Perry A. Newman

Approximation and Model Management in Aerodynamic Optimization with Variable-Fidelity Models

This workdiscussesan approach,e rst-orderapproximation and modelmanagementoptimization (AMMO), for solving design optimization problems that involve computationally expensive simulations. AMMO maximizes the use of lower-e delity, cheaper models in iterative procedures with occasional, but systematic, recourse to highere delity, more expensive models for monitoring the progress of design optimization. A distinctive feature of the approach is thatit is globally convergent to a solution oftheoriginal, high-e delity problem. VariantsofAMMObased on three nonlinear programming algorithms are demonstrated on a three-dimensional aerodynamic wing optimization problemand atwo-dimensionalairfoiloptimizationproblem. Euleranalysisonmeshesof varying degrees of ree nement provides a suite of variable-e delity models. Preliminary results indicate threefold savings in terms of high-e delity analyses for the three-dimensional problem and twofold savings for the two-dimensional problem.

Optimization with variable-fidelity models applied to wing design

This work discusses an approach, the Approximation Management Framework (AMF), for solving optimization problems that involve computationally expensive simulations. AMF aims to maximize the use of lower-fidelity, cheaper models in iterative procedures with occasional, but systematic, recourse to higher-fidelity, more expensive models for monitoring the progress of the algorithm. The method is globally convergent to a solution of the original, high-fidelity problem. Three versions of AMF, based on three nonlinear programming algorithms, are demonstrated on a 3D aerodynamic wing optimization problem and a 2D airfoil optimization problem. In both cases Euler analysis solved on meshes of various refinement provides a suite of variable-fidelity models. Preliminary results indicate threefold savings in terms of high-fidelity analyses in case of the 3D problem and twofold savings for the 2D problem.

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

Automatic Differentiation of Advanced CFD Codes for Multidisciplinary Design

Automated multidisciplinary design of aircraft and other flight vehicles requires the optimization of complex performance objectives with respect to a number of design parameters and constraints. The effect of these independent design variables on the system performance criteria can be quantified in terms of sensitivity derivatives which must be calculated and propagated by the individual discipline simulation codes. Typical advanced CFD analysis codes do not provide such derivatives as part of a flow solution; these derivatives are very expensive to obtain by divided (finite) differences from perturbed solutions. It is shown that sensitivity derivatives can be obtained accurately and efficiently using the ADIFOR source translator for automatic differentiation. In particular, it is demonstrated that the 3-D, thin-layer Navier- Stokes, multigrid flow solver called TLNS3D is amenable to automatic differentiation in the forward mode even with its implicit iterative solution algorithm and complex turbulence modeling. It is significant that by using computational differentiation, consistent discrete nongeometric sensitivity derivatives have been obtained from an aerodynamic 3-D CFD code in a relatively short time, e.g., O(man- week) not O(man-year). DOE

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

Simultaneous Aerodynamic Analysis and Design Optimization (SAADO) for a 3-D Flexible Wing

The formulation and implementation of an optimization method called Simultaneous Aerodynamic Analysis and Design Optimization (SAADO) are extended from single discipline analysis (aerodynamics only) to multidisciplinary analysis - in this case, static aero-structural analysis - and applied to a simple 3-D wing problem. The method aims to reduce the computational expense incurred in performing shape optimization using state-of-the-art Computational Fluid Dynamics (CFD) flow analysis, Finite Element Method (FEM) structural analysis and sensitivity analysis tools. Results for this small problem show that the method reaches the same local optimum as conventional optimization. However, unlike its application to the rigid wing (single discipline analysis), the method, as implemented here, may not show significant reduction in the computational cost. Similar reductions were seen in the two-design-variable (DV) problem results but not in the 8-DV results given here.

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.

EFFECT OF RANDOM GEOMETRIC UNCERTAINTY ON THE COMPUTATIONAL DESIGN OF A 3-D FLEXIBLE WING

The effect of geometric uncertainty due to statistically independent, random, normally distributed shape parameters is demonstrated in the computational design of a 3-D flexible wing. A first-order second-moment statistical approximation method is used to propagate the assumed input uncertainty through coupled Euler CFD aerodynamic / finite element structural codes for both analysis and sensitivity analysis. First-order sensitivity derivatives obtained by automatic differentiation are used in the input uncertainty propagation. These propagated uncertainties are then used to perform a robust design of a simple 3-D flexible wing at supercritical flow conditions. The effect of the random input uncertainties is shown by comparison with conventional deterministic design results. Sample results are shown for wing planform, airfoil section, and structural sizing variables.

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.