Multi-Model Multi-Domain Computational Methods: Full Potential

Introduction

The full potential equation describes the flow of an inviscid, irrotational, isentropic fluid, in a single nonlinear conservation law. More general than the linear potential equation, the nonlinear full potential equation accommodates reversible compressibility effects. In practice, full potential is also extended to flows with mild shocks (upstream Mach numbers of 1.2 or less). Coupled with a model for the boundary layer, the full potential formulation has sufficient fidelity for the prediction of flows over streamlined aircraft at near optimal cruise conditions. The full potential code TRANAIR is, in fact, the principal analysis code employed in the design of Boeing commercial aircraft, such as the 777, in the design of which computational simulation played a revolutionary role. One of the advantages of full potential over primitive variables is in discrete problem size. Whereas a 3D Euler or Navier-Stokes code stores 25 or more nonzeros per incident vertex in the block row of the the Jacobian matrix corresponding to each vertex, a full potential code stores just 1. This permits much denser grids for a fixed memory resource, and therefore, more adaptive resolution of geometry and flow features. Another attractive feature of the full potential model as a demonstration vehicle for our project is that in the subsonic regime, its Jacobians satisfy the hypotheses of the two-level Additive Schwarz theory.

Research Overview

We have prototyped the TRANAIR solver in an academic full potential code in order to demonstrate the utility of the Schwarz approach in the parallelization of the full code. Our code represents the standard test case of a zero-angle of attack NACA 0012 airfoil through transpiration boundary conditions on a highly resolved uniform grid. The standard device of "density upwinding" is used to stabilize the shock in transonic cases.

Figure showing test geometry and illustrating the geometric parameters of the Schwarz preconditioner for the case of nine subdomains. The NACA 0012 profile is shown on the symmetry plane.

Figures showing converged airfoil pressure distributions for subsonic and transonic cases.

Figures showing airfoil convergence histories for subsonic and transonic cases. Both problems converge at a rapid asymptotic rate; however, the transonic case stalls in residual norm reduction while the location of the shock is converging. (Convergence is not superlinear because we terminate the Krylov iterations prematurely at each Newton step; this improves overall execution time.)

We employ the Newton-Krylov-Schwarz domain decomposition method, using one subdomain per processor in the parallelization. We have extensively tested Schwarz "tuning parameters", such as the density of the coarse grid, the degree of overlap, and the degree of fill in the approximate solutions on the subdomains and shown that modest investments in each of these three areas yield almost all of the convergence rate benefit obtainable from a fuller investment at a fraction of the parallel overhead. Parallel efficiencies in excess of 60% are available together with sustained computation rates on a par with the best sequential sparse matrix implementations.

Papers