Truss optimization with discrete design variables: a critical review

This review presents developed models, theory, and numerical methods for structural optimization of trusses with discrete design variables in the period 1968 – 2014. The comprehensive reference list collects, for the first time, the articles in the field presenting deterministic optimization methods and meta heuristics. The field has experienced a shift in focus from deterministic methods to meta heuristics, i.e. stochastic search methods. Based on the reported numerical results it is however not possible to conclude that this shift has improved the competences to solve application relevant problems. This, and other, observations lead to a set of recommended research tasks and objectives to bring the field forward. The development of a publicly available benchmark library is urgently needed to support development and assessment of existing and new heuristics and methods. Combined with this effort, it is recommended that the field begins to use modern methods such as performance profiles for fair and accurate comparison of optimization methods. Finally, theoretical results are rare in this field. This means that most recent methods and heuristics are not supported by mathematical theory. The field should therefore re-focus on theoretical issues such as problem analysis and convergence properties of new methods.

[1]  Liyong Tong,et al.  Improved genetic algorithm for design optimization of truss structures with sizing, shape and topology variables , 2005 .

[2]  Stephen P. Boyd,et al.  Semidefinite Programming , 1996, SIAM Rev..

[3]  R. Horst,et al.  Global Optimization: Deterministic Approaches , 1992 .

[4]  J. A. Bland.  OPTIMAL STRUCTURAL DESIGN BY ANT COLONY OPTIMIZATION , 2001 .

[5]  Extension of the generalized benders' decomposition , 1986 .

[6]  Harvey J. Greenberg,et al.  Automatic design of optimal structures , 1964 .

[7]  David A. Kendrick,et al.  GAMS : a user's guide, Release 2.25 , 1992 .

[8]  A. Kaveh,et al.  A DISCRETE BIG BANG - BIG CRUNCH ALGORITHM FOR OPTIMAL DESIGN OF SKELETAL STRUCTURES , 2010 .

[9]  Erik Lund,et al.  On structural optimization of composite shell structures using a discrete constitutive parametrization , 2004 .

[10]  D. F. Yates,et al.  The complexity of procedures for determining minimum weight trusses with discrete member sizes , 1982 .

[11]  Osman A. Shallan,et al.  Optimization of Plane and Space Trusses Using Genetic Algorithms , 2014 .

[12]  G. Rozvany.  On design-dependent constraints and singular topologies , 2001 .

[13]  Arthur M. Geoffrion,et al.  Elements of Large-Scale Mathematical Programming Part I: Concepts , 1970 .

[14]  M. J. Box.  A New Method of Constrained Optimization and a Comparison With Other Methods , 1965, Comput. J..

[15]  Yoshihiro Kanno,et al.  TOPOLOGY OPTIMIZATION OF TENSEGRITY STRUCTURES UNDER SELF-WEIGHT LOADS , 2012 .

[16]  Makoto Ohsaki.  Optimization of Finite Dimensional Structures , 2010 .

[17]  Saeed Shojaee,et al.  A HYBRID ALGORITHM FOR SIZING AND LAYOUT OPTIMIZATION OF TRUSS STRUCTURES COMBINING DISCRETE PSO AND CONVEX APPROXIMATION , 2013 .

[18]  Martin P. Bendsøe,et al.  Equivalent displacement based formulations for maximum strength truss topology design , 1992, IMPACT Comput. Sci. Eng..

[19]  W. Gutkowski,et al.  A DISCRETE METHOD FOR LATTICE STRUCTURES OPTIMIZATION , 1981 .

[20]  O. Hasançebi,et al.  Optimal design of planar and space structures with genetic algorithms , 2000 .

[21]  D. F. Yates,et al.  A SEGMENTAL METHOD FOR THE DISCRETE OPTIMUM DESIGN OF STRUCTURES , 1983 .

[22]  I. Grossmann.  Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques , 2002 .

[23]  Anikó Csébfalvi.  Discrete optimal weight design of geometrically nonlinear truss-structures , 1999 .

