Browse ORBi by ORBi project

- Background
- Content
- Benefits and challenges
- Legal aspects
- Functions and services
- Team
- Help and tutorials

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)Indirect quadrangular mesh generation and validation of curved finite elements Johnen, Amaury Doctoral thesis (2016) Among the different types of 3D finite element meshes, hexahedral meshes present properties that can be highly desirable, such as alignment with physical features or a lower computational cost. For this ... [more ▼] Among the different types of 3D finite element meshes, hexahedral meshes present properties that can be highly desirable, such as alignment with physical features or a lower computational cost. For this reason and despite the maturity of the tetrahedral mesh generators, hexahedral mesh generation has always been a prolific research domain. Yet, there exists currently no robust algorithm capable of generating conformal all-hexahedral meshes with prescribed input size field on any arbitrary geometry. One difficulty that remains is that there exists no method to robustly assert that a hexahedron is valid. Indeed, linear hexahedra can be folded (tangled) in the same way than curvilinear tetrahedra. This thesis addresses two subjects. First, two original quadrangular mesh generation techniques are investigated, with the aim to generalize them to 3D. Both are indirect methods and thus consider the problem of combining pairs of triangles of an initial input triangular mesh. The first technique, called Blossom-Quad, computes the optimal solution of this problem with respect to a given quality criterion. As for any indirect method, the quality of the solution strongly depends on the location of the nodes in the initial triangular mesh. The generalization to 3D is however unclear and a second technique is investigated. This one aims at computing a near-optimal solution by using a look-ahead tree technique. The corresponding algorithm allows tuning the quality of the final mesh by choosing the depth of the tree as a parameter. This technique gives a promising way forward, especially as it is directly applicable in 3D. The second subject concerns the development of a method that permits to compute, with respect to any prescribed tolerance, the extrema of Jacobian-based quantities defined on finite elements of any order and type. Applied to the Jacobian determinant, this method allows to assert the validity of any (curvi-)linear finite element. This method is also applied to a quality measure that quantifies the pointwise anisotropy of the elements. Besides being very attractive for hexahedral mesh generation, this method is especially useful for the analysis of curvilinear finite element meshes. It can moreover be an important component of optimization techniques for achieving robustness. [less ▲] Detailed reference viewed: 41 (8 ULg)Geometrical validity of curvilinear pyramidal finite elements Johnen, Amaury ; Geuzaine, Christophe in Journal of Computational Physics (2015), 299 Detailed reference viewed: 22 (3 ULg)Sequential decision-making approach for quadrangular mesh generation Johnen, Amaury ; Ernst, Damien ; Geuzaine, Christophe in Engineering with Computers (2015), 31(4), 729-735 A new indirect quadrangular mesh generation algorithm which relies on sequential decision-making techniques to search for optimal triangle recombinations is presented. In contrast to the state-of-art ... [more ▼] A new indirect quadrangular mesh generation algorithm which relies on sequential decision-making techniques to search for optimal triangle recombinations is presented. In contrast to the state-of-art Blossom-quad algorithm, this new algorithm is a good candidate for addressing the 3D problem of recombining tetrahedra into hexahedra. [less ▲] Detailed reference viewed: 390 (70 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)Geometrical validity of high-order pyramidal finite elements Johnen, Amaury ; Geuzaine, Christophe Conference (2014, July 24) Detailed reference viewed: 38 (15 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)Computing bounds on the geometrical quality of 2D curvilinear finite elements Johnen, Amaury ; ; et al Conference (2013, April 25) The development of high-order computational methods for solving partial differen- tial equations on unstructured grids has been underway for many years. Such meth- ods critically depend on the ... [more ▼] The development of high-order computational methods for solving partial differen- tial equations on unstructured grids has been underway for many years. Such meth- ods critically depend on the availability of high-quality curvilinear meshes, as one badly-shaped element can degrade the solution in the whole domain (J. Shewchuk, “What Is a Good Linear Finite Element? Interpolation, Conditioning, Anisotropy, and Quality Measures”, Preprint, 2002). The usual way of generating curved meshes is to first generate a straight sided mesh and to curve mesh entities that are classified on the boundaries of the domain. The latter operation introduces a “shape-distortion” that should be controlled if we suppose that the straight sided mesh is composed of well-shaped elements. 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 opera- tions. Many quality measures has been proposed for quadratic triangular finite element. Recently, X. Roca et al. (“Defining Quality Measures for High-Order Planar Triangles and Curved Mesh Generation”, Proceedings of the 20th Interna- tional Meshing Roundtable, 2011) proposed a technique that allows extending any Jacobian based quality measure for linear elements to high-order iso-parametric planar triangles of any interpolation degree. In this work we propose an efficient method to provide accurate bounds on the mag- nitude of the shape distortion of any triangular and quadrangular curved element. The shape distortion is measured with respect to an ideal element, which can e.g. be an equilateral triangle or the element from the original straight-sided mesh. The key feature of the method is that we can adaptively expand functions based on the Jacobian matrix and its determinant in terms of Be ́zier functions. Be ́zier functions have both properties of boundedness and positivity, which allow sharp computation of minimum or maximum of the interpolated functions. [less ▲] Detailed reference viewed: 81 (20 ULg)Geometrical validity of curvilinear finite elements Johnen, Amaury ; ; Geuzaine, Christophe in Journal of Computational Physics (2013), 233 In this paper, we describe a way to compute accurate bounds on Jacobian determinants of curvilinear polynomial finite elements. Our condition enables to guarantee that an element is geometrically valid, i ... [more ▼] In this paper, we describe a way to compute accurate bounds on Jacobian determinants of curvilinear polynomial finite elements. Our condition enables to guarantee that an element 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 distortion of curvilinear elements. 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: 94 (32 ULg)New mesh generation developments in GMSH ; Johnen, Amaury ; et al Conference (2013) Detailed reference viewed: 28 (11 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)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)Génération de maillages éléments finis d’ordre élevé Johnen, Amaury Master's dissertation (2010) Detailed reference viewed: 16 (0 ULg) |
||