate curved high-order meshes on CAD surfaces with independence of parameterization. Int. J. Numer. Meth. Engng, 106: 1100–1130. doi: 10.1002/nme.5162], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5162/abstractA framework to validate and generate curved nodal high-order meshes on Computer-Aided Design (CAD) surfaces is presented. The proposed framework is of major interest to generate meshes suitable for thin-shell and 3D finite element analysis with unstructured high-order methods. First, we define a distortion (quality) measure for high-order meshes on parameterized surfaces that we prove to be independent of the surface parameterization. Second, we derive a smoothing and untangling procedure based on the minimization of a regularization of the proposed distortion measure. The minimization is performed in terms of the parametric coordinates of the nodes to enforce that the nodes slide on the surfaces. Moreover, the proposed algorithm repairs invalid curved meshes (untangling), deals with arbitrary polynomial degrees (high-order), and handles with low-quality CAD parameterizations (independence of parameterization). Third, we use the optimization procedure to generate curved nodal high-order surface meshes by means of an a posteriori approach. Given a linear mesh, we increase the polynomial degree of the elements, curve them to match the geometry, and optimize the location of the nodes to ensure mesh validity. Finally, we present several examples to demonstrate the features of the optimization procedure, and to illustrate the surface mesh generation process.Peer ReviewedPostprint (author's final draft

Distortion and quality measures for validating and generating high-order tetrahedral meshes

A procedure to quantify the distortion (quality) of a high-order mesh composed of curved tetrahedral elements is presented. The proposed technique has two main applications. First, it can be used to check the validity and quality of a high-order tetrahedral mesh. Second, it allows the generation of curved meshes composed of valid and high-quality high-order tetrahedral elements. To this end, we describe a method to smooth and untangle high-order tetrahedral meshes simultaneously by minimizing the proposed mesh distortion. Moreover, we present a -continuation procedure to improve the initial configuration of a high-order mesh for the optimization process. Finally, we present several results to illustrate the two main applications of the proposed technique.Peer ReviewedPostprint (author’s final draft

On curving high-order hexahedral meshes

We present a new definition of distortion and quality measures for high-order hexahedral (quadrilateral) elements. This definition leads to two direct applications. First, it can be used to check the validity and quality of a high-order hexahedral (quadrilateral) mesh. Second, it allows the generation of high-order curved meshes composed of valid and high-quality hexahedral (quadrilateral) elements. We describe a method to simultaneously smooth and untangle high-order hexahedral (quadrilateral) meshes by minimizing the proposed mesh distortion. Finally, we analyze the behavior of the proposed distortion measure and we present several results to illustrate the benefits of the mesh generation framework.Peer ReviewedPostprint (author's final draft

Generation of Curved High-order Meshes with Optimal Quality and Geometric Accuracy

We present a novel methodology to generate curved high-order meshes featuring optimal mesh quality and geometric accuracy. The proposed technique combines a distortion measure and a geometric Full-size image (<1 K)-disparity measure into a single objective function. While the element distortion term takes into account the mesh quality, the Full-size image (<1 K)-disparity term takes into account the geometric error introduced by the mesh approximation to the target geometry. The proposed technique has several advantages. First, we are not restricted to interpolative meshes and therefore, the resulting mesh approximates the target domain in a non-interpolative way, further increasing the geometric accuracy. Second, we are able to generate a series of meshes that converge to the actual geometry with expected rate while obtaining high-quality elements. Third, we show that the proposed technique is robust enough to handle real-case geometries that contain gaps between adjacent entities.This research was partially supported by the Spanish Ministerio de Economía y Competitividad under grand contract CTM2014-55014-C3-3-R, and by the Government of Catalonia under grand contract 2014-SGR-1471. The work of the last author was supported by the European Commission through the Marie Sklodowska-Curie Actions (HiPerMeGaFlows project).Peer ReviewedPostprint (published version

A Thermo-elastic Analogy for High-order Curvilinear Meshing with Control of Mesh Validity and Quality

