Research Papers

Decomposition Templates and Joint Morphing Operators for Genetic Algorithm Optimization of Multicomponent Structural Topology

[+] Author and Article Information
Zebin Zhou

CAE Engineering,
Safety Engineering
and Virtual Technology Technical Center,
SAIC Motor Passenger Vehicle Co.,
Jiading, Shanghai 201804, China
e-mail: zhouzebin@saicmotor.com

Karim Hamza

Senior Research Fellow
e-mail: khamza@umich.edu

Kazuhiro Saitou

e-mail: kazu@umich.edu
Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109-2102

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received September 14, 2012; final manuscript received November 1, 2013; published online December 11, 2013. Assoc. Editor: Michael Kokkolaras.

J. Mech. Des 136(2), 021004 (Dec 11, 2013) (13 pages) Paper No: MD-12-1456; doi: 10.1115/1.4026030 History: Received September 14, 2012; Revised November 01, 2013

This paper presents a continuum-based approach for multi-objective topology optimization of multicomponent structures. Objectives include minimization of compliance, weight, and cost of assembly and manufacturing. Design variables are partitioned into two main groups: those pertaining to material allocation within a design domain (base topology problem), and those pertaining to decomposition of a monolithic structure into multiple components (joint allocation problem). Generally speaking, the two problems are coupled in the sense that the decomposition of an optimal monolithic structure is not always guaranteed to produce an optimal multicomponent structure. However, for spot-welded sheet-metal structures (such as those often found in automotive applications), certain assumptions can be made about the performance of a monolithic structure that favor the adoption of a two-stage approach that decouples the base topology and joint allocation problems. A multi-objective genetic algorithm (GA) is used throughout the studies in this paper. While the problem decoupling in two-stage approaches significantly reduces the size of the search space and allows better performance of the GA, the size of the search space can still be quite enormous in the second stage. To further improve the performance, a new mutation operator based on decomposition templates and localized joints morphing is proposed. A cantilever-loaded structure is used to study and compare various setups of single and two-stage GA approaches and establish the merit of the proposed GA operators. The approach is then applied to a simplified model of an automotive vehicle floor subject to global bending loading condition.

Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.


