Abstract
We investigate and computationally solve a shape optimization problem constrained by a variational inequality of the first kind, a so-called obstacle-type problem, with a gradient descent and a BFGS algorithm in the space of smooth shapes. In order to circumvent the numerical problems related to the non-linearity of the shape derivative, we consider a regularization strategy leading to novel possibilities to numerically exploit structures, as well as possible treatment of the regularized variational inequality constrained shape optimization in the context of optimization on infinite dimensional Riemannian manifolds.
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References
Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)
Alnæs, M.S., et al.: The FEniCS project version 1.5. Arch. Numer. Softw. 3(100), 9–23 (2015)
Bock, H.: Numerical treatment of inverse problems in chemical reaction kinetics. In: Ebert, K., Deuflhard, P., Jäger, W. (eds.) Modelling of Chemical Reaction Systems. Springer Series Chemical Physics, vol. 18, pp. 102–125. Springer, Heidelberg (1981). https://doi.org/10.1007/978-3-642-68220-9_8
Denkowski, Z., Migorski, S.: Optimal shape design for hemivariational inequalities. Universitatis Iagellonicae Acta Matematica 36, 81–88 (1998)
Führ, B., Schulz, V., Welker, K.: Shape optimization for interface identification with obstacle problems. Vietnam. J. Math. (2018). https://doi.org/10.1007/s10013-018-0312-0
Gangl, P., Laurain, A., Meftahi, H., Sturm, K.: Shape optimization of an electric motor subject to nonlinear magnetostatics. SIAM J. Sci. Comput. 37(6), B1002–B1025 (2015)
Gasiński, L.: Mapping method in optimal shape design problems governed by hemivariational inequalities. In: Cagnol, J., Polis, M., Zolésio, J.-P. (eds.) Shape Optimization and Optimal Design. Lecture Notes in Pure and Applied Mathematics, vol. 216, pp. 277–288. Marcel Dekker, New York (2001)
Hintermüller, M., Laurain, A.: Optimal shape design subject to elliptic variational inequalities. SIAM J. Control. Optim. 49(3), 1015–1047 (2011)
Ito, K., Kunisch, K.: Semi-smooth Newton methods for variational inequalities and their applications. M2AN Math. Model. Numer. Anal. 37, 41–62 (2003)
Ito, K., Kunisch, K.: Semi-smooth Newton methods for variational inequalities of the first kind. ESAIM Math. Model. Numer. Anal. 37(1), 41–62 (2003)
Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications, vol. 31. SIAM, Philadelphia (1980)
Kocvara, M., Outrata, J.: Shape optimization of elasto-plastic bodies governed by variational inequalities. In: Zolésio, J.-P. (ed.) Boundary Control and Variation. Lecture Notes in Pure and Applied Mathematics, vol. 163, pp. 261–271. Marcel Dekker, New York (1994)
Kriegl, A., Michor, P.: The Convient Setting of Global Analysis. Mathematical Surveys and Monographs, vol. 53. American Mathematical Society, Providence (1997)
Liu, W., Rubio, J.: Optimal shape design for systems governed by variational inequalities, Part 1: existence theory for the elliptic case. J. Optim. Theory Appl. 69(2), 351–371 (1991)
Luft, D., Schulz, V.H., Welker, K.: Efficient techniques for shape optimization with variational inequalities using adjoints. arXiv:1904.08650 (2019)
Michor, P., Mumford, D.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. 8, 1–48 (2006)
Michor, P., Mumford, D.: An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach. Appl. Comput. Harmon. Anal. 23, 74–113 (2007)
Myśliński, A.: Level set method for shape and topology optimization of contact problems. In: Korytowski, A., Malanowski, K., Mitkowski, W., Szymkat, M. (eds.) CSMO 2007. IAICT, vol. 312, pp. 397–410. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04802-9_23
Schmidt, S., Ilic, C., Schulz, V., Gauger, N.: Three dimensional large scale aerodynamic shape optimization based on the shape calculus. AIAA J. 51(11), 2615–2627 (2013)
Schulz, V., Welker, K.: On optimization transfer operators in shape spaces. In: Schulz, V., Seck, D. (eds.) Shape Optimization. Homogenization and Optimal Control, pp. 259–275. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-90469-6_13
Schulz, V.H., Siebenborn, M., Welker, K.: Efficient PDE constrained shape optimization based on Steklov-Poincaré type metrics. SIAM J. Optim. 26(4), 2800–2819 (2016)
Sokolowski, J., Zolésio, J.-P.: An Introduction to Shape Optimization. Springer, Heidelberg (1992). https://doi.org/10.1007/978-3-642-58106-9
Udawalpola, R., Berggren, M.: Optimization of an acoustic horn with respect to efficiency and directivity. Int. J. Numer. Methods Eng. 73(11), 1571–1606 (2007)
Welker, K.: Optimization in the space of smooth shapes. In: Nielsen, F., Barbaresco, F. (eds.) GSI 2017. LNCS, vol. 10589, pp. 65–72. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-68445-1_8
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Luft, D., Welker, K. (2019). Computational Investigations of an Obstacle-Type Shape Optimization Problem in the Space of Smooth Shapes. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science(), vol 11712. Springer, Cham. https://doi.org/10.1007/978-3-030-26980-7_60
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DOI: https://doi.org/10.1007/978-3-030-26980-7_60
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