A finite element solution of an ion channel dielectric continuum model such as Poisson-Boltzmann equation (PBE) and a system of Poisson-Nernst-Planck equations (PNP) requires tetrahedral meshes for an ion channel protein region, a membrane region, and an ionic solvent region as well as an interface fitted irregular tetrahedral mesh of a simulation box domain. However, generating these meshes is very difficult and highly technical due to the related three regions having very complex geometrical shapes. Currently, an ion channel mesh generation software package developed in Lu's research group is one available in the public domain. To significantly improve its mesh quality and computer performance, in this paper, new numerical schemes for generating membrane and solvent meshes are presented and implemented in Python, resulting in a new ion channel mesh generation software package. Numerical results are then reported to demonstrate the efficiency of the new numerical schemes and the quality of meshes generated by the new package for ion channel proteins with ion channel pores having different geometric complexities.
A family of quadratic finite volume method (FVM) schemes are constructed and analyzed over tetrahedral meshes. In order to prove stability and error estimate, we propose the minimum V-angle condition on tetrahedral meshes, and the surface and volume orthogonal conditions on dual meshes. Through the element analysis technique, the local stability is equivalent to a positive definiteness of a $9\times9$ element matrix, which is difficult to analyze directly or even numerically. With the help of the surface orthogonal condition and congruent transformation, this element matrix is reduced into a b…
Let $Ω$ be a Lipschitz polyhedral (can be nonconvex) domain in $\mathbb{R}^{3}$, and $V_{h}$ denotes the finite element space of continuous piecewise linear polynomials. On non-obtuse quasi-uniform tetrahedral meshes, we prove that the finite element projection $R_{h}u$ of $u \in H^{1}(Ω) \cap C(\overlineΩ)$ (with $R_{h} u$ interpolating $u$ at the boundary nodes) satisfies \begin{align*} \Vert R_{h} u\Vert_{L^{\infty}(Ω)} \leq C \vert \log h \vert \Vert u\Vert_{L^{\infty}(Ω)}. \end{align*} If we further assume $u \in W^{1,\infty}(Ω)$, then \begin{align*} \Vert R_{h} u\Vert_{W^{1, \infty}(Ω)} …
Adaptive mesh refinement is a key component of efficient unstructured space-time finite element methods. Underlying any adaptive mesh refinement scheme is, of course, a method for local refinement of simplices. However, simplex bisection algorithms in dimension greater than three have strict mesh preconditions which can be hard to satisfy. We prove that certain four-dimensional simplex space-time meshes can be handled with a relaxed precondition. Namely, we prove that if a tetrahedral mesh is 4-colorable, then we can produce a 4D simplex mesh which always satisfies the bisection precondition. …
Taming Slivers: A Robust TFEM Framework for Reliable Computations on Degenerate Tetrahedral Meshes
UCLouvain, Institute of Mechanics, Materials and Civil Engineering (iMMC)（瓦隆天主教大学，力学、材料与土木工程研究所）；Charles University, Department of Mathematics and Physics（查理大学，数学与物理学院）
In recent papers the author introduced a simple alternative to isoparametric finite elements of the n-simplex type, to enhance the accuracy of approximations of second-order boundary value problems with Dirichlet conditions, posed in smooth curved domains. This technique is based upon trial-functions consisting of piecewise polynomials defined on straight-edged triangular or tetrahedral meshes, interpolating the Dirichlet boundary conditions at points of the true boundary. In contrast the test-functions are defined upon the standard degrees of freedom associated with the underlying method for …
In this paper, we use the first-order virtual element method (VEM) to investigate the effect of shape quality of polyhedra in the estimation of the critical time step for explicit three-dimensional elastodynamic finite element (FE) simulations. Low-quality finite elements are common when meshing realistic complex components, and while tetrahedral meshing technology is generally robust, meshing algorithms cannot guarantee high-quality meshes for arbitrary geometries or for non-water-tight computer-aided design models. For reliable simulations on such meshes, we consider FE meshes with tetrahedr…
A High-Order Surface Finite Element Method Based on Intersections with Background Tetrahedral Meshes
Tsinghua University（清华大学）；Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心，清华大学）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京雁栖湖应用数学研究院）
Two important classes of three-dimensional elements in computational meshes are hexahedra and tetrahedra. While several efficient methods exist that convert a hexahedral element to a tetrahedral elements, the existing algorithm for tetrahedralization of a hexahedral complex is the marching tetrahedron algorithm which limits pre-selection of face divisions. We generalize a procedure for tetrahedralizing triangular prisms to tetrahedralizing cubes, and combine it with certain heuristics to design an algorithm that can triangulate any hexahedra.
A staggered space-time discontinuous Galerkin method for the three-dimensional incompressible Navier-Stokes equations on unstructured tetrahedral meshes
University of Houston（休斯顿大学）；Moscow State University（莫斯科国立大学）；RWTH-Aachen University（亚琛工业大学）；Institute of Computational Mathematics and Scientific/Engineering Computing, NCMIS, AMSS, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所，国家数学与交叉研究中心）
In this paper, we develop two fully nonconforming (both H(grad curl)-nonconforming and H(curl)-nonconforming) finite elements on cubical meshes which can fit into the Stokes complex. The newly proposed elements have 24 and 36 degrees of freedom, respectively. Different from the fully H(grad curl)-nonconforming tetrahedral finite elements in [9], the elements in this paper lead to a robust finite element method to solve the singularly perturbed quad-curl problem. To confirm this, we prove the optimal convergence of order $O(h)$ for a fixed parameter $ε$ and the uniform convergence of order $O(h…
A finite element elasticity complex on tetrahedral meshes is devised. The $H^1$ conforming finite element is the smooth finite element developed by Neilan for the velocity field in a discrete Stokes complex. The symmetric div-conforming finite element is the Hu-Zhang element for stress tensors. The construction of an $H(\textrm{inc})$-conforming finite element for symmetric tensors is the main focus of this paper. The key tools of the construction are the decomposition of polynomial tensor spaces and the characterization of the trace of the $\textrm{inc}$ operator. The polynomial elasticity co…
This paper introduces a novel method for the efficient and accurate computation of volume fractions on unstructured polyhedral meshes, where the phase boundary is an orientable hypersurface, implicitly given as the iso-contour of a sufficiently smooth level-set function. Locally, i.e. in each mesh cell, we compute a principal coordinate system in which the hypersurface can be approximated as the graph of an osculating paraboloid. A recursive application of the \textsc{Gaussian} divergence theorem then allows to analytically transform the volume integrals to curve integrals associated to the po…
A simple but successful strategy for building a discrete diffusion operator in finite volume schemes of industrial use is to correct the standard two-point flux approximation with a term accounting for the local mesh non-orthogonality. Practical experience with a variety of different mesh typologies, including non-orthogonal tetrahedral, hexahedral and polyhedral meshes, has shown that this discrete diffusion operator is accurate and robust whenever the mesh is not too distorted and sufficiently regular. In this work, we show that this approach can be interpreted as equivalent to introducing a…
Uniform Stability of Scott-Vogelius Elements on Three-Dimensional Freudenthal Meshes in Degrees Four and Five: Resolving the Farrell-Mitchell-Scott Conjecture
Nanophotonics and Biophotonics Key Laboratory of Jilin Province, School of Physics, Changchun University of Science and Technology（吉林省纳米光子学与生物光子学重点实验室，长春理工大学物理学院）
We introduce an immersed high-order discontinuous Galerkin method for solving the compressible Navier-Stokes equations on non-boundary-fitted meshes. The flow equations are discretised with a mixed discontinuous Galerkin formulation and are advanced in time with an explicit time marching scheme. The discretisation meshes may contain simplicial (triangular or tetrahedral) elements of different sizes and need not be structured. On the discretisation mesh the fluid domain boundary is represented with an implicit signed distance function. The cut-elements partially covered by the solid domain are …
h-Adaptive FV Subcell Shock-Capturing for DGSEM on Heterogeneous Curvilinear Meshes
Institute of Aerodynamics and Gas Dynamics, University of Stuttgart（斯图加特大学空气动力学与气体动力学研究所）；Institute of Applied Analysis and Numerical Simulation, University of Stuttgart（斯图加特大学应用分析与数值模拟研究所）
Continuous 3D Finite Element Subgrid Basis Functions for Discontinuous Galerkin Methods on Polyhedral Meshes
De Vinci Higher Education（德文奇高等教育）；De Vinci Research Center（德文奇研究中心）；CNR IMATI（意大利国家研究委员会应用数学研究所）；Laboratoire de Mathématiques UMR 5127 CNRS, Université Savoie Mont Blanc（萨瓦省-蒙布朗大学CNRS第5127号数学实验室）；Department of Mathematics and Computer Science, University of Ferrara（费拉拉大学数学与计算机科学系）
We construct finite element Stokes complexes on tetrahedral meshes in three-dimensional space. In the lowest order case, the finite elements in the complex have 4, 18, 16, and 1 degrees of freedom, respectively. As a consequence, we obtain gradcurl-conforming finite elements and inf-sup stable Stokes pairs on tetrahedra which fit into complexes. We show that the new elements lead to convergent algorithms for solving a gradcurl model problem as well as solving the Stokes system with precise divergence-free condition. We demonstrate the validity of the algorithms by numerical experiments.
