# American Institute of Mathematical Sciences

May  2020, 25(5): 1729-1755. doi: 10.3934/dcdsb.2019249

## A gradient-type algorithm for constrained optimization with application to microstructure optimization

 1 CMAF-CIO, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal 2 Dep. Area of Mathematics, ISEL, Instituto Politécnico de Lisboa, Rua Conselheiro Emídio Navarro, 1959-007 Lisboa, Portugal

* Corresponding author: cabarbarosie@fc.ul.pt

Received  May 2019 Revised  June 2019 Published  November 2019

Fund Project: Supported by National Funding from FCT - Fundação para a Ciência e a Tecnologia, under the project UID/MAT/04561/2019.

We propose a method to optimize periodic microstructures for obtaining homogenized materials with negative Poisson ratio, using shape and/or topology variations in the model hole. The proposed approach employs worst case design in order to minimize the Poisson ratio of the (possibly anisotropic) homogenized elastic tensor in several prescribed directions. We use a minimization algorithm for inequality constraints based on an active set strategy and on a new algorithm for solving minimization problems with equality constraints, belonging to the class of null-space gradient methods. It uses first order derivatives of both the objective function and the constraints. The step is computed as a sum between a steepest descent step (minimizing the objective functional) and a correction step related to the Newton method (aiming to solve the equality constraints). The linear combination between these two steps involves coefficients similar to Lagrange multipliers which are computed in a natural way based on the Newton method. The algorithm uses no projection and thus the iterates are not feasible; the constraints are only satisfied in the limit (after convergence). A local convergence result is proven for a general nonlinear setting, where both the objective functional and the constraints are not necessarily convex functions.

Citation: Cristian Barbarosie, Anca-Maria Toader, Sérgio Lopes. A gradient-type algorithm for constrained optimization with application to microstructure optimization. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1729-1755. doi: 10.3934/dcdsb.2019249
##### References:
 [1] G. Allaire, E. Bonnetier, G. Francfort and F. Jouve, Shape optimization by the homogenization method, Numer. Math., 76 (1997), 27-68.  doi: 10.1007/s002110050253.  Google Scholar [2] G. Allaire, Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, 146, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4684-9286-6.  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] F. Feppon, G. Allaire and C. Dapogny, Null space gradient flows for constrained optimization with applications to shape optimization, preprint, 2019. Google Scholar [5] K. Atkinson and W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework, Texts in Applied Mathematics, 39, Springer, Dordrecht, 2009. doi: 10.1007/978-1-4419-0458-4.  Google Scholar [6] C. Barbarosie, Shape optimization of periodic structures, Computational Mechanics, 30 (2003), 235–246. doi: 10.1007/s00466-002-0382-3.  Google Scholar [7] C. Barbarosie and A.-M. Toader, Shape and topology optimization for periodic problems, part I: The shape and the topological derivative, Struct. Multidiscip. Optim., 40 (2010), 381-391.  doi: 10.1007/s00158-009-0378-0.  Google Scholar [8] C. Barbarosie and A.-M. Toader, Shape and topology optimization for periodic problems, part II: Optimization algorithm and numerical examples, Struct. Multidiscip. Optim., 40 (2010), 393-408.  doi: 10.1007/s00158-009-0377-1.  Google Scholar [9] C. Barbarosie and S. Lopes, A gradient-type algorithm for optimization with constraints, preprint Pre-2011-001, available from http://cmaf.fc.ul.pt/preprints.html, 2011. Google Scholar [10] C. Barbarosie and S. Lopes, A generalized notion of compliance, Comptes Rendus Mécanique, 339 (2011), 641–648. doi: 10.1016/j.crme.2011.07.002.  Google Scholar [11] D. Bertsekas, Nonlinear Programming, 2$^\rm { nd }$ edition, Athena Scientific Optimization and Computation Series, Athena Scientific, Belmont, MA 1999.  