The Linear Interpolation Approach (LInA), an approach to speed up the Successive Response Surface Method
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RESEARCH PAPER
The Linear Interpolation Approach (LInA), an approach to speed up the Successive Response Surface Method Alexander Timmer 1,2 & Rainer Immel 1
&
Lothar Harzheim 1,2
Received: 19 December 2019 / Revised: 15 June 2020 / Accepted: 9 July 2020 # Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Metamodel-based approaches are very often used for nonlinear optimization. The Successive Response Surface Method (SRSM) (Roux et al. in Int J Numer Methods Eng 42:517–534, 1998; Stander and Craig in Eng Comput 19:431–450, 2002) is one example for the application of such an approach. SRSM computes a smoothing approximation for each system response in each iteration based on a set of sample points generated in a defined domain around the current optimum. This domain is shifted and contracted in each iteration according to the resulting current best point. By default, a linear polynomial is used as the approximation function. A new Linear Interpolation Approach (LInA) has been developed to reduce the number of nonlinear analyses required during the optimization. In contrast to SRSM, LInA uses interpolating linear polynomials rather than regressions. Linear approximation is used to exclude artificial local minima between sample points. Due to the fact that no oversampling is required anymore, interpolation results in a lower number of nonlinear analyses without losing accuracy. A second reduction of the required analyses is achieved by avoiding a complete resampling in each iteration. It is assumed that the vectors defined from the current best point to the remaining sample points should be as orthogonal as possible. Therefore, LInA starts with initial sample points from a Koshal design of experiments (DOE) locally around the initial design point. After having computed the new best point based on the linear approximation, only a few existing sample points are deleted in each iteration and replaced by new ones. Complete QR decomposition is used to generate the new sample points in an orthogonal complementary subspace. First studies with two nonlinear structural optimization problems have shown that the LInA approach leads to a noteworthy reduction of the number of required nonlinear analyses in comparison to SRSM. Furthermore, LInA is compared to two kriging approaches using a subdomain similar as LInA does. LInA shows results comparable to kriging with respect to the number of analysis runs. Keywords Nonlinear optimization . Metamodel . SRSM . Linear interpolating polynomials . Reuse of sample points . QR decomposition . Local kriging
1 Introduction Simulation-based structural optimization using the finite element method (FEM) to compute system responses has been established as an important tool in product development (Harzheim 2014), especially if it is based on linear analysis. Responsible Editor: Pingfeng Wang * Rainer Immel [email protected] 1
Opel Automobile GmbH, 65423 Rüsselsheim, Germany
2
Technische Universität Darmstadt, fnb, Darmstadt, Germany
For this purpose, many
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