[24]  V. K. Koumousis,et al.  Reliability-Based Optimal Design of Truss Structures Using Particle Swarm Optimization , 2009 .

[25]  Mathias Stolpe,et al.  On the reformulation of topology optimization problems as linear or convex quadratic mixed 0–1 programs , 2007 .

[26]  Mathias Stolpe.  Models and Methods for Structural Topology Optimization with Discrete Design Variables , 2013 .

[27]  Mathias Stolpe,et al.  On some fundamental properties of structural topology optimization problems , 2010 .

[28]  A. K. Dhingra,et al.  Optimization of truss topology using tabu search , 1995 .

[29]  Kristo Mela,et al.  Resolving issues with member buckling in truss topology optimization using a mixed variable approach , 2014 .

[30]  Gui-Rong Liu,et al.  An optimization procedure for truss structures with discrete design variables and dynamic constraints , 2001 .

[31]  K. V. John,et al.  Discrete optimal design of trusses with stress and frequency constraints , 1990 .

[32]  D. S. Juang,et al.  A revised discrete Lagrangian-based search algorithm for the optimal design of skeletal structures using available sections , 2006 .

[33]  Sven Leyffer,et al.  Solving mixed integer nonlinear programs by outer approximation , 1994, Math. Program..

[34]  Prabhat Hajela,et al.  Design optimization with advanced genetic search strategies , 1994 .

[35]  G. Guerlement,et al.  Discrete minimum weight design of steel structures using EC3 code , 2001 .

[36]  Luca Maria Gambardella,et al.  Ant colony system: a cooperative learning approach to the traveling salesman problem , 1997, IEEE Trans. Evol. Comput..

[37]  M. Galante,et al.  GENETIC ALGORITHMS AS AN APPROACH TO OPTIMIZE REAL‐WORLD TRUSSES , 1996 .

[38]  Yoshihiro Kanno,et al.  Global optimization of trusses with constraints on number of different cross-sections: a mixed-integer second-order cone programming approach , 2016, Comput. Optim. Appl..

[39]  Hojjat Adeli,et al.  Hybrid CPN–Neural Dynamics Model for Discrete Optimization of Steel Structures , 1996 .

[40]  Y. Kanno.  Topology optimization of tensegrity structures under compliance constraint: a mixed integer linear programming approach , 2013 .

[41]  C. Fleury,et al.  Dual methods and approximation concepts in structural synthesis , 1980 .

[42]  M. Bendsøe,et al.  Topology Optimization: "Theory, Methods, And Applications" , 2011 .

[43]  Ardeshir Bahreininejad,et al.  Mine blast algorithm for optimization of truss structures with discrete variables , 2012 .

[44]  O. Hasançebi,et al.  Performance evaluation of metaheuristic search techniques in the optimum design of real size pin jointed structures , 2009 .

[45]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[46]  M. Ohsaki.  Random search method based on exact reanalysis for topology optimization of trusses with discrete cross-sectional areas , 2001 .

[47]  Mustafa Sonmez,et al.  Discrete optimum design of truss structures using artificial bee colony algorithm , 2011 .

[48]  Wolfgang Achtziger,et al.  Structural Topology Optimization with Eigenvalues , 2007, SIAM J. Optim..

[49]  Ayse T. Daloglu,et al.  An improved genetic algorithm with initial population strategy and self-adaptive member grouping , 2008 .

[50]  Aleksander Muc,et al.  Optimal design of engineering structures , 2017, 2017 13th International Conference on Natural Computation, Fuzzy Systems and Knowledge Discovery (ICNC-FSKD).

[51]  Yongcun Zhang,et al.  A new method of discrete optimization for cross-section selection of truss structures , 2014 .

[52]  Samuel L. Lipson,et al.  DISCRETE SIZING OF TRUSSES FOR OPTIMAL GEOMETRY , 1977 .

[53]  N. Katoh,et al.  Semi-definite programming for topology optimization of trusses under multiple eigenvalue constraints , 1999 .

