Compression for Smooth Shape Analysis

Most 3D shape analysis methods use triangular meshes to discretize both the shape and functions on it as piecewise linear functions. With this representation, shape analysis requires fine meshes to represent smooth shapes and geometric operators like normals, curvatures, or Laplace-Beltrami eigenfunctions at large computational and memory costs. We avoid this bottleneck with a compression technique that represents a smooth shape as subdivision surfaces and exploits the subdivision scheme to parametrize smooth functions on that shape with a few control parameters. This compression does not affect the accuracy of the Laplace-Beltrami operator and its eigenfunctions and allow us to compute shape descriptors and shape matchings at an accuracy comparable to triangular meshes but a fraction of the computational cost. Our framework can also compress surfaces represented by point clouds to do shape analysis of 3D scanning data

Parallel implementation of an optimal two level additive Schwarz preconditioner for the 3-D finite element solution of elliptic partial differential equations

This paper presents a description of the extension and parallel implementation of a new two level additive Schwarz (AS) preconditioner for the solution of 3-D elliptic partial differential equations (PDEs). This preconditioner, introduced in Bank et al. (SIAM J. Sci. Comput. 2002; 23: 1818), is based upon the use of a novel form of overlap between the subdomains which makes use of a hierarchy of meshes: with just a single layer of overlapping elements at each level of the hierarchy. The generalization considered here is based upon the restricted AS approach reported in (SIAM J. Sci. Comput. 1999; 21: 792) and the parallel implementation is an extension of work in two dimensions (Concurrency Comput. Practice Experience 2001; 13: 327)

A multilevel approach for obtaining locally optimal finite element meshes

In this paper we consider the adaptive finite element solution of a general class of variational problems using a combination of node insertion, node movement and edge swapping. The adaptive strategy that is proposed is based upon the construction of a hierarchy of locally optimal meshes starting with a coarse grid for which the location and connectivity of the nodes is optimized. This grid is then locally refined and the new mesh is optimized in the same manner. Results presented indicate that this approach is able to produce better meshes than those possible by more conventional adaptive strategies and in a relatively efficient manner

Well-Centered Triangulation

Meshes composed of well-centered simplices have nice orthogonal dual meshes (the dual Voronoi diagram). This is useful for certain numerical algorithms that prefer such primal-dual mesh pairs. We prove that well-centered meshes also have optimality properties and relationships to Delaunay and minmax angle triangulations. We present an iterative algorithm that seeks to transform a given triangulation in two or three dimensions into a well-centered one by minimizing a cost function and moving the interior vertices while keeping the mesh connectivity and boundary vertices fixed. The cost function is a direct result of a new characterization of well-centeredness in arbitrary dimensions that we present. Ours is the first optimization-based heuristic for well-centeredness, and the first one that applies in both two and three dimensions. We show the results of applying our algorithm to small and large two-dimensional meshes, some with a complex boundary, and obtain a well-centered tetrahedralization of the cube. We also show numerical evidence that our algorithm preserves gradation and that it improves the maximum and minimum angles of acute triangulations created by the best known previous method.Comment: Content has been added to experimental results section. Significant edits in introduction and in summary of current and previous results. Minor edits elsewher