Over the last three decades Computational Fluid Dynamics (CFD) has gradually joined the wind tunnel and flight test as a primary flow analysis tool for aerodynamic designers. CFD has had its most favorable impact on the aerodynamic design of the high-speed cruise configuration of a transport. This success has raised expectations among aerodynamicists that the applicability of CFD can be extended to the full flight envelope. However, the complex nature of the flows and geometries involved places substantially increased demands on the solution methodology and resources required. Currently most simulations involve Reynolds-Averaged Navier-Stokes (RANS) codes although Large Eddy Simulation (LES) and Detached Eddy Suimulation (DES) codes are occasionally used for component analysis or theoretical studies. Despite simplified underlying assumptions, current RANS turbulence models have been spectacularly successful for analyzing attached, transonic flows. Whether or not these same models are applicable to complex flows with smooth surface separation is an open question. A prerequisite for answering this question is absolute confidence that the CFD codes employed reliably solve the continuous equations involved. Too often, failure to agree with experiment is mistakenly ascribed to the turbulence model rather than inadequate numerics. Grid convergence in three dimensions is rarely achieved. Even residual convergence on a given grid is often inadequate. This paper discusses issues involved in residual and especially grid convergence.

Ordering and incomplete factorization issues for matrices arising from the TRANAIR CFD code. A locally reened rectangular grid nite element method: Application to computational uid dynamics and computational physics. A comparison of domain decomposition techniques for elliptic partial diierential equations and their parallel implementation. Inexact Newton's method solutions to the incompressible Navier-Stokes and energy equations using standard and matrix-free implementations. 16 eration counts and/or memory consumption. However, the plethora of parameters can be exploited, in principle, to produce optimal tradeoos in space and time for a given problem class. Though parametric tuning is important to performance, conservative robust choices are not diicult. methods for the unsteady compressible Navier-Stokes equations on unstructured meshes. A comparison of some domain decomposition and ILU preconditioned iterative methods for nonsymmetric elliptic problems. 15 Table 8: Wall-clock performance and relative parallel eeciency for unstructured Euler code on an Intel Paragon. We have shown that steady aerodynamics problems in two diierent formulations (full potential and Euler) can be eeectively solved, and cost-eeectively solved in parallel, by NKS methods. The NK technique has been compared with V-cycle multigrid on Euler and Navier-Stokes problems without parallelizing the preconditioning in 14, 20]. For a subsonic unstructured grid example, NK trails multigrid in execution time by a factor of only about 1.5. This penalty can be accepted when it is realized that the NK method has the advantage of doing all of its computation without generation of a family of coarse unstructured grids (which is diicult for three-dimensional unstructured grids). This work has been extended to three-dimensional problems in 20]. Large-scale time-dependent problems suuering from multiple scales often require parallel implicit algorithms. The KS technique has been shown eeective in the unsteady Navier-Stokes context in 3]. In 3], two of the same parameters explored herein (level of ll in the local ILU factorizations and subdomain overlap) are varied to produce a Schwarz precondi-tioner whose strength can be adjusted to adapt to the varying time-evolving ill-conditioning of the linear system arising at each implicit time step. A variety of CFD applications are (or have inner) nonlinear elliptically-dominated problems amenable to solution by NKS algorithms, which are characterized by low storage requirements (for an implicit method) and locally concentrated data dependencies with small overlaps between the preconditioner blocks. The addition of a global coarse grid in the Schwarz preconditioner is often eeective, where architecturally convenient. A deterrent to the widespread adoption of NKS …

Recent emphasis on reduction of design cycle time and cost in the design of commercial aircraft has sparked a renewed interest in design optimization in aerodynamic s, structures, and aeroelastics. The constrained aerodynamic optimization problem is closely related to the problem of solving nonlinear systems of equations. In applying Newton's method to steady-state compressible CFD analysis problems, the nonlinear elimination method has been remarkably successful. In this paper we consider the implications of this experience for design optimization formulations in the general case of state equation equality constraints. This relationship between nonlinear equation solving and design optimization is illustrated by drawing on computational examples from the TRANAIR compressible CFD code. We first discuss various formulations of the PDE constrained optimization problem related to the Lagrange Newton method and the multiplier free version implementation in TRANAIR. We then discuss the nonlinear elimination method and its application to a simple nozzle problem. This method is then applied to derive various globalization methods in design optimization which are illustrated by a computational example in airfoil design. Finally, we discuss some remaining limitations and issues.

Recent emphasis on reduction of design cycle time and cost in the design of commercial aircraft (P.E. Rubbert, 'CFD and the changing world of airplane design', AIAA Wright Brothers Lecture, September, 1994) has sparked a renewed interest in design optimization in aerodynamics, structures and aeroelastics. In this paper, recent developments in the use of design optimization in aerodynamics using the TRANAIR code are considered. Globalization techniques and the extension of the methodology to multipoint design will be discussed. Copyright (C) 1999 John Wiley & Sons, Ltd.

The TRANAIR aerodynamics code is used as a testbed for developing advanced iterative methods that lend themselves to parallel or distributed computing. In the case of aerodynamics design optimization, a significant part of the computational cost is in the solution of large sparse nonsymmetric linear systems. We have implemented a two-level method for solving such linear systems and present preliminary computational results for problems in two space dimensions. We observe a lime-space trade-off mediated by parameters that govern the amount of fill-in in incomplete factorizations on each level. In both subsonic and transonic flow regimes, parameter choices exist that lead to substantial improvements in runtime for comparable memory requirements, relative to a single grid method.

The adaptive grid method implemented in the 3D general geometry CFD code TRANAIR is described. Adaptive gridding in TRANAIR has been used to help solve many industrial aerospace analysis and design problems. Grids are adapted to numerical solutions by refining and coarsening local rectangular finite elements according to values of error indicators computed for the elements. The underlying finite element method is summarized and details of the adaptive gridding approach are given. Emphasis is placed on several principles used in developing and applying the approach. In particular, many computational comparisons are presented to explain the reasoning behind the local error indicator and gridding strategy used.

A new finite element method for solving important linear and nonlinear boundary value problems arising in computational physics is described in this paper. The method is designed to handle general three-dimensional regions, boundary conditions, and material properties. The boundaries are described by piecewise planar surfaces on which boundary conditions are imposed. The method uses box finite elements defined by a Cartesian grid that is independent of the boundary definition. Local refinements are performed by dividing a box element into eight similar box elements. The discretization uses trilinear approximations on the box elements with special element stiffness matrices for boxes cut by any boundary surface. This discretization process is automated and does not require the generation of a boundary conforming grid. The resulting (possibly nonlinear) discrete system is solved using a preconditioned GMRES algorithm. The primary preconditioner is a sparse matrix solver using a dynamic drop tolerance in the decomposition phase. Results are presented for aerodynamics problems with up to 400,000 elements, demonstrating the accuracy and efficiency of the method.

In this paper the use of a new out-of-core sparse matrix package for the numerical solution of partial differential equations involving complex geometries arising from aerospace applications is discussed. The sparse matrix solver accepts contributions to the matrix elements in random order and assembles the matrix using fast sort/merge routines. Fill-in is reduced through the use of a physically based nested dissection ordering. For very large problems a drop tolerance is used during the matrix decomposition phase. The resulting incomplete factorization is an effective preconditioner for Krylov subspace methods, such as GMRES. Problems involving 200,000 unknowns routinely are solved on the Cray X-MP using 64MW of solid-state storage device (SSD).