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All issues Volume 59 / No 4 (July-August 2025) ESAIM: M2AN, 59 4 (2025) 2141-2170 References
Open Access
Issue
ESAIM: M2AN
Volume 59, Number 4, July-August 2025
Page(s) 2141 - 2170
DOI https://doi.org/10.1051/m2an/2025058
Published online 23 July 2025
  1. M. Abdelbari, K. Nachi, J. Sokolowski and K. Szulc, Shape sensitivity of optimal control for the Stokes problem. Control Cybernet. 49 (2020) 1–30. [Google Scholar]
  2. R. Adams and J. Fournier, Sobolev Spaces. Elsevier (2003). [Google Scholar]
  3. G. Allaire, F. Jouve and A. Toader, Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194 (2004) 363–393. [Google Scholar]
  4. M.P. Bendsøe and O. Sigmund, Topology Optimization. Theory, Methods and Applications. Springer, Berlin (2003). [Google Scholar]
  5. M. Berggren, A unified discrete-continuous sensitivity analysis method for shape optimization, in Applied and Numerical Partial Differential Equations, in Computational Methods in Applied Sciences, edited by W. Fitzgibbon, et al. Vol. 15. Springer (2010) 2539. [Google Scholar]
  6. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer (2008). [Google Scholar]
  7. E. Burman, D. Elfverson, P. Hansbo, M.G. Larson and K. Larsson. A cut finite element method for the Bernoulli free boundary value problem. Comput. Methods Appl. Mech. Eng. 317 (2017) 598–618. [Google Scholar]
  8. E. Burman, D. Elfverson, P. Hansbo, M.G. Larson and K. Larsson, Shape optimization using the cut finite element method. Comput. Methods Appl. Mech. Eng. 328 (2018) 242–261. [Google Scholar]
  9. M.C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 2nd edition. SIAM, Philadelphia (2011). [Google Scholar]
  10. Y. Deng, Z. Liu, P. Zhang, L. Liu and Y. Wu, Topology optimization of unsteady incompressible Navier–Stokes flows. J. Comput. Phys. 230 (2011) 6688–6708. [Google Scholar]
  11. K. Deckelnick, P. Herbert and M. Hinze, A novel W1,∞ approach to shape optimization with Lipschitz domains. ESAIM Control Optim. Calc. Var. 22 (2016) 309–337. [Google Scholar]
  12. A. Ern and J. Guermond, Theory and Practice of Finite Elements. Springer, New York (2004). [Google Scholar]
  13. F. Feppon, G. Allaire, C. Dapogny and P. Jolivet, Topology optimization of thermal fluid-structure systems using body-fitted meshes and parallel computing. J. Comput. Phys. 417 (2020) 109574. [Google Scholar]
  14. W. Gong, J. Li and S. Zhu, Improved discrete boundary type shape gradients for PDE-constrained shape optimization. SIAM J. Sci. Comput. 44 (2022) A2464–A2505. [Google Scholar]
  15. W. Gong, B. Li and Q. Rao, Convergent evolving finite element approximations of boundary evolution under shape gradient flow. IMA J. Numer. Anal. 44 (2023) 2667–2697. [Google Scholar]
  16. H. Harbrecht and J. Tausch, An efficient numerical method for a shape-identification problem arising from the heat equation. Inv. Prob. 27 (2011) 065013. [Google Scholar]
  17. J. Haslinger and R. M¨akinen, Introduction to Shape Optimization: Theory, Approximation, and Computation. SIAM (2003). [Google Scholar]
  18. F. Hecht, New development in FreeFem++. J. Numer. Math. 20 (2012) 251–265. [CrossRef] [MathSciNet] [Google Scholar]
  19. F. Hettlich and W. Rundell, Identification of a discontinuous source in the heat equation. Inv. Prob. 17 (2001) 1465–1482. [Google Scholar]
  20. M. Hintermuller, A. Laurain and I. Yousept, Shape sensitivities for an inverse problem in magnetic induction tomography based on the eddy current model. Inv. Prob. 31 (2015) 065006. [Google Scholar]
  21. R. Hiptmair and A. Paganini, Shape optimization by pursuing diffeomorphisms. Comput. Methods Appl. Math. 15 (2015) 291–305. [CrossRef] [MathSciNet] [Google Scholar]
  22. R. Hiptmair, A. Paganini and S. Sargheini, Comparison of approximate shape gradients. BIT Numer. Math. 55 (2015) 459–485. [CrossRef] [Google Scholar]
  23. X. Hu and S. Zhu, On geometric inverse problems in time-fractional subdiffusion. SIAM J. Sci. Comput. 44 (2022) 3560–A3591. [Google Scholar]
  24. A. Klein, P.B. Nair and M. Yano, A priori error analysis of shape derivatives of linear functionals in structural topology optimization. Comput. Methods Appl. Mech. Eng. 395 (2022) 114991. [Google Scholar]
