Numerical Homogenization of the Acoustic Wave Equations with a Continuum of Scales

In this paper, we consider numerical homogenization of acoustic wave equations with heterogeneous coefficients, namely, when the bulk modulus and the density of the medium are only bounded. We show that under a Cordes type condition the second order derivatives of the solution with respect to harmonic coordinates are $L^2$ (instead $H^{-1}$ with respect to Euclidean coordinates) and the solution itself is in $L^{\infty}(0,T,H^2(\Omega))$ (instead of $L^{\infty}(0,T,H^1(\Omega))$ with respect to Euclidean coordinates). Then, we propose an implicit time stepping method to solve the resulted linear system on coarse spatial scales, and present error estimates of the method. It follows that by pre-computing the associated harmonic coordinates, it is possible to numerically homogenize the wave equation without assumptions of scale separation or ergodicity.Comment: 27 pages, 4 figures, Submitte

Tutte Embeddings of Tetrahedral Meshes

Tutte's embedding theorem states that every 3-connected graph without a $K_5$ or $K_{3,3}$ minor (i.e. a planar graph) is embedded in the plane if the outer face is in convex position and the interior vertices are convex combinations of their neighbors. We show that this result extends to simply connected tetrahedral meshes in a natural way: for the tetrahedral mesh to be embedded if the outer polyhedron is in convex position and the interior vertices are convex combination of their neighbors it is sufficient (but not necessary) that the graph of the tetrahedral mesh contains no $K_6$ and no $K_{3,3,1}$, and all triangles incident on three boundary vertices are boundary triangles

Bijective Mappings Of Meshes With Boundary And The Degree In Mesh Processing

This paper introduces three sets of sufficient conditions, for generating bijective simplicial mappings of manifold meshes. A necessary condition for a simplicial mapping of a mesh to be injective is that it either maintains the orientation of all elements or flips all the elements. However, these conditions are known to be insufficient for injectivity of a simplicial map. In this paper we provide additional simple conditions that, together with the above mentioned necessary conditions guarantee injectivity of the simplicial map. The first set of conditions generalizes classical global inversion theorems to the mesh (piecewise-linear) case. That is, proves that in case the boundary simplicial map is bijective and the necessary condition holds then the map is injective and onto the target domain. The second set of conditions is concerned with mapping of a mesh to a polytope and replaces the (often hard) requirement of a bijective boundary map with a collection of linear constraints and guarantees that the resulting map is injective over the interior of the mesh and onto. These linear conditions provide a practical tool for optimizing a map of the mesh onto a given polytope while allowing the boundary map to adjust freely and keeping the injectivity property in the interior of the mesh. The third set of conditions adds to the second set the requirement that the boundary maps are orientation preserving as-well (with a proper definition of boundary map orientation). This set of conditions guarantees that the map is injective on the boundary of the mesh as-well as its interior. Several experiments using the sufficient conditions are shown for mapping triangular meshes. A secondary goal of this paper is to advocate and develop the tool of degree in the context of mesh processing

Multilevel Solvers for Unstructured Surface Meshes

Parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing. Such parameterization approaches give rise to large and exceedingly ill-conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner

Analysis of and workarounds for element reversal for a finite element-based algorithm for warping triangular and tetrahedral meshes

We consider an algorithm called FEMWARP for warping triangular and tetrahedral finite element meshes that computes the warping using the finite element method itself. The algorithm takes as input a two- or three-dimensional domain defined by a boundary mesh (segments in one dimension or triangles in two dimensions) that has a volume mesh (triangles in two dimensions or tetrahedra in three dimensions) in its interior. It also takes as input a prescribed movement of the boundary mesh. It computes as output updated positions of the vertices of the volume mesh. The first step of the algorithm is to determine from the initial mesh a set of local weights for each interior vertex that describes each interior vertex in terms of the positions of its neighbors. These weights are computed using a finite element stiffness matrix. After a boundary transformation is applied, a linear system of equations based upon the weights is solved to determine the final positions of the interior vertices. The FEMWARP algorithm has been considered in the previous literature (e.g., in a 2001 paper by Baker). FEMWARP has been succesful in computing deformed meshes for certain applications. However, sometimes FEMWARP reverses elements; this is our main concern in this paper. We analyze the causes for this undesirable behavior and propose several techniques to make the method more robust against reversals. The most successful of the proposed methods includes combining FEMWARP with an optimization-based untangler.Comment: Revision of earlier version of paper. Submitted for publication in BIT Numerical Mathematics on 27 April 2010. Accepted for publication on 7 September 2010. Published online on 9 October 2010. The final publication is available at http://www.springerlink.co

A One-step Image Retargeing Algorithm Based on Conformal Energy

The image retargeting problem is to find a proper mapping to resize an image to one with a prescribed aspect ratio, which is quite popular these days. In this paper, we propose an efficient and orientation-preserving one-step image retargeting algorithm based on minimizing the harmonic energy, which can well preserve the regions of interest (ROIs) and line structures in the image. We also give some mathematical proofs in the paper to ensure the well-posedness and accuracy of our algorithm.Comment: 24 pages, 10 figure