Most existing cell-centered finite volume schemes need to introduce auxiliary unknowns in order to maintain the second-order accuracy when the mesh is distorted or the problem is discontinuous,so interpolation algorit...Most existing cell-centered finite volume schemes need to introduce auxiliary unknowns in order to maintain the second-order accuracy when the mesh is distorted or the problem is discontinuous,so interpolation algorithms of auxiliary unknowns are required.Interpolation algorithms are not only difficult to construct,but also bring extra computation.In this paper,an interpolation-free cell-centered finite volume scheme is proposed for the heterogeneous and anisotropic convectiondiffusion problems on arbitrary polyhedral meshes.We propose a new interpolationfree discretization method for diffusion term,and two new second-order upwind algorithms for convection term.Most interestingly,the scheme can be adapted to any mesh topology and can handle any discontinuity strictly.Numerical experiments show that this new scheme is robust,possesses a small stencil,and has approximately secondorder accuracy for both diffusion-dominated and convection-dominated problems.展开更多
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...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.展开更多
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...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.展开更多
基金partially supported by the National Natural Science Foundation of China(Nos.11871009,12271055,12171048)the foundation of CAEP(CX20210044)the Foundation of LCP.
文摘Most existing cell-centered finite volume schemes need to introduce auxiliary unknowns in order to maintain the second-order accuracy when the mesh is distorted or the problem is discontinuous,so interpolation algorithms of auxiliary unknowns are required.Interpolation algorithms are not only difficult to construct,but also bring extra computation.In this paper,an interpolation-free cell-centered finite volume scheme is proposed for the heterogeneous and anisotropic convectiondiffusion problems on arbitrary polyhedral meshes.We propose a new interpolationfree discretization method for diffusion term,and two new second-order upwind algorithms for convection term.Most interestingly,the scheme can be adapted to any mesh topology and can handle any discontinuity strictly.Numerical experiments show that this new scheme is robust,possesses a small stencil,and has approximately secondorder accuracy for both diffusion-dominated and convection-dominated problems.
基金supported by the NSF of China(Nos.61672482,11626253),‘100 Talents Project’of Chinese Academy of Sciences.
文摘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.
基金Regione Puglia(Italy)through the research programme REFIN-Research for Innovation(protocol code 901D2CAA,project No.UNISAL026)MF acknowledges support from the Italian National Institute of High Mathematics(INdAM)through the INdAM-GNCS Project no.CUP E55F22000270001+3 种基金the Global Challenges Research Fund through the Engineering and Physical Sciences Research Council grant number EP/T00410X/1:UK-Africa Postgraduate Advanced Study Institute in Mathematical Sciences,the Health Foundation(1902431)the NIHR(NIHR133761)and by the Discovery Grant awarded by Canadian Natural Sciences and Engineering Research Council(2023-2028)AM acknowledges support from the Royal Society Wolfson Research Merit Award funded generously by the Wolfson Foundation(2016-2021)AM is a Distinguished Visiting Scholar to the Department of Mathematics,University of Johannesburg,South Africa,and the University of Pretoria in South Africa.IS and MF are members of the INdAM-GNCS activity group.The work of IS is supported by the PRIN 2020 research project(no.2020F3NCPX)”Mathematics for Industry 4.0”,and from the”National Centre for High Performance Computing,Big Data and Quantum Computing”funded by European Union-NextGenerationEU,PNRR project code CN00000013,CUP F83C22000740001.
文摘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.