  25. T. Lassila, A. Manzoni, A. Quateroni and G. Rozza, Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty. ESAIM Math. Model. Numer. Anal. 47 (2013) 1107–1131. [Google Scholar]
  26. A. Laurain, Distributed and boundary expressions of first and second-order shape derivatives in nonsmooth domains. J. Math. Pures. Appl. 134 (2020) 328–368. [Google Scholar]
  27. A. Laurain and K. Sturm, Distributed shape derivative via averaged adjoint method and applications. ESAIM Math. Model. Numer. Anal. 50 (2016) 1241–1267. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  28. B. Li, Maximum-norm stability and maximal Lp regularity of FEMs for parabolic equations with Lipschitz continuous coefficients. Numer. Math. 131 (2015) 489–516. [Google Scholar]
  29. B. Li, Maximal regularity of multistep fully discrete finite element methods for parabolic equations. IMA J. Numer. Anal. 42 (2022) 1700–1734. [Google Scholar]
  30. B. Li and W. Sun, Maximal Lp analysis of FEM for parabolic equations with nonsmooth coefficients in convex polyhedra. Math. Comp. 86 (2017) 1071–1102. [Google Scholar]
  31. B. Li and W. Sun, Maximal regularity of fully discrete finite element solutions of parabolic equations. SIAM J. Numer. Anal. 55 (2017) 521–542. [Google Scholar]
  32. J. Li and S. Zhu, Shape optimization of the Stokes eigenvalue problem. SIAM J. Sci. Comput. 45 (2023) A798–A828. [Google Scholar]
  33. J. Li and S. Zhu, Shape optimization of Navier–Stokes flows by a two-grid method. Comput. Methods Appl. Mech. Eng. 400 (2022) 115531. [Google Scholar]
  34. B. Mohammadi and O. Pironneau, Applied Shape Optimization for Fluids. Clarendon Press, Oxford (2001). [Google Scholar]
  35. M. Moubachir and J.P. Zolesio, Moving Shape Analysis and Control: Applications to Fluid Structure Interactions. Pure Appl. Math. Vol. 277. Chapman and Hall/CRC, Boca Raton (2006). [Google Scholar]
  36. 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. [Google Scholar]
  37. I.P.A. Papadopoulos and E. Süli, Numerical analysis of a topology optimization problem for Stokes flow. J. Comput. Appl. Math. 412 (2022) 114295. [Google Scholar]
  38. P. Plotnikov and J. Sokolowski, Boundary control of the motion of a heavy piston in viscous gas. SIAM J. Control Optim. 53 (2015) 3319–3338. [Google Scholar]
  39. A.H. Schatz, V. Thomée and L.B. Wahlbin, Maximum norm stability and error estimates in parabolic finite element equations. Comm. Pure Appl. Math. 33 (1980) 265–304. [CrossRef] [MathSciNet] [Google Scholar]
  40. V.H. Schulz, M. Siebenborn and K. Welker, Structured inverse modeling in parabolic diffusion problems. SIAM J. Control Optim. 53 (2015) 3319–3338. [Google Scholar]
  41. J. Sokolowski and J.P. Zolésio, Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer, Heidelberg (1992). [Google Scholar]
  42. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer, Heidelberg (2006). [Google Scholar]
  43. H. Wang and Y. Li, Numerical solution of an inverse boundary value problem for the heat equation with unknown inclusions. J. Comput. Phys. 369 (2018) 1–15. [Google Scholar]
  44. K. Zhang, S. Zhu, J. Li and W. Yan, Shape gradient methods for shape optimization of an unsteady multiscale fluid-structure interaction model. J. Geom. Anal. 34 (2024) 245. [Google Scholar]
  45. S. Zhu, Effective shape optimization of Laplace eigenvalue problems using domain expressions of Eulerian derivatives. J. Optim. Theory Appl. 176 (2018) 17–34. [Google Scholar]
  46. S. Zhu and Z. Gao, Convergence analysis of mixed finite element approximations to shape gradients in the Stokes equation. Comput. Methods Appl. Mech. Eng. 343 (2019) 127–150. [CrossRef] [Google Scholar]
  47. S. Zhu, X. Hu and Q. Liao, Convergence analysis of Galerkin finite element approximations to shape gradients in eigenvalue optimization. BIT 60 (2020) 853–878. [Google Scholar]
  48. J. Li, W. Li, S. Zhu, Convergence analysis of approximate shape gradients for shape optimization in parabolic problems. Code GitHub: https://github.com/JiajieLiOptimization/ParabolicShapeGradient.git. (2025). [Google Scholar]

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