Model reduction and level set methods for shape optimization problems

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Aalto-yliopiston teknillinen korkeakoulu | Doctoral thesis (article-based)
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Verkkokirja (366 KB, 31 s.)
Research reports / Helsinki University of Technology, Institute of Mathematics. A, 593
In this work two topics related to mathematical shape optimization are considered. Topological optimization methods need not know the correct topology (number of connected components and possible holes) of the optimal shape beforehand. Shape optimization can be performed by a topological gradient descent algorithm. Computational applications of topological optimization and level set based shape optimization involve the optimal damping of vibrating structures and an inverse problem of reconstructing a shape based on noisy interferogram measurements. For parametric shape optimization with partial differential constraints the model reduction approach of reduced basis methods is considered. In the reduced basis method a basis of snapshot solutions is used to construct a problem-dependent approximation space that has much smaller dimension than the underlying finite element approximations. The state constraints for optimization are then replaced with their reduced basis approximation, leading to efficient shape optimization methods. Computational examples involve the optimal engineering design of airfoils in potential and thermal flow.
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Eirola, Timo, Prof.
shape optimization, topological optimization, level set method, model reduction, reduced basis method, partial differential equations, 35R35, 49Q10, 65K10
Other note
  • [Publication 1]: Toni Lassila. 2009. Optimal damping of a membrane and topological shape optimization. Structural and Multidisciplinary Optimization, volume 38, number 1, pages 43-52.
  • [Publication 2]: Timo Eirola and Toni Lassila. 2009. Optimization of convex shapes: An approach to crystal shape identification. In: Xue-Cheng Tai, Knut Mørken, Marius Lysaker, and Knut-Andreas Lie (editors). Proceedings of the 2nd International Conference on Scale Space and Variational Methods in Computer Vision (SSVM 2009). Voss, Norway. 1-5 June 2009. Berlin, Heidelberg, Germany. Springer. Lecture Notes in Computer Science, volume 5567, pages 660-671. ISBN 978-3-642-02255-5.
  • [Publication 3]: Toni Lassila and Gianluigi Rozza. 2010. Parametric free-form shape design with PDE models and reduced basis method. Computer Methods in Applied Mechanics and Engineering, volume 199, numbers 23-24, pages 1583-1592. © 2010 Elsevier Science. By permission.
  • [Publication 4]: Gianluigi Rozza, Toni Lassila, and Andrea Manzoni. 2011. Reduced basis approximation for shape optimization in thermal flows with a parametrized polynomial geometric map. In: Jan S. Hesthaven and Einar M. Rønquist (editors). Spectral and High Order Methods for Partial Differential Equations. Selected Papers from the 8th International Conference on Spectral and High Order Methods (ICOSAHOM 2009). Trondheim, Norway. 22-26 June 2009. Berlin, Heidelberg, Germany. Springer. Lecture Notes in Computational Science and Engineering, volume 76, pages 307-315. ISBN 978-3-642-15336-5. © 2011 by authors and © 2011 Springer Science+Business Media. By permission.
  • [Publication 5]: Toni Lassila and Gianluigi Rozza. 2010. Reduced formulation of a steady fluid-structure interaction problem with parametric coupling. arXiv:1005.3384v1 [math.NA]. Previously published in: R. A. E. Mäkinen, P. Neittaanmäki, T. Tuovinen, and K. Valpe (editors). Proceedings of the 10th Finnish Mechanics Days. Jyväskylä, Finland. 3-4 December 2009. 10 pages. ISBN 978-951-39-3738-6. © 2010 by authors.