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NASA NTRS2004

Adjoint Algorithm for CAD-Based Shape Optimization Using a Cartesian Method

Adjoint solutions of the governing flow equations are becoming increasingly important for the development of efficient analysis and optimization algorithms. A well-known use of the adjoint method is gradient-based shape optimization. Given an objective function that defines some measure of performance, such as the lift and drag functionals, its gradient is computed at a cost that is essentially independent of the number of design variables (geometric parameters that control the shape). More recently, emerging adjoint applications focus on the analysis problem, where the adjoint solution is used to drive mesh adaptation, as well as to provide estimates of functional error bounds and corrections. The attractive feature of this approach is that the mesh-adaptation procedure targets a specific functional, thereby localizing the mesh refinement and reducing computational cost. Our focus is on the development of adjoint-based optimization techniques for a Cartesian method with embedded boundaries.12 In contrast t o implementations on structured and unstructured grids, Cartesian methods decouple the surface discretization from the volume mesh. This feature makes Cartesian methods well suited for the automated analysis of complex geometry problems, and consequently a promising approach to aerodynamic optimization. Melvin et developed an adjoint formulation for the TRANAIR code, which is based on the full-potential equation with viscous corrections. More recently, Dadone and Grossman presented an adjoint formulation for the Euler equations. In both approaches, a boundary condition is introduced to approximate the effects of the evolving surface shape that results in accurate gradient computation. Central to automated shape optimization algorithms is the issue of geometry modeling and control. The need to optimize complex, "real-life" geometry provides a strong incentive for the use of parametric-CAD systems within the optimization procedure. In previous work, we presented an effective optimization framework that incorporates a direct-CAD interface. In this work, we enhance the capabilities of this framework with efficient gradient computations using the discrete adjoint method. We present details of the adjoint numerical implementation, which reuses the domain decomposition, multigrid, and time-marching schemes of the flow solver. Furthermore, we explain and demonstrate the use of CAD in conjunction with the Cartesian adjoint approach. The final paper will contain a number of complex geometry, industrially relevant examples with many design variables to demonstrate the effectiveness of the adjoint method on Cartesian meshes.

Nemec, Marian↗

NASA NTRS1993

A homotopy algorithm for digital optimal projection control GASD-HADOC

The linear-quadratic-gaussian (LQG) compensator was developed to facilitate the design of control laws for multi-input, multi-output (MIMO) systems. The compensator is computed by solving two algebraic equations for which standard closed-loop solutions exist. Unfortunately, the minimal dimension of an LQG compensator is almost always equal to the dimension of the plant and can thus often violate practical implementation constraints on controller order. This deficiency is especially highlighted when considering control-design for high-order systems such as flexible space structures. This deficiency motivated the development of techniques that enable the design of optimal controllers whose dimension is less than that of the design plant. A homotopy approach based on the optimal projection equations that characterize the necessary conditions for optimal reduced-order control. Homotopy algorithms have global convergence properties and hence do not require that the initializing reduced-order controller be close to the optimal reduced-order controller to guarantee convergence. However, the homotopy algorithm previously developed for solving the optimal projection equations has sublinear convergence properties and the convergence slows at higher authority levels and may fail. A new homotopy algorithm for synthesizing optimal reduced-order controllers for discrete-time systems is described. Unlike the previous homotopy approach, the new algorithm is a gradient-based, parameter optimization formulation and was implemented in MATLAB. The results reported may offer the foundation for a reliable approach to optimal, reduced-order controller design.

Collins, Emmanuel G., Jr.↗

NASA NTRS2004

Employing Sensitivity Derivatives for Robust Optimization under Uncertainty in CFD

A robust optimization is demonstrated on a two-dimensional inviscid airfoil problem in subsonic flow. Given uncertainties in statistically independent, random, normally distributed flow parameters (input variables), an approximate first-order statistical moment method is employed to represent the Computational Fluid Dynamics (CFD) code outputs as expected values with variances. These output quantities are used to form the objective function and constraints. The constraints are cast in probabilistic terms; that is, the probability that a constraint is satisfied is greater than or equal to some desired target probability. Gradient-based robust optimization of this stochastic problem is accomplished through use of both first and second-order sensitivity derivatives. For each robust optimization, the effect of increasing both input standard deviations and target probability of constraint satisfaction are demonstrated. This method provides a means for incorporating uncertainty when considering small deviations from input mean values.

