Multi-Model Multi-Domain Computational Methods: Unsteady Euler and Navier-Stokes

Introduction

The unsteady Navier-Stokes equations provide the most general description of the conservation of mass, momentum, and energy in a viscous Newtonian fluid. Their solution to high resolution, in complex geometry, is the ultimate goal of computational fluid dynamics, and is a subsidiary goal of important engineering applications, such as the simulation of fluid-structure interaction and aerodynamic design. The presence of boundary layers endows Navier-Stokes problems with multiple scales, and puts a premium on the development of robust implicit time marching methods, as well as preconditioners that are sensitive to and exploit the natural anisotropies of the problem.

Research Overview

We have ported to large-scale parallelism a 3D unstructured grid NS code. A 2D illustration of the unstructured subdomains , partitioned by the recursive spectral bisection technique in TOP DOMDEC , is shown here.

We employ the Newton-Krylov-Schwarz domain decomposition method using a family of Schwarz methods to as the Variable Degree Schwarz methods (VDS) on the overlapping submeshes. In VDS, the subdomain preconditioner is constructed by using a polynomial in two matrix variables, namely the matrix in unfactored form at the current time step and another matrix in factored form from at a previous time step. The degree of the matrix polynomial in each subdomain is determined automatically so that extra preconditioning is performed only in subdomains whose associated local matrices have large condition numbers. The extra preconditioning occurs often near the body of the airfoil. Unlike the well-known elliptic theory, we observe that the convergence rate of VDS preconditioned GMRES degenerates only mildly without a coarse space for reasonably large number of subdomains.

A time series of pressure (contours) showing an impulsively started airfoil at a reference transonic Mach number of 0.84 was computed in parallel on an IBM SP2. For 3D Navier-Stokes computations, we have developed a scalable and robust solution method based on a defect-correction nonlinear solver and an overlapping restricted additive Schwarz preconditioned linear iterative solver with incomplete factorization type subdomain solvers. With this Defect Correction-Krylov-Schwarz, we have successfully calculated 3D subsonic, transonic and supersonic flows around complete air planes that are accelerating (including changing their orientation angles) with respect to the freestream.

Papers