Open Access
Issue
ESAIM: COCV
Volume 28, 2022
Article Number 2
Number of page(s) 29
DOI https://doi.org/10.1051/cocv/2021108
Published online 11 January 2022
  1. G. Allaire, C. Dapogny and F. Jouve, Shape and topology optimization, in Differential Geometric Partial Differential Equations: Part II, vol. 22 of Handbook of Numerical Analysis. Elsevier, Amsterdam, Netherlands (2021) 3–124. [Google Scholar]
  2. U. Ayachit, The ParaView Guide: A Parallel Visualization Application. Kitware, Inc., Clifton Park, NY, USA (2015). [Google Scholar]
  3. P. Bastian, M. Blatt, A. Dedner, N.-A. Dreier, C. Engwer, R. Fritze, C. Gräser, C. Grüninger, D. Kempf, R. Klöfkorn, M. Ohlberger and O. Sander, The dune framework: Basic concepts and recent developments. Comput. Math. Appl. 81 (2021) 75–112. [CrossRef] [MathSciNet] [Google Scholar]
  4. J.A. Bello, E. Fernández-Cara, J. Lemoine and J. Simon, The differentiability of the drag with respect to the variations of a Lipschitz domain in a Navier–Stokes flow. SIAM J. Control Optim. 35 (1997) 626–640. [CrossRef] [MathSciNet] [Google Scholar]
  5. A. Boulkhemair, A. Chakib and A. Sadik, On a shape derivative formula for a family of star-shaped domains (2020). [Google Scholar]
  6. C. Brandenburg, F. Lindemann, M. Ulbrich and S. Ulbrich, A continuous adjoint approach to shape optimization for Navier stokes flow, in Optimal Control of Coupled Systems of Partial Differential Equations. vol. 158 of Int. Ser. Numer. Math., Basel, Birkhäuser (2015) 35–56. [CrossRef] [Google Scholar]
  7. V.I. Burenkov, Sobolev spaces on domains. vol. 137, Springer (1998). [CrossRef] [Google Scholar]
  8. K. Deckelnick, G. Dziuk and C.M. Elliott, Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14 (2005) 139–232. [Google Scholar]
  9. M. Delfour and J. Zolesio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Second Edition, Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104) (2011). [Google Scholar]
  10. M. Eigel and K. Sturm, Reproducing kernel Hilbert spaces and variable metric algorithms in PDE-constrained shape optimization. Optim. Methods Softw. 33 (2018) 268–296. [CrossRef] [MathSciNet] [Google Scholar]
  11. K. Eppler and H. Harbrecht, Shape optimization for free boundary problems, in Proceedings of the International Conference Systems Theory: Modelling, Analysis and Control. Vol. 160 of Internat. Ser. Numer. Math., Basel, Birkhäuser (2012) 277–288. [CrossRef] [Google Scholar]
  12. K. Eppler, H. Harbrecht and R. Schneider, On convergence in elliptic shape optimization. SIAM J. Control Optim. 46 (2007) 61–83. [CrossRef] [MathSciNet] [Google Scholar]
  13. L. Evansand R. Gariepy, Measure Theory and Fine Properties of Functions, Revised Edition, Textbooks in Mathematics, CRC Press (2015). [Google Scholar]
  14. M. Fischer, F. Lindemann, M. Ulbrich and S. Ulbrich, Fréchet differentiability of unsteady incompressible Navier–Stokes flow withrespect to domain variations of low regularity by using a general analytical framework. SIAM J. Control Optim. 55 (2017) 3226–3257. [CrossRef] [MathSciNet] [Google Scholar]
  15. H. Garcke, C. Hecht, M. Hinze and C. Kahle, Numerical approximation of phase field based shape and topology optimization for fluids. SIAM J. Sci. Comput. 37 (2015) A1846–A1871. [CrossRef] [Google Scholar]
  16. H. Garcke, M. Hinze, C. Kahle and K. Lam, A phase field approach to shape optimization in Navier–Stokes flow with integral state constraints. Adv. Comput. Math. 44 (2018) 1345–1383. [CrossRef] [MathSciNet] [Google Scholar]
  17. P. Guillaume and M. Masmoudi, Computation of high order derivatives in optimal shape design. Numer. Math. 67 (1994) 231–250. [CrossRef] [MathSciNet] [Google Scholar]
  18. J. Haubner, M. Siebenborn and M. Ulbrich, A continuous perspective on shape optimization via domain transformations. To appear Siam J. Sci. Comput. (2021). [Google Scholar]