Newman, Perry A.↗

NASA NTRS1995

Implementation of a multiblock sensitivity analysis method in numerical aerodynamic shape optimization

A multiblock sensitivity analysis method is applied in a numerical aerodynamic shape optimization technique. The Sensitivity Analysis Domain Decomposition (SADD) scheme which is implemented in this study was developed to reduce the computer memory requirements resulting from the aerodynamic sensitivity analysis equations. Discrete sensitivity analysis offers the ability to compute quasi-analytical derivatives in a more efficient manner than traditional finite-difference methods, which tend to be computationally expensive and prone to inaccuracies. The direct optimization procedure couples CFD analysis based on the two-dimensional thin-layer Navier-Stokes equations with a gradient-based numerical optimization technique. The linking mechanism is the sensitivity equation derived from the CFD discretized flow equations, recast in adjoint form, and solved using direct matrix inversion techniques. This investigation is performed to demonstrate an aerodynamic shape optimization technique on a multiblock domain and its applicability to complex geometries. The objectives are accomplished by shape optimizing two aerodynamic configurations. First, the shape optimization of a transonic airfoil is performed to investigate the behavior of the method in highly nonlinear flows and the effect of different grid blocking strategies on the procedure. Secondly, shape optimization of a two-element configuration in subsonic flow is completed. Cases are presented for this configuration to demonstrate the effect of simultaneously reshaping interfering elements. The aerodynamic shape optimization is shown to produce supercritical type airfoils in the transonic flow from an initially symmetric airfoil. Multiblocking effects the path of optimization while providing similar results at the conclusion. Simultaneous reshaping of elements is shown to be more effective than individual element reshaping due to the inclusion of mutual interference effects.

Lacasse, James M.↗

NASA NTRS1997

Aerodynamic Shape Sensitivity Analysis and Design Optimization of Complex Configurations Using Unstructured Grids

A three-dimensional unstructured grid approach to aerodynamic shape sensitivity analysis and design optimization has been developed and is extended to model geometrically complex configurations. The advantage of unstructured grids (when compared with a structured-grid approach) is their inherent ability to discretize irregularly shaped domains with greater efficiency and less effort. Hence, this approach is ideally suited for geometrically complex configurations of practical interest. In this work the nonlinear Euler equations are solved using an 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 for the two-dimensional geometry and a Gauss-Seidel algorithm for the three-dimensional; similar procedures are used to solve the accompanying linear aerodynamic sensitivity equations in incremental iterative form. As shown, this particular form of the sensitivity equation makes large-scale gradient-based aerodynamic optimization possible by taking advantage of memory efficient methods to construct exact Jacobian matrix-vector products. Simple parameterization techniques are utilized for demonstrative purposes. 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-differentiation software tool). To demonstrate the ability of this procedure to analyze and design complex configurations of practical interest, the sensitivity analysis and shape optimization has been performed for a two-dimensional high-lift multielement airfoil and for a three-dimensional Boeing 747-200 aircraft.

Taylor, Arthur C., III↗

NASA NTRS2000

Global Design Optimization for Aerodynamics and Rocket Propulsion Components

Modern computational and experimental tools for aerodynamics and propulsion applications have matured to a stage where they can provide substantial insight into engineering processes involving fluid flows, and can be fruitfully utilized to help improve the design of practical devices. In particular, rapid and continuous development in aerospace engineering demands that new design concepts be regularly proposed to meet goals for increased performance, robustness and safety while concurrently decreasing cost. To date, the majority of the effort in design optimization of fluid dynamics has relied on gradient-based search algorithms. Global optimization methods can utilize the information collected from various sources and by different tools. These methods offer multi-criterion optimization, handle the existence of multiple design points and trade-offs via insight into the entire design space, can easily perform tasks in parallel, and are often effective in filtering the noise intrinsic to numerical and experimental data. However, a successful application of the global optimization method needs to address issues related to data requirements with an increase in the number of design variables, and methods for predicting the model performance. In this article, we review recent progress made in establishing suitable global optimization techniques employing neural network and polynomial-based response surface methodologies. Issues addressed include techniques for construction of the response surface, design of experiment techniques for supplying information in an economical manner, optimization procedures and multi-level techniques, and assessment of relative performance between polynomials and neural networks. Examples drawn from wing aerodynamics, turbulent diffuser flows, gas-gas injectors, and supersonic turbines are employed to help demonstrate the issues involved in an engineering design context. Both the usefulness of the existing knowledge to aid current design practices and the need for future research are identified.

