Algorithmic Aspects of Isogeometric Shape Optimization
Shape optimization is concerned about finding optimal designs under the aspect of some cost criteria often involving the solution of a partial differential equation (PDE) over the afore said unknown shape. In general, industrial cases involve a geometric model from Computer Aided Design (CAD). However, solving PDEs requires an analysis suitable working model, typically a Finite Element (FEM) triangulation. Hence, some of the geometric properties known from the CAD model may be lost during this format change. Therefore, we employ isogeometric analysis (IGA) instead, which has a tighter connection between geometry, simulation and shape optimization. In this paper, we present a self-contained treatment of gradient based shape optimization method with isogeometric analysis, focusing on algorithmic and practical aspects like computation of shape gradients in an IGA formulation and updating B-spline and NURBS geometries.
KeywordsShape Optimization Variable Weight Sequential Quadratic Programming Isogeometric Analysis Shape Optimization Problem
We thank Utz Wever from Siemens AG, Corporate Technology, for many helpful discussions on the subject.
This work was supported by the European Union within the Project 284981 “TERRIFIC” (7th Framework Program).
- 5.M.C. Delfour, J.P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization. Advances in Design and Control, 2nd edn. (SIAM, Philadelphia, 2011)Google Scholar
- 10.S.G. Johnson, The NLOPT nonlinear-optimization package (2014), http://ab-initio.mit.edu/nlopt. [Online; Accessed 10 Oct 2014]
- 12.MATLAB, Release 2012a (2012), “The MathWorks Inc.”Google Scholar
- 14.D.M. Nguyen, Isogeometric analysis and shape optimization in electromagnetism. Ph.D. thesis, Technical University of Denmark, Feb 2012Google Scholar
- 16.J. Nocedal, S.J. Write, Numerical Optimization. Springer Series in Operations Research and Financial Engineering, 2nd edn. (Springer, New York, 2006)Google Scholar
- 22.J. Sokolowski, J.P. Zolésio, Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer Series in Computational Mathematics, vol. 16 (Springer, Berlin/New York, 1992)Google Scholar