Arthur C. Taylor

Overview of Sensitivity Analysis and Shape Optimization for Complex Aerodynamic Configurations

This paper presents a brief overview of some of the more recent advances in steady aerodynamic shape-design sensitivity analysis and optimization, based on advanced computational fluid dynamics (CFD). The focus here is on those methods particularly well-suited to the study of geometrically complex configurations and their potentially complex associated flow physics. When nonlinear state equations are considered in the optimization process, difficulties are found in the application of sensitivity analysis. Some techniques for circumventing such difficulties are currently being explored and are included here. Attention is directed to methods that utilize automatic differentiation to obtain aerodynamic sensitivity derivatives for both complex configurations and complex flow physics. Various examples of shape - design sensitivity analysis for unstructured-grid CFD algorithms are demonstrated for different formulations of the sensitivity equations. Finally, the use of advanced, unstructured-grid CFDs in multidisciplinary analyses and multidisciplinary sensitivity analyses within future optimization processes is recommended and encouraged.

Approach for uncertainty propagation and robust design in CFD using sensitivity derivatives

This paper presents an implementation of the approximate statistical moment method for uncertainty propagation and robust optimization for a quasi I-D Euler CFD code. Given uncertainties in statistically independent, random, normally distributed input variables, a firstand second-order statistical moment matching procedure is performed to approximate the uncertainly in the CFD output. Efficient calculation of both firstand second-order sensitivity derivatives is required. In order to assess the validity of the approximations, the moments are compared with statistical moments generated through Monte Carlo simulations. The uncertainties in the CFD input variables are also incorporated into a robust optimization procedure. For this optimization, statistical moments involving firstorder sensitivity derivatives appear in the objective function and system constraints. Second-order sensitivity derivatives are used in a gradient-based search to successfully execute a robust optimization. The approximate methods used throughout the analyses are found to be valid when considering robustness about input parameter mean values. A a b b F g k M M Nomenclature nozzle area Minf geometric shape parameter Mt geometric shape parameter N vector of independent input variables Pb vector of CFD output functions Q vector of conventional optimization constraints q number of standard deviations qt Mach number at nozzle inlet R vector of Mach number at each grid point V Vt x (Y * LTC, US Army, Ph.D. Candidate, Department of Mechanical Engineering, mputko @ tabdemo.larc.nasa.gov tSenior Research Scientist, Muhidisciplinary Optimization Branch, M/S 159, p.a.newman@larc.nasa.gov -: Associate Professor, Deparlment of Mechanical Engineering, ataylor @lions.odu.edu §Research Scientist, Multidisciplinary Optimization Branch, M/S 159, AIAA senior member, l.l.grecn@larc.nasa.gov This paper is declared a work of the U.S. Government and is not subject to copyright protection in the United Stales. free-stream Mach number target inlet Mach number sample size normalized nozzle static back (outlet) pressure vector of flow-field variables (state variables) mass flux through nozzle target mass flux through nozzle vector of state equation residuals nozzle volume target nozzle volume used for optimization normalized axial position within nozzle standard deviation variance

