## Topological Methods in Nonlinear Analysis

### Topological optimization via cost penalization

#### Abstract

We consider general shape optimization problems governed by Dirichlet boundary value problems. The proposed approach may be extended to other boundary conditions as well. It is based on a recent representation result for implicitly defined manifolds, due to the authors, and it is formulated as an optimal control problem. The discretized approximating problem is introduced and we give an explicit construction of the associated discrete gradient. Some numerical examples are also indicated.

#### Article information

Source
Topol. Methods Nonlinear Anal., Advance publication (2019), 28 pp.

Dates
First available in Project Euclid: 11 November 2019

https://projecteuclid.org/euclid.tmna/1573441227

Digital Object Identifier
doi:10.12775/TMNA.2019.080

#### Citation

Murea, Cornel Marius; Tiba, Dan. Topological optimization via cost penalization. Topol. Methods Nonlinear Anal., advance publication, 11 November 2019. doi:10.12775/TMNA.2019.080. https://projecteuclid.org/euclid.tmna/1573441227

#### References

• E.G. Allaire, Shape optimization by the homogenization method, Springer, New York, 2002.
• V. Barbu and Th. Precupanu, Convexity and optimization in Banach spaces. Mathematics and its Applications, vol. 10, D. Reidel Publishing Co., Dordrecht; Editura Academiei, Bucharest, 1986.
• M. Bendsoe and O. Sigmund, Topology Optimization: Theory, Methods and Application, 2-nd edition, Engineering Online Library, Springer, Berlin, 2003.
• F. Bouchut and L. Desvillets, On two-dimensional Hamiltonian transport equations with continuous coefficients Differential Integral Equations 14 (2001), no. 8, 1015–1024. https://projecteuclid.org/euclid.die/1356123178
• D. Bucur and G. Buttazzo, Variational methods in shape optimization problems. Progress in Nonlinear Differential Equations and their Applications, vol. 65, Birkhäuser Boston, Inc., Boston, MA, 2005.
• P.G. Ciarlet, The finite element method for elliptic problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002.
• F.H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley – Interscience Publication, John Wiley & Sons, Inc., New York, 1983.
• M.C. Delfour and J.P. Zolesio, Shapes and Geometries, Analysis, Differential Calculus and Optimization, SIAM, Philadelphia, 2001.
• F. Hecht, New development in FreeFem$++$, J. Numer. Math. 20 (2012), 251–265. http://www.freefem.org
• A. Henrot and M. Pierre, Variations et optimisation de formes. Une analyse géométrique, Springer, 2005.
• M.W. Hirsch, S. Smale and R.L. Devaney, Differential Equations, Dynamical Systems and an Introduction to Chao, Elsevier, Academic Press, San Diego, 2004.
• J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier–Villars, Paris 1969.
• R. Mäkinen, P. Neittaanmäki and D. Tiba, On a fixed domain approach for a shape optimization problem, Computational and Applied Mathematics II: Differential Equations, (W.F. Ames and P.J. van der Houwen, eds) North-Holland, Amsterdam, 1992, pp. 317–326.
• P. Neittaanmäki, A. Pennanen and D. Tiba, Fixed domain approaches in shape optimization problems with Dirichlet boundary conditions, Inverse Problems 25 (2009), 1–18.
• P. Neittaanmäki, J. Sprekels and D. Tiba, Optimization of elliptic systems. Theory and applications, Springer, New York, 2006.
• P. Neittaanmäki and D. Tiba, Optimal control of nonlinear parabolic systems. Theory, algorithms, and applications, Monographs and Textbooks in Pure and Applied Mathematics, vol. 179, Marcel Dekker, Inc., New York, 1994.
• P. Neittaanmäki and D. Tiba, Fixed domain approaches in shape optimization problems, Inverse Problems, 28 (2012), 1–35.
• M.R. Nicolai and D. Tiba, Implicit functions and parametrizations in dimension three: generalized solutions, Discrete Contin. Dyn. Syst. 35 (2015), no. 6, 2701–2710.
• S. Osher and J. Sethian, Fronts propagating with curvature-dependent speed, J. Comput. Phys.79 (1988), 12–49.
• O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer, Berlin, 1984.
• P.-A. Raviart and J.-M. Thomas, Introduction à l'Analyse Numérique des Équations aux Dérivées Partielles, Dunod, 2004.
• T.C. Sideris, Ordinary Differential Equations and Dynamical Systems, Atlantis Press, Paris, 2013.
• J. Sokolowski and J.P. Zolesio, Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer, Berlin, 1992.
• D. Tiba, The implicit function theorem and implicit parametrizations, Ann. Acad. Rom. Sci. Ser. Math. Appl. 5 (2013), no. 1–2, 193–208.
• D. Tiba, Iterated Hamiltonian type systems and applications J. Differential Equations 264 (2018), no. 8, 5465–5479.
• D. Tiba, A penalization approach in shape optimization, Atti Accad. Peloritana Pericolanti Cl. Sci. Fis. Mat. Natur. 96 (2018), no. 1, A8, DOI: 10.1478/AAPP.961A8.