Multi-Model Multi-Domain Computational Methods: Steady Euler

Introduction

The Euler equations describe the conservation of mass, momentum, and energy in an inviscid fluid. A key parameter in an Euler description is the Mach number, which ranges from zero in an incompressible fluid through unity for transonic flows into the supersonic regime. The Euler model permits entropy generation and rotational flow, and is hence more general than the full potential model. In particular, surfaces of discontinuity known as "shocks" are permitted, which pose challenges for finite discretization schemes that make use of local Taylor expansions or polynomial approximations, and sophisticated conservative control volume schemes with flux limiters may be required in such problems. Boundary conditions at solid surfaces for Euler flow are free-slip, thus boundary layers do not arise, and Euler flows can be represented on grids with cells of near unit aspect ratio.

Research Overview

We have ported to large-scale parallelism two different "legacy" Euler codes, written in FORTRAN by recognized authorities in external aerodynamics and hydrodynamics. One code uses a mapped structured grid based on a conforming hexahedral mesh, and the other uses an unstructured grid based on a tetrahedralization of space. We use both codes to compute the flow past an M6 wing, a standard aerodynamics test case. The unstructured code, FUN3D, is a research code that is commercially used in the aircraft and automotive industries, e.g., as in this visualization of streamlines over a slotted wing by Kyle Anderson of NASA.

We employ the Newton-Krylov-Schwarz domain decomposition method with pseudo-transient continuation, using one subdomain per processor in the parallelization. Load-balanced partitioning of the structured grid is a trivial matter of partitioning the Cartesian index space. For the unstructured grid, we use the MeTiS partitioner, so far without node or edge weights. Some algorithmic comparisons are shown in these graphs.

We have employed up to 2.8 million grid points on up to 128 processors of state of the art parallel computers such as the IBM SP and the Cray T3E. Parallel efficiencies in excess of 75% are available together with computation rates on a par with the best sequential sparse matrix implementations.

Papers