Some Advanced Concepts in Discrete Aerodynamic Sensitivity Analysis

Abstract 1.0 Introduction An efficient incremental-iterative approach for dif-ferentiating advanced flow codes is successfully demon-strated on a 2D inviscid model problem. The methodemploys the reverse-mode capability of the automatic-differentiation software tool ADIFOR 3.0, and isproven to yield accurate first-order aerodynamic sensi-tivity, derivatives. A substantial reduction in CPU timeand computer memory is demonstrated in comparisonwith results from a straight-forward, black-box t:everse-mode application of ADIFOR 3.0 to the same flowcode. An ADIFOR-assisted procedure for accurate sec-ond-order aerodynamic sensitivity derivatives is suc-cessfidly verified on an inviscid transonic lifting airfoilexample problem. The method requires that first-orderderivatives are calculated first using both the fonvard(direct) and reverse (adjoint) procedures; then, a veryefficient non-iterative calculation of all second-orderderivatives can be accomplished. Accurate second de-rivatives (Le., the complete Hessian matrices) of lift,wave-drag, and pitching-moment coefficients are calcu-lated with respect to geometric-shape, angle-of-attack,and freestream Mach numberComputing sensitivity derivatives (SDs) from high-fidelity, nonlinear CFD codes is an enabling technologyfor design of advanced concept vehicles. In recent yearssignificant progress has been achieved in the efficientcalculation of accurate SDs from these CFD codes _.-The automatic differentiation (AD) software toolADIFOR (Automatic Differentiation of FORTRAN)has been proven an effective tool for extracting aerody-namlc

Sensitivity derivatives for three dimensional supersonic Euler code using incremental iterative strategy

In a recent work, an incremental strategy was proposed to iteratively solve the very large systems of linear equations that are required to obtain quasianalytical sensitivity derivatives from advanced computational fluid dynamics (CFD) codes. The technique was sucessfully demonstrated for two large two-dimensional problems: a subsonic and a transonic airfoil. The principal feature of this incremental iterative stategy is that it allows the use of the identical approximate coefficient matrix operator and algorithm to solve the nonlinear flow and the linear sensitivity equations; at convergence, the accuracy of the sensitivity derivatives is not compromised. This feature allows a comparatively straightforward extension of the methodology to three-dimensional problems; this extension is successfully demonstrated in the present study for a space-marching solution of the three-dimensional Euler equations over a Mach 2.4 blended wing-body configuration.

Three-dimensional aerodynamic shape sensitivity analysis and design optimization using the Euler equations on unstructured grids

Developed is an approach whereby aerodynamic shape sensitivity analysis and design optimization are pedormed on three-dimensional unstructured meshes. The advantage of unstructured grids (when compared with a structured-grid approach) is their inherent ability to discretize irregularly shaped domains with greater eficiency and less effort. Hence, this approach is ideally suited for geometrically complex configurations of practical interest. The nonlinear Euler equations are solved using a fullyimplicit, upwind, cell-centered, finite-volume scheme. The discrete, linearized systems which result from this scheme are solved iteratively by a preconditioned conjugate-gradient-like algorithm known as GMRES; a similar procedure is also used to solve the accompanying linear aerodynamic sensitivity equalions in incremental iterative form. As shown, this particular form of the sensitivity equations makes large-scale gradient-based aerodynamic optimization possible by taking advantage of memory eSJicient methods to construct exact matrix-vector products. Wing-planform parameterization is trccornplished via scaling and translation factors at pre-selected locations along the wing span, then linearly varying these factors between locations. Once the surface has been deformed, the unstructured grid is adapted by considering the mesh as a system of interconnected springs. Grid sensitivities are obtained by differentiating the surface parameterization and the grid adaptation algorithms with ADIFOR (which is an advanced automatic-diferentiation software tool). To evaluate this shape optimization procedure, the planform shape of an initially rectangular wing with uniform NACA-0012 cross-sections is optimized in a compressible, inviscid flow. 1. Introduct ion As recently noted by Reuther et al. [ 11 “while flow analysis has maturecl to the extent that Navier-Stokes calculations are routinely carried out over very complex configurations, direct CFD based design is only just beginning to be used in the treatment of moderately complex three-dimensional coizfgurations”. This is primarily due to the fact that to generate a single structured grid about such a configuration is difficult, if not impossible. Thus, to handle geometry of practical interest, some sort of domain decomposition scheme must be incorporated into the design code. For structured grid solvers, these techniques would include multiblocked, zonally patched, and overlapped (sometimes referred to as Chimera) grid algorithms. However, as the geometric flexibility of the method increases, so does the complexity of the underlying algorithm. Since the use of sensitivity analysis, to evaluate the needed gradients for a numerical optimizer, is still evolving, little work has been done toward extending these algorithms to include these domain decomposition methods. The research which has been accomplished has mostly concentrated on the use of niultiblocked grids. On this, Reuther et al. [ I ] have developed a multiblock-multigrid adjoint solver (“variational” or “control theory” approach [ 2 ] ) which was applied for the wing redesign of a transonic business jet. Eleshaky and Baysal [3] developed a multiblock “discrete” adjoint solver which was applied to a simple axisynlmetric nozzle near a flat plat. As for the use of the more advanced domain decomposition methods (zonal and overlapped grids), and combinations of the three various types, Taylor [4] has differentiated an advanced flow-analysis code to perform the discrete sensitivity analysis. * Graduate Research Assistant. Student Member, AIM # Associate Professor. Member, AIAA. Copyright

