Four‐dimensional elastically deformed simplex space‐time meshes for domains with time‐variant topology

Abstract

Considering the flow through biological or engineered valves as an example, 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 behavior, but poses a particular challenge in computer simulations. A way to overcome this challenge is to consider the application‐specific space‐time geometry 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 complex geometries, the widely used elastic mesh update method is extended to four‐dimensional meshes. In the resulting workflow, the topology change is elegantly included in the pentatope mesh and does not require any additional treatment during the simulation. The potential of simplex space‐time meshes for domains with time‐variant topology is demonstrated in a valve simulation, and a flow simulation inspired by a clamped artery.

A conforming sliding mesh technique for an embedded‐hybridized discontinuous Galerkin discretization for fluid‐rigid body interaction

“…We will assume that the number of spatial elements $$$ {K}^n $$$ is constant for all $$$ n=0,\dots, N $$$ and we will denote the (global) identifiers of the three vertices of an element $$$ {K}^n $$$ by $$$ {p}_i^n $$$ with $$$ i\in \left\{1,\dots, {N}_v\right\} $$$ with $$$ {N}_v $$$ the total number of vertices in the spatial mesh. For more on four‐dimensional space‐time simplicial meshes, we refer to References 61‐63.…”

Time‐continuous and time‐discontinuous space‐time finite elements for advection‐diffusion problems

“…[4][5][6][7] In particular, simplex space-time finite elements 8 can provide a boundary conforming space-time mesh for spatial domains that change topology over time. 9 To benefit from these advantages, space-time finite elements have been used to perform simulations in various fields of computational fluid dynamics (CFD). Recent examples of simplex space-time simulations include the computation of complex fluid flows in production engineering applications 10,11 and the computation of dense granular flows.…”

“…[13][14][15] Note that the solution of transient three-dimensional problems with space-time finite elements requires four-dimensional meshes. Recent advances in generation, 9,16,17 adaptation, 18 and numerical handling 19,20 of four-dimensional meshes mark the state-of-the-art in this active research field.…”

“…The simulation is repeated on finer meshes with 2 6 = 64 elements in space and time (medium) and 2 9 = 512 elements (fine), respectively. Figure 5 shows the differences to the analytical solution at the nodes of the two finer meshes.…”

Surface and hypersurface meshing techniques for space-time finite element methods

“…The prism extrusion method of Behr was expanded upon by von Danwitz et al [9]. In particular, they extend the method to accommodate time-variant topology through what they call a 4D-elastic mesh update method.…”