Lyu, N., and Saitou, K., 2005, “Topology Optimization of Multi-Component Structures Via Decomposition-Based Assembly Synthesis,” ASME J. Mech. Des., 127(2), pp. 170–183. [CrossRef]
Yildiz, A., and Saitou, K., 2011, “Topology Synthesis of Multi-Component Structural Assemblies in Continuum Domains,” ASME J. Mech. Des., 133, p. 011008. [CrossRef]
Zhou, Z., Hamza, K., and Saitou, K., 2011, “Multi-Objective Topology Optimization of Spot-Welded Planar Multi-Component Continuum Structures,” 9th World Congress on Structural and Multidisciplinary Optimization, Shizuoka, Japan.
Dorn, W. C., Gomory, R. E., and Greenberg, H. J., 1964, “Automatic Design of Optimal Structures,” J. Mec., 3, pp. 25–52.
Bendsøe, M. P., and Kikuchi, N., 1988, “Generating Optimal Topologies in Structural Design Using a Homogenization Method,” Comput. Methods Appl. Mech. Eng., 71, pp. 197–224. [CrossRef]
Bendsøe, M. P., Ben-Tal, A., and Zowe, J., 1994, “Optimization Methods for Truss Geometry and Topology Design,” Struct. Optim., 7, pp. 141–159. [CrossRef]
Kirsch, U., 1989, “Optimal Topologies of Structures,” Appl. Mech. Rev., 42(8), pp. 223–238. [CrossRef]
Rozvany, G. I. N., Bendsøe, M. P., and Kirsch.U., 1995, “Layout Optimization of Structures,” Appl. Mech. Rev., 48, pp. 41–119. [CrossRef]
Shea, K., Cagan, J., and Fenves, S. J., 1997, “A Shape Annealing Approach to Optimal Truss Design With Dynamic Grouping of Members,” ASME J. Mech. Des., 119(3), pp. 388–394. [CrossRef]
Reddy, G., and Cagan, J., 1995, “Optimally Directed Truss Topology Generation Using Shape Annealing,” ASME J. Mech. Des., 117(1), pp. 206–209. [CrossRef]
Gil, L., and Andreu, A., 2001, “Shape and Cross-Section Optimisation of a Truss Structure,” Comput. Struct., 79, pp. 681–689. [CrossRef]
Pedersen, N. L., and Nielsen, A. K., 2003, “Optimization of Practical Trusses With Constraints on Eigenfrequencies, Displacements, Stresses, and Buckling,” Struct. Multidiscip. Optim., 25(5–6), pp. 436–445. [CrossRef]
Suzuki, K., and Kikuchi, N., 1991, “A Homogenization Method for Shape and Topology Optimization,” Comput. Methods Appl. Mech. Eng., 93, pp. 291–318. [CrossRef]
Xie, Y. M., and Steven, G. P., 1993, “A Simple Evolutionary Procedure for Structural Optimization,” Comput. Struct., 49(5), pp. 885–896. [CrossRef]
Chapman, C., Saitou, K., and Jakiela, M., 1994, “Genetic Algorithms as an Approach to Configuration and Topology Design,” ASME J. Mech. Des., 116(4), pp. 1005–1012. [CrossRef]
Chapman, C. D., and Jakiela, M. J., 1996, “Genetic Algorithm-Based Structural Topology Design With Compliance and Topology Simplification Considerations,” ASME J. Mech. Des., 118(1), pp. 89–98. [CrossRef]
Diaz, A. R., and Kikuchi, N., 1992, “Solutions to Shape and Topology Eigenvalue Optimization Using a Homogenization Method,” Int. J. Numer. Methods Eng., 35, pp. 487–1502.
Ma, Z.-D., Kikuchi, N., and Cheng, H.-C., 1995, “Topological Design for Vibrating Structures,” Comput. Methods Appl. Mech. Eng., 121(1–4), pp. 259–280. [CrossRef]
Mayer, R. R., Kikuchi, N., and Scott, R. A., 1996, “Application of Topological Optimization Techniques to Structural Crashworthiness,” Int. J. Numer. Methods Eng., 39(8), pp. 1383–1403. [CrossRef]
Luo, J., Gea, H. C., and Yang, R. J., 2000, “Topology Optimization for Crush Design,” 8th AIAA/USAF/NASA/ ISSMO Symposium on Multidisciplinary Analysis and Optimization, Long Beach, CA, Paper No. AIAA-2000-4770.
Mayer, R. R., Maurer, D., and Bottcher, C., 2000, “Application of Topological Optimization Program to the Danner Test Simulation,” ASME DETC 2000/DAC-14292.
Gea, H. C., and Luo, J., 2001, “Design for Energy Absorption: A Topology Optimization Approach,” ASME DETC2001/DAC-21060.
Soto, C. A., 2001, “Optimal Structural Topology Design for Energy Absorption: A Heurtistic Approach,” ASME DETC 2001/DAC-21126.
Soto, C. A., 2001, “Structural Topology for Crashworthiness Design by Matching Plastic Strain and Stress Levels,” ASME IMECE 2001 /AMD-25455.
Bae, K.-K., Wang, S. W., and Choi, K. K., 2002, “Reliability-Based Topology Optimization With Uncertainties,” China-Japan-Korea Joint Symposium on Optimization of Structural and Mechanical Systems, Busan, Korea, pp. 647–653.
Kharmanda, G., Olhoff, N., Mohamed, A., and Lemaire, M., 2004, “Reliability-Based Topology Optimization,” Struct. Multidiscip. Optim., 26(5), pp. 295–307. [CrossRef]
Johanson, R., Kikuchi, N., and Papalambros, P., 1994, “Simultaneous Topology and Material Microstructure Design,” Advances in Structural Optimization, Topping, B. H. V., and Papadrakakis, M., eds., Civil-Comp Ltd., Edinburgh, Scotland, pp. 143–149.
Jiang, T., and Chirehdast, M., 1997, “A Systems Approach to Structural Topology Optimization: Designing Optimal Connections,” ASME J. Mech. Des., 119(1), pp. 40–47. [CrossRef]
Chickermane, H., and Gea, H. C., 1997, “Design of Multi-Component Structural System for Optimal Layout Topology and Joint Locations,” Eng. Comput., 13, pp. 235–243. [CrossRef]
Li, Q., Steven, G. P., and Xie, Y. M., 2001, “Evolutionary Structural Optimization for Connection Topology Design of Multi-Component Systems,” Eng. Comput., 18(3–4), pp. 460–479. [CrossRef]
Yetis, A., and Saitou, K., 2002, “Decomposition-Based Assembly Synthesis Based on Structural Considerations,” ASME J. Mech. Des., 124(4), pp. 593–601. [CrossRef]
Lyu, N., and Saitou, K., 2003, “Decomposition-Based Assembly Synthesis for Structural Stiffness,” ASME J. Mech. Des., 125(3), pp. 452–463. [CrossRef]
Lyu, N., and Saitou, K., 2005, “Decomposition-Based Assembly Synthesis of a Three-Dimensional Body-in-White Model for Structural Stiffness,” ASME J. Mech. Des., 127(1), pp. 34–48. [CrossRef]
Lyu, N., and Saitou, K., 2006, “Decomposition-Based Assembly Synthesis of Space Frame Structures Using Joint Library,” ASME J. Mech. Des., 128(1), pp. 57–65. [CrossRef]
Sigmund, O., 2011, “On the Usefulness of Non-Gradient Approaches in Topology Optimization,” Struct. Multidiscip. Optim., 43, pp. 589–596. [CrossRef]
Boothroyd, G., Dewhurst, P., and Knight, W., 1994, Product Design for Manufacturing and Assembly, Marcel Dekker, New York.
Coello, C