Gerhard Venter

Construction of Response Surface Approximations for Design Optimization

Using response surface approximations for design constraints in design optimization provides the designer with an overall perspective of the system response within the design space. Response surface approximations also reduce the numerical noise inherent in many numerical models and simplify the process of integrating several design codes, as is typically required in multidisciplinary optimization. Procedures are discussed for constructing accurate response surface approximations to represent design constraints in design optimization. Response surface approximations are constructed for the stresses and buckling loads of an isotropic plate with an abrupt change of thickness. These response surface approximations are constructed from numerical experiments conducted with a finite element analysis procedure and are used for minimum-weight optimum design of the plate. Nondimensional variahles and stepwise regression are used to reduce the complexity and increase the accuracy of the response surface approximations. Additionally, higher-order polynomials (cubic and quartic instead of the more traditional quadratic) are used as response surface approximations, and a detailed error analysis, using an independent data set, is performed. Finally, it is shown that, by making use of response surface approximations, the optimum weight of the plate may be presented in the form of a design chart for a wide range of geometric, loading, and material constants.

Construction of response surfaces for design optimization applications

Using response surface approximations in design optimization provides the designer with an overall view of the response. Response surface approximations also reduce the numerical noise inherent in many numerical models and simplify the process of integrating several design codes, as is typically required in the multidisciplanary optimization process. The present paper discusses procedures for constructing accurate response surface approximations to be used in design optimization, by tailoring the response surface to the specific design problem. A homogeneous, isotropic plate with a change in thickness is the design problem considered in the present and response surface approximations are constructed for the stress concentration factor at the thickness discontinuity and for the buckling load of the plate. These response surfaces are constructed from the results of numerical experiments conducted with a finite element analysis. It is shown that by using the proposed procedures, it is possible to obtain response surfaces with a high degree of accuracy. Graduate Research Assistant 1 Professor, Associate Fellow AIAA * Head, Structural Mechanics Branch. Fellow, AIAA Copyright

Multi-Fidelity Optimization with High-Fidelity Analysis and Low-Fidelity Gradients

The paper introduces a new approach to multi-fidelit y optimization. The approach employs gradient-based optimization, where the one-dimensional search points are evaluated using high-fidelity analysis, while the gradients are evaluated using low-fidelity analysis. Correlation between the results of the high- and low-fidelity analyses is not required. The approach is demonstrated using two example problems. Computational savings in terms of time and the number of high-fidelity analyses are discussed. I. Introduction NE of the obstacles in practical implementation of optimization in industry is a potential high computational cost. An analysis of a complex system may take several hours and even days to complete and optimization requires performing many of these analyses. The number of design variables in optimization directly affects the number of analyses: the more design variables in the problem, the more analyses should be performed. This is especially true for a gradient-based optimization, where the gradients are evaluated using finite-difference calculations. A partial answer to the computational cost problem is Response Surface optimization methods 1-7 ,12 , which do not require gradient information for optimization, thus reducing the required number of analyses. One difficulty with the Response Surface optimization methods is that their range of application is typically limited by about 20 design variables. Another approach to reducing the computational cost is multi-fidelity optimization methods 8-12 . These methods combine high and low-fidelity analyses. One example of employing multi-fidelity optimization is creating a response surface from a relatively small number of high-fidelity analyses, then performing low-fidelity analyses for the same points and creating a response surface for low-fidelity analyses. Next, a correction factor is introduced that helps converting low-fidelity analysis results into the high-fidelity analysis results. The correction may be done for the response surfaces or for the analysis results themselves. Finally, when optimization is performed using the lowfidelity analysis, the results of each low-fidelity analysis is updated using the obtained correction factor. At some intermediate stage of the optimization and at the optimum a high-fidelity analysis is performed to verify the results. If the correlation is not satisfactory, the response surfaces for high and low-fidelity analyses are recreated and the correction factor is reevaluated. The process may be repeated several times. And the correction factor itself may constitute a response surface 12 . One of the disadvantages of this approach is that the results of high and low-fidelity analyses have to be correlated periodically during the course of optimization. For a relatively large number of design variables and responses the correlation may become rather involved, particularly, if each response employs its own correction factor, bringing up the limitation in the number of design variables and responses used. The current paper proposes a modified approach to multi-fidelit y optimization, where the one-dimensional search points in gradient-base d optimization are evaluated using high-fidelity analysis and the finite difference gradient calculations are performed using low-fidelity analysis. One of the advantages of the proposed approach is that with the proper selection of high and low-fidelity analysis models there is no need to correlate the results of the two during optimization. Another advantage is that such an approach removes the potential limitation on the number of design variables and responses employed in response surface based multi-fidelity optimization.

VisualDOC: A Software System for General Purpose Integration and Design Optimization

The main purpose of this paper is to draw attention to existing commercial general-purpose optimization tools. The representative capabilities of such tools are discussed using VisualDOC by Vanderplaats Research and Development, Inc. as an example. The ease of use of VisualDOC allows a person without an optimization background to start applying optimization to his particular problem within a couple of hours after first encountering VisualDOC. This is emphasized by discussing main VisualDOC features. Particular attention is paid to several ways VisualDOC can be interfaced and/or integrated with almost any analysis program. Practical examples of applying VisualDOC to actual industrial problems are presented to emphasize the benefits of applying optimization in any field.

