Comptes Rendus
Short paper
Large-scale smooth plastic topology optimization using domain decomposition
Comptes Rendus. Mécanique, Volume 349 (2021) no. 2, pp. 323-344.

A domain decomposition procedure based on overlapping partitions of the design domain is proposed for solving large problems of smooth topology optimization of plastic continua. The procedure enables the solution of problems with sizes exceeding the available computational and storage capacities. It takes advantage of the favorable features of the integrated limit analysis and design formulation of the smooth topology design problem. The integrated approach preserves the mathematical structure and properties of the underlying static, lower bound problem of limit analysis. In particular, the formulation is characterized by weak coupling between subproblems because it does not involve a stress–strain relationship. The decomposition strategy begins by solving a reduced design problem, using a coarse finite element mesh, followed by an iterative process using a fine discretization. At each iteration, an independent topology optimization subproblem is associated with each subdomain, considered as a substructure. The traction vectors acting on the subdomain boundaries are updated at each iteration as the overlapping partitions are switched. The numerical tests showed that as early as the first iteration, the decomposition process generates a feasible, near optimal design with a weight less than 0.1% above the direct solution.

Published online:
DOI: 10.5802/crmeca.88
Keywords: Topology optimization, Direct method, Limit analysis, Domain decomposition, Plastic design

Mohamed Fourati 1; Zied Kammoun 2; Jamel Neji 1; Hichem Smaoui 3, 1

1 Université de Tunis El Manar, Ecole Nationale d’Ingénieurs de Tunis, LR11ES16 Laboratoire de Matériaux, Optimisation et Energie pour la Durabilité, 1002, Tunis, Tunisie
2 Université de Carthage, Institut Supérieur des Technologies de l’Environnement de l’Urbanisme et du Bâtiment, 2 Rue de l’Artisanat Charguia 2, 2035 Tunis, Tunisie
3 Faculty of Engineering, King Abdulaziz University, P.O. Box 80230, Jeddah, Kingdom of Saudi Arabia
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Mohamed Fourati and Zied Kammoun and Jamel Neji and Hichem Smaoui},
     title = {Large-scale smooth plastic topology optimization using domain decomposition},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {323--344},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {349},
     number = {2},
     year = {2021},
     doi = {10.5802/crmeca.88},
     language = {en},
AU  - Mohamed Fourati
AU  - Zied Kammoun
AU  - Jamel Neji
AU  - Hichem Smaoui
TI  - Large-scale smooth plastic topology optimization using domain decomposition
JO  - Comptes Rendus. Mécanique
PY  - 2021
SP  - 323
EP  - 344
VL  - 349
IS  - 2
PB  - Académie des sciences, Paris
DO  - 10.5802/crmeca.88
LA  - en
ID  - CRMECA_2021__349_2_323_0
ER  - 
%0 Journal Article
%A Mohamed Fourati
%A Zied Kammoun
%A Jamel Neji
%A Hichem Smaoui
%T Large-scale smooth plastic topology optimization using domain decomposition
%J Comptes Rendus. Mécanique
%D 2021
%P 323-344
%V 349
%N 2
%I Académie des sciences, Paris
%R 10.5802/crmeca.88
%G en
%F CRMECA_2021__349_2_323_0
Mohamed Fourati; Zied Kammoun; Jamel Neji; Hichem Smaoui. Large-scale smooth plastic topology optimization using domain decomposition. Comptes Rendus. Mécanique, Volume 349 (2021) no. 2, pp. 323-344. doi : 10.5802/crmeca.88.

[1] G. Costa; M. Montemurro; J. Pailhès NURBS hyper-surfaces for 3D topology optimization problems, Mech. Adv. Mater. Struct., Volume 28 (2021) no. 7, pp. 665-684 | DOI

[2] K.-F. Seitz; J. Grabe Three-dimensional topology optimization for geotechnical foundations in granular soil, Comput. Geotech., Volume 80 (2016), pp. 41-48 | DOI

[3] P. G. Coelho; J. B. Cardoso; P. R. Fernandes; H. C. Rodrigues Parallel computing techniques applied to the simultaneous design of structure and material, Adv. Eng. Softw., Volume 42 (2011) no. 5, pp. 219-227 | DOI | Zbl

