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28
Flexible Simulation of Deformable Models Using Discontinuous Galerkin FEM
 IN ACM SIGGRAPH / EUROGRAPHICS SYMPOSIUM ON COMPUTER ANIMATION
, 2008
"... We propose a simulation technique for elastically deformable objects based on the discontinuous Galerkin finite element method (DG FEM). In contrast to traditional FEM, it overcomes the restrictions of conforming basis functions by allowing for discontinuous elements with weakly enforced continuity ..."
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Cited by 23 (4 self)
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We propose a simulation technique for elastically deformable objects based on the discontinuous Galerkin finite element method (DG FEM). In contrast to traditional FEM, it overcomes the restrictions of conforming basis functions by allowing for discontinuous elements with weakly enforced continuity constraints. This added flexibility enables the simulation of arbitrarily shaped, convex and nonconvex polyhedral elements, while still using simple polynomial basis functions. For the accurate strain integration over these elements we propose an analytic technique based on the divergence theorem. Being able to handle arbitrary elements eventually allows us to derive simple and efficient techniques for volumetric mesh generation, adaptive mesh refinement, and robust cutting.
K.: Framebased elastic models
 ACM Trans. Graph
, 2011
"... We present a new type of deformable model which combines the realism of physically based continuum mechanics models and the usability of framebased skinning methods. The degrees of freedom are coordinate frames. In contrast with traditional skinning, frame positions are not scripted but move in reac ..."
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Cited by 18 (6 self)
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We present a new type of deformable model which combines the realism of physically based continuum mechanics models and the usability of framebased skinning methods. The degrees of freedom are coordinate frames. In contrast with traditional skinning, frame positions are not scripted but move in reaction to internal body forces. The displacement field is smoothly interpolated using dual quaternion blending. The deformation gradient and its derivatives are computed at each sample point of a deformed object and used in the equations of Lagrangian mechanics to achieve physical realism. This allows easy and very intuitive definition of the degrees of freedom of the deformable object. The meshless discretization allows onthefly insertion of frames to create local deformations where needed. We formulate the dynamics of these models in detail and describe some precomputations that can be used for speed. We show that our method is effective for behaviors ranging from simple unimodal deformations to complex realistic deformations comparable with Finite Element simulations. To encourage its use, the software will be freely available in the simulation platform SOFA.
R.: A hexahedral multigrid approach for simulating cuts in deformable objects
 IEEE TVCG
"... Abstract—We present a hexahedral finite element method for simulating cuts in deformable bodies using the corotational formulation of strain at high computational efficiency. Key to our approach is a novel embedding of adaptive element refinements and topological changes of the simulation grid into ..."
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Cited by 11 (4 self)
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Abstract—We present a hexahedral finite element method for simulating cuts in deformable bodies using the corotational formulation of strain at high computational efficiency. Key to our approach is a novel embedding of adaptive element refinements and topological changes of the simulation grid into a geometric multigrid solver. Starting with a coarse hexahedral simulation grid, this grid is adaptively refined at the surface of a cutting tool until a finest resolution level, and the cut is modeled by separating elements along the cell faces at this level. To represent the induced discontinuities on successive multigrid levels, the affected coarse grid cells are duplicated and the resulting connectivity components are distributed to either side of the cut. Drawing upon recent work on octree and multigrid schemes for the numerical solution of partial differential equations, we develop efficient algorithms for updating the systems of equations of the adaptive finite element discretization and the multigrid hierarchy. To construct a surface that accurately aligns with the cuts, we adapt the splitting cubes algorithm to the specific linked voxel representation of the simulation domain we use. The paper is completed by a convergence analysis of the finite element solver and a performance comparison to alternative numerical solution methods. These investigations show that our approach offers high computational efficiency and physical accuracy, and that it enables cutting of deformable bodies at very high resolutions. Index Terms—Deformable objects, cutting, finite elements, multigrid, octree meshes. F 1
Error estimates for generalized barycentric interpolation
 ADV COMPUT MATH
, 2011
"... We prove the optimal convergence estimate for firstorder interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress approach explicitly constructs rational functions, the Sibson ..."
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Cited by 11 (3 self)
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We prove the optimal convergence estimate for firstorder interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress approach explicitly constructs rational functions, the Sibson approach uses Voronoi diagrams on the vertices of the polygon to define the functions, and the Harmonic approach defines the functions as the solution of a PDE. We show that given certain conditions on the geometry of the polygon, each of these constructions can obtain the optimal convergence estimate. In particular, we show that the wellknown maximum interior angle condition required for interpolants over triangles is still required for Wachspress functions but not for Sibson functions.
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.
Optimizing voronoi diagrams for polygonal finite element computations
 In Proceedings of the 19th International Meshing Roundtable
, 2010
"... Summary. We present a 2D mesh improvement technique that optimizes Voronoi diagrams for their use in polygonal finite element computations. Starting from a centroidal Voronoi tessellation of the simulation domain we optimize the mesh by minimizing a carefully designed energy functional that effectiv ..."
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Cited by 8 (0 self)
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Summary. We present a 2D mesh improvement technique that optimizes Voronoi diagrams for their use in polygonal finite element computations. Starting from a centroidal Voronoi tessellation of the simulation domain we optimize the mesh by minimizing a carefully designed energy functional that effectively removes the major reason for numerical instabilities—short edges in the Voronoi diagram. We evaluate our method on a 2D Poisson problem and demonstrate that our simple but effective optimization achieves a significant improvement of the stiffness matrix condition number. 1
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