[54]  Arkadi Nemirovski,et al.  Lectures on modern convex optimization - analysis, algorithms, and engineering applications , 2001, MPS-SIAM series on optimization.

[55]  Siamak Talatahari,et al.  Ant Colony Optimization for Design of Space Trusses , 2008 .

[56]  A. Dhingra,et al.  Single and multiobjective structural optimization in discrete‐continuous variables using simulated annealing , 1995 .

[57]  P. Hajela,et al.  Genetic algorithms in truss topological optimization , 1995 .

[58]  S. Wu,et al.  Steady-state genetic algorithms for discrete optimization of trusses , 1995 .

[59]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[60]  Agostinho Agra,et al.  New branch and bound approaches for truss topology design with discrete areas , 2010 .

[61]  P. Pardalos,et al.  Handbook of global optimization , 1995 .

[62]  Mathias Stolpe,et al.  Global optimization of discrete truss topology design problems using a parallel cut-and-branch method , 2008 .

[63]  A. R. Toakley,et al.  Optimum Design Using Available Sections , 1968 .

[64]  O. Hasançebi,et al.  Layout optimization of trusses using improved GA methodologies , 2001 .

[65]  W. H. Zhang,et al.  Combined shape and sizing optimization of truss structures , 2002 .

[66]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[67]  S. Chai,et al.  An application of relative difference quotient algorithm to topology optimization of truss structures with discrete variables , 1999 .

[68]  Ibrahim Eksin,et al.  A new optimization method: Big Bang-Big Crunch , 2006, Adv. Eng. Softw..

[69]  I. Grossmann,et al.  Convergence properties of generalized benders decomposition , 1991 .

[70]  Zafer Gürdal,et al.  A penalty approach for nonlinear optimization with discrete design variables , 1990 .

[71]  Helio J. C. Barbosa,et al.  Rank‐based ant colony algorithms for truss weight minimization with discrete variables , 2006 .

[72]  K. Svanberg.  The method of moving asymptotes—a new method for structural optimization , 1987 .

[73]  Zong Woo Geem,et al.  A New Heuristic Optimization Algorithm: Harmony Search , 2001, Simul..

[74]  Mathias Stolpe,et al.  Generalized Benders’ Decomposition for topology optimization problems , 2011, J. Glob. Optim..

[75]  Matteo Fischetti,et al.  Projected Chvátal–Gomory cuts for mixed integer linear programs , 2008, Math. Program..

[76]  Mathias Stolpe,et al.  Truss topology optimization with discrete design variables by outer approximation , 2015, J. Glob. Optim..

[77]  Mathias Stolpe,et al.  To bee or not to bee—comments on “Discrete optimum design of truss structures using artificial bee colony algorithm” , 2011 .

[78]  A heuristic method for the design of minimum weight trusses using discrete member sizes , 1983 .

[79]  V. Toğan,et al.  Optimization of 3d trusses with adaptive approach in genetic algorithms , 2006 .

[80]  Charles V. Camp,et al.  Design of space trusses using modified teaching–learning based optimization , 2014 .

[81]  Feng Liu,et al.  A heuristic particle swarm optimization method for truss structures with discrete variables , 2009 .

[82]  Yoshihiro Kanno,et al.  Topology design of tensegrity structures via mixed integer programming , 2010 .

[83]  R. Haftka.  Simultaneous analysis and design , 1985 .

[84]  Arkadi Nemirovski,et al.  Robust Truss Topology Design via Semidefinite Programming , 1997, SIAM J. Optim..

[85]  Moacir Kripka,et al.  Discrete optimization of trusses by simulated annealing , 2004 .

[86]  Helio J. C. Barbosa,et al.  An adaptive penalty scheme for genetic algorithms in structural optimization , 2004 .

[87]  Atsushi Kawamoto,et al.  Design of planar articulated mechanisms using branch and bound , 2004, Math. Program..

[88]  Joaquim J. Júdice,et al.  An integer programming model for truss topology optimization , 2006 .

[89]  A. Kaveh,et al.  A novel heuristic optimization method: charged system search , 2010 .

