Parameter optimization in multiquadric response surface approximations

Multiquadric (MQ) response surface approximation uses the function C_i|X-X_i|²+h^1/2 to interpolate a given set of data. The performance of MQ approximation depends on the shift parameter h. Efficient methods of computing the optimal shift parameter based on the leave-one-out cross validation technique are presented in this paper. We also proved that the condition number of the MQ coefficient matrix is an increasing function of the shift parameter h. Two numerical examples are included to illustrate the proposed formulation.     

Design optimization through parallel-generated surrogate models, optimization methodologies and the utility of legacy simulation software

This paper presents the motivation development and an application of a unique methodology to solve industrial optimization problems, using existing legacy simulation software programs. The methodology is based on approximation models generated with the utility of design of experiments methodologies and response surface methods applied on high-fidelity simulations, coupled together with classical optimization methodologies. Several DOE plans are included, in order to be able to adopt the appropriate level of detail. The approximations are based on stochastic interpolation techniques, or on classical least squares methods. The optimization methods include both local and global techniques. Finally, an application from the plastic molding industry (process simulation) demonstrates the methodology and the software package. Received December 30, 2000     

Multipoint cubic surrogate function for sequential approximate optimization

Multipoint cubic approximations are investigated as surrogate functions for nonlinear objective and constraint functions in the context of sequential approximate optimization. The proposed surrogate functions match actual function and gradient values, including the current expansion point, thus satisfying the zero and first-order necessary conditions for global convergence to a local minimum of the original problem. Function and gradient information accumulated from multiple design points during the iteration history is used in estimating a reduced Hessian matrix and selected cubic terms in a design subspace appropriate for problems with many design variables. The resulting approximate response surface promises to accelerate convergence to an optimal design within the framework of a trust region algorithm. The hope is to realize computational savings in solving large numerical optimization problems. Numerical examples demonstrate the effectiveness of the new multipoint surrogate function in reducing errors over large changes in design variables.     

Two-level optimization of airframe structures using response surface approximation

In this paper, a two-level optimization approach is developed for the preliminary and conceptual design of airframe structures. The preliminary design, involving a single objective multidisciplinary optimization, constitutes the lower level where ASTROS (Automated STRuctural Optimization System) is employed for multidisciplinary optimization. The conceptual design, which is carried out at the upper level, aims mainly at configuration design. The multiple objectives are incorporated as a single objective function by using the K-S function formulation. The objective function and constraints at the upper level are modelled through response surface approximation. During the upper level optimization process, the branch and bound method is applied for solving the problem with discrete design variables. The proposed strategy is demonstrated by the optimization of an Intermediate Complexity Wing (ICW) model. Received June 23, 1999     

一种新颖的混合响应面优化方法

为减少黑箱优化过程中的评估次数,提出了一种新颖的混合响应面优化方法(HRSO),利用混合响应面建立高精度的近似模型作为代理模型,通过迭代更新响应面不断接近真实模型,从而完成优化。以Dixon-Szego函数类作为测试函数,以评估次数为方法性能优劣的评价指标,实验结果表明,与Gutmann-RBF、CORS-RBF两种方法相比,HRSO能够在较少的评估次数内满足相同的收敛条件,且向全局快速收敛,是一种适合求解黑箱优化问题的方法。     

Update strategies for kriging models used in variable fidelity optimization

Many optimization methods for simulation-based design rely on the sequential use of metamodels to reduce the associated computational burden. In particular, kriging models are frequently used in variable fidelity optimization. Nevertheless, such methods may become computationally inefficient when solving problems with large numbers of design variables and/or sampled data points due to the expensive process of optimizing the kriging model parameters in each iteration. One solution to this problem would be to replace the kriging models with traditional Taylor series response surface models. Kriging models, however, were shown to provide good approximations of computer simulations that incorporate larger amounts of data, resulting in better global accuracy. In this paper, a metamodel update management scheme (MUMS) is proposed to reduce the cost of using kriging models sequentially by updating the kriging model parameters only when they produce a poor approximation. The scheme uses the trust region ratio (TR-MUMS), which is a ratio that compares the approximation to the true model. Two demonstration problems are used to evaluate the proposed method: an internal combustion engine sizing problem and a control-augmented structural design problem. The results indicate that the TR-MUMS approach is very effective; on the demonstration problems, it reduced the number of likelihood evaluations by three orders of magnitude compared to using a global optimizer to find the kriging parameters in every iteration. It was also found that in trust region-based method, the kriging model parameters need not be updated using a global optimizer—local methods perform just as well in terms of providing a good approximation without affecting the overall convergence rate, which, in turn, results in a faster execution time.     

