Exact shape derivatives with unfitted finite element methods
-
Jeremy T. Shahan
Abstract
We present an approach to shape optimization problems that uses an unfitted finite element method (FEM). The domain geometry is represented, and optimized, using a (discrete) level set function and we consider objective functionals that are defined over bulk domains. For a discrete objective functional, defined in the unfitted FEM framework, we show that the exact discrete shape derivative essentially matches the shape derivative at the continuous level. In other words, our approach has the benefits of both optimize-then-discretize and discretize-then-optimize approaches. Specifically, we establish the shape Fréchet differentiability of discrete (unfitted) bulk shape functionals using both the perturbation of the identity approach and direct perturbation of the level set representation. The latter approach is especially convenient for optimizing with respect to level set functions. Moreover, our Fréchet differentiability results hold for any polynomial degree used for the discrete level set representation of the domain. We illustrate our results with some numerical accuracy tests, a simple model (geometric) problem with known exact solution, as well as shape optimization of structural designs.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-2111474
-
Research ethics: Not applicable.
-
Informed consent: Not applicable.
-
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
-
Use of Large Language Models, AI and Machine Learning Tools: None declared.
-
Conflict of interest: The authors state no conflict of interest.
-
Research funding: National Science Foundation (NSF): DMS-2111474.
-
Data availability: Not applicable.
References
[1] J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation, vol. 7 of Advances in Design and Control, Philadelphia, PA, SIAM, 2003.10.1137/1.9780898718690Suche in Google Scholar
[2] A. Henrot and M. Pierre, Shape Variation and Optimization, vol. 28 of EMS Tracts in Mathematics, Zürich, European Mathematical Society (EMS), 2018.10.4171/178Suche in Google Scholar
[3] O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer Series in Computational Physics, New York, NY, Springer-Verlag, 1984.10.1007/978-3-642-87722-3Suche in Google Scholar
[4] S. Schmidt and V. H. Schulz, “A linear view on shape optimization,” SIAM J. Control Optim., vol. 61, no. 4, pp. 2358–2378, 2023. https://doi.org/10.1137/22m1488910.Suche in Google Scholar
[5] J. Sokolowski and J.-P. Zolésio, Introduction to Shape Optimization, Springer Series in Computational Mathematics, Springer-Verlag, 1992.10.1007/978-3-642-58106-9Suche in Google Scholar
[6] S. W. Walker, The Shapes of Things: A Practical Guide to Differential Geometry and the Shape Derivative, vol. 28 of Advances in Design and Control, 1st ed. Philadelphia, PA, SIAM, 2015.10.1137/1.9781611973969Suche in Google Scholar
[7] M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with PDE Constraints, vol. 23 of Mathematical Modeling: Theory and Applications, New York, Springer, 2009.Suche in Google Scholar
[8] F. Tröltzsch, Optimal Control of Partial Differential Equations, Graduate Studies in Mathematics, Providence, RI, American Mathematical Society, 2010.10.1090/gsm/112Suche in Google Scholar
[9] M. D. Gunzburger and H. Kim, “Existence of an optimal solution of a shape control problem for the stationary Navier-Stokes equations,” SIAM J. Control Optim., vol. 36, no. 3, pp. 895–909, 1998. https://doi.org/10.1137/s0363012994276123.Suche in Google Scholar
[10] B. Mohammadi and O. Pironneau, Applied Shape Optimization for Fluids, Numerical Mathematics and Scientific Computation, New York, NY, The Clarendon Press Oxford University Press, 2001.Suche in Google Scholar
