Topological design with multiple constraints is of great importance in practical engineering design problems. The present work extends the bi-directional evolutionary structural optimization (BESO) method to multiple constraints of displacement and frequency in addition to the amount of material usage. Besides the binary design variables, the Lagrange multipliers for constraints are considered as additional continuous variables and determined by a search scheme. The enhanced approach can include a number of constraints besides the simple volume constraint. To demonstrate the effectiveness of the proposed BESO approach, several examples are presented for the maximization of structural overall stiffness subject to the material volume, displacement and frequency constraints.
BendsøeM.P. (1989). “Optimal shape design as a material distribution problem”, Structural and Multidisciplinary Optimizition, Vol. 1, No. 4, pp. 193–202.
2.
BendsøeM.P. and SigmundO. (2003). Topology Optimization: Theory, Methods and Applications, Springer, Berlin, Heidelberg, German.
3.
BendsøeM.P. and SigmundO. (1999). “Material interpolation schemes in topology optimization”, Archive of Applied Mechanics, Vol. 69, No. 9–10, pp. 635–654.
4.
EdwardsC.S.KimH.A. and BuddC.J. (2007). “An evaluative study on ESO and SIMP for optimizing a cantilever tie-beam”, Structural and Multidisciplinary Optimizition, Vol. 34, No. 5, pp. 403–414.
5.
HuangX. and XieY.M. (2007). “Convergent and meshindependent solutions for the bi-directional evolutionary structural optimization method”, Finite Element in Analysis and Design, Vol 43, No. 14, pp. 1039–1049.
6.
HuangX. and XieY.M. (2009). “Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials”, Computational Mechanics, Vol. 43, No. 3, pp. 393–401.
7.
HuangX. and XieY.M. (2010). “Evolutionary topology optimization of continuum structures with an additional displacement constraint”, Structural and Multidisciplinary Optimizition, Vol. 40, No. 1–6, pp. 409–416.
8.
HuangX. and XieY.M. (2010). Evolutionary Topology Optimization of Continuum Structures: Methods and Applications, John Wiley & Sons, Ltd, Chichester, England.
9.
HuangX.ZuoZ.H. and XieY.M. (2010). “Evolutionary topological optimization of vibrating continuum structures for natural frequencies”, Computers and Structures, Vol. 88, No. 5–6, pp. 357–364.
10.
LiQ.StevenG.P.QuerinO.M. and XieY.M. (1999a). “Evolutionary shape optimization for stress minimization”, Mechanics Research Communications, Vol. 26, No. 6, pp. 657–664.
11.
LiQ.StevenG.P. and XieY.M. (1999b). “Shape and topology design for heat conduction by evolutionary structural optimization”, International Journal of Heat and Mass Transfer, Vol. 42, No. 17, pp. 3361–3371.
12.
ManicharajahD. and XieY.M. (1998). “An evolutionary method for optimization of plate buckling resistance”, Finite Element in Analysis and Design, Vol. 29, No. 3–4, pp. 205–230.
13.
PedersenN.L. (2000). “Maximization of eigenvalues using topology optimization”, Structural and Multidisciplinary Optimization, Vol. 20, No. 1, pp. 2–11.
14.
PragerW. and RozvanyG.I.N. (1977). “Optimization of structural geometry”, Dynamical Systems, Academic Press, New York, USA, pp. 265–293.
15.
QuerinO.M.StevenG.P. and XieY.M. (1998). “Evolutionary structural optimization (ESO) using a bi-directional algorithm”, Engineering Computations, Vol. 15, No. 8, pp. 1034–1048.
16.
RongJ.H.XieY.M.YangX.Y. and LiangQ.Q. (2000). “Toplogy optimization of structures under dynamic response constraints”, Journal of Sound and Vibration, Vol. 234, No. 2, pp. 177–189.
17.
RozvanyG.I.N. (2008). “A critical review of established methods of structural topology optimisation”, Structural and Multidisciplinary Optimizition, Vol. 37, No. 3, pp. 217–237.
18.
RozvanyG.I.N. and QuerinO.M. (2002). “Combining ESO with rigorous optimality criteria”, International Journal of Vehicle Design, Vol. 28, No. 4, pp. 294–299.
19.
RozvanyG.I.N.ZhouM. and BirkerT. (1992). “Generalized shape optimization without homogenization”, Structural and Multidisciplinary Optimization, Vol. 4, No. 3–4, pp. 250–254.
20.
SigmundO. and PetersonJ. (1998). “Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima”, Structural and Multidisciplinary Optimization, Vol. 16, No. 1, pp. 68–75.
21.
SvanbergK. (1987). “The method of moving asymtotes – a new method for structural optimization”, International Journal of Numerical Methods in Engineering, Vol. 24, No. 2, pp. 359–373.
22.
TanskanenP. (2002). “The evolutionary structural optimization method: Theoretical aspects”, Computer Methods in Applied Mechanics and Engineering, Vol. 191, No. 47–48, pp. 5485–5498.
23.
TanskanenP. (2006). “A multiobjective and fixed elements based modification of the evolutionary structural optimization method”, Computer Methods in Applied Mechanics and Engineering, Vol. 196, No. 1–3, pp. 76–90.
24.
XieY.M. and StevenG.P. (1993). “A simple evolutionary procedure for structural optimization”, Computers and Structures, Vol. 49, No. 5, pp. 885–886.
25.
XieY.M. and StevenG.P. (1997). Evolutionary Structural Optimization, Springer, London, UK.
26.
YangX.Y.XieY.M.StevenG.P. and QuerinO.M. (1999). “Topology optimization for frequencies using an evolutionary method”, Journal of Structural Engineering, ASCE, Vol. 125, No. 12, pp. 1432–1438.
27.
ZhaoC.B.StevenG.P. and XieY.M. (1997). “Evolutionary natural frequency optimization of two-dimensional structures with additional non-structural lumped masses”, Engineering Computations, Vol. 14, No. 2, pp. 233–251.
28.
ZhouM. and RozvanyG.I.N. (1991). “The COC algorithm, Part II: Topological geometry and generalized shape optimization”, Computational Methods in Applied Mechanics and Engineering, Vol. 89, No. 1–3, pp. 197–224.
29.
ZhouM. and RozvanyG.I.N. (2001). “On the validity of ESO type methods in topology optimization”, Structural and Multidisciplinary Optimization, Vol. 21, No. 1, pp. 80–83.
30.
ZuoZ.H.XieY.M. and HuangX.D. (2010). “An improved bi-directional evolutionary topology optimization method for frequencies”, International Journal of Structural Stability and Dynamics, Vol. 10, No. 1, pp. 55–75.
