Manifold Construction Over Polyhedral Mesh

We present a smooth parametric surface construction method over polyhe-dral mesh with arbitrary topology based on manifold construction theory.The surface is automatically generated with any required smoothness,and it has an explicit form.As prior methods that build manifolds from meshes need some preprocess to get poly-hedral meshes with specialtypes of connectivity,such as quad mesh and triangle mesh,the preprocess will result in more charts.By a skillful use of a kind of bivariate spline function which defines on arbitrary shape of 2D polygon,we introduce an approach that directly works on the input mesh without such preprocess.Fornon-closedpolyhe-dral mesh,we apply a global parameterization and directly divide it into several charts.As for closed polyhedral mesh,we propose to segment the mesh into a sequence of quadrilateral patches without any overlaps.As each patch is an non-closed polyhedral mesh,the non-closed surface construction method can be applied.And all the patches are smoothly stitched with a special process on the boundary charts which define on the boundary vertex of each patch.Thus,the final constructed surface can also achieve any required smoothness.

The Bulk-Surface Virtual Element Method for Reaction-Diffusion PDEs:Analysis and Applications

Bulk-surface partial differential equations(BS-PDEs)are prevalent in manyapplications such as cellular,developmental and plant biology as well as in engineeringand material sciences.Novel numerical methods for BS-PDEs in three space dimensions(3D)are sparse.In this work,we present a bulk-surface virtual elementmethod(BS-VEM)for bulk-surface reaction-diffusion systems,a form of semilinearparabolic BS-PDEs in 3D.Unlike previous studies in two space dimensions(2D),the3D bulk is approximated with general polyhedra,whose outer faces constitute a flatpolygonal approximation of the surface.For this reason,the method is restricted tothe lowest order case where the geometric error is not dominant.The BS-VEM guaranteesall the advantages of polyhedral methods such as easy mesh generation andfast matrix assembly on general geometries.Such advantages are much more relevantthan in 2D.Despite allowing for general polyhedra,general nonlinear reaction kineticsand general surface curvature,the method only relies on nodal values without needingadditional evaluations usually associated with the quadrature of general reactionkinetics.This latter is particularly costly in 3D.The BS-VEM as implemented in thisstudy retains optimal convergence of second order in space.