SGP08: Eurographics Symposium on Geometry Processing
https://diglib.eg.org:443/handle/10.2312/439
2019-06-26T20:03:37ZPolyhedral Finite Elements Using Harmonic Basis Functions
https://diglib.eg.org:443/handle/10.2312/CGF.v27i5pp1521-1529
Polyhedral Finite Elements Using Harmonic Basis Functions
Martin, Sebastian; Kaufmann, Peter; Botsch, Mario; Wicke, Martin; Gross, Markus
Finite element simulations in computer graphics are typically based on tetrahedral or hexahedral elements, which enables simple and efficient implementations, but in turn requires complicated remeshing in case of topological changes or adaptive refinement. We propose a flexible finite element method for arbitrary polyhedral elements, thereby effectively avoiding the need for remeshing. Our polyhedral finite elements are based on harmonic basis functions, which satisfy all necessary conditions for FEM simulations and seamlessly generalize both linear tetrahedral and trilinear hexahedral elements. We discretize harmonic basis functions using the method of fundamental solutions, which enables their flexible computation and efficient evaluation. The versatility of our approach is demonstrated on cutting and adaptive refinement within a simulation framework for corotated linear elasticity.
2008-01-01T00:00:00ZMaximum Entropy Coordinates for Arbitrary Polytopes
https://diglib.eg.org:443/handle/10.2312/CGF.v27i5pp1513-1520
Maximum Entropy Coordinates for Arbitrary Polytopes
Hormann, K.; Sukumar, N.
Barycentric coordinates can be used to express any point inside a triangle as a unique convex combination of the triangle s vertices, and they provide a convenient way to linearly interpolate data that is given at the vertices of a triangle. In recent years, the ideas of barycentric coordinates and barycentric interpolation have been extended to arbitrary polygons in the plane and general polytopes in higher dimensions, which in turn has led to novel solutions in applications like mesh parameterization, image warping, and mesh deformation. In this paper we introduce a new generalization of barycentric coordinates that stems from the maximum entropy principle. The coordinates are guaranteed to be positive inside any planar polygon, can be evaluated efficiently by solving a convex optimization problem with Newton s method, and experimental evidence indicates that they are smooth inside the domain. Moreover, the construction of these coordinates can be extended to arbitrary polyhedra and higher-dimensional polytopes.
2008-01-01T00:00:00ZPointwise radial minimization: Hermite interpolation on arbitrary domains
https://diglib.eg.org:443/handle/10.2312/CGF.v27i5pp1505-1512
Pointwise radial minimization: Hermite interpolation on arbitrary domains
Floater, M. S.; Schulz, C.
In this paper we propose a new kind of Hermite interpolation on arbitrary domains, matching derivative data of arbitrary order on the boundary. The basic idea stems from an interpretation of mean value interpolation as the pointwise minimization of a radial energy function involving first derivatives of linear polynomials. We generalize this and minimize over derivatives of polynomials of arbitrary odd degree. We analyze the cubic case, which assumes first derivative boundary data and show that the minimization has a unique, infinitely smooth solution with cubic precision. We have not been able to prove that the solution satisfies the Hermite interpolation conditions but numerical examples strongly indicate that it does for a wide variety of planar domains and that it behaves nicely.
2008-01-01T00:00:00ZA Local/Global Approach to Mesh Parameterization
https://diglib.eg.org:443/handle/10.2312/CGF.v27i5pp1495-1504
A Local/Global Approach to Mesh Parameterization
Liu, Ligang; Zhang, Lei; Xu, Yin; Gotsman, Craig; Gortler, Steven J.
We present a novel approach to parameterize a mesh with disk topology to the plane in a shape-preserving manner. Our key contribution is a local/global algorithm, which combines a local mapping of each 3D triangle to the plane, using transformations taken from a restricted set, with a global stitch operation of all triangles, involving a sparse linear system. The local transformations can be taken from a variety of families, e.g. similarities or rotations, generating different types of parameterizations. In the first case, the parameterization tries to force each 2D triangle to be an as-similar-as-possible version of its 3D counterpart. This is shown to yield results identical to those of the LSCM algorithm. In the second case, the parameterization tries to force each 2D triangle to be an as-rigid-as-possible version of its 3D counterpart. This approach preserves shape as much as possible. It is simple, effective, and fast, due to pre-factoring of the linear system involved in the global phase. Experimental results show that our approach provides almost isometric parameterizations and obtains more shape-preserving results than other state-of-the-art approaches.We present also a more general hybrid parameterization model which provides a continuous spectrum of possibilities, controlled by a single parameter. The two cases described above lie at the two ends of the spectrum. We generalize our local/global algorithm to compute these parameterizations. The local phase may also be accelerated by parallelizing the independent computations per triangle.
2008-01-01T00:00:00Z