[11] O. Pironneau, “On optimum profiles in Stokes flow,” J. Fluid Mech., vol. 59, no. 1, pp. 117–128, 1973. https://doi.org/10.1017/s002211207300145x.Suche in Google Scholar
[12] O. Pironneau, “On optimum design in fluid mechanics,” J. Fluid Mech., vol. 64, no. 1, pp. 97–110, 1974. https://doi.org/10.1017/s0022112074002023.Suche in Google Scholar
[13] G. Doğan, P. Morin, and R. H. Nochetto, “A variational shape optimization approach for image segmentation with a Mumford–Shah functional,” SIAM J. Sci. Comput., vol. 30, no. 6, pp. 3028–3049, 2008. https://doi.org/10.1137/070692066.Suche in Google Scholar
[14] M. Hintermüller and W. Ring, “A second order shape optimization approach for image segmentation,” SIAM J. Appl. Math., vol. 64, no. 2, pp. 442–467, 2004. https://doi.org/10.1137/s0036139902403901.Suche in Google Scholar
[15] A. Laurain and S. W. Walker, “Droplet footprint control,” SIAM J. Control Optim., vol. 53, no. 2, pp. 771–799, 2015. https://doi.org/10.1137/140979721.Suche in Google Scholar
[16] S. W. Walker and E. E. Keaveny, “Analysis of shape optimization for magnetic microswimmers,” SIAM J. Control Optim., vol. 51, no. 4, pp. 3093–3126, 2013. https://doi.org/10.1137/110845823.Suche in Google Scholar
[17] S. W. Walker and M. J. Shelley, “Shape optimization of peristaltic pumping,” J. Comput. Phys., vol. 229, no. 4, pp. 1260–1291, 2010. https://doi.org/10.1016/j.jcp.2009.10.030.Suche in Google Scholar
[18] 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., vol. 328, pp. 242–261, 2018, https://doi.org/10.1016/j.cma.2017.09.005.Suche in Google Scholar
[19] D. Chenais, B. Rousselet, and R. Benedict, “Design sensitivity for arch structures with respect to midsurface shape under static loading,” J. Optim. Theor. Appl., vol. 58, no. 2, pp. 225–239, 1988. https://doi.org/10.1007/bf00939683.Suche in Google Scholar
[20] M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus, and Optimization, vol. 4 of Advances in Design and Control, 2nd ed. Philadelphia, PA, SIAM, 2011.10.1137/1.9780898719826Suche in Google Scholar
[21] G. Iancu and E. Schnack, Shape Optimization with FEM, Vienna, Springer Vienna, 1992, pp. 411–430.10.1007/978-3-7091-2788-9_20Suche in Google Scholar
[22] M. Berggren, A Unified Discrete–Continuous Sensitivity Analysis Method for Shape Optimization, Dordrecht, Springer Netherlands, 2010, pp. 25–39.10.1007/978-90-481-3239-3_4Suche in Google Scholar
[23] P. Gangl and M. H. Gfrerer, “A unified approach to shape and topological sensitivity analysis of discretized optimal design problems,” Appl. Math. Optim., vol. 88, no. 2, p. 46, 2023. https://doi.org/10.1007/s00245-023-10016-2.Suche in Google Scholar PubMed PubMed Central
[24] R. Herzog and E. Loayza-Romero, “A discretize-then-optimize approach to pde-constrained shape optimization,” ESAIM: COCV, vol. 30, p. 11, 2024, https://doi.org/10.1051/cocv/2023071.Suche in Google Scholar
[25] E. Laporte and P. L. Tallec, Numerical Methods in Sensitivity Analysis and Shape Optimization, Modeling and Simulation in Science, Boston, MA, Engineering and Technology, Birkhäuser, 2003.10.1007/978-1-4612-0069-7Suche in Google Scholar
[26] G. Allaire, C. Dapogny, and F. Jouve, “Chapter 1 – shape and topology optimization,” in Geometric Partial Differential Equations – Part II, A. Bonito and R. H. Nochetto, Eds., vol. 22 of Handbook of Numerical Analysis, Amsterdam, Netherlands, Elsevier, 2021, pp. 1–132.10.1016/bs.hna.2020.10.004Suche in Google Scholar
[27] N. Lachenmaier, D. Baumgärtner, H. P. Schiffer, and J. Kech, Gradient-Based Optimization of a Radial Turbine Volute and a Downstream Bend Using Vertex Morphing, vol. 2E: Turbomachinery of Turbo Expo: Power for Land, Sea, and Air, 2020.10.1115/GT2020-14145Suche in Google Scholar
