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Efficient Computation of the Extrema of Algebraic Quality Measures for Curvilinear Finite Elements Johnen, Amaury ; Geuzaine, Christophe ; Conference (2016, June) The development of high-order computational methods for solving partial differential equations on unstructured grids has been underway for many years. Such methods critically depend on the availability of ... [more ▼] The development of high-order computational methods for solving partial differential equations on unstructured grids has been underway for many years. Such methods critically depend on the availability of high-quality curvilinear meshes, as one bad element can degrade the solution in the whole domain. The usual way of generating curved meshes is to first generate a (high-quality) straight-sided mesh. Then, mesh entities that are classified on the boundaries of the domain are curved. This operation introduces a "shape-distortion" that should be controlled. Quality measures allow to quantify to which point an element is well-shaped. They also provide tools to improve the quality of meshes through optimization. In this work we propose an efficient method to compute several quality measures for curved elements, based on the Jacobian of the mapping between the straight-sided elements and the curved ones. Contrary to the approach presented in "A. Gargallo-Peiró, X. Roca, J. Peraire, and J. Sarrate. Distortion and quality measures for validating and generating high-order tetrahedral meshes. Engineering with Computers, pages 1–15, 2014.", which relies on an L2-norm over the elements, we compute the actual minimum and maximum of the local quality measure for each element. The method is an extension of previous works on the validity of those elements (A. Johnen et al., 2013). The key feature is that we can adaptively expand functions based on the Jacobian matrix and its determinant in terms of Bézier functions. Bézier functions have both properties of boundedness and positivity, which allow sharp computation of minimum or maximum of the interpolated functions. [less ▲] Detailed reference viewed: 33 (3 ULg)The generation of valid curvilinear meshes Geuzaine, Christophe ; Johnen, Amaury ; et al in IDIHOM: Industrialization of High-Order Methods - A Top-Down Approach (2015) Detailed reference viewed: 25 (3 ULg)Optimizing the geometrical accuracy of 2D curvilinear finite element meshes ; ; et al Conference (2014, June 27) Detailed reference viewed: 27 (6 ULg)High-order mesh generation for CFD ; Geuzaine, Christophe ; Johnen, Amaury et al Conference (2014) Detailed reference viewed: 19 (4 ULg)Geometrical validity of high-order triangular finite elements Johnen, Amaury ; ; Geuzaine, Christophe in Engineering with Computers (2014), 30(3), 375-382 This paper presents a method to compute accurate bounds on Jacobian determinants of high-order (curvilinear) triangular finite elements. This method can be used to guarantee that a curvilinear triangle is ... [more ▼] This paper presents a method to compute accurate bounds on Jacobian determinants of high-order (curvilinear) triangular finite elements. This method can be used to guarantee that a curvilinear triangle is geometrically valid, i.e., that its Jacobian determinant is strictly positive everywhere in its reference domain. It also provides an efficient way to measure the quality the triangles. The key feature of the method is to expand the Jacobian determinant using a polynomial basis, built using Bézier functions, that has both properties of boundedness and positivity. Numerical results show the sharpness of our estimates. [less ▲] Detailed reference viewed: 110 (30 ULg)High-order mesh generation for CFD with aeronautical applications ; Geuzaine, Christophe ; Johnen, Amaury et al Conference (2014) Detailed reference viewed: 17 (1 ULg)Optimizing the geometrical accuracy of 2D curvilinear meshes ; ; Geuzaine, Christophe et al in Procedia Engineering (2014), 82 Detailed reference viewed: 6 (0 ULg)Optimal parametrizations for surface remeshing ; ; Geuzaine, Christophe in Engineering with Computers (2014), 30(3), 383-402 Detailed reference viewed: 6 (0 ULg)A frontal approach to hex-dominant mesh generation ; ; et al in Advanced Modeling and Simulation in Engineering Sciences (2014), 1(1), 1-30 Detailed reference viewed: 31 (5 ULg)Fifth International Conference on Advanced COmputational Methods in ENgineering (ACOMEN 2011) Béchet, Eric ; ; Geuzaine, Christophe et al in International Journal of Computational & Applied Mathematics (2013), 246 Detailed reference viewed: 169 (38 ULg)New mesh generation developments in GMSH ; Johnen, Amaury ; et al Conference (2013) Detailed reference viewed: 28 (11 ULg)Numerical simulation of CAD thin structures using the eXtended Finite Element Method and Level Sets ; Geuzaine, Christophe ; et al in Finite Elements in Analysis and Design (2013), 77 Detailed reference viewed: 9 (0 ULg)Robust untangling of curvilinear meshes ; Geuzaine, Christophe ; et al in Journal of Computational Physics (2013), 254 Detailed reference viewed: 9 (1 ULg)Cardiovascular and lung mesh generation based on centerlines ; Geuzaine, Christophe ; in International journal for numerical methods in biomedical engineering (2013), 29(6), 665-682 Detailed reference viewed: 20 (1 ULg)ONELAB: Open Numerical Engineering LABoratory Geuzaine, Christophe ; ; et al in Actes du 11e Colloque National en Calcul des Structures (CSMA 2013), Giens, France (2013) Detailed reference viewed: 19 (1 ULg)Generation of provably correct high-order meshes ; Geuzaine, Christophe ; et al in Advances in Computational Mechanics (ACM 2013) - Finite Elements in Flow Problems (FEF 2013) (2013) Detailed reference viewed: 12 (0 ULg)Blossom-Quad: a non-uniform quadrilateral mesh generator using a minimum cost perfect matching algorithm ; ; et al in International Journal for Numerical Methods in Engineering (2012), 89(9), 1102-1119 A new indirect way of producing all-quad meshes is presented. The method takes advantage of a well-known algorithm of the graph theory, namely the Blossom algorithm, that computes the minimum-cost perfect ... [more ▼] A new indirect way of producing all-quad meshes is presented. The method takes advantage of a well-known algorithm of the graph theory, namely the Blossom algorithm, that computes the minimum-cost perfect matching in a graph in polynomial time. The new Blossom-Quad algorithm is compared with standard indirect procedures. Meshes produced by the new approach are better both in terms of element shape and in terms of size field efficiency. [less ▲] Detailed reference viewed: 136 (14 ULg)Efficient evaluation of the geometrical validity of curvilinear finite elements Johnen, Amaury ; ; Geuzaine, Christophe Conference (2011, November 14) The development of high-order numerical techniques on unstructured grids has been underway for many years. The accuracy of these methods strongly depends of the accuracy of the geometrical discretization ... [more ▼] The development of high-order numerical techniques on unstructured grids has been underway for many years. The accuracy of these methods strongly depends of the accuracy of the geometrical discretization, and thus depends on the availability of quality curvilinear meshes. The usual way of building such curvilinear meshes is to first generate a straight sided mesh. Then, mesh entities that are classified on the curved boundaries of the domain are curved accordingly. Some internal mesh entities may be curved as well. If we assume that the straight sided mesh is composed of well shaped elements, curving elements introduces a kind of "shape distortion" that should be controlled so that the final curvilinear mesh is also composed of well shaped elements. In this work we propose a method to analyze curvilinear meshes in terms of their elementary jacobians. The method does not deal with the actual generation of the high order mesh. Instead, it provides an efficient way to guarantee that a curvilinear element is geometrically valid, i.e., that its jacobian is strictly positive in all its reference domain. It also provides a way to measure the distortion of the curvilinear element. The key feature of the method is to adaptively expand the elementary jacobians in a polynomial basis, built using Bézier functions, that has both properties of boundedness and positivity. The algorithm has been implemented in the open-source mesh generator Gmsh, and allows to control the geometrical validity of curvilinear meshes made of triangles, quadrangles, tetrahedra, hexahedra and prisms of any order. [less ▲] Detailed reference viewed: 45 (16 ULg)Proceedings of the 5th International Conference on Advanded COmputational Methods in Engineering (ACOMEN2011) Hogge, Michel ; ; et al Book published by Université de Liège - Dépôt légal: D/2011/0480/31 (2011) Detailed reference viewed: 46 (7 ULg)Geometrical validity of curvilinear finite elements Johnen, Amaury ; ; Geuzaine, Christophe in William Roshan, Quadros (Ed.) Proceedings of the 20th International Meshing Roundtable (2011, October 25) In this paper, we describe a way to compute accurate bounds on Jacobians of curvilinear finite elements of all kinds. Our condition enables to guarantee that an element is geometrically valid, i.e., that ... [more ▼] In this paper, we describe a way to compute accurate bounds on Jacobians of curvilinear finite elements of all kinds. Our condition enables to guarantee that an element is geometrically valid, i.e., that its Jacobian is strictly positive everywhere in its reference domain. It also provides an efficient way to measure the distortion of curvilinear elements. The key feature of the method is to expand the Jacobian using a polynomial basis, built using Bézier functions, that has both properties of boundedness and positivity. Numerical results show the sharpness of our estimates. [less ▲] Detailed reference viewed: 34 (11 ULg) |
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