A two species genetic algorithm for designing composite laminates subjected to uncertainty

The high computational cost associated with genetic algorithms make the use thereof impractical in most two level optimization problems. A two species genetic algorithm is introduced as a way of overcoming this problem by effectively reducing the two level problem to a single level. The present paper describes features that had to be introduced for successful implementation of the two species genetic algorithm. These features concern the fitness evaluation as well as the rate of evolution for each of the two species. Finally the paper shows a successful application of the two species genetic algorithm to the design of a composite laminate plate, subject to uncertainty. Significant savings in computational cost compared to the corresponding two level problem are demonstrated.

Using Response Surface Methodology in Fuzzy Set Based Design Optimization

The paper focuses on the use of fuzzy set theory to model uncertainty typical of the aircraft industry, associated with design with future materials. The design problem involves maximizing the safety level of a structure for a fixed weight budget, where the structure will be built from materials not yet available. In this case little information is available regarding the uncertainty, which is described based on expert opinion and assumptions made by the designer. Therefore, fuzzy set models are appropriate for this type of uncertainty. Response surface methodology is used throughout the design process described in the present paper, mainly to reduce the computational burden associated with designing for uncertainty. Response surface methodology is also used to integrate the analysis code with the optimization algorithm and to eliminate numerical noise, which is inherent to the response function. The elimination of noise in the response function, allows the use of a derivative based optimization algorithm. An isotropic plate with a change in thickness across its width is considered as a design problem. All problem parameters are uncertain and both yield stress and buckling load failure criteria are considered. The optimum design obtained from fuzzy set theory is compared to a traditional deterministic design, which uses a safety factor to account for the uncertainty. It is shown that for the example problem considered, the fuzzy set based design is superior to the equivalent deterministic design. Also, substantial savings in computational cost are realized when using response surface methodology during the design process.

Combined Kriging and Gradient-Based Optimization Method

This paper presents a new Kriging-based optimization method. The goal of this research is to develop a practical and robust general-purpose Kriging-based optimization tool for general design problems. The proposed optimization method efficiently combines Kriging approximations with a gradient-based optimizer. The proposed method is applied to several test problems to examine its efficiency and versatility.

Accounting for Proof Test Data in a Reliability-Based Design Optimization Framework

This paper investigates the use of proof (or acceptance) test data during the reliability based design optimization of structural components. It is assumed that every component will be proof tested and that the component will only enter into service if it passes the proof test. The goal is to reduce the component weight, while maintaining high reliability, by exploiting the proof test results during the design process. The proposed procedure results in the simultaneous design of the structural component and the proof test itself and provides the designer with direct control over the probability of failing the proof test. The procedure is illustrated using two analytical example problems and the results indicate that significant weight savings are possible when exploiting the proof test results during the design process.

Response Surface Optimization with Discrete Variables

Recent advances in non-gradient based optimization methods (e.g., Genetic Algorithms, Particle Swarm Optimization) enhanced the abilities of discrete, integer, and mixed optimization problems. However, the very nature of non-gradient based algorithms is that the number of analyses required to get to an optimal solution is several orders of magnitude higher than for traditional gradient based optimization methods or response surface optimization methods, when considering continuous problems. Instead of these approaches we propose to use a response surface approximate optimization method modified to work with discrete design variables. In this case whenever it is required to perform the actual analysis of responses for the purpose of fitting a response surface approximation, the design variables will be converted to corresponding discrete values. Two discretization techniques are proposed. We demonstrate that although lacking global search properties like Genetic Algorithms and Particle Swarm Optimization, the discrete response surface optimization provides a computationally efficient way of improving an initial design and getting into a region of an optimum using only discrete points for analysis.

Efficient Global Optimization with Experimental Data: Revisiting the Paper Helicopter Design

Experimental optimization has been used since the early 20th century to help farmers maximize yields (defining inputs such as water and fertilizer). The traditional approach iterates in cycles consisting of fitting a polynomial to samples (that differed in the set of input variables) and optimizing the fitted surrogate. In each cycle, a set of designs is defined and tested. Although engineering design relies mostly on computer experiments, there are cases where simulations are expensive enough and the system is cheap enough to manufacture and test to favor experimental over analytical optimization. In this paper, we use the design of a paper helicopter to illustrate how we can adapt the modern efficient global optimization (EGO) algorithm to handle experimental data. The objective is to maximize the time a simple paper helicopter takes to fall from a specific height. We propose running EGO with multiple surrogates (MSEGO) for generating not only one, but multiple candidate designs per optimization cycle. Here, we use kriging, radial basis neural network, linear Shepard, and support vector regression. We also heavily penalize regions of the design space where designs are predicted to fail, using support vector classification to define the failure region. We found MSEGO reduced the impact of failed designs, allowed for exploration of the design space, and improved the fall time by 10% .