401-3640-13L  Seminar in Applied Mathematics: Shape Calculus

SemesterAutumn Semester 2018
LecturersR. Hiptmair
Periodicitynon-recurring course
Language of instructionEnglish
CommentNumber of participants limited to 10


AbstractShape calculus studies the dependence of solutions of partial differential equations on deformations of the domain and/or interfaces. It is the foundation of gradient methods for shape optimization. The seminar will rely on several sections of monographs and research papers covering analytical and numerical aspects of shape calculus.
Objective* Understanding of concepts like shape derivative, shape gradient, shape Hessian, and adjoint problem.
* Ability to derive analytical formulas for shape gradients
* Knowledge about numerical methods for the computation of shape gradients.
ContentTopics:

1. The velocity method and Eulerian shape gradients: Main reference [SZ92, Sect. 2.8–2.11, 2.1, 2.18], covers the “velocity method”, the Hadamard structure theorem and formulas for shape gradients of particular functionals. Several other sections of [SZ92,Ch. 2] provide foundations and auxiliary results and should be browsed, too.

2. Material derivatives and shape derivatives, based on [SZ92, Sect. 2.25–2.32].

3. Shape calculus with exterior calculus, following [HL13] (without Sections 5 & 6). Based on classical vector analysis the formulas are also derived in [SZ92, Sects 2,19,2.20] and [DZ10, Ch. 9, Sect. 5]. Important background and supplementary information about the shape Hessian can be found in [DZ91, BZ97] and [DZ10, Ch. 9, Sect. 6].

4. Shape derivatives of solutions of PDEs using exterior calculus [HL17], see also [HL13,Sects. 5 & 6]. From the perspective of classical calculus the topic is partly covered in [SZ92, Sects. 3.1-3.2].

5. Shape gradients under PDE constraints according to [Pag16, Sect. 2.1] including a presentation of the adjoint method for differentiating constrained functionals [HPUU09, Sect. 1.6]. Related information can be found in [DZ10, Ch. 10, Sect. 2.5] and [SZ92, Sect. 3.3].

6. Approximation of shape gradients following [HPS14]. Comparison of discrete shape gradients based on volume and boundary formulas, see also [DZ10, Ch. 10, Sect. 2.5].

7. Optimal shape design based on boundary integral equations following [Epp00b], with some additional information provided in [Epp00a].

8. Convergence in elliptic shape optimization as discussed in [EHS07]. Relies on results reported in [Epp00b] and [DP00]. Discusses Ritz-Galerkin discretization of optimality conditions for normal displacement parameterization.

9. Shape optimization by pursuing diffeomorphisms according to [HP15], see also [Pag16,Ch. 3] for more details, and [PWF17] for extensions.

10. Distributed shape derivative via averaged adjoint method following [LS16].
LiteratureReferences:

[BZ97] Dorin Bucur and Jean-Paul Zolsio. Anatomy of the shape hessian via
lie brackets. Annali di Matematica Pura ed Applicata, 173:127–143, 1997.
10.1007/BF01783465.

[DP00] Marc Dambrine and Michel Pierre. About stability of equilibrium shapes. M2AN Math. Model. Numer. Anal., 34(4):811–834, 2000.

[DZ91] Michel C. Delfour and Jean-Paul Zolésio. Velocity method and Lagrangian formulation for the computation of the shape Hessian. SIAM J. Control Optim.,
29(6):1414–1442, 1991.

[DZ10] M.C. Delfour and J.-P. Zolésio. Shapes and Geometries, volume 22 of Advances in Design and Control. SIAM, Philadelphia, 2nd edition, 2010.

[EHS07] Karsten Eppler, Helmut Harbrecht, and Reinhold Schneider. On convergence in elliptic shape optimization. SIAM J. Control Optim., 46(1):61–83 2007.

[Epp00a] Karsten Eppler. Boundary integral representations of second derivatives in shape optimization. Discuss. Math. Differ. Incl. Control Optim., 20(1):63–78, 2000.
German-Polish Conference on Optimization—Methods and Applications (Żagań,
1999).

[Epp00b] Karsten Eppler. Optimal shape design for elliptic equations via BIE-methods. Int. J. Appl. Math. Comput. Sci., 10(3):487–516, 2000.

[HL13] Ralf Hiptmair and Jingzhi Li. Shape derivatives in differential forms I: an intrinsic perspective. Ann. Mat. Pura Appl. (4), 192(6):1077–1098, 2013.

[HL17] R. Hiptmair and J.-Z. Li. Shape derivatives in differential forms II: Application
to scattering problems. Report 2017-24, SAM, ETH Zürich, 2017. To appear in
Inverse Problems.

[HP15] Ralf Hiptmair and Alberto Paganini. Shape optimization by pursuing diffeomorphisms. Comput. Methods Appl. Math., 15(3):291–305, 2015.

[HPS14] R. Hiptmair, A. Paganini, and S. Sargheini. Comparison of approximate shape gradients. BIT Numerical Mathematics, 55:459–485, 2014.

[HPUU09] M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich. Optimization with PDE constraints, volume 23 of Mathematical Modelling: Theory and Applications. Springer, New York, 2009.

[LS16] Antoine Laurain and Kevin Sturm. Distributed shape derivative via averaged adjoint method and applications. ESAIM Math. Model. Numer. Anal., 50(4):1241–1267,2016.

[Pag16] A. Paganini. Numerical shape optimization with finite elements. Eth dissertation 23212, ETH Zurich, 2016.

[PWF17] A. Paganini, F. Wechsung, and P.E. Farell. Higher-order moving mesh methods for pde-constrained shape optimization. Preprint arXiv:1706.03117 [math.NA], arXiv, 2017.

[SZ92] J. Sokolowski and J.-P. Zolesio. Introduction to shape optimization, volume 16 of Springer Series in Computational Mathematics. Springer, Berlin, 1992.
Prerequisites / NoticeKnowledge of analysis and functional analysis; knowledge of PDEs is an advantage and so is some familiarity with numerical methods for PDEs