Google Scholar [12] J. Bonnans, J. Gilbert, C. Lemaréchal and C. Sagastizábal, Numerical Optimization – Theoretical and Practical Aspects, Universitext, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-05078-1.  Google Scholar [13] A. Boresi, R. Schmidt and O. Sidebottom, Advanced Mechanics of Materials, Wiley, 1993. Google Scholar [14] P. W. Christensen and A. Klarbring, An Introduction to Structural Optimization, Solid Mechanics and Its Applications, Springer, New York, 2009. doi: 10.1007/978-1-4020-8666-3.  Google Scholar [15] P. G. Ciarlet, Introduction à l'Analyse Numérique Matricielle et à l'Optimisation, Masson, Paris, 1990.  Google Scholar [16] R. Fletcher, Practical Methods of Optimization, Constrained optimization, John Wiley & Sons, Chichester, 2013. doi: 10.1002/9781118723203.  Google Scholar [17] J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation and Computation, Advances in Design and Control, 7, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003. doi: 10.1137/1.9780898718690.  Google Scholar [18] A. Henrot and M. Pierre, Shape Variation and Optimization: A Geometrical Analysis, European Mathematical Society (EMS), Zürich, 2018. doi: 10.4171/178.  Google Scholar [19] E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, 1978.  Google Scholar [20] T. C. Lim, Auxetic Materials and Structures, Engineering Materials, Springer, 2015. doi: 10.1007/978-981-287-275-3.  Google Scholar [21] D. G. Luenberger and Y. Ye, Linear and Nonlinear Programming, 3$^\rm { rd }$ edition, International Series in Operations Research & Management Science, 116, Springer, New York, 2008.  Google Scholar [22] G. Milton, The Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics, 6, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511613357.  Google Scholar [23] J. Nocedal and S. Wright, Numerical Optimization, 2$^\rm { nd }$ edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006. doi: 10.1007/b98874.  Google Scholar [24] A. A. Novotny and J. Sokołowski, Topological Derivatives in Shape Optimization, Interaction of Mechanics and Mathematics, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-35245-4.  Google Scholar [25] A. A. Novotny, J. Sokołowski and A. Żochowski, Applications of the Topological Derivative Method, Studies in Systems, Decision and Control, 188, Springer, 2019. doi: 10.1007/978-3-030-05432-8.  Google Scholar [26] A. Rothwell, Optimization Methods in Structural Design, Solid Mechanics and its Applications, 242, Springer, 2017. doi: 10.1007/978-3-319-55197-5.  Google Scholar [27] W. R. Spillers and K. M. MacBain, Structural Optimization, in Computers & Structures, Springer, Dordrecht, 2009. doi: 10.1016/j.compstruc.2011.05.006.  Google Scholar [28] C. Van Hooricks, O. Sigmund, M. Schevenels, B. S. Lazarov and G. Lombaert, Topology optimization of two-dimensional elastic wave barriers, Journal of Sound and Vibration, 376 (2016), 95–111. Google Scholar [29] J. Sokołowski and J.P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer Series in Computational Mathematics, 16, Springer-Verlag, Berlin, 1992.  Google Scholar [30] A.-M. Toader, The topological derivative of homogenized elastic coefficients of periodic microstructures, J. Control Optim., 49 (2011), 1607-1628.  doi: 10.1137/100782772.  Google Scholar [31] H. Walker and L. Watson, Least-change secant update methods for underdetermined systems, SIAM J. Numer. Anal., 27 (1990), 1227-1262.  doi: 10.1137/0727071.  Google Scholar [32] M. Wormser, F. Wein, M. Stingl and C. Korner, Design and additive manufacturing of 3D phononic band gap structures based on gradient based optimization, Materials, 10 (2017). doi: 10.3390/ma10101125.  Google Scholar [33] H. Yamashita, A differential equation approach to nonlinear programming, Math. Programming, 18 (1980), 155-168.  doi: 10.1007/BF01588311.  Google Scholar [34] Y. Yuan, A review of trust region algorithms for optimization, ICIAM, Oxford Univ. Press, Oxford, 2000.   Google Scholar [35] Z. Zhu, X. Cai and J. Jian, An improved SQP algorithm for solving minimax problems, Appl. Math. Lett., 22 (2009), 464-469.  doi: 10.1016/j.aml.2008.06.017.  Google Scholar

