The Diamond Laplace for Polygonal and Polyhedral Meshes

Abstract

We introduce a construction for discrete gradient operators that can be directly applied to arbitrary polygonal surface as well as polyhedral volume meshes. The main idea is to associate the gradient of functions defined at vertices of the mesh with diamonds: the region spanned by a dual edge together with its corresponding primal element — an edge for surface meshes and a face for volumetric meshes. We call the operator resulting from taking the divergence of the gradient Diamond Laplacian. Additional vertices used for the construction are represented as affine combinations of the original vertices, so that the Laplacian operator maps from values at vertices to values at vertices, as is common in geometry processing applications. The construction is local, exactly the same for all types of meshes, and results in a symmetric negative definite operator with linear precision. We show that the accuracy of the Diamond Laplacian is similar or better compared to other discretizations. The greater versatility and generally good behavior come at the expense of an increase in the number of non‐zero coefficients that depends on the degree of the mesh elements.

A Survey on Discrete Laplacians for General Polygonal Meshes

“…Therefore, we analyze a selection of values besides the recommended choices of the authors. Concerning the placement of the virtual vertices, we follow the recommendations of Bunge et al [BHKB20, BBA21] and use the squared triangle area and squared tetrahedra volume minimizer, respectively. For the harmonic shape functions [MKB*08], we noticed that the number of chosen kernels and control points strongly affects the results on our chosen test meshes.…”

“…The virtual cell vertices x l and x r of the adjacent polyhedra form the respective tips of the diamond. The figure was taken from [BBA21].…”

“…The first steps are analogous to the surface case, meaning that all faces f ∈ ℱ of a given polyhedron c ∈ 𝒞 are refined into triangles with virtual vertices placed at the point that minimizes the sum of squared triangle areas (see Equation (80)). To span the virtual tetrahedra, Bunge et al [BBA21] introduce an additional vertex x c inside of each cell, which is, similar to the surface case, the affine combination of the original cells' vertex positions …”

“…However, the basic derivation of the 2D Laplace operator with FV assumes a Delaunay triangle mesh, more specifically orthogonal dual and primal edges, in order to prevent negative coefficients. To avoid this restriction, Bunge et al [BBA21] used a special polygonal variant of the FV, called Discrete Duality Finite Volume (DDFV) [Her00, Her09, DO05, CH11] and combined this technique with their previously described virtual triangle refinement to define a gradient, divergence and Laplacian operator for polygonal and polyhedral meshes.…”

“…Bunge et al [BBA21] adapt the previously defined DDFV gradient operator in two ways to extend it to polygon surface meshes embedded in 3D: First, they reduce the systems' dimension by expressing the values at the dual vertices as affine combinations of the original primal vertices with the help of the previously introduced face prolongation matrix P (see Equation (75)). As in Section 5.2, the position of the dual vertices is the minimizer of the sum of squared triangle areas of the refined face. Second, they define an intrinsic gradient operator associated with the (not necessarily planar) diamond cells, allowing them to directly apply the formula defined in Equation (95).…”