# American Institute of Mathematical Sciences

December  2020, 9(4): 1115-1131. doi: 10.3934/eect.2020043

## Topological optimization and minimal compliance in linear elasticity

 1 Département de Mathématiques, IRIMAS, Université de Haute Alsace, France 2 Institute of Mathematics (Romanian Academy) and Academy of Romanian Scientists, Bucharest, Romania

* Corresponding author: Cornel Marius Murea

Received  August 2019 Revised  November 2019 Published  March 2020

We investigate a fixed domain approach in shape optimization, using a regularization of the Heaviside function both in the cost functional and in the state system. We consider the compliance minimization problem in linear elasticity, a well known application in this area of research. The optimal design problem is approached by an optimal control problem defined in a prescribed domain including all the admissible unknown domains. This approximating optimization problem has good differentiability properties and a gradient algorithm can be applied. Moreover, the paper also includes several numerical experiments that demonstrate the descent of the obtained cost values and show the topological and the boundary variations of the computed domains. The proposed approximation technique is new and can be applied to state systems given by various boundary value problems.

Citation: Cornel Marius Murea, Dan Tiba. Topological optimization and minimal compliance in linear elasticity. Evolution Equations & Control Theory, 2020, 9 (4) : 1115-1131. doi: 10.3934/eect.2020043
##### References:
 [1] G. Allaire, Conception Optimale de Structures, With the Collaboration of Marc Schoenauer (INRIA) in the Writing of Chapter 8. Mathématiques & Applications, 58. Springer-Verlag, Berlin, 2007.  Google Scholar [2] G. Allaire, C. Dapogny and P. Frey, Shape optimization with a level set based mesh evolution method, Comput. Methods Appl. Mech. Engrg., 282 (2014), 22-53.  doi: 10.1016/j.cma.2014.08.028.  Google Scholar [3] G. Allaire, F. Jouve and A.-M. Toader, Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys., 194 (2004), 363-393.  doi: 10.1016/j.jcp.2003.09.032.  Google Scholar [4] G. Allaire, https://portail.polytechnique.edu/cmap/fr/boite-outils-freefem-pour-loptimisation-de-formes, the files $\texttt{levelset-cantilever.edp}$ and $\texttt{pont.homog.struct.edp}$ . Google Scholar [5] M. P. Bendsoe, Optimization of Structural Topology, Shape, and Material, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-662-03115-5.  Google Scholar [6] M. P. Bendsoe and O. Sigmund, Topology Optimization. Theory, Methods and Applications, Springer-Verlag, Berlin, 2003.  Google Scholar [7] M. Burger, A framework for the construction of level set methods for shape optimization and reconstruction, Interfaces Free Bound., 5 (2003), 301-329.  doi: 10.4171/IFB/81.  Google Scholar [8] P. G. Ciarlet, Mathematical Elasticity. Vol. I. Three-Dimensional Elasticity, Studies in Mathematics and its Applications, 20. North-Holland Publishing Co., Amsterdam, 1988.  Google Scholar [9] G. Delgado, Optimization of Composite Structures: A Shape and Topology Sensitivity Analysis, PhD, École Polytechnique, 2017. Google Scholar [10] J. E. Dennis, Jr. and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Classics in Applied Mathematics, 16. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1996. doi: 10.1137/1.9781611971200.  Google Scholar [11] A. Halanay, C. M. Murea and D. Tiba, Existence of a steady flow of Stokes fluid past a linear elastic structure using fictitious domain, J. Math. Fluid Mech., 18 (2016), 397-413.  doi: 10.1007/s00021-015-0247-0.  Google Scholar [12] F. Hecht, New development in freefem++, J. Numer. Math., 20 (2012), 251–265. http://www.freefem.org. doi: 10.1515/jnum-2012-0013.  Google Scholar [13] H. Kawarada and M. Natori, An application of the integrated penalty method to free boundary problems of Laplace equation, Numer. Funct. Anal. Optim., 3 (1981), 1-17.  doi: 10.1080/01630568108816076.  Google Scholar [14] R. A. E. Makinen, P. Neittaanmäki and D. Tiba, On a fixed domain approach for a shape optimization problem, Computational and Applied Mathematics, North-Holland, Amsterdam, 2 (1992), 317-326.   Google Scholar [15] P. Neittaanmäki, A. Pennanen and D. Tiba, Fixed domain approaches in shape optimization problems with Dirichlet boundary conditions, Inverse Problems, 25 (2009), 055003, 18 pp. doi: 10.1088/0266-5611/25/5/055003.  Google Scholar [16] P. Neittaanmäki and D. Tiba, Fixed domain approaches in shape optimization problems, Inverse Problems, 28 (2012), 093001, 35 pp. doi: 10.1088/0266-5611/28/9/093001.  Google Scholar [17] P. Philip and D. Tiba, A penalization and regularization technique in shape optimization problems, Siam J. Control Optim., 51 (2013), 4295-4317.  doi: 10.1137/120892131.  Google Scholar [18] P.-A. Raviart and J.-M. Thomas, Introduction à L'analyse Numérique des Équations aux Dérivées Partielles, Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.  Google Scholar [19] D. Tiba and C. M. Murea, Optimization of a plate with holes, Comput. Math. Appl., 77 (2019), 3010-3020.  doi: 10.1016/j.camwa.2018.08.037.  Google Scholar [20] D. Tiba, A penalization approach in shape optimization, Atti Accad. Pelorit. Pericol. Cl. Sci. Fis. Mat. Nat., 96 (2018), A8, 10 pp. http://dx.doi.org/10.1478/AAPP.961A8.  Google Scholar