[4] M. P. Scardaoni; M. Montemurro A general global-local modelling framework for the deterministic optimisation of composite structures, Struct. Multidiscipl. Optim., Volume 62 (2020), pp. 1927-1949 | DOI | MR

[5] J. Sobieszczanski-Sobieski; B. B. James; A. R. Dovi Structural optimization by multilevel decomposition, AIAA J., Volume 23 (1985) no. 11, pp. 1775-1782 | DOI | MR | Zbl

[6] T. Woo; L. Schmit Decomposition in optimal plastic design of structures, Int. J. Solids Struct., Volume 17 (1981) no. 1, pp. 39-56 | DOI | MR | Zbl

[7] B. H. V. Topping; A. I. Khan Parallel Finite Element Computations, Saxe-Coburg Publications, Edinburgh, 1996

[8] L. S. Lasdon Optimization Theory for Large Systems, Courier Corporation, USA, 2002 | Zbl

[9] T. Borrvall; J. Petersson Large-scale topology optimization in 3D using parallel computing, Comput. Methods Appl. Mech. Eng., Volume 190 (2001) no. 46–47, pp. 6201-6229 | DOI | Zbl

[10] A. Mahdavi; R. Balaji; M. Frecker; E. M. Mockensturm Topology optimization of 2D continua for minimum compliance using parallel computing, Struct. Multidiscipl. Optim., Volume 32 (2006) no. 2, pp. 121-132 | DOI

[11] Z. Kammoun; H. Smaoui A direct approach for continuous topology optimization subject to admissible loading, C. R. Méc., Volume 342 (2014) no. 9, pp. 520-531 | DOI

[12] Z. Kammoun A formulation for multiple loading cases in plastic topology design of continua, C. R. Méc., Volume 344 (2016) no. 10, pp. 725-735 | DOI

[13] Z. Kammoun; M. Fourati; H. Smaoui Direct limit analysis based topology optimization of foundations, Soils Found., Volume 59 (2019) no. 4, pp. 1063-1072 | DOI

[14] M. A. Herfelt; P. N. Poulsen; L. C. Hoang Strength-based topology optimisation of plastic isotropic von Mises materials, Struct. Multidiscipl. Optim., Volume 59 (2019) no. 3, pp. 893-906 | DOI | MR

[15] J. Fin; L. A. Borges; E. A. Fancello Structural topology optimization under limit analysis, Struct. Multidiscipl. Optim., Volume 59 (2019) no. 4, pp. 1355-1370 | DOI | MR

[16] Z. Kammoun; H. Smaoui A direct method formulation for topology plastic design of continua, Direct Methods for Limit and Shakedown Analysis of Structures (P. Fuschi; A. Pisano; D. Weichert, eds.), Springer, Cham, 2015, pp. 47-63

[17] H. Smaoui; Z. Kammoun Convergence of the direct limit analysis design method for discrete topology optimization, Optim. Eng. (2020) | DOI

[18] A. Lyamin; S. Sloan Lower bound limit analysis using non-linear programming, Int. J. Numer. Methods Eng., Volume 55 (2002) no. 5, pp. 573-611 | DOI | Zbl

[19] A. Lyamin; S. Sloan Upper bound limit analysis using linear finite elements and non-linear programming, Int. J. Numer. Anal. Methods Geomech., Volume 26 (2002) no. 2, pp. 181-216 | DOI | Zbl

[20] J. Fin; L. A. Borges; E. A. Fancello Structural topology optimization under limit analysis, Struct. Multidiscipl. Optim., Volume 59 (2019) no. 4, pp. 1355-1370 | DOI | MR

[21] M. P. Bendsøe; O. Sigmund Material interpolation schemes in topology optimization, Arch. Appl. Mech., Volume 69 (1999) no. 9, pp. 635-654 | Zbl

[22] P. Duysinx; M. P. Bendsøe Topology optimization of continuum structures with local stress constraints, Int. J. Numer. Methods Eng., Volume 43 (1998) no. 8, pp. 1453-1478 | DOI | MR | Zbl

[23] M. Bruggi On an alternative approach to stress constraints relaxation in topology optimization, Struct. Multidiscipl. Optim., Volume 36 (2008) no. 2, pp. 125-141 | DOI | MR | Zbl

