A multiobjective and fixed elements based modification of the evolutionary structural optimization method. (English) Zbl 1120.74708

Summary: In 1993, Y.M. Xie and G.P. Steven introduced an approach called evolutionary structural optimization (ESO). ESO is based on the simple idea that the optimal structure (maximum stiffness, minimum weight) can be produced by gradually removing the ineffectively used material from the design domain. The design domain is constructed by the FE method, and furthermore, external loads and support conditions are applied to the element model. Considering the engineering aspects, ESO seems to have some attractive features: the ESO method is very simple to program via the FEA packages and requires a relatively small amount of FEA time. On the other hand, different constraints cannot be added into the problem. In the ESO optimization the results supposedly approach truss-like, fully stressed topologies, which have the maximum stiffness with respect to the volume. Generally, these types of structures correspond to least-weight trusses.
Although ESO is not capable of handling general stress or displacement constraints, the design problems are often such that these constraints do not need to be included in the topology optimization, especially if the design optimization task is divided into two stages. In the first stage, only the overall geometry is outlined, and for that reason, the actual constraints do not have to be activated. In the second stage, the sizing optimization is performed. It can be concluded that ESO is well suited to solve the first stage optimization problems.
In some design problems it may occur that the structure cannot attain the fully stressed state because of geometrical constraints. It follows that the topology having the maximum stiffness with respect to the volume does not necessarily produce the least-weight structure when the stress constraints are applied in the second stage optimization. The geometrical constraints may force some structural components to be subject to a understressed state, i.e. to carry some “waste material”. As a consequence, the aim of this paper was to study whether ESO can be modified so that some geometrical constraints can be taken into account already in the first stage topology optimization. The modification was based on the assumption that if the stress level of otherwise understressed structural components can be increased during the \(compliance-volume\) product minimization, a lighter topology may be obtained. This new approach, the multiobjective and fixed elements based modification of the evolutionary structural optimization (MESO) utilizes a new optimization objective in which the overall stiffness of the structure and the loading of some parts of the structure are increased simultaneously. The gradient vector of the MESO objective function was determined by the FE method. Some of the partial derivatives involved were first presolved and then approximated. This approach was justified by large savings in the analysis time. Yet, MESO cannot take the general constraints into account.
To study the performance of the MESO optimization, two numerical examples were evaluated. The main purpose of these examples was to study whether MESO can produce structures lighter than the ESO results for problems having both stress and geometrical constraints.
These example were based on the two-stage optimization approach. In both example the MESO truss turned out to be lighter than the corresponding ESO truss. However, the ESO truss was the stiffest one also having a smaller overall stress value. In the question of shallow structures the deflection criteria may be predominant, and as a consequence, the ESO optimization may yield lighter structures than MESO.


74P10 Optimization of other properties in solid mechanics
74P15 Topological methods for optimization problems in solid mechanics
65K10 Numerical optimization and variational techniques


