Simplex Space-Time Meshes in Compressible Flow Simulations

TL;DR

Simplex space-time meshes enable simultaneous space-time discretization of compressible flows, extending unstructured meshes to temporal evolution and supporting topology-changing domains and local temporal refinement. The authors couple a SUPG-stabilized finite-element formulation for the compressible Navier–Stokes equations with a pressure-primitive variable formulation and an invariant simplex-based metric to produce a stable, permutation-invariant stabilization. They develop and compare flat space-time, simplex space-time, and unstructured space-time discretizations, deriving the weak form and the SUPG operator, and validating through cases including supersonic flow over a flat plate, a pressure pulse, valve dynamics, and blow-by past piston rings; results show comparable temporal accuracy to conventional FST with notable gains from temporal refinement and 4D adaptivity. The work demonstrates practical benefits for simulations of flows with evolving topology and strong transient behavior, offering scalable, accurate methods for engineering problems with moving boundaries and topology changes.

Abstract

Employing simplex space-time meshes enlarges the scope of compressible flow simulations. The simultaneous discretization of space and time with simplex elements extends the flexibility of unstructured meshes from space to time. In this work, we adopt a finite element formulation for compressible flows to simplex space-time meshes. The method obtained allows, e.g., flow simulations on spatial domains that change topology with time. We demonstrate this with the two-dimensional simulation of compressible flow in a valve that fully closes and opens again. Furthermore, simplex space-time meshes facilitate local temporal refinement. A three-dimensional transient simulation of blow-by past piston rings is run in parallel on 120 cores. The timings point out savings of computation time gained from local temporal refinement in space-time meshes.

Figures (14)

• Figure 1: Examples of Simplex Finite Elements for $d=1,2,3,$ and $4$ previously published in karyofylli2018simplex.
• Figure 2: Mappings between $\mathbb{P}_1$ reference element $\bigtriangleup_{*}$, physical element $\bigtriangleup_{k}$ and regular element $\bigtriangleup_{r}$ in the two-dimensional case.
• Figure 3: Space-time discretization methods.
• Figure 4: Supersonic flow over flat plate. Influence of the viscous terms in the boundary integral $\int_{P_n} \mathbf{W}^h \cdot {\boldsymbol{\mathit{h}}}^h \, dP$ along the outflow boundary on the velocity vectors next to the wall.
• Figure 5: Supersonic flow over a flat plate. Flow field solutions and pressure coefficient.