Approximation methods in multidisciplinary analysis and optimization: a panel discussion

This paper summarizes the discussion at the Approximation Methods Panel that was held at the 9^thAIAA/ISSMO Symposium on Multidisciplinary Analysis & Optimization in Atlanta, GA on September 2–4, 2002. The objective of the panel was to discuss the current state-of-the-art of approximation methods and identify future research directions important to the community. The panel consisted of five representatives from industry and government: (1) Andrew J. Booker from The Boeing Company, (2) Dipankar Ghosh from Vanderplaats Research & Development, (3) Anthony A. Giunta from Sandia National Laboratories, (4) Patrick N. Koch from Engineous Software, Inc., and (5) Ren-Jye Yang from Ford Motor Company. Each panelist was asked to (i) give one or two brief examples of typical uses of approximation methods by his company, (ii) describe the current state-of-the-art of these methods used by his company, (iii) describe the current challenges in the use and adoption of approximation methods within his company, and (iv) identify future research directions in approximation methods. Several common themes arose from the discussion, including differentiating between design of experiments and design and analysis of computer experiments, visualizing experimental results and data from approximation models, capturing uncertainty with approximation methods, and handling problems with large numbers of variables. These are discussed in turn along with the future directions identified by the panelists, which emphasized educating engineers in using approximation methods.     

Polynomial genetic programming for response surface modeling Part 2: adaptive approximate models with probabilistic optimization problems

This is the second in a series of papers. The first deals with polynomial genetic programming (PGP) adopting the directional derivative-based smoothing (DDBS) method, while in this paper, an adaptive approximate model (AAM) based on PGP is presented with the partial interpolation strategy (PIS). The AAM is sequentially modified in such a way that the quality of fitting in the region of interest where an optimum point may exist can be gradually enhanced, and accordingly the size of the learning set is gradually enlarged. If the AAM uses a smooth high-order polynomial with an interpolative capability, it becomes more and more difficult for PGP to obtain smooth polynomials, whose size should be larger than or equal to the number of the samples, because the order of the polynomial becomes unnecessarily high according to the increase in its size. The PIS can avoid this problem by selecting samples belonging to the region of interest and interpolating only those samples. Other samples are treated as elements of the extended data set (EDS). Also, the PGP system adopts a multiple-population approach in order to simultaneously handle several constraints. The PGP system with the variable-fidelity response surface method is applied to reliability-based optimization (RBO) problems in order to significantly cut the high computational cost of RBO. The AAMs based on PGP are responsible for fitting probabilistic constraints and the cost function while the variable-fidelity response surface method is responsible for fitting limit state equations. Three numerical examples are presented to show the performance of the AAM based on PGP.     

Multi-criteria shape optimization of a funnel in cathode ray tubes using a response surface model

One purpose of simulation describing the behaviors of structures is to optimize the performances within specific functional requirements and customers needs with respect to the design variables. For reduction of the volume of cathode ray tubes, the design of the glass geometry, especially funnel geometry, is essential while maintaining the internal vacuum pressure of the cathode ray tube. In order to describe the three-dimensional geometry of the funnel in cathode ray tubes, a higher-order response surface model is employed in the simulation model instead of non-uniform rational B-splines (NURBS) or Bezier curves because the response surface model is more robust for understanding the geometry change in finite element analysis. We formulate the design problem as a multi-criteria optimization because minimization of both volume and maximum stress is required. Using the response surface model of the geometry of the funnel and sequential quadratic programming within the process integration framework, the shape optimization of a funnel is successfully performed and the maximum stress level of the funnel is decreased by almost half.     

Sequential kriging optimization using multiple-fidelity evaluations

When cost per evaluation on a system of interest is high, surrogate systems can provide cheaper but lower-fidelity information. In the proposed extension of the sequential kriging optimization method, surrogate systems are exploited to reduce the total evaluation cost. The method utilizes data on all systems to build a kriging metamodel that provides a global prediction of the objective function and a measure of prediction uncertainty. The location and fidelity level of the next evaluation are selected by maximizing an augmented expected improvement function, which is connected with the evaluation costs. The proposed method was applied to test functions from the literature and a metal-forming process design problem via finite element simulations. The method manifests sensible search patterns, robust performance, and appreciable reduction in total evaluation cost as compared to the original method.     

Sequential approximate optimization using dual subproblems based on incomplete series expansions

In this paper, dual formulations for nonlinear multipoint approximations with diagonal approximate Hessian matrices are proposed; these approximations for example derive from the incomplete series expansion (ISE) proposed previously. A salient feature of the ISE is that it may be used to formulate strictly convex and separable (recast) primal approximate subproblems for use in sequential approximate optimization (SAO). In turn, this allows for the formulation of highly efficient dual formulations, and different combinations of direct, reciprocal, and exponential intervening variables for the objective and the constraint functions may be used. Two frequently encountered problems in structural optimization, namely the weight minimization problem with sizing design variables and the minimum compliance topology optimization problem, are degenerate cases of the formulations we present. Computational experiments confirm the efficiency of our proposed methodology; to this end, comparative results for the method of moving asymptotes (MMA) are presented. Based on the paper entitled “Duality in Convex Nonlinear Multipoint Approximations with Diagonal Approximate Hessian Matrices Deriving from Incomplete Series Expansions,” presented at the 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Portsmouth, VA, USA, September 2006, paper no. AIAA-2006-7090.     

