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OC1: Optimisation and Control Location: A702 Session Chair: Behzad Azmi | |
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Riemannian gradient descent along geodesic paths for shape optimization using isogeometric analysis 1: Helmut Schmidt University / University of the Federal Armed Forces Hamburg, Germany; 2: RPTU University Kaiserslautern-Landau, Germany A common procedure in isogeometric shape optimization is to use a NURBS description of the geometry and consider a subset of control point coordinates as design variables [1, 2, 3], which corresponds to the first discretize–then optimize approach. While this is a fairly straightforward way to perform shape optimization in the context of isogeometric analysis, the results will usually depend on the particular representation of the geometry, since the problem is reduced to a finite-dimensional one that often disregards the inherent geometric structure of shapes. We present the Riemannian framework for PDE constrained shape optimization [4] in an attempt to provide suitable descent paths for control points that evolve the shape in a consistent manner independent of the chosen representation. The idea is to endow the shape space with a reparametrization-invariant Riemannian metric [5] and move the shapes along geodesic paths corresponding to the metric, which allows for mesh-independent shape optimization. This first optimize–then discretize approach has been applied to the compliance minimization of thin elastic shells using a first-order Sobolev-type metric in [6]. We now investigate the effects of using geodesic retractions for optimization on Riemannian shape spaces of planar curves and surfaces, where isogeometric finite element methods are used for the discretization of the equations to efficiently obtain accurate numerical approximations of the Riemannian shape gradient. References: [1] W. Wall, M. Frenzel, and C. Cyron. Isogeometric structural shape optimization. Computer Methods in Applied Mechanics and Engineering, 197:2976–2988, 2008. [2] K.-U. Bletzinger, J. Kiendl, R. Schmidt, and R. Wüchner. Isogeometric shape optimization of shells using semi-analytical sensitivity analysis and sensitivity weighting. Computer Methods in Applied Mechanics and Engineering, 274:148–167, 2014. [3] J. Lopez, C. Anitescu, and T. Rabczuk. Isogeometric structural shape optimization using automatic sensitivity analysis. Applied Mathematical Modelling, 89:1004–1024, 2021. [4] V. H. Schulz, M. Siebenborn, and K. Welker. PDE constrained shape optimization as optimization on shape manifolds. Geometric Science of Information, volume 9389 of Lecture Notes in Computer Science, pages 499–508. Springer, 2015. [5] M. Bauer, P. Harms, and P. W. Michor. Sobolev metrics on shape space of surfaces. Journal of Geometric Mechanics, 3(4):389–438, 2011. [6] R. Rosandi and B. Simeon. Riemannian shape optimization of thin shells using isogeometric analysis. Proceedings in Applied Mathematics and Mechanics, 25(1):e202400204, 2025. | |
