Figure 1.
Periodicity cell with model hole (zoomed)
-
Previous Article
Stability analysis of traveling wave solutions for lattice reaction-diffusion equations
- DCDS-B Home
- This Issue
-
Next Article
The effect of noise intensity on parabolic equations
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 |
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.
Keywords:
Nonlinear programming,
constrained minimization,
worst case design,
optimization of microstructures,
porous materials,
microstructure,
auxetic materials.
Mathematics Subject Classification:
Primary: 65K10, 49M15; Secondary: 90C52.
Citation:
Cristian Barbarosie, Anca-Maria Toader, Sérgio Lopes. A gradient-type algorithm for constrained optimization with application to microstructure optimization. Discrete and 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. |
| [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. |
| [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. |
| [4] |
F. Feppon, G. Allaire and C. Dapogny, Null space gradient flows for constrained optimization with applications to shape optimization, preprint, 2019. |
| [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. |
| [6] |
C. Barbarosie, Shape optimization of periodic structures, Computational Mechanics, 30 (2003), 235–246.
doi: 10.1007/s00466-002-0382-3. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
D. Bertsekas, Nonlinear Programming, 2$^\rm { nd }$ edition, Athena Scientific Optimization and Computation Series, Athena Scientific, Belmont, MA 1999. |
| [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. |
| [13] |
A. Boresi, R. Schmidt and O. Sidebottom, Advanced Mechanics of Materials, Wiley, 1993. |
| [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. |
| [15] |
P. G. Ciarlet, Introduction à l'Analyse Numérique Matricielle et à l'Optimisation, Masson, Paris, 1990. |
| [16] |
R. Fletcher, Practical Methods of Optimization, Constrained optimization, John Wiley & Sons, Chichester, 2013.
doi: 10.1002/9781118723203. |
| [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. |
| [18] |
A. Henrot and M. Pierre, Shape Variation and Optimization: A Geometrical Analysis, European Mathematical Society (EMS), Zürich, 2018.
doi: 10.4171/178. |
| [19] |
E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, 1978. |
| [20] |
T. C. Lim, Auxetic Materials and Structures, Engineering Materials, Springer, 2015.
doi: 10.1007/978-981-287-275-3. |
| [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. |
| [22] |
G. Milton, The Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics, 6, Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511613357.
|
| [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. |
| [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. |
| [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. |
| [26] |
A. Rothwell, Optimization Methods in Structural Design, Solid Mechanics and its Applications, 242, Springer, 2017.
doi: 10.1007/978-3-319-55197-5. |
| [27] |
W. R. Spillers and K. M. MacBain, Structural Optimization, in Computers & Structures, Springer, Dordrecht, 2009.
doi: 10.1016/j.compstruc.2011.05.006. |
| [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. |
| [29] |
J. Sokołowski and J.P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer Series in Computational Mathematics, 16, Springer-Verlag, Berlin, 1992. |
| [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. |
| [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. |
| [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. |
| [33] |
H. Yamashita,
A differential equation approach to nonlinear programming, Math. Programming, 18 (1980), 155-168.
doi: 10.1007/BF01588311. |
| [34] |
Y. Yuan, A review of trust region algorithms for optimization, ICIAM, Oxford Univ. Press, Oxford, 2000.
|
| [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. |
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. |
| [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. |
| [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. |
| [4] |
F. Feppon, G. Allaire and C. Dapogny, Null space gradient flows for constrained optimization with applications to shape optimization, preprint, 2019. |
| [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. |
| [6] |
C. Barbarosie, Shape optimization of periodic structures, Computational Mechanics, 30 (2003), 235–246.
doi: 10.1007/s00466-002-0382-3. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
D. Bertsekas, Nonlinear Programming, 2$^\rm { nd }$ edition, Athena Scientific Optimization and Computation Series, Athena Scientific, Belmont, MA 1999. |
| [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. |
| [13] |
A. Boresi, R. Schmidt and O. Sidebottom, Advanced Mechanics of Materials, Wiley, 1993. |
| [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. |
| [15] |
P. G. Ciarlet, Introduction à l'Analyse Numérique Matricielle et à l'Optimisation, Masson, Paris, 1990. |
| [16] |
R. Fletcher, Practical Methods of Optimization, Constrained optimization, John Wiley & Sons, Chichester, 2013.
doi: 10.1002/9781118723203. |
| [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. |
| [18] |
A. Henrot and M. Pierre, Shape Variation and Optimization: A Geometrical Analysis, European Mathematical Society (EMS), Zürich, 2018.
doi: 10.4171/178. |
| [19] |
E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, 1978. |
| [20] |
T. C. Lim, Auxetic Materials and Structures, Engineering Materials, Springer, 2015.
doi: 10.1007/978-981-287-275-3. |
| [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. |
| [22] |
G. Milton, The Theory of Composites, Cambridge Monographs on Applied and Computational Mathematics, 6, Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511613357.
|
| [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. |
| [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. |
| [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. |
| [26] |
A. Rothwell, Optimization Methods in Structural Design, Solid Mechanics and its Applications, 242, Springer, 2017.
doi: 10.1007/978-3-319-55197-5. |
| [27] |
W. R. Spillers and K. M. MacBain, Structural Optimization, in Computers & Structures, Springer, Dordrecht, 2009.
doi: 10.1016/j.compstruc.2011.05.006. |
| [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. |
| [29] |
J. Sokołowski and J.P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer Series in Computational Mathematics, 16, Springer-Verlag, Berlin, 1992. |
| [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. |
| [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. |
| [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. |
| [33] |
H. Yamashita,
A differential equation approach to nonlinear programming, Math. Programming, 18 (1980), 155-168.
doi: 10.1007/BF01588311. |
| [34] |
Y. Yuan, A review of trust region algorithms for optimization, ICIAM, Oxford Univ. Press, Oxford, 2000.
|
| [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. |
Figure 2.
