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Structural and Multidisciplinary Optimization

, Volume 60, Issue 6, pp 2305–2323 | Cite as

A modified search direction method for inequality constrained optimization problems using the singular-value decomposition of normalized response gradients

  • Long ChenEmail author
  • Kai-Uwe Bletzinger
  • Armin Geiser
  • Roland Wüchner
Research Paper
  • 537 Downloads

Abstract

Motivated by the applications of the node-based shape optimization problems, where various response evaluations are often considered in constrained optimization, we propose a modified search direction method for optimization design updates. There exist numerous methods for solving constrained optimization problems. The methods of the class active-set strategy try to travel along the active constraints to solve constrained optimization problems. Contrary to the active-set methods, the algorithms of the class interior-point method reach an optimal solution by traversing the interior of the feasible domain. This characteristic is considered to be beneficial for shape optimization because the usual zig-zagging behavior when traveling along the active constraints is avoided. However, the interior-point methods require the solution of a Newton problem in each iteration step, which is generally considered to be difficult for the shape optimization problem. In the present work, we propose a modified search direction method that applies the singular-value decomposition method for the normalized objective and constraint gradients. The modified search direction is a descent direction of the objective function. Using this search direction for the design update in each iteration, we observe that the centrality conditions for the interior-point method are approached iteratively. We are able to achieve a local minimum by traversing the interior of the feasible domain by only using the gradient information of the objective and constraint functions. The results are shown first with analytical 2D problems, and then the results of shape optimization problems with a large number of design variables are discussed. To robustly deal with complex geometries, the Vertex Morphing method is used.

Keywords

Gradient-based constrained optimization Centrality conditions Singular-value decomposition Shape optimization Vertex Morphing 

Notes

Funding information

This work was supported by the Deutsche Forschungsgemeinschaft (DFG) as part of the SPP project “Polymorphic uncertainty modelling for the numerical design of structures.”

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Technical University of MunichMunichGermany

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