Shyy, Wei↗

NASA NTRS1996

Observations Regarding Use of Advanced CFD Analysis, Sensitivity Analysis, and Design Codes in MDO

Observations regarding the use of advanced computational fluid dynamics (CFD) analysis, sensitivity analysis (SA), and design codes in gradient-based multidisciplinary design optimization (MDO) reflect our perception of the interactions required of CFD and our experience in recent aerodynamic design optimization studies using CFD. Sample results from these latter studies are summarized for conventional optimization (analysis - SA codes) and simultaneous analysis and design optimization (design code) using both Euler and Navier-Stokes flow approximations. The amount of computational resources required for aerodynamic design using CFD via analysis - SA codes is greater than that required for design codes. Thus, an MDO formulation that utilizes the more efficient design codes where possible is desired. However, in the aerovehicle MDO problem, the various disciplines that are involved have different design points in the flight envelope; therefore, CFD analysis - SA codes are required at the aerodynamic 'off design' points. The suggested MDO formulation is a hybrid multilevel optimization procedure that consists of both multipoint CFD analysis - SA codes and multipoint CFD design codes that perform suboptimizations.

Newman, Perry A.↗

NASA NTRS2003

Aerodynamic Design of Complex Configurations Using Cartesian Methods and CAD Geometry

The objective for this paper is to present the development of an optimization capability for the Cartesian inviscid-flow analysis package of Aftosmis et al. We evaluate and characterize the following modules within the new optimization framework: (1) A component-based geometry parameterization approach using a CAD solid representation and the CAPRI interface. (2) The use of Cartesian methods in the development Optimization techniques using a genetic algorithm. The discussion and investigations focus on several real world problems of the optimization process. We examine the architectural issues associated with the deployment of a CAD-based design approach in a heterogeneous parallel computing environment that contains both CAD workstations and dedicated compute nodes. In addition, we study the influence of noise on the performance of optimization techniques, and the overall efficiency of the optimization process for aerodynamic design of complex three-dimensional configurations. of automated optimization tools. rithm and a gradient-based algorithm.

Nemec, Marian↗

NASA NTRS2001

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 1-D Euler CFD (computational fluid dynamics) code. Given uncertainties in statistically independent, random, normally distributed input variables, a first- and second-order statistical moment matching procedure is performed to approximate the uncertainty in the CFD output. Efficient calculation of both first- and 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 first-order 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.

Putko, Michele M.↗

NASA NTRS2002

Approach for Input Uncertainty Propagation and Robust Design in CFD Using Sensitivity Derivatives

An implementation of the approximate statistical moment method for uncertainty propagation and robust optimization for quasi 3-D Euler CFD code is presented. Given uncertainties in statistically independent, random, normally distributed input variables, first- and second-order statistical moment procedures are performed to approximate the uncertainty in the CFD output. Efficient calculation of both first- and second-order sensitivity derivatives is required. In order to assess the validity of the approximations, these 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 first-order 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.

Putko, Michele M.↗

NASA NTRS2002

Turbopump Performance Improved by Evolutionary Algorithms

The development of design optimization technology for turbomachinery has been initiated using the multiobjective evolutionary algorithm under NASA's Intelligent Synthesis Environment and Revolutionary Aeropropulsion Concepts programs. As an alternative to the traditional gradient-based methods, evolutionary algorithms (EA's) are emergent design-optimization algorithms modeled after the mechanisms found in natural evolution. EA's search from multiple points, instead of moving from a single point. In addition, they require no derivatives or gradients of the objective function, leading to robustness and simplicity in coupling any evaluation codes. Parallel efficiency also becomes very high by using a simple master-slave concept for function evaluations, since such evaluations often consume the most CPU time, such as computational fluid dynamics. Application of EA's to multiobjective design problems is also straightforward because EA's maintain a population of design candidates in parallel. Because of these advantages, EA's are a unique and attractive approach to real-world design optimization problems.

Oyama, Akira↗

NASA NTRS2007

Topology Synthesis of Structures Using Parameter Relaxation and Geometric Refinement

Typically, structural topology optimization problems undergo relaxation of certain design parameters to allow the existence of intermediate variable optimum topologies. Relaxation permits the use of a variety of gradient-based search techniques and has been shown to guarantee the existence of optimal solutions and eliminate mesh dependencies. This Technical Publication (TP) will demonstrate the application of relaxation to a control point discretization of the design workspace for the structural topology optimization process. The control point parameterization with subdivision has been offered as an alternative to the traditional method of discretized finite element design domain. The principle of relaxation demonstrates the increased utility of the control point parameterization. One of the significant results of the relaxation process offered in this TP is that direct manufacturability of the optimized design will be maintained without the need for designer intervention or translation. In addition, it will be shown that relaxation of certain parameters may extend the range of problems that can be addressed; e.g., in permitting limited out-of-plane motion to be included in a path generation problem.