基于响应面的概率集群优化方法

面对实际应用中的大规模优化问题,基于响应面估计的概率集群优化方法以设计变量的概率分布作为优化对象,而非直接对设计变量值进行优化,可适应连续、离散及混合的设计变量类型。采用响应面构建概率集群评估函数的近似模型,并采用置信区间方法在迭代优化过程中不断更新响应曲面以确保近似精度。实验结果表明算法对解决复杂优化问题有效。     

Nonlinear diffusion process modeling using response surface methodology and variable transformation

The nonlinear process modeling is investigated using statistical design method and response surface methodology. Three input factors are examined with respect to the response factor. In order to minimize the joint confidence region of fabrication process with varying conditions, D-optimal experimental design technique is performed and diffusion rate is characterized by response model. Then, the statistical results are used to verify the fitness of the nonlinear process model. Based on the results, this modeling methodology can be optimized process condition for semiconductor manufacturing.     

Expedited simulation‐driven design optimization of UWB antennas by means of response features

In this work, a method for fast design optimization of broadband antennas is considered. The approach is based on a feature‐based optimization (FBO) concept where reflection characteristics of the structure at hand are formulated in terms of suitably defined feature points. Redefinition of the design problem allows for reducing the design optimization cost, because the dependence of feature point coordinates on antenna dimensions is less nonlinear than for the original frequency characteristics (here, S‐parameters). This results in faster convergence of the optimization algorithm. The cost of the design process is further reduced using variable‐fidelity electromagnetic (EM) simulation models. In case of UWB antennas, the feature points are defined, among others, as the levels of the reflection characteristic at its local in‐band maxima, as well as location of the frequency point which corresponds to acceptable reflection around the lower corner frequency within the UWB band. Also, the number of characteristic points depends on antenna topology and its dimensions. Performance of FBO‐based design optimization is demonstrated using two examples of planar UWB antennas. Moreover, the computational cost of the approach is compared with conventional optimization driven by a pattern search algorithm. Experimental validation of the numerical results is also provided.     

Material identification in structural optimization using response surfaces

This paper evaluates the use of a response surface optimization algorithm for structural material or parameter identification. The algorithm used is the successive response surface method (SRSM) as implemented in LS-OPT. Two methods are used in the formulation of the optimization problem. The first is to minimize the maximum deviation of the distance function between the simulated and experimental results at selected points, while the second approach minimizes the more standard least squares residual form of the distance function, effectively providing a compromised match over all the parameters selected. SRSM uses a trust region that is adapted using a heuristic contraction and panning approach. The method has only one user-required parameter, the size of the initial trust region. To illustrate the robustness of SRSM as a material identification tool, three test cases are presented. The first concerns the identification of the power-law material parameters of a simple tensile test specimen. The second test case determines the leakage coefficient-pressure load curve of an airbag given experimental kinematic data of a chest form impacting the airbag. In the third test case the material identification of a rate-dependent low-density foam material is conducted. It is shown that SRSM essentially converges within 10 iterations for all the test cases, and that the two distance function minimization approaches produce similar results.     

An optimal design of magnetic actuators using topology optimization and the response surface method

The magnetic actuator is a device that transforms electric energy to mechanical energy. By mechanical energy transformation, some part of the electric energy creates force, and the other part is stored within the ferrous material. An actuator with improved magnetic force can be designed by reducing the stored energy within the ferrous material at the core or the armature. Topology optimization based on the homogenization design method (HDM) is used for the initial design by determining the porous hole size of each element created. The homogenized magnetic permeability is applied in calculation of the magnetic energy stored. The magnetic energy is calculated by finite element analysis and the sensitivity is calculated mathematically by determining the effects of the magnetic energy according to the permeability change at each element. Repeating the process of the porous hole size determination by the sequential linear programming (SLP), eventually leads to a design of an actuator that makes the most improved magnetic force within the limited volume. The initial actuator model derived from topology optimization uses parameter optimization for detail designs. In parameter optimization design, the response surface method (RSM) based on the central composite design is used to obtain a clear final design.     

Reliability-based design optimization using probabilistic sufficiency factor

A probabilistic sufficiency factor approach is proposed that combines safety factor and probability of failure. The probabilistic sufficiency factor approach represents a factor of safety relative to a target probability of failure. It provides a measure of safety that can be used more readily than the probability of failure or the safety index by designers to estimate the required weight increase to reach a target safety level. The probabilistic sufficiency factor can be calculated from the results of Monte Carlo simulation with little extra computation. The paper presents the use of probabilistic sufficiency factor with a design response surface approximation, which fits it as a function of design variables. It is shown that the design response surface approximation for the probabilistic sufficiency factor is more accurate than that for the probability of failure or for the safety index. Unlike the probability of failure or the safety index, the probabilistic sufficiency factor does not suffer from accuracy problems in regions of low probability of failure when calculated by Monte Carlo simulation. The use of the probabilistic sufficiency factor accelerates the convergence of reliability-based design optimization.