  19. J. Haubner, M. Ulbrich and S. Ulbrich, Analysis of shape optimization problems for unsteady fluid-structure interaction. Inverse Probl. 36 (2020) 1–38. [Google Scholar]
  20. A. Henrot and M. Pierre, Shape Variation and Optimization: A Geometrical Analysis, EMS tracts in mathematics, European Mathematical Society (2018). [Google Scholar]
  21. M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30 (2005) 45–63. [CrossRef] [MathSciNet] [Google Scholar]
  22. R. Hiptmair and A. Paganini, Shape optimization by pursuing diffeomorphisms. Comput. Methods Appl. Math. 15 (2015) 291–305. [CrossRef] [MathSciNet] [Google Scholar]
  23. R. Hiptmair, A. Paganini and S. Sargheini, Comparison of approximate shape gradients. BIT Numer. Math. 55 (2015) 459–485. [CrossRef] [Google Scholar]
  24. J.A. Iglesias, K. Sturm and F. Wechsung, Two-dimensional shape optimization with nearly conformal transformations. SIAM J. Sci. Comput. 40 (2018) A3807–A3830. [CrossRef] [Google Scholar]
  25. H. Ishii and P. Loreti, Limits of solutions of p-Laplace equations as p goes to infinity and related variational problems. SIAM J. Math. Anal. 37 (2005) 411–437. [CrossRef] [MathSciNet] [Google Scholar]
  26. H. Jylhä, An optimal transportation problem related to the limits of solutions of local and nonlocal p-Laplace-type problems. Revista matemática complutense 28 (2015) 85–121. [CrossRef] [MathSciNet] [Google Scholar]
  27. N. Kühl, P. Müller, M. Hinze and T. Rung, Decoupling of control and force objective in adjoint-based fluid dynamic shape optimization. AIAA J. 57 (2019) 4110. [CrossRef] [Google Scholar]
  28. P.M. Müller, N. Kühl, M. Siebenborn, K. Deckelnick, M. Hinze and T. Rung, A novel p-harmonic descent approach applied to fluid dynamic shape optimization. Struct. Multidisc Optim. 64 (2021) 3489–3503. [CrossRef] [Google Scholar]
  29. F. Murat and J. Simon, Etude de problemes d’optimal design, in Optimization Techniques Modeling and Optimization in the Service of Man Part 2. Springer Berlin Heidelberg (1976) 54–62. [CrossRef] [Google Scholar]
  30. A. Paganini, F. Wechsung and P.E. Farrell, Higher-order moving mesh methods for PDE-constrained shape optimization. SIAM J. Sci. Comput. 40 (2018) A2356–A2382. [CrossRef] [Google Scholar]
  31. G. Peyré and M. Cuturi, Computational optimal transport. Found. Trends Mach. Learn. 11 (2019) 355–607. [CrossRef] [Google Scholar]
  32. L. Radke, J. Heners, M. Hinze and A. Düster, A partitioned approach for adjoint shape optimization in fluid-structure interaction. J. Comput. Mech. 61 (2018) 259–276. [CrossRef] [Google Scholar]
  33. F. Santambrogio, Optimal transport for applied mathematicians. Birkäuser, NY, 55 (2015), 94. [Google Scholar]
  34. S. Schmidt, C. Ilic, V. Schulz and N. Gauger, Three dimensional large scale aerodynamic shape optimization based on the shape calculus. AIAA J. 51 (2013) 2615–2627. [CrossRef] [Google Scholar]
  35. V. Schulz, M. Siebenborn and K. Welker, PDE constrained shape optimization as optimization on shape manifolds, in Geometric Science of Information. Vol. 9389 of Lecture Notes in Computer Science. Springer, New York (2015) 499–508. [CrossRef] [Google Scholar]
  36. V. Schulz, M. Siebenborn and K. Welker, Efficient PDE constrained shape optimization based on Steklov-Poincaré type metrics. SIAM J. Optim. 26 (2016) 2800–2819. [CrossRef] [MathSciNet] [Google Scholar]
  37. M. Siebenborn and K. Welker, Algorithmic aspects of multigrid methods for optimization in shape spaces. SIAM J. Sci. Comput. 39 (2017) B1156–B1177. [CrossRef] [Google Scholar]
  38. J. Simon, Differentiation with respect to the domain in boundary value problems. Numer. Funct. Anal. Optim. 2 (1980) 649–687. [Google Scholar]
  39. J. Sokołowski and J. Zolésio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Lecture Notes in Computer Science. Springer-Verlag (1992). [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.