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##### References:
 [1] G. Allaire, Conception Optimale de Structures, With the Collaboration of Marc Schoenauer (INRIA) in the Writing of Chapter 8. Mathématiques & Applications, 58. Springer-Verlag, Berlin, 2007.  Google Scholar [2] G. Allaire, C. Dapogny and P. Frey, Shape optimization with a level set based mesh evolution method, Comput. Methods Appl. Mech. Engrg., 282 (2014), 22-53.  doi: 10.1016/j.cma.2014.08.028.  Google Scholar [3] G. Allaire, F. Jouve and A.-M. Toader, Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys., 194 (2004), 363-393.  doi: 10.1016/j.jcp.2003.09.032.  Google Scholar [4] G. Allaire, https://portail.polytechnique.edu/cmap/fr/boite-outils-freefem-pour-loptimisation-de-formes, the files $\texttt{levelset-cantilever.edp}$ and $\texttt{pont.homog.struct.edp}$ . Google Scholar [5] M. P. Bendsoe, Optimization of Structural Topology, Shape, and Material, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-662-03115-5.  Google Scholar [6] M. P. Bendsoe and O. Sigmund, Topology Optimization. Theory, Methods and Applications, Springer-Verlag, Berlin, 2003.  Google Scholar [7] M. Burger, A framework for the construction of level set methods for shape optimization and reconstruction, Interfaces Free Bound., 5 (2003), 301-329.  doi: 10.4171/IFB/81.  Google Scholar [8] P. G. Ciarlet, Mathematical Elasticity. Vol. I. Three-Dimensional Elasticity, Studies in Mathematics and its Applications, 20. North-Holland Publishing Co., Amsterdam, 1988.  Google Scholar [9] G. Delgado, Optimization of Composite Structures: A Shape and Topology Sensitivity Analysis, PhD, École Polytechnique, 2017. Google Scholar [10] J. E. Dennis, Jr. and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Classics in Applied Mathematics, 16. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1996. doi: 10.1137/1.9781611971200.  Google Scholar [11] A. Halanay, C. M. Murea and D. Tiba, Existence of a steady flow of Stokes fluid past a linear elastic structure using fictitious domain, J. Math. Fluid Mech., 18 (2016), 397-413.  doi: 10.1007/s00021-015-0247-0.  Google Scholar [12] F. Hecht, New development in freefem++, J. Numer. Math., 20 (2012), 251–265. http://www.freefem.org. doi: 10.1515/jnum-2012-0013.  Google Scholar [13] H. Kawarada and M. Natori, An application of the integrated penalty method to free boundary problems of Laplace equation, Numer. Funct. Anal. Optim., 3 (1981), 1-17.  doi: 10.1080/01630568108816076.  Google Scholar [14] R. A. E. Makinen, P. Neittaanmäki and D. Tiba, On a fixed domain approach for a shape optimization problem, Computational and Applied Mathematics, North-Holland, Amsterdam, 2 (1992), 317-326.   Google Scholar [15] P. Neittaanmäki, A. Pennanen and D. Tiba, Fixed domain approaches in shape optimization problems with Dirichlet boundary conditions, Inverse Problems, 25 (2009), 055003, 18 pp. doi: 10.1088/0266-5611/25/5/055003.  Google Scholar [16] P. Neittaanmäki and D. Tiba, Fixed domain approaches in shape optimization problems, Inverse Problems, 28 (2012), 093001, 35 pp. doi: 10.1088/0266-5611/28/9/093001.  Google Scholar [17] P. Philip and D. Tiba, A penalization and regularization technique in shape optimization problems, Siam J. Control Optim., 51 (2013), 4295-4317.  doi: 10.1137/120892131.  Google Scholar [18] P.-A. Raviart and J.-M. Thomas, Introduction à L'analyse Numérique des Équations aux Dérivées Partielles, Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983.  Google Scholar [19] D. Tiba and C. M. Murea, Optimization of a plate with holes, Comput. Math. Appl., 77 (2019), 3010-3020.  doi: 10.1016/j.camwa.2018.08.037.  Google Scholar [20] D. Tiba, A penalization approach in shape optimization, Atti Accad. Pelorit. Pericol. Cl. Sci. Fis. Mat. Nat., 96 (2018), A8, 10 pp. http://dx.doi.org/10.1478/AAPP.961A8.  Google Scholar