This is the final version of the article. Available from Elsevier via the DOI in this record.In recent years, techniques for the generation of high-order curvilinear mesh have frequently adopted mesh deformation procedures to project the curvature of the surface onto the mesh, thereby introducing curvature into the interior of the domain and lessening the occurrence of self-intersecting elements. In this article, we propose an extension of this approach whereby thermal stress terms are incorporated into the state equation to provide control on the validity and quality of the mesh, thereby adding an extra degree of robustness which is lacking in current approaches

Generation of curved high-order meshes with optimal quality and geometric accuracy

We present a novel methodology to generate curved high-order meshes featuring optimal mesh quality and geometric accuracy. The proposed technique combines a distortion measure and a geometric L2-disparity measure into a single objective function. While the element distortion term takes into account the mesh quality, the L2-disparity term takes into account the geometric error introduced by the mesh approximation to the target geometry. The proposed technique has several advantages. First, we are not restricted to interpolative meshes and therefore, the resulting mesh approximates the target domain in a non-interpolative way, further increasing the geometric accuracy. Second, we are able to generate a series of meshes that converge to the actual geometry with expected rate while obtaining high-quality elements. Third, we show that the proposed technique is robust enough to handle real-case geometries that contain gaps between adjacent entities.Peer ReviewedPostprint (published version

A Thermo-elastic Analogy for High-order Curvilinear Meshing with Control of Mesh Validity and Quality

AbstractIn recent years, techniques for the generation of high-order curvilinear mesh have frequently adopted mesh deformation procedures to project the curvature of the surface onto the mesh, thereby introducing curvature into the interior of the domain and lessening the occurrence of self-intersecting elements. In this article, we propose an extension of this approach whereby thermal stress terms are incorporated into the state equation to provide control on the validity and quality of the mesh, thereby adding an extra degree of robustness which is lacking in current approaches

Optimizing the geometrical accuracy of curvilinear meshes

This paper presents a method to generate valid high order meshes with optimized geometrical accuracy. The high order meshing procedure starts with a linear mesh, that is subsequently curved without taking care of the validity of the high order elements. An optimization procedure is then used to both untangle invalid elements and optimize the geometrical accuracy of the mesh. Standard measures of the distance between curves are considered to evaluate the geometrical accuracy in planar two-dimensional meshes, but they prove computationally too costly for optimization purposes. A fast estimate of the geometrical accuracy, based on Taylor expansions of the curves, is introduced. An unconstrained optimization procedure based on this estimate is shown to yield significant improvements in the geometrical accuracy of high order meshes, as measured by the standard Haudorff distance between the geometrical model and the mesh. Several examples illustrate the beneficial impact of this method on CFD solutions, with a particular role of the enhanced mesh boundary smoothness.Comment: Submitted to JC

Finding Hexahedrizations for Small Quadrangulations of the Sphere

This paper tackles the challenging problem of constrained hexahedral meshing. An algorithm is introduced to build combinatorial hexahedral meshes whose boundary facets exactly match a given quadrangulation of the topological sphere. This algorithm is the first practical solution to the problem. It is able to compute small hexahedral meshes of quadrangulations for which the previously known best solutions could only be built by hand or contained thousands of hexahedra. These challenging quadrangulations include the boundaries of transition templates that are critical for the success of general hexahedral meshing algorithms. The algorithm proposed in this paper is dedicated to building combinatorial hexahedral meshes of small quadrangulations and ignores the geometrical problem. The key idea of the method is to exploit the equivalence between quad flips in the boundary and the insertion of hexahedra glued to this boundary. The tree of all sequences of flipping operations is explored, searching for a path that transforms the input quadrangulation Q into a new quadrangulation for which a hexahedral mesh is known. When a small hexahedral mesh exists, a sequence transforming Q into the boundary of a cube is found; otherwise, a set of pre-computed hexahedral meshes is used. A novel approach to deal with the large number of problem symmetries is proposed. Combined with an efficient backtracking search, it allows small shellable hexahedral meshes to be found for all even quadrangulations with up to 20 quadrangles. All 54,943 such quadrangulations were meshed using no more than 72 hexahedra. This algorithm is also used to find a construction to fill arbitrary domains, thereby proving that any ball-shaped domain bounded by n quadrangles can be meshed with no more than 78 n hexahedra. This very significantly lowers the previous upper bound of 5396 n.Comment: Accepted for SIGGRAPH 201