Hull, P. V.↗

NASA NTRS2006

Single- and Multiple-Objective Optimization with Differential Evolution and Neural Networks

Genetic and evolutionary algorithms have been applied to solve numerous problems in engineering design where they have been used primarily as optimization procedures. These methods have an advantage over conventional gradient-based search procedures became they are capable of finding global optima of multi-modal functions and searching design spaces with disjoint feasible regions. They are also robust in the presence of noisy data. Another desirable feature of these methods is that they can efficiently use distributed and parallel computing resources since multiple function evaluations (flow simulations in aerodynamics design) can be performed simultaneously and independently on ultiple processors. For these reasons genetic and evolutionary algorithms are being used more frequently in design optimization. Examples include airfoil and wing design and compressor and turbine airfoil design. They are also finding increasing use in multiple-objective and multidisciplinary optimization. This lecture will focus on an evolutionary method that is a relatively new member to the general class of evolutionary methods called differential evolution (DE). This method is easy to use and program and it requires relatively few user-specified constants. These constants are easily determined for a wide class of problems. Fine-tuning the constants will off course yield the solution to the optimization problem at hand more rapidly. DE can be efficiently implemented on parallel computers and can be used for continuous, discrete and mixed discrete/continuous optimization problems. It does not require the objective function to be continuous and is noise tolerant. DE and applications to single and multiple-objective optimization will be included in the presentation and lecture notes. A method for aerodynamic design optimization that is based on neural networks will also be included as a part of this lecture. The method offers advantages over traditional optimization methods. It is more flexible than other methods in dealing with design in the context of both steady and unsteady flows, partial and complete data sets, combined experimental and numerical data, inclusion of various constraints and rules of thumb, and other issues that characterize the aerodynamic design process. Neural networks provide a natural framework within which a succession of numerical solutions of increasing fidelity, incorporating more realistic flow physics, can be represented and utilized for optimization. Neural networks also offer an excellent framework for multiple-objective and multi-disciplinary design optimization. Simulation tools from various disciplines can be integrated within this framework and rapid trade-off studies involving one or many disciplines can be performed. The prospect of combining neural network based optimization methods and evolutionary algorithms to obtain a hybrid method with the best properties of both methods will be included in this presentation. Achieving solution diversity and accurate convergence to the exact Pareto front in multiple objective optimization usually requires a significant computational effort with evolutionary algorithms. In this lecture we will also explore the possibility of using neural networks to obtain estimates of the Pareto optimal front using non-dominated solutions generated by DE as training data. Neural network estimators have the potential advantage of reducing the number of function evaluations required to obtain solution accuracy and diversity, thus reducing cost to design.

Rai, Man Mohan↗

NASA NTRS2004

On the Use of CAD and Cartesian Methods for Aerodynamic Optimization

The objective for this paper is to present the development of an optimization capability for Curt3D, a Cartesian inviscid-flow analysis package. We present the construction of a new optimization framework and we focus on the following issues: 1) Component-based geometry parameterization approach using parametric-CAD models and CAPRI. A novel geometry server is introduced that addresses the issue of parallel efficiency while only sparingly consuming CAD resources; 2) The use of genetic and gradient-based algorithms for three-dimensional aerodynamic design problems. The influence of noise on the optimization methods is studied. Our goal is to create a responsive and automated framework that efficiently identifies design modifications that result in substantial performance improvements. In addition, we examine the architectural issues associated with the deployment of a CAD-based approach in a heterogeneous parallel computing environment that contains both CAD workstations and dedicated compute engines. We demonstrate the effectiveness of the framework for a design problem that features topology changes and complex geometry.