[90]  P. Hajela,et al.  Genetic search strategies in multicriterion optimal design , 1991 .

[91]  R. Cook,et al.  Concepts and Applications of Finite Element Analysis , 1974 .

[92]  M. Bendsøe,et al.  Material interpolation schemes in topology optimization , 1999 .

[93]  Jan A. Snyman,et al.  A regional genetic algorithm for the discrete optimal design of truss structures , 1999 .

[94]  J. Arora,et al.  Methods for optimization of nonlinear problems with discrete variables: A review , 1994 .

[95]  Ali Kaveh,et al.  Genetic algorithm for discrete‐sizing optimal design of trusses using the force method , 2002 .

[96]  Fred Glover,et al.  Tabu Search - Part II , 1989, INFORMS J. Comput..

[97]  W. M. Jenkins.  A decimal-coded evolutionary algorithm for constrained optimization , 2002 .

[98]  Subramaniam Rajan,et al.  Sizing, Shape, and Topology Design Optimization of Trusses Using Genetic Algorithm , 1995 .

[99]  Yoshihiro Kanno,et al.  Exploring new tensegrity structures via mixed integer programming , 2013 .

[100]  James Kennedy,et al.  Particle swarm optimization , 2002, Proceedings of ICNN'95 - International Conference on Neural Networks.

[101]  Jan A. Snyman,et al.  A pseudo-discrete rounding method for structural optimization , 1996 .

[102]  J. Cai,et al.  A parallel evolution strategy for solving discrete structural optimization , 1996 .

[103]  Wolfgang Achtziger,et al.  Global optimization of truss topology with discrete bar areas—Part I: theory of relaxed problems , 2008, Comput. Optim. Appl..

[104]  G. Vanderplaats,et al.  Method for nonlinear optimization with discrete design variables , 1989 .

[105]  Andrew B. Templeman,et al.  Discrete optimum structural design , 1988 .

[106]  Ali Kaveh,et al.  Colliding Bodies Optimization method for optimum discrete design of truss structures , 2014 .

[107]  Jorge J. Moré,et al.  Digital Object Identifier (DOI) 10.1007/s101070100263 , 2001 .

[108]  Ulf Ringertz,et al.  ON METHODS FOR DISCRETE STRUCTURAL OPTIMIZATION , 1988 .

[109]  Charles V. Camp.  DESIGN OF SPACE TRUSSES USING BIG BANG–BIG CRUNCH OPTIMIZATION , 2007 .

[110]  Mehmet Polat Saka,et al.  OPTIMUM DESIGN OF PIN-JOINTED STEEL STRUCTURES WITH PRACTICAL APPLICATIONS , 1990 .

[111]  Ali Kaveh,et al.  Size/geometry optimization of trusses by the force method and genetic algorithm , 2004 .

[112]  K. Lee,et al.  The harmony search heuristic algorithm for discrete structural optimization , 2005 .

[113]  K. Deb,et al.  Design of truss-structures for minimum weight using genetic algorithms , 2001 .

[114]  Arkadi Nemirovski,et al.  Potential Reduction Polynomial Time Method for Truss Topology Design , 1994, SIAM J. Optim..

[115]  E. Salajegheh,et al.  Optimum design of trusses with discrete sizing and shape variables , 1993 .

[116]  Charles V. Camp,et al.  Design of Space Trusses Using Ant Colony Optimization , 2004 .

[117]  A. Kaveh,et al.  Sizing, geometry and topology optimization of trusses via force method and genetic algorithm , 2008 .

[118]  Du Ming-zhu,et al.  An improved Templeman's algorithm for the optimum design of trusses with discrete member sizes , 1986 .

[119]  Hiroshi Ohmori,et al.  Truss topology optimization by a modified genetic algorithm , 2002 .

[120]  E. Balas.  An Additive Algorithm for Solving Linear Programs with Zero-One Variables , 1965 .

[121]  M. Ohsaki.  Genetic algorithm for topology optimization of trusses , 1995 .

[122]  W. Gutkowski,et al.  An effective method for discrete structural optimization , 2000 .

