A Massively Parallel Hybrid Finite Volume/Finite Element Scheme for Computational Fluid Dynamics

Abstract

In this paper, we propose a novel family of semi-implicit hybrid finite volume/finite element schemes for computational fluid dynamics (CFD), in particular for the approximate solution of the incompressible and compressible Navier-Stokes equations, as well as for the shallow water equations on staggered unstructured meshes in two and three space dimensions. The key features of the method are the use of an edge-based/face-based staggered dual mesh for the discretization of the nonlinear convective terms at the aid of explicit high resolution Godunov-type finite volume schemes, while pressure terms are discretized implicitly using classical continuous Lagrange finite elements on the primal simplex mesh. The resulting pressure system is symmetric positive definite and can thus be very efficiently solved at the aid of classical Krylov subspace methods, such as a matrix-free conjugate gradient method. For the compressible Navier-Stokes equations, the schemes are by construction asymptotic preserving in the low Mach number limit of the equations, hence a consistent hybrid FV/FE method for the incompressible equations is retrieved. All parts of the algorithm can be efficiently parallelized, i.e., the explicit finite volume step as well as the matrix-vector product in the implicit pressure solver. Concerning parallel implementation, we employ the Message-Passing Interface (MPI) standard in combination with spatial domain decomposition based on the free software package METIS. To show the versatility of the proposed schemes, we present a wide range of applications, starting from environmental and geophysical flows, such as dambreak problems and natural convection, over direct numerical simulations of turbulent incompressible flows to high Mach number compressible flows with shock waves. An excellent agreement with exact analytical, numerical or experimental reference solutions is achieved in all cases. Most of the simulations are run with millions of degrees of freedom on thousands of CPU cores. We show strong scaling results for the hybrid FV/FE scheme applied to the 3D incompressible Navier-Stokes equations, using millions of degrees of freedom and up to 4096 CPU cores. The largest simulation shown in this paper is the well-known 3D Taylor-Green vortex benchmark run on 671 million tetrahedral elements on 32,768 CPU cores, showing clearly the suitability of the presented algorithm for the solution of large CFD problems on modern massively parallel distributed memory supercomputers.

Staggered Semi-Implicit Hybrid Finite Volume/Finite Element Schemes for Turbulent and Non-Newtonian Flows

“…To solve the incompressible RANS equations in combination with the k − ε turbulence model, we extend the family of hybrid finite volume/finite element methods described in [27][28][29][30][31][32]. This methodology relies on a specific combination of explicit and implicit FV and FE methods to solve the subsystems obtained from the flux splitting introduced in the previous section.…”

“…Throughout this section, capital letters refer to the discrete variables, and the super-index * denotes intermediate approximations, that is, if the state vector is defined as w = (w v , w k , w ε ) T = (ρu, ρv, ρw, ρk, ρε) then, W n is the discrete approximation of w(x, t n ) = (ρu(x, t n ), ρv(x, t n ), ρw(x, t n ), ρk(x, t n ), ρε(x, t n )) T , and W * and W * * are the intermediate approximations. For the sake of simplicity, the hybrid FV/FE method is in the following only presented in two space dimensions, but it has already been implemented as well for the unstructured three-dimensional case, see [27,28,30,31].…”

“…Throughout this paper the international system of units (SI) is employed. The governing equations are presented for the most general case in three space dimensions, and the unstructured hybrid FV/FE method has also been implemented in 2D and 3D, see [27,28,30,31]. However, the numerical test problems shown later are restricted to the two-dimensional case, i.e., assuming planar problems with ∂/∂z = 0.…”

“…To the best knowledge of the authors, this is the first time that a semi-implicit hybrid finite volume/finite element scheme is proposed on staggered Cartesian meshes, since previous work on hybrid FV/FE schemes was focused on unstructured simplex meshes in two and three space dimensions, see [27,28,[30][31][32]. Here, we propose to use a staggered Cartesian mesh similar to the one used in [8,9,12,25,88].…”

“…A careful discretization of the governing PDE system is therefore needed, which is able to deal with both low and high Reynolds number flows. In this paper, we aim at extending the hybrid FV/FE methodology presented in the series of papers [27][28][29][30][31][32], to the case of turbulent flows and non-Newtonian power-law fluids with yield stress.…”