Nemec, M.↗

NASA NTRS2004

Adjoint Formulation for an Embedded-Boundary Cartesian Method

Many problems in aerodynamic design can be characterized by smooth and convex objective functions. This motivates the use of gradient-based algorithms, particularly for problems with a large number of design variables, to efficiently determine optimal shapes and configurations that maximize aerodynamic performance. Accurate and efficient computation of the gradient, however, remains a challenging task. In optimization problems where the number of design variables dominates the number of objectives and flow- dependent constraints, the cost of gradient computations can be significantly reduced by the use of the adjoint method. The problem of aerodynamic optimization using the adjoint method has been analyzed and validated for both structured and unstructured grids. The method has been applied to design problems governed by the potential, Euler, and Navier-Stokes equations and can be subdivided into the continuous and discrete formulations. Giles and Pierce provide a detailed review of both approaches. Most implementations rely on grid-perturbation or mapping procedures during the gradient computation that explicitly couple changes in the surface shape to the volume grid. The solution of the adjoint equation is usually accomplished using the same scheme that solves the governing flow equations. Examples of such code reuse include multistage Runge-Kutta schemes coupled with multigrid, approximate-factorization, line-implicit Gauss-Seidel, and also preconditioned GMRES. The development of the adjoint method for aerodynamic optimization problems on Cartesian grids has been limited. In contrast to implementations on structured and unstructured grids, Cartesian grid methods decouple the surface discretization from the volume grid. This feature makes Cartesian methods well suited for the automated analysis of complex geometry problems, and consequently a promising approach to aerodynamic optimization. Melvin e t al. developed an adjoint formulation for the TRANAIR code, which is based on the full-potential equation with viscous corrections. More recently, Dadone and Grossman presented an adjoint formulation for the Euler equations. In both approaches, a boundary condition is introduced to approximate the effects of the evolving surface shape that results in accurate gradient computation.

Nemec, Marian↗

NASA NTRS2004

A Study of Penalty Function Methods for Constraint Handling with Genetic Algorithm

COMETBOARDS (Comparative Evaluation Testbed of Optimization and Analysis Routines for Design of Structures) is a design optimization test bed that can evaluate the performance of several different optimization algorithms. A few of these optimization algorithms are the sequence of unconstrained minimization techniques (SUMT), sequential linear programming (SLP) and the sequential quadratic programming techniques (SQP). A genetic algorithm (GA) is a search technique that is based on the principles of natural selection or "survival of the fittest". Instead of using gradient information, the GA uses the objective function directly in the search. The GA searches the solution space by maintaining a population of potential solutions. Then, using evolving operations such as recombination, mutation and selection, the GA creates successive generations of solutions that will evolve and take on the positive characteristics of their parents and thus gradually approach optimal or near-optimal solutions. By using the objective function directly in the search, genetic algorithms can be effectively applied in non-convex, highly nonlinear, complex problems. The genetic algorithm is not guaranteed to find the global optimum, but it is less likely to get trapped at a local optimum than traditional gradient-based search methods when the objective function is not smooth and generally well behaved. The purpose of this research is to assist in the integration of genetic algorithm (GA) into COMETBOARDS. COMETBOARDS cast the design of structures as a constrained nonlinear optimization problem. One method used to solve constrained optimization problem with a GA to convert the constrained optimization problem into an unconstrained optimization problem by developing a penalty function that penalizes infeasible solutions. There have been several suggested penalty function in the literature each with there own strengths and weaknesses. A statistical analysis of some suggested penalty functions is performed in this study. Also, a response surface approach to robust design is used to develop a new penalty function approach. This new penalty function approach is then compared with the other existing penalty functions.

Ortiz, Francisco↗

NASA NTRS1991

The effect of model uncertainty on some optimal routing problems

The effect of model uncertainties on optimal routing in a system of parallel queues is examined. The uncertainty arises in modeling the service time distribution for the customers (jobs, packets) to be served. For a Poisson arrival process and Bernoulli routing, the optimal mean system delay generally depends on the variance of this distribution. However, as the input traffic load approaches the system capacity the optimal routing assignment and corresponding mean system delay are shown to converge to a variance-invariant point. The implications of these results are examined in the context of gradient-based routing algorithms. An example of a model-independent algorithm using online gradient estimation is also included.

Mohanty, Bibhu↗

NASA NTRS1992

Observations on computational methodologies for use in large-scale, gradient-based, multidisciplinary design incorporating advanced CFD codes

How a combination of various computational methodologies could reduce the enormous computational costs envisioned in using advanced CFD codes in gradient based optimized multidisciplinary design (MdD) procedures is briefly outlined. Implications of these MdD requirements upon advanced CFD codes are somewhat different than those imposed by a single discipline design. A means for satisfying these MdD requirements for gradient information is presented which appear to permit: (1) some leeway in the CFD solution algorithms which can be used; (2) an extension to 3-D problems; and (3) straightforward use of other computational methodologies. Many of these observations have previously been discussed as possibilities for doing parts of the problem more efficiently; the contribution here is observing how they fit together in a mutually beneficial way.

Newman, P. A.↗