[24] C. Le; J. Norato; T. Bruns; C. Ha; D. Tortorelli Stress-based topology optimization for continua, Struct. Multidiscipl. Optim., Volume 41 (2010) no. 4, pp. 605-620 | DOI

[25] D. Yang; H. Liu; W. Zhang; S. Li Stress-constrained topology optimization based on maximum stress measures, Comput. Struct., Volume 198 (2018), pp. 23-39 | DOI

[26] C. Fleury; H. Smaoui Convex approximation strategies in structural optimization, Discretization Methods and Structural Optimization—Procedures and Applications (H. A. Eschenauer; G. Thierauf, eds.) (Lecture Notes in Engineering), Volume 42, Springer, Berlin, Heidelberg, 1989, pp. 118-126 | DOI

[27] K. Svanberg A class of globally convergent optimization methods based on conservative convex separable approximations, SIAM J. Optim., Volume 12 (2002) no. 2, pp. 555-573 | DOI | MR | Zbl

[28] G. Xu; C. Gengdong Epsilon-continuation approach for truss topology optimization, Acta Mech. Sin., Volume 20 (2004) no. 5, pp. 526-533 | DOI

[29] Y. M. Xie; G. P. Steven A simple evolutionary procedure for structural optimization, Comput. Struct., Volume 49 (1993) no. 5, pp. 885-896 | DOI

[30] M. P. Bendsøe; N. Kikuchi Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Eng., Volume 71 (1988) no. 2, pp. 197-224 | DOI | MR | Zbl

[31] G. Strang; R. V. Kohn Optimal design in elasticity and plasticity, Int. J. Numer. Methods Eng., Volume 22 (1986) no. 1, pp. 183-188 | DOI | MR | Zbl

[32] G. Costa; M. Montemurro; J. Pailhès Minimum length scale control in a NURBS-based SIMP method, Comput. Methods Appl. Mech. Eng., Volume 354 (2019), pp. 963-989 | DOI | MR | Zbl

[33] R. Zakhama; M. Abdalla; H. Smaoui; Z. Gürdal Multigrid implementation of cellular automata for topology optimization, First International Conference on Multidisciplinary Design and Applications (2007) | Zbl

[34] G. Costa; M. Montemurro; J. Pailhès A 2D topology optimisation algorithm in NURBS framework with geometric constraints, Int. J. Mech. Mater. Des., Volume 14 (2018) no. 4, pp. 669-696 | DOI

[35] K. Maute; S. Schwarz; E. Ramm Adaptive topology optimization of elastoplastic structures, Struct. Optim., Volume 15 (1998) no. 2, pp. 81-91 | DOI

[36] M. Wallin; V. Jönsson; E. Wingren Topology optimization based on finite strain plasticity, Struct. Multidiscipl. Optim., Volume 54 (2016) no. 4, pp. 783-793 | DOI | MR

[37] G. Zhang; L. Li; K. Khandelwal Topology optimization of structures with anisotropic plastic materials using enhanced assumed strain elements, Struct. Multidiscipl. Optim., Volume 55 (2017) no. 6, pp. 1965-1988 | DOI | MR

[38] J. Salençon Yield Design, John Wiley & Sons, 2013 | DOI | Zbl

[39] D. C. Drucker; W. Prager Soil mechanics and plastic analysis or limit design, Q. Appl. Math., Volume 10 (1952) no. 2, pp. 157-165 | DOI | MR | Zbl

[40] A. Gvozdev The determination of the value of the collapse load for statically indeterminate systems undergoing plastic deformation, Int. J. Mech. Sci., Volume 1 (1960) no. 4, pp. 322-335 | DOI | MR

[41] G. Maier Quadratic programming and theory of elastic-perfectly plastic structures, Meccanica, Volume 3 (1968) no. 4, pp. 265-273 | DOI | MR | Zbl

[42] J. Lysmer Limit analysis of plane problems in soil mechanics, J. Soil Mech. Found. Div., Volume 96 (1970) no. 4, pp. 1311-1334 | DOI

[43] J. Pastor Analyse limit determination numerique de solutions statistique completes, Application au talus vertical, J. Mech. Appl., Volume 2 (1978), pp. 167-196