Full Text: DOI


[1] Baumgartner, A.; Harzheim, L.; Mattheck, C., SKO (soft kill option): the biological way to find an optimum structure topology, Int. J. fatigue, 14, 387-393, (1992)
[2] Baumgartner, A.; Mattheck, C., A new design of a bicycle frame: an example for an effective layout procedure based on FE-simulation of biological growth, (), 335-340
[3] Bendsøe, M.P., Optimal shape as a material distribution problem, Struct. optimiz., 1, 193-202, (1989)
[4] Bendsøe, M.P., Optimization of structural topology, shape, and material, (1995), Springer-Verlag Berlin · Zbl 0822.73001
[5] Bendsøe, M.P.; Kikuchi, N., Generating optimal topologies in structural design using a homogenization method, Comput. methods appl. mech. engrg., 71, 197-224, (1988) · Zbl 0671.73065
[6] Bendsøe, M.P.; Dı´az, A.R.; Taylor, J.E., On the prediction of extremal material properties and optimal material distribution for multiple loading conditions, (), 213-220
[7] Bendsøe, M.P.; Rasmussen, J.; Rodrigues, H.C., Topology and boundary shape optimization as an integrated tool for computer aided design, (), 27-34
[8] Chapman, C.D.; Jakiela, M.J., Genetic algorithm-based structural topology design with compliance and manufacturability considerations, (), 309-322
[9] Chapman, C.D.; Saitou, K.; Jakiela, M.J., Genetic algorithms as an approach to configuration and topology design, J. mech. des., 116, 1005-1012, (1994)
[10] Chu, D.N.; Xie, Y.M.; Hira, A.; Steven, G.P., Evolutionary structural optimization for problems with stiffness constraints, Finite elem. anal. des., 21, 239-251, (1996) · Zbl 0875.73149
[11] Chu, D.N.; Xie, Y.M.; Hira, A.; Steven, G.P., On various aspects of evolutionary structural optimization for problems with stiffness constraints, Finite elem. anal. des., 24, 197-212, (1997) · Zbl 0914.73037
[12] Dorn, W.C.; Gomory, R.E.; Greenberg, H.J., Automatic design of optimal structures, J. mecanique, 3, 25-52, (1964)
[13] Falzon, B.G.; Steven, G.P.; Xie, Y.M., Shape optimization of interior cutouts in composite panels, Struct. optimiz., 11, 43-49, (1996)
[14] Gallagher, R.H., Fully stressed design, (), 19-32
[15] Gea, H.C., Topology optimization: a new micro-structure based design domain method, (), 283-290
[16] Goldberg, D.E., Genetic algorithms in search, optimization and machine learning, (1989), Addison-Wesley Publishing Company Inc. USA · Zbl 0721.68056
[17] Haber, R.B.; Jog, C.S.; Bendsøe, M.P., Variable-topology shape optimization with a control of perimeter, (), 261-272
[18] Hajela, P.; Lee, E.; Lin, C.Y., Genetic algorithms in structural topology optimization, (), 117-133
[19] Harzheim, L.; Mattheck, C., 3D-shape optimization: different ways to an optimized design, (), 173-177
[20] Hinton, E.; Sienz, J., Fully stressed topological design of structures using an evolutionary procedure, Engrg. comput., 12, 229-244, (1995) · Zbl 0825.73468
[21] Holland, H.J., Adaption in natural and artificial systems, (1992), MIT Press USA
[22] Huber-Betzer, H.; Mattheck, C., Computer simulated self-optimization of bony structures, (), 293-300
[23] Mattheck, C., Design and growth rules for biological structures and their application to engineering, Fatigue fract. engrg. mater. struct., 13, 5, 535-550, (1990)
[24] Mattheck, C.; Burkhardt, S., A new method of structural shape optimization based on biological growth, Int. J. fatigue, 12, 3, 185-190, (1990)
[25] Mattheck, C.; Baumgartner, A.; Walther, F., Optimization procedures by use of the finite element method, (), 27-32
[26] Michell, A.G.M., The limits of economy of material in frame-structures, Philos. mag., 8, 589-597, (1904) · JFM 35.0828.01
[27] Olhoff, N.; Taylor, J.E., On structural optimization, J. appl. mech., 50, 1139-1151, (1983) · Zbl 0526.73090
[28] N. Olhoff, M.P. Bendsøe, J. Rasmussen, On CAD-integrated structural topology and design optimization, Report no. 27, Institute of Mechanical Engineering, Aalborg University, 1990. · Zbl 0802.73048
[29] Pedersen, P., Topology optimization of three dimensional trusses, (), 19-30
[30] Prager, W., Optimality criteria in structural design, Proc. nat. acad. sci., 61, 794-796, (1968)
[31] Rajan, S.D., Sizing, shape and topology design optimization of trusses using genetic algorithm, J. struct. engrg., 121, 10, 1480-1487, (1995)
[32] Rautaruukin putkipalkkikäsikirja, Rautaruukki Oyj, Otava, Keuruu, 1991.
[33] Ringertz, U.T., On topology optimization of trusses, Engrg. optimiz., 9, 209-218, (1985)
[34] Rodriguez-Velazquez, J.; Seireg, A.A., Optimizing the shapes of structures via a rule-based computer program, Comput. mech. engrg., 4, 20-28, (1985)
[35] Roško, P., Three-dimensional topology design of structures using crystal model, Comput. struct., 55, 6, 1077-1083, (1995) · Zbl 0919.73079
[36] Rozvany, G.I.N., Layout theory for grid-type structures, (), 251-272
[37] Rozvany, G.I.N.; Zhou, M., Application of the COC algorithm in layout optimization, (), 59-70
[38] Rozvany, G.I.N.; Zhou, M., Layout and generalized shape optimization by iterative COC methods, (), 103-120
[39] Sankaranarayanan, S.; Haftka, R.T.; Kapania, R.K., Truss topology optimization with stress and displacement constraints, (), 71-78
[40] ()
[41] Schmit, L.R., Structural optimization – some key ideas and insights, (), 1-45
[42] Suzuki, K.; Kikuchi, N., A homogenization method for shape topology optimization, Comput. methods appl. mech. engrg., 93, 291-318, (1991) · Zbl 0850.73195
[43] Tanskanen, P., The evolutionary structural optimization method: theoretical aspects, Comput. methods appl. mech. engrg., 191, 5485-5498, (2002) · Zbl 1083.74572
[44] Tenek, L.H.; Hagiwara, I., Optimal rectangular plate and shallow shell topologies using thickness distribution or homogenization, Comput. methods appl. mech. engrg., 115, 111-124, (1994)
[45] Teräsrakenteet, ohjeet 1996, Ympäristöministeriö, rakentamismääräyskokoelma B7, Finland, 1996.
[46] Vanderplaats, G.N., Thirty years of modern structural optimization, Adv. engrg. softw., 16, 81-88, (1993)
[47] Venkayya, V.B., Structural optimization: a review and some recommendations, Int. J. numer. methods engrg., 13, 203-228, (1978) · Zbl 0389.73079
[48] Xie, Y.M.; Steven, G.P., A simple evolutionary procedure for structural optimization, Comput. struct., 49, 885-896, (1993)
[49] Xie, Y.M.; Steven, G.P., Optimal design of multiple load case structures using an evolutionary procedure, Engrg. comput., 11, 295-302, (1994) · Zbl 0941.74524
[50] Xie, Y.M.; Steven, G.P., A simple approach to structural frequency optimization, Comput. struct., 53, 6, 1487-1491, (1994)
[51] Xie, Y.M.; Steven, G.P., Evolutionary structural optimization, (1997), Springer-Verlag London Limited London · Zbl 0898.73003
[52] Yang, R.J.; Chuang, C.H., Optimal topology design using linear programming, Comput. struct., 52, 2, 265-275, (1994) · Zbl 0900.73493
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.