show all references

##### References:
 [1] G. Allaire, E. Bonnetier, G. Francfort and F. Jouve, Shape optimization by the homogenization method, Numer. Math., 76 (1997), 27-68.  doi: 10.1007/s002110050253.  Google Scholar [2] G. Allaire, Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, 146, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4684-9286-6.  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] F. Feppon, G. Allaire and C. Dapogny, Null space gradient flows for constrained optimization with applications to shape optimization, preprint, 2019. Google Scholar [5] K. Atkinson and W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework, Texts in Applied Mathematics, 39, Springer, Dordrecht, 2009. doi: 10.1007/978-1-4419-0458-4.  Google Scholar [6] C. Barbarosie, Shape optimization of periodic structures, Computational Mechanics, 30 (2003), 235–246. doi: 10.1007/s00466-002-0382-3.  Google Scholar [7] C. Barbarosie and A.-M. Toader, Shape and topology optimization for periodic problems, part I: The shape and the topological derivative, Struct. Multidiscip. Optim., 40 (2010), 381-391.  doi: 10.1007/s00158-009-0378-0.  Google Scholar [8] C. Barbarosie and A.-M. Toader, Shape and topology optimization for periodic problems, part II: Optimization algorithm and numerical examples, Struct. Multidiscip. Optim., 40 (2010), 393-408.  doi: 10.1007/s00158-009-0377-1.  Google Scholar [9] C. Barbarosie and S. Lopes, A gradient-type algorithm for optimization with constraints, preprint Pre-2011-001, available from http://cmaf.fc.ul.pt/preprints.html, 2011. Google Scholar [10] C. Barbarosie and S. Lopes, A generalized notion of compliance, Comptes Rendus Mécanique, 339 (2011), 641–648. doi: 10.1016/j.crme.2011.07.002.  Google Scholar [11] D. Bertsekas, Nonlinear Programming, 2$^\rm { nd }$ edition, Athena Scientific Optimization and Computation Series, Athena Scientific, Belmont, MA 1999.  Google Scholar [12] J. Bonnans, J. Gilbert, C. Lemaréchal and C. Sagastizábal, Numerical Optimization – Theoretical and Practical Aspects, Universitext, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-05078-1.  Google Scholar [13] A. Boresi, R. Schmidt and O. Sidebottom, Advanced Mechanics of Materials, Wiley, 1993. Google Scholar [14] P. W. Christensen and A. Klarbring, An Introduction to Structural Optimization, Solid Mechanics and Its Applications, Springer, New York, 2009. doi: 10.1007/978-1-4020-8666-3.  Google Scholar [15] P. G. Ciarlet, Introduction à l'Analyse Numérique Matricielle et à l'Optimisation, Masson, Paris, 1990.  Google Scholar [16] R. Fletcher, Practical Methods of Optimization, Constrained optimization, John Wiley & Sons, Chichester, 2013. doi: 10.1002/9781118723203.  Google Scholar [17] J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation and Computation, Advances in Design and Control, 7, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003. doi: 10.1137/1.9780898718690.  Google Scholar [18] A. Henrot and M. Pierre, Shape Variation and Optimization: A Geometrical Analysis, European Mathematical Society (EMS), Zürich, 2018. doi: 10.4171/178.  Google Scholar [19] E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, 1978.  Google Scholar [20] T. C. Lim, Auxetic Materials and Structures, Engineering Materials, Springer, 2015. doi: 10.1007/978-981-287-275-3.  Google Scholar [21] D. G. Luenberger and Y. Ye, Linear and Nonlinear Programming, 3$^\rm { rd }$ edition, International Series in Operations Research & Management Science, 116, Springer, New York, 2008.  Google Scholar [22] G. Milton, The Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics, 6, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511613357.  Google Scholar [23] J. Nocedal and S. Wright, Numerical Optimization, 2$^\rm { nd }$ edition, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006. doi: 10.1007/b98874.  Google Scholar [24] A. A. Novotny and J. Sokołowski, Topological Derivatives in Shape Optimization, Interaction of Mechanics and Mathematics, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-35245-4.  Google Scholar [25] A. A. Novotny, J. Sokołowski and A. Żochowski, Applications of the Topological Derivative Method, Studies in Systems, Decision and Control, 188, Springer, 2019. doi: 10.1007/978-3-030-05432-8.  Google Scholar [26] A. Rothwell, Optimization Methods in Structural Design, Solid Mechanics and its Applications, 242, Springer, 2017. doi: 10.1007/978-3-319-55197-5.  Google Scholar [27] W. R. Spillers and K. M. MacBain, Structural Optimization, in Computers & Structures, Springer, Dordrecht, 2009. doi: 10.1016/j.compstruc.2011.05.006.  Google Scholar [28] C. Van Hooricks, O. Sigmund, M. Schevenels, B. S. Lazarov and G. Lombaert, Topology optimization of two-dimensional elastic wave barriers, Journal of Sound and Vibration, 376 (2016), 95–111. Google Scholar [29] J. Sokołowski and J.P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer Series in Computational Mathematics, 16, Springer-Verlag, Berlin, 1992.  Google Scholar [30] A.-M. Toader, The topological derivative of homogenized elastic coefficients of periodic microstructures, J. Control Optim., 49 (2011), 1607-1628.  doi: 10.1137/100782772.  Google Scholar [31] H. Walker and L. Watson, Least-change secant update methods for underdetermined systems, SIAM J. Numer. Anal., 27 (1990), 1227-1262.  doi: 10.1137/0727071.  Google Scholar [32] M. Wormser, F. Wein, M. Stingl and C. Korner, Design and additive manufacturing of 3D phononic band gap structures based on gradient based optimization, Materials, 10 (2017). doi: 10.3390/ma10101125.  Google Scholar [33] H. Yamashita, A differential equation approach to nonlinear programming, Math. Programming, 18 (1980), 155-168.  doi: 10.1007/BF01588311.  Google Scholar [34] Y. Yuan, A review of trust region algorithms for optimization, ICIAM, Oxford Univ. Press, Oxford, 2000.   Google Scholar [35] Z. Zhu, X. Cai and J. Jian, An improved SQP algorithm for solving minimax problems, Appl. Math. Lett., 22 (2009), 464-469.  doi: 10.1016/j.aml.2008.06.017.  Google Scholar
Periodicity cell with model hole (zoomed)
Periodically perforated plane ${\mathbb{R}}^2 _{\hbox{ perf}}$
Optimized microstructures with respect to one direction only
Algorithm
Initial guess and final microstructure for square periodicity
Initial guess and final microstructure for hexagonal periodicity
History of convergence, zoom of the first 40 iterations and zoom of the last 6 iterations
History of convergence, zoom of the first 40 iterations, zoom of the last 8 iterations
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