Gene Hou

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.

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.

Shape-sensitivity analysis and design optimization of linear, thermoelastic solids

In this study, a numerical scheme is developed for shape-sensitivity analysis and design optimization of linear, quasistatic, thermoelastic solids. In this scheme, the finite-element method is used as the analyzer for analyzing stress, temperature, shape sensitivity, and design velocity field. Based upon the method of material derivatives, both the techniques of the direct-differentiation method and the adjoint-variable method are applied to derive the shape-sensitivity equations. The shape-optimization formulations discussed here include boundary integrals of displacements and heat fluxes as well as domain integrals of stresses and areas. Numerical results show that the proposed scheme works well in terms of accuracy.

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.

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

Reliability Assessment of a Robust Design Under Uncertainty for a 3-D Flexible Wing

The paper presents reliability assessment results for the robust designs under uncertainty of a 3-D flexible wing previously reported by the authors. Reliability assessments (additional optimization problems) of the active constraints at the various probabilistic robust design points are obtained and compared with the constraint values or target constraint probabilities specified in the robust design. In addition, reliability-based sensitivity derivatives with respect to design variable mean values are also obtained and shown to agree with finite difference values. These derivatives allow one to perform reliability based design without having to obtain second-order sensitivity derivatives. However, an inner-loop optimization problem must be solved for each active constraint to find the most probable point on that constraint failure surface.

Eigenvalue and Eigenvector Approximate Analysis for Repeated Eigenvalue Problems

The method developed for approximate analysis involves a reparamaterization of the multivariable structural eigenvalue problems in terms of a single positive-valued parameter. The resulting equations yield first order approximations of changes in both the eigenvalues and eigenvectors associated with the repeated eigenvalue problem.