[123]  A. M. Geoffrion.  Generalized Benders decomposition , 1972 .

[124]  Srinivas Bollapragada,et al.  Optimal Design of Truss Structures by Logic-Based Branch and Cut , 2001, Oper. Res..

[125]  L. A. Schmit,et al.  Discrete-continuous variable structural synthesis using dual methods , 1980 .

[126]  D. Karaboga,et al.  On the performance of artificial bee colony (ABC) algorithm , 2008, Appl. Soft Comput..

[127]  G. Sved,et al.  Structural optimization under multiple loading , 1968 .

[128]  K. Svanberg,et al.  On the convexity and concavity of compliances , 1994 .

[129]  M. Stolpe,et al.  Truss topology optimization with discrete design variables—Guaranteed global optimality and benchmark examples , 2007 .

[130]  A. Kaveh,et al.  Size optimization of space trusses using Big Bang-Big Crunch algorithm , 2009 .

[131]  Panos Y. Papalambros,et al.  Solution of mixed-discrete structural optimization problems with a new sequential linearization algorithm , 1990 .

[132]  Ole Sigmund,et al.  On the usefulness of non-gradient approaches in topology optimization , 2011 .

[133]  Prabhat Hajela,et al.  Handling variable string lengths in GA-based structural topology optimization , 2004 .

[134]  J. Cassis,et al.  An efficient algorithm for the optimum design of trusses with discrete variables , 1986 .

[135]  Ignacio E. Grossmann,et al.  Mixed-integer linear programming reformulations for some nonlinear discrete design optimization problems , 1992 .

[136]  Nicholas I. M. Gould,et al.  CUTEr and SifDec: A constrained and unconstrained testing environment, revisited , 2003, TOMS.

[137]  Andrea Lodi,et al.  MIPLIB 2010 , 2011, Math. Program. Comput..

[138]  Xu Guo,et al.  A mixed integer programming for robust truss topology optimization with stress constraints , 2010 .

[139]  Samuel L. Lipson,et al.  WEIGHT OPTIMIZATION OF PLANE TRUSSES , 1974 .

[140]  Wolfgang Achtziger,et al.  Global optimization of truss topology with discrete bar areas—Part II: Implementation and numerical results , 2009, Comput. Optim. Appl..

[141]  Siamak Talatahari,et al.  A CHARGED SYSTEM SEARCH WITH A FLY TO BOUNDARY METHOD FOR DISCRETE OPTIMUM DESIGN OF TRUSS STRUCTURES , 2010 .

[142]  Siamak Talatahari,et al.  A particle swarm ant colony optimization for truss structures with discrete variables , 2009 .

[143]  Christian Kanzow,et al.  Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications , 2008, Math. Program..

[144]  M. Pyrz,et al.  Discrete optimization of geometrically nonlinear truss structures under stability constraints , 1990 .

[145]  S. Rajeev,et al.  Discrete Optimization of Structures Using Genetic Algorithms , 1992 .

[146]  Brian W. Kernighan,et al.  AMPL: A Modeling Language for Mathematical Programming , 1993 .

[147]  Erik Lund,et al.  Discrete material optimization of general composite shell structures , 2005 .

[148]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[149]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[150]  Mathias Stolpe,et al.  Global optimization of large-scale topology design problems using branch-and-bound methods , 2006 .

[151]  Bartlomiej Blachowski,et al.  Discrete structural optimization by removing redundant material , 2008 .

[152]  Hyo Seon Park,et al.  Optimization of steel structures using distributed simulated annealing algorithm on a cluster of personal computers , 2002 .

[153]  Ignacio E. Grossmann,et al.  An outer-approximation algorithm for a class of mixed-integer nonlinear programs , 1986, Math. Program..

[154]  W. Achtziger.  Multiple-Load Truss Topology and Sizing Optimization: Some Properties of Minimax Compliance , 1998 .

[155]  E. Hinton,et al.  Optimization of trusses using genetic algorithms for discrete and continuous variables , 1999 .