[28] F. Mohebbi and M. Sellier, “Three-dimensional optimal shape design in heat transfer based on body-fitted grid generation,” Int. J. Comput. Methods Eng. Sci. Mech., vol. 14, no. 6, pp. 473–490, 2013. https://doi.org/10.1080/15502287.2013.784384.Suche in Google Scholar
[29] R. Bergmann, R. Herzog, E. Loayza-Romero, and K. Welker, “Shape optimization: what to do first, optimize or discretize?,” PAMM, vol. 19, no. 1, p. e201900067, 2019. https://doi.org/10.1002/pamm.201900067.Suche in Google Scholar
[30] M. D. Gunzburger, Perspectives in Flow Control and Optimization, Philadelphia, PA, SIAM, 2003.10.1137/1.9780898718720Suche in Google Scholar
[31] P. Duysinx, L. Van Miegroet, T. Jacobs, and C. Fleury, “Generalized shape optimization using X-FEM and level set methods,” in IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials, M. P. Bendsøe, N. Olhoff, and O. Sigmund, Eds., Dordrecht, Springer Netherlands, 2006, pp. 23–32.10.1007/1-4020-4752-5_3Suche in Google Scholar
[32] J. S. Dokken, S. W. Funke, A. Johansson, and S. Schmidt, “Shape optimization using the finite element method on multiple meshes with Nitsche coupling,” SIAM J. Sci. Comput., vol. 41, no. 3, pp. A1923–A1948, 2019. https://doi.org/10.1137/18m1189208.Suche in Google Scholar
[33] M. Berggren, “Shape calculus for fitted and unfitted discretizations: domain transformations vs. boundary-face dilations,” Commun. Optim. Theory, vol. 2023, pp. 1–33, 2023.10.23952/cot.2023.27Suche in Google Scholar
[34] E. Burman, C. He, and M. G. Larson, “Comparison of shape derivatives using CutFEM for ill-posed Bernoulli free boundary problem,” J. Sci. Comput., vol. 88, no. 2, p. 35, 2021. https://doi.org/10.1007/s10915-021-01544-6.Suche in Google Scholar
[35] E. G. Birgin, A. Laurain, R. Massambone, and A. G. Santana, “A shape-Newton approach to the problem of covering with identical balls,” SIAM J. Sci. Comput., vol. 44, no. 2, pp. A798–A824, 2022. https://doi.org/10.1137/21m1426067.Suche in Google Scholar
[36] E. G. Birgin, A. Laurain, and T. C. Menezes, “Sensitivity analysis and tailored design of minimization diagrams,” Math. Comput., vol. 92, no. 4, pp. 2715–2768, 2023. https://doi.org/10.1090/mcom/3839.Suche in Google Scholar
[37] E. Burman, “Ghost penalty,” C. R. Math., vol. 348, no. 21, pp. 1217–1220, 2010. https://doi.org/10.1016/j.crma.2010.10.006.Suche in Google Scholar
[38] E. Burman, S. Claus, P. Hansbo, M. G. Larson, and A. Massing, “CutFEM: discretizing geometry and partial differential equations,” Int. J. Numer. Methods Eng., vol. 104, no. 7, pp. 472–501, 2014. https://doi.org/10.1002/nme.4823.Suche in Google Scholar
[39] E. Burman, P. Hansbo, M. G. Larson, and S. Zahedi, “Cut finite element methods for coupled bulk–surface problems,” Numer. Math., vol. 133, no. 2, pp. 203–231, 2016. https://doi.org/10.1007/s00211-015-0744-3.Suche in Google Scholar
[40] A. Hansbo, P. Hansbo, and M. G. Larson, “A finite element method on composite grids based on Nitsche’s method,” ESAIM: M2AN, vol. 37, no. 3, pp. 495–514, 2003. Available at: https://doi.org/10.1051/m2an:2003039.10.1051/m2an:2003039Suche in Google Scholar
[41] C. Lehrenfeld and M. Olshanskii, “An Eulerian finite element method for PDEs in time-dependent domains,” ESAIM Math. Model. Numer. Anal., vol. 53, no. 2, pp. 585–614, 2019. https://doi.org/10.1051/m2an/2018068.Suche in Google Scholar
[42] P. Hansbo, M. G. Larson, and K. Larsson, “Cut finite element methods for linear elasticity problems,” in Geometrically Unfitted Finite Element Methods and Applications, vol. 121 of Lecture Notes in Computational Science and Engineering, S. Bordas, E. Burman, M. Larson, and M. Olshanskii, Eds., New York, NY, Springer International Publishing, 2017, pp. 25–63.10.1007/978-3-319-71431-8_2Suche in Google Scholar
[43] S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, New York, NY, Springer-Verlag, 2003.10.1007/b98879Suche in Google Scholar
[44] S. A. Sethian, Level Set Methods and Fast Marching Methods, 2nd ed. New York, NY, Cambridge University Press, 1999.10.1137/S0036144598347059Suche in Google Scholar
[45] D. Baumgärtner, J. Wolf, R. Rossi, P. Dadvand, and R. Wüchner, “A robust algorithm for implicit description of immersed geometries within a background mesh,” Adv. Model. Simul. Eng. Sci., vol. 5, no. 1, p. 21, 2018. https://doi.org/10.1186/s40323-018-0113-8.Suche in Google Scholar
[46] E. Burman and P. Hansbo, “Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method,” Appl. Numer. Math., vol. 62, no. 4, pp. 328–341, 2012. https://doi.org/10.1016/j.apnum.2011.01.008.Suche in Google Scholar
[47] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, vol. 15 of Texts in Applied Mathematics, 3rd ed. New York, NY, Springer, 2008.10.1007/978-0-387-75934-0Suche in Google Scholar
[48] S. Gross, M. A. Olshanskii, and A. Reusken, “A trace finite element method for a class of coupled bulk-interface transport problems,” ESAIM: M2AN, vol. 49, no. 5, pp. 1303–1330, 2015. https://doi.org/10.1051/m2an/2015013.Suche in Google Scholar
[49] S. J. Osher and F. Santosa, “Level set methods for optimization problems involving geometry and constraints I. frequencies of a two-density inhomogeneous drum,” J. Comput. Phys., vol. 171, no. 1, pp. 272–288, 2001. https://doi.org/10.1006/jcph.2001.6789.Suche in Google Scholar
[50] A. Laurain, “Analyzing smooth and singular domain perturbations in level set methods,” SIAM J. Math. Anal., vol. 50, no. 4, pp. 4327–4370, 2018. https://doi.org/10.1137/17m1118956.Suche in Google Scholar
[51] J. K. Hale, Ordinary Differential Equations, Dover Books on Mathematics, Garden City, New York, Dover Publications, 1980.Suche in Google Scholar
[52] E. M. Stein, Singular Integrals and Differentiability Properties of Functions (PMS-30), Princeton, NJ, Princeton University Press, 1970.10.1515/9781400883882Suche in Google Scholar
[53] J. Schöberl, “C++11 implementation of finite elements in NGSolve,” Institute for Analysis and Scientific Computing, Tech. Rep. ASC-2014-30, 2014.Suche in Google Scholar
[54] C. Lehrenfeld, F. Heimann, J. Preuss, and H. von Wahl, “ngsxfem: Add-on to NGSolve for geometrically unfitted finite element discretizations,” J. Open Source Softw., vol. 6, no. 64, p. 3237, 2021. https://doi.org/10.21105/joss.03237.Suche in Google Scholar
[55] C. Li, C. Xu, C. Gui, and M. D. Fox, “Distance regularized level set evolution and its application to image segmentation,” IEEE Trans. Image Process., vol. 19, no. 12, pp. 3243–3254, 2010. https://doi.org/10.1109/tip.2010.2069690.Suche in Google Scholar
[56] N. Parolini and E. Burman, “A local projection reinitialization procedure for the level set equation on unstructured grids,” École Polytechnique Fédérale de Lausanne, Tech. Rep. CMCS-REPORT-2007-004, 2007.Suche in Google Scholar
[57] C. Basting and D. Kuzmin, “A minimization-based finite element formulation for interface-preserving level set reinitialization,” Computing, vol. 95, no. 1, pp. 13–25, 2013. https://doi.org/10.1007/s00607-012-0259-z.Suche in Google Scholar
[58] R. I. Saye, “High-order methods for computing distances to implicitly defined surfaces,” Commun. Appl. Math. Comput. Sci., vol. 9, no. 1, pp. 107–141, 2014. https://doi.org/10.2140/camcos.2014.9.107.Suche in Google Scholar
[59] A. Jaklič, A. Leonardis, and F. Solina, Segmentation and Recovery of Superquadrics, Computational Imaging and Vision, 1st ed. Dordrecht, Springer, 2010.Suche in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