Periodically perforated plane $ {\mathbb{R}}^2 _{\hbox{ perf}} $
Figure 3.
Optimized microstructures with respect to one direction only
Figure 4.
Algorithm
Figure 5.
Initial guess and final microstructure for square periodicity
Figure 7.
Initial guess and final microstructure for hexagonal periodicity
Figure 6.
History of convergence, zoom of the first 40 iterations and zoom of the last 6 iterations
Figure 8.
History of convergence, zoom of the first 40 iterations, zoom of the last 8 iterations
| [1] |
Stan Chiriţă. Spatial behavior in the vibrating thermoviscoelastic porous materials. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2027-2038. doi: 10.3934/dcdsb.2014.19.2027 |
| [2] |
K. A. Ariyawansa, Leonid Berlyand, Alexander Panchenko. A network model of geometrically constrained deformations of granular materials. Networks and Heterogeneous Media, 2008, 3 (1) : 125-148. doi: 10.3934/nhm.2008.3.125 |
| [3] |
Behrooz Yousefzadeh. Computation of nonreciprocal dynamics in nonlinear materials. Journal of Computational Dynamics, 2022 doi: 10.3934/jcd.2022010 |
| [4] |
Haolei Wang, Lei Zhang. Energy minimization and preconditioning in the simulation of athermal granular materials in two dimensions. Electronic Research Archive, 2020, 28 (1) : 405-421. doi: 10.3934/era.2020023 |
| [5] |
Ying Ji, Shaojian Qu, Yeming Dai. A new approach for worst-case regret portfolio optimization problem. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 761-770. doi: 10.3934/dcdss.2019050 |
| [6] |
Faustino Maestre, Pablo Pedregal. Dynamic materials for an optimal design problem under the two-dimensional wave equation. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 973-990. doi: 10.3934/dcds.2009.23.973 |
| [7] |
Paolo Paoletti. Acceleration waves in complex materials. Discrete and Continuous Dynamical Systems - B, 2012, 17 (2) : 637-659. doi: 10.3934/dcdsb.2012.17.637 |
| [8] |
Edward Della Torre, Lawrence H. Bennett. Analysis and simulations of magnetic materials. Conference Publications, 2005, 2005 (Special) : 854-861. doi: 10.3934/proc.2005.2005.854 |
| [9] |
Qun Lin, Antoinette Tordesillas. Towards an optimization theory for deforming dense granular materials: Minimum cost maximum flow solutions. Journal of Industrial and Management Optimization, 2014, 10 (1) : 337-362. doi: 10.3934/jimo.2014.10.337 |
| [10] |
Zhiqiang Yang, Junzhi Cui, Qiang Ma. The second-order two-scale computation for integrated heat transfer problem with conduction, convection and radiation in periodic porous materials. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 827-848. doi: 10.3934/dcdsb.2014.19.827 |
| [11] |
John Murrough Golden. Constructing free energies for materials with memory. Evolution Equations and Control Theory, 2014, 3 (3) : 447-483. doi: 10.3934/eect.2014.3.447 |
| [12] |
Luca Deseri, Massiliano Zingales, Pietro Pollaci. The state of fractional hereditary materials (FHM). Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2065-2089. doi: 10.3934/dcdsb.2014.19.2065 |
| [13] |
Mariano Giaquinta, Paolo Maria Mariano, Giuseppe Modica. A variational problem in the mechanics of complex materials. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 519-537. doi: 10.3934/dcds.2010.28.519 |
| [14] |
Merab Svanadze. On the theory of viscoelasticity for materials with double porosity. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2335-2352. doi: 10.3934/dcdsb.2014.19.2335 |
| [15] |
Yongjian Yang, Zhiyou Wu, Fusheng Bai. A filled function method for constrained nonlinear integer programming. Journal of Industrial and Management Optimization, 2008, 4 (2) : 353-362. doi: 10.3934/jimo.2008.4.353 |
| [16] |
Donald L. Brown, Vasilena Taralova. A multiscale finite element method for Neumann problems in porous microstructures. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1299-1326. doi: 10.3934/dcdss.2016052 |
| [17] |
Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control and Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023 |
| [18] |
Songqiang Qiu, Zhongwen Chen. An adaptively regularized sequential quadratic programming method for equality constrained optimization. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2675-2701. doi: 10.3934/jimo.2019075 |
| [19] |
Bin Li, Kok Lay Teo, Cheng-Chew Lim, Guang Ren Duan. An optimal PID controller design for nonlinear constrained optimal control problems. Discrete and Continuous Dynamical Systems - B, 2011, 16 (4) : 1101-1117. doi: 10.3934/dcdsb.2011.16.1101 |
| [20] |
David G. Ebin. Global solutions of the equations of elastodynamics for incompressible materials. Electronic Research Announcements, 1996, 2: 50-59. |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]