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HOT: HodgeOptimized Triangulations
 ACM Trans. Graph
, 2011
"... We introduce Hodgeoptimized triangulations (HOT), a family of wellshaped primaldual pairs of complexes designed for fast and accurate computations in computer graphics. Previous work most commonly employs barycentric or circumcentric duals; while barycentric duals guarantee that the dual of each ..."
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Cited by 17 (4 self)
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We introduce Hodgeoptimized triangulations (HOT), a family of wellshaped primaldual pairs of complexes designed for fast and accurate computations in computer graphics. Previous work most commonly employs barycentric or circumcentric duals; while barycentric duals guarantee that the dual of each simplex lies within the simplex, circumcentric duals are often preferred due to the induced orthogonality between primal and dual complexes. We instead promote the use of weighted duals (“power diagrams”). They allow greater flexibility in the location of dual vertices while keeping primaldual orthogonality, thus providing a valuable extension to the usual choices of dual by only adding one additional scalar per primal vertex. Furthermore, we introduce a family of functionals on pairs of complexes that we derive from bounds on the errors induced by diagonal Hodge stars, commonly used in discrete computations. The minimizers of these functionals, called HOT meshes, are shown to be generalizations of Centroidal Voronoi Tesselations and Optimal Delaunay Triangulations, and to provide increased accuracy and flexibility for a variety of computational purposes.
Quadratic Serendipity Finite Elements on Polygons Using Generalized Barycentric Coordinates
, 2011
"... We introduce a finite element construction for use on the class of convex, planar polygons and show it obtains a quadratic error convergence estimate. On a convex ngon satisfying simple geometric criteria, our construction produces 2n basis functions, associated in a Lagrangelike fashion to each v ..."
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Cited by 10 (3 self)
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We introduce a finite element construction for use on the class of convex, planar polygons and show it obtains a quadratic error convergence estimate. On a convex ngon satisfying simple geometric criteria, our construction produces 2n basis functions, associated in a Lagrangelike fashion to each vertex and each edge midpoint, by transforming and combining a set of n(n + 1)/2 basis functions known to obtain quadratic convergence. The technique broadens the scope of the socalled ‘serendipity’ elements, previously studied only for quadrilateral and regular hexahedral meshes, by employing the theory of generalized barycentric coordinates. Numerical evidence is provided on a trapezoidal quadrilateral mesh, previously not amenable to serendipity constructions, and applications to adaptive meshing are discussed.
New perspectives on polygonal and polyhedral finite element methods
 MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
, 2014
"... Generalized barycentric coordinates such as Wachspress and mean value coordinates have been used in polygonal and polyhedral finite element methods. Recently, mimetic finitedifference schemes were cast within a variational framework, and a consistent and stable finite element method on arbitrary ..."
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Cited by 5 (2 self)
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Generalized barycentric coordinates such as Wachspress and mean value coordinates have been used in polygonal and polyhedral finite element methods. Recently, mimetic finitedifference schemes were cast within a variational framework, and a consistent and stable finite element method on arbitrary polygonal meshes was devised. The method was coined as the Virtual Element Method (VEM), since it did not require the explicit construction of basis functions. This advance provides a more indepth understanding of mimetic schemes, and also endows polygonalbased Galerkin methods with greater flexibility than threenode and fournode finite element methods. In the VEM, a projection operator is used to realize the decomposition of the stiffness matrix into two terms: a consistent matrix that is known, and a stability matrix that must be positive semidefinite and which is only required to scale like the consistent matrix. In this paper, we first present an overview of previous developments on conforming polygonal and polyhedral finite elements, and then appeal to the exact decomposition in the VEM to obtain a robust and efficient generalized barycentric coordinatebased Galerkin method on polygonal and polyhedral elements. The consistent matrix of the VEM is adopted, and numerical quadrature with generalized barycentric coordinates is used to compute the stability matrix. This facilitates postprocessing of field variables and
Quadratic maximumentropy serendipity shape functions
"... for arbitrary planar polygons ..."
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NCND license (http://creativecommons.org/licenses/byncnd/3.0/). Selection and peerreview under responsibility of the Optimisation of sizecontrollable centroidal voronoi tessellation for FEM simulation of micro forming processes
"... Abstract Voronoi tessellation has been employed to characterise material features in Finite Element Method (FEM) simulation, however, a poor mesh quality of the voronoi tessellations causes problems in explicit dynamic simulation of forming processes. Although centroidal voronoi tessellation can pa ..."
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Abstract Voronoi tessellation has been employed to characterise material features in Finite Element Method (FEM) simulation, however, a poor mesh quality of the voronoi tessellations causes problems in explicit dynamic simulation of forming processes. Although centroidal voronoi tessellation can partly improve the mesh quality by homogenisation of voronoi tessellations, small features, such as short edges and small facets, lead to an inferior mesh quality. Further, centroidal voronoi tessellation cannot represent all real micro structures of materials because of the almost equal tessellation shape and size. In this paper, a density function is applied to control the size and distribution of voronoi tessellations and then a Laplacian operator is employed to optimise the centroidal voronoi tessellations. After optimisation, the small features can be eliminated and the elements are quadrilateral in 2D and hexahedral in 3D cases. Moreover, the mesh quality is significantly higher than that of the mesh generated on the original voronoi or centroidal voronoi tessellation. This work is beneficial for explicit dynamic simulation of forming processes, such as micro deep drawing processes.
A hybrid TTrefftz polygonal finite element for linear elasticity
, 2014
"... In this paper, we construct hybrid TTrefftz polygonal finite elements. The displacement field within the polygon is represented by the homogeneous solution to the governing differential equation, also called as the Tcomplete set. On the boundary of the polygon, a conforming displacement field is ..."
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In this paper, we construct hybrid TTrefftz polygonal finite elements. The displacement field within the polygon is represented by the homogeneous solution to the governing differential equation, also called as the Tcomplete set. On the boundary of the polygon, a conforming displacement field is independently defined to enforce continuity of the displacements across the element boundary. An optimal number of Tcomplete functions are chosen based on the number of nodes of the polygon and degrees of freedom per node. The stiffness matrix is computed by the hybrid formulation with auxiliary displacement frame. Results from the numerical studies presented for a few benchmark problems in the context of linear elasticity shows that the proposed method yield highly accurate results.
ProjectTeam Geometrica Geometric Computing
"... c t i v it y e p o r t 2009 Table of contents ..."
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