Efficient nonlinear static aeroelastic wing analysis

Abstract The objective of this work is to demonstrate a computationally efficient, high-fidelity, integrated static aeroelastic analysis procedure. The aerodynamic analysis consists of solving the nonlinear Euler equations by using an upwind cell-centered finite-volume scheme on unstructured tetrahedral meshes. The use of unstructured grids enhances the discretization of irregularly shaped domains and the interaction compatibility with the wing structure. The structural analysis utilizes finite elements to model the wing so that accurate structural deflections are obtained and allows the capability for computing detailed stress information for the configuration. Parameters are introduced to control the interaction of the computational fluid dynamics and structural analyses; these control parameters permit extremely efficient static aeroelastic computations. To demonstrate and evaluate this procedure, static aeroelastic analysis results for a flexible wing in low subsonic, high subsonic (subcritical), transonic (supercritical), and supersonic flow conditions are presented.

Observations on computational methodologies for use in large-scale, gradient-based, multidisciplinary design

Various computational methodologies relevant to large-scale multidisciplinary gradient-based optimization for engineering systems design problems are examined with emphasis on the situation where one or more discipline responses required by the optimized design procedure involve the solution of a system of nonlinear partial differential equations. Such situations occur when advanced CFD codes are applied in a multidisciplinary procedure for optimizing an aerospace vehicle design. A technique for satisfying the multidisciplinary design requirements for gradient information is presented. The technique is shown to permit some leeway in the CFD algorithms which can be used, an expansion to 3D problems, and straightforward use of other computational methodologies.

Methodology for Calculating Aerodynamic Sensitivity Derivatives

A general procedure is developed for calculating aerodynamic sensitivity coefficients using the full equations of inviscid fluid flow, where the focus of the work is the treatment of geometric shape design variables. Using an upwind cell-centered finite volume approximation to represent the Euler equations, sensitivity derivatives are determined by direct differentiation of the resulting set of coupled nonlinear algebraic equations that model the fluid flow

Transonic turbulent airfoil design optimization with automatic differentiation in incremental iterative forms

The primary effort in this study is to develop a practical approach for incorporating a validated computational fluid dynamics (CFD) code into an aerodynamic design optimization procedure with minimal code modification. The approach relies on the automatic differentiation tool ADTFOR to generate the grid sensitivities and derivative terms required by the incremental iterative method for aerodynamic sensitivity analysis. Transonic turbulent airfoil design optimization, based on the Navier-Stokes equations, is chosen as a sample problem to facilitate discussion of the code implementation procedure and the factors that affect the computational efficiency of aerodynamic design optimization.

Recent advances in steady compressible aerodynamic sensitivity analysis

An overview is given of some recent accomplishments by different researchers in calculating gradient information of interest from modern flow-analysis codes. Of particular interest here is advanced computational fluid dynamics (CFD) software, which solves the nonlinear multidimensional Euler and/or Navier-Stokes equations. The accurate, efficient calculation of aerodynamic sensitivity derivatives is very important in design-oriented applications of these CFD codes to single discipline and multidisciplinary problems [1, 2].