[44] E. D. Andersen; C. Roos; T. Terlaky On implementing a primal-dual interior-point method for conic quadratic optimization, Math. Program., Volume 95 (2003) no. 2, pp. 249-277 | DOI | MR | Zbl

[45] H. Smaoui Finite element limit load analysis by the static approach using nonlinear programming, Computing in Civil Engineering, ASCE, 1994, pp. 1936-1943

[46] J. Pastor; T.-H. Thai; P. Francescato Interior point optimization and limit analysis: an application, Int. J. Numer. Methods Biomed. Eng., Volume 19 (2003) no. 10, pp. 779-785 | Zbl

[47] Z. Kammoun; F. Pastor; H. Smaoui; J. Pastor Large static problem in numerical limit analysis: a decomposition approach, Int. J. Numer. Anal. Methods Geomech., Volume 34 (2010) no. 18, pp. 1960-1980 | DOI | Zbl

[48] K. Anoukou; F. Pastor; P. Dufrenoy; D. Kondo Limit analysis and homogenization of porous materials with Mohr–Coulomb matrix. Part I: Theoretical formulation, J. Mech. Phys. Solids, Volume 91 (2016), pp. 145-171 | DOI | MR

[49] R. Sun; J. Yang Axisymmetric adaptive lower bound limit analysis for Mohr–Coulomb materials using semidefinite programming, Comput. Geotech., Volume 130 (2021), 103906

[50] K. Krabbenhoft; L. Damkilde A general non-linear optimization algorithm for lower bound limit analysis, Int. J. Numer. Methods Eng., Volume 56 (2003) no. 2, pp. 165-184 | DOI | Zbl

[51] R. T. Haftka Integrated nonlinear structural analysis and design, AIAA J., Volume 27 (1989) no. 11, pp. 1622-1627 | DOI | Zbl

[52] H. Smaoui; L. Schmit An integrated approach to the synthesis of geometrically non-linear structures, Int. J. Numer. Methods Eng., Volume 26 (1988) no. 3, pp. 555-570 | DOI | Zbl

[53] D. Yang; H. Liu; W. Zhang; S. Li Stress-constrained topology optimization based on maximum stress measures, Comput. Struct., Volume 198 (2018), pp. 23-39 | DOI

[54] J. F. Benders Partitioning procedures for solving mixed-variables programming problems, Numer. Math., Volume 4 (1962), pp. 238-252 | DOI | MR | Zbl

[55] B. Maar; V. Schulz Interior point multigrid methods for topology optimization, Struct. Multidiscipl. Optim., Volume 19 (2000) no. 3, pp. 214-224 | DOI

[56] F. Pastor; E. Loute Limit analysis decomposition and finite element mixed method, J. Comput. Appl. Math., Volume 234 (2010) no. 7, pp. 2213-2221 | DOI | MR | Zbl

[57] J. J. Muñoz; N. Rabiei; A. Lyamin; A. Huerta Computation of bounds for anchor problems in limit analysis and decomposition techniques, Direct Methods for Limit States in Structures and Materials, Springer, 2014, pp. 79-99 | DOI

[58] J. Salençon Théorie des charges limites: poinçonnement d’une plaque par deux poinçons symétriques en déformation plane, C. R. Méc. Acad. Sci. Paris, Volume 265 (1967), pp. 869-872 | Zbl

[59] F. Alizadeh; D. Goldfarb Second-order cone programming, Math. Program., Volume 95 (2003) no. 1, pp. 3-51 | DOI | MR | Zbl

[60] A. Mosek The MOSEK optimization software, 2020 (Online at)

[61] Q. Q. Liang; Y. M. Xie; G. P. Steven Optimal topology selection of continuum structures with displacement constraints, Compos. Struct., Volume 77 (2000) no. 6, pp. 635-644 | DOI

[62] G. Allaire; M. Schoenauer Conception Optimale de Structures, 58, Springer, 2007 | MR

[63] M. Abdi; I. Ashcroft; R. Wildman Topology optimization of geometrically nonlinear structures using an evolutionary optimization method, Eng. Optim., Volume 50 (2018) no. 11, pp. 1850-1870 | DOI | MR

Cited by Sources:

Comments - Policy