The geometrical configuration
The fixed domain $D$ including the unknown domain
Cantilever. Left: Geometrical configuration of $D$. Right: Convergence history of the objective functions for descent directions $i)$ given by (2.30) and $ii)$ given by (2.31)
Cantilever. Initial (top, left), intermediate and optimal (bottom, right, after 50 iterations) domains using descent direction $i)$
Cantilever. Optimal domains using descent directions $ii)$ given by (2.31) (left, after 38 iterations) and $iii)$ given by (2.32) (right, after 3 iterations) for the initial domain as in Figure 4
Bridge. Left: Geometrical configuration of $D$. Right: Convergence history of the objective functions for descent directions $i)$ and $ii)$
Bridge. Initial (top, left), intermediate and optimal (bottom, right, after 100 iterations) domains using descent direction $i)$
Bridge. Optimal domain using descent directions $ii)$ after 80 iterations, for the initial domain as in Figure 7
Bridge. Optimal domain using descent directions $i)$ after 100 iterations, for the initial domain $\Omega_0 = ]-1, 1[ \times ]0, 0.6[$, the bottom half of $D$
Bridge. The meshes after the elastic deformations for the initial domain $\Omega_0 = ]-1, 1[ \times ]0, 0.6[$ and for the optimal domain $\Omega_{100}$ presented in Figure 9. The displacements were reduced by a factor $0.1$. The cost (1.7) decreases from $0.378632$ (left image) to $0.297857$ (right image)
The dependence of the cost given by (1.7) and of the displacement $\mathbf{y}^\epsilon(g_0)$ solution of (2.10) in the initial domain $\Omega_0 = ]-1, 1[ \times ]0, 0.6[$. The cost for $\mathbf{y}^*$ is $0.378727$
 $\epsilon$ $J$ $\left\| \mathbf{y}^\epsilon(g_0)-\mathbf{y}^*\right\|_{L^2(\Omega_0)}$ $\left\| \mathbf{y}^\epsilon(g_0)-\mathbf{y}^*\right\|_{H^1(\Omega_0)}$ 0.01 0.353644 0.097841 0.302369 0.005 0.369480 0.035030 0.112965 0.001 0.378150 0.002096 0.027376 0.0005 0.378506 0.000799 0.026536
 $\epsilon$ $J$ $\left\| \mathbf{y}^\epsilon(g_0)-\mathbf{y}^*\right\|_{L^2(\Omega_0)}$ $\left\| \mathbf{y}^\epsilon(g_0)-\mathbf{y}^*\right\|_{H^1(\Omega_0)}$ 0.01 0.353644 0.097841 0.302369 0.005 0.369480 0.035030 0.112965 0.001 0.378150 0.002096 0.027376 0.0005 0.378506 0.000799 0.026536
The dependence of the cost given by (1.7) and of the displacement $\mathbf{y}^\epsilon(g_{100})$ solution of (2.10) in the final domain $\Omega_{100}$. The cost for $\mathbf{y}^*$ is $0.298536$
 $\epsilon$ $J$ $\left\| \mathbf{y}^\epsilon(g_{100})-\mathbf{y}^*\right\|_{L^2(\Omega_{100})}$ $\left\| \mathbf{y}^\epsilon(g_{100})-\mathbf{y}^*\right\|_{H^1(\Omega_{100})}$ 0.01 0.296596 0.018279 0.105998 0.005 0.297032 0.005071 0.084750 0.002 0.297813 0.002263 0.076559 0.001 0.298063 0.001925 0.073167
 $\epsilon$ $J$ $\left\| \mathbf{y}^\epsilon(g_{100})-\mathbf{y}^*\right\|_{L^2(\Omega_{100})}$ $\left\| \mathbf{y}^\epsilon(g_{100})-\mathbf{y}^*\right\|_{H^1(\Omega_{100})}$ 0.01 0.296596 0.018279 0.105998 0.005 0.297032 0.005071 0.084750 0.002 0.297813 0.002263 0.076559 0.001 0.298063 0.001925 0.073167
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