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. We also briefly discuss strategies to handle tetrahedral meshes which are not 4-colorable.
We define a weak compatibility condition for the Newest Vertex Bisection algorithm on simplex grids of any dimension and show that using this condition the iterative algorithm terminates successfully. Additionally we provide an O(n) algorithm that renumbers any simplex grid to fulfil this condition. Furthermore we conduct experiments to estimate the distance to the standard compatibility and also the geometric quality of the produced meshes.
Thinking of the flow through biological or technical valves, there is a variety of applications in which the topology of a fluid domain changes over time. This topology change is characteristic for the physical behaviour, but poses a particular challenge in computer simulations. A way to overcome this challenge is to consider the space-time extent of the application as a contiguous computational domain. In this work, we obtain a boundary conforming discretization of the space-time domain with four-dimensional simplex elements (pentatopes). To facilitate the construction of pentatope meshes for…
We devise a generalization of tree approximation that generates conforming meshes, i.e., meshes with a particular structure like edge-to-edge triangulations. A key feature of this generalization is that the choices of the cells to be subdivided are affected by that particular structure. As main result, we prove near best approximation with respect to conforming meshes, independent of constants like the completion constant for newest-vertex bisection. Numerical experiments complement the theoretical results and indicate better approximation properties than previous approaches.
On the involution preserving properties of semi-discrete Active Flux methods on simplex
Institute of Mathematics, University of Zürich（苏黎世大学数学研究所）；University of Trento（特伦托大学）；University of Science and Technology, Shenzhen（深圳理工大学）；INRIA, Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251（法国国家信息与自动化研究所、波尔多大学、法国国家科学研究中心、波尔多理工学院、IMB）
A simple and general framework for the construction of exactly div-curl-grad compatible discontinuous Galerkin finite element schemes on unstructured simplex meshes
This paper proposes some efficient and accurate adaptive two-grid (ATG) finite element algorithms for linear and nonlinear partial differential equations (PDEs). The main idea of these algorithms is to utilize the solutions on the $k$-th level adaptive meshes to find the solutions on the $(k+1)$-th level adaptive meshes which are constructed by performing adaptive element bisections on the $k$-th level adaptive meshes. These algorithms transform non-symmetric positive definite (non-SPD) PDEs (resp., nonlinear PDEs) into symmetric positive definite (SPD) PDEs (resp., linear PDEs). The proposed …
Consejo Nacional de Investigaciones Científicas y Técnicas（国家科学技术研究委员会）；Universidad Nacional del Litoral（利托拉尔国立大学）；IMAL - Güemes 3450（IMAL - 盖勒斯街3450号）
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）
We present a novel second-order semi-implicit hybrid finite volume / finite element (FV/FE) scheme for the numerical solution of the incompressible and weakly compressible Navier-Stokes equations on moving unstructured meshes using an Arbitrary-Lagrangian-Eulerian (ALE) formulation. The scheme is based on a suitable splitting of the governing PDE into subsystems and employs staggered grids, where the pressure is defined on the primal simplex mesh, while the velocity and the remaining flow quantities are defined on an edge-based staggered dual mesh. The key idea of the scheme is to discretize t…
A conforming triangular mixed element recently proposed by Hu and Zhang for linear elasticity is extended by rearranging the global degrees of freedom. More precisely, adaptive meshes $\mathcal{T}_1$, $\cdots$, $\mathcal{T}_N$ which are successively refined from an initial mesh $\mathcal{T}_0$ through a newest vertex bisection strategy, admit a crucial hierarchical structure, namely, a newly added vertex $\boldsymbol{x}$ of the mesh $\mathcal{T}_\ell$ is the midpoint of an edge $e$ of the coarse mesh $\mathcal{T}_{\ell-1}$. Such a hierarchical structure is explored to partially relax the $C^0$…
In this paper a new hybrid semi-implicit finite volume / finite element (FV/FE) scheme is presented for the numerical solution of the compressible Euler and Navier-Stokes equations at all Mach numbers on unstructured staggered meshes in two and three space dimensions. The chosen grid arrangement consists of a primal simplex mesh composed of triangles or tetrahedra, and an edge-based / face-based staggered dual mesh. The governing equations are discretized in conservation form. The nonlinear convective terms of the equations, as well as the viscous stress tensor and the heat flux, are discretiz…
This paper discusses the practical development of space-time boundary element methods for the wave equation in three spatial dimensions. The employed trial spaces stem from simplex meshes of the lateral boundary of the space-time cylinder. This approach conforms genuinely to the distinguished structure of the solution operators of the wave equation, so-called retarded potentials. Since the numerical evaluation of the arising integrals is intricate, the bulk of this work is constituted by ideas about quadrature techniques for retarded layer potentials and associated energetic bilinear forms. Fi…
A space-time framework is applied to simulate dense granular flow. Two different numerical experiments are performed: a column collapse and a dam break on an inclined plane. The experiments are modeled as two-phase flows. The dense granular material is represented by a constitutive model, the $μ$($I$)-rheology, that is based on the Coulomb's friction law, such that the normal stress applied by the pressure is related to the tangential stress. The model represents a complex shear thinning viscoplastic material behavior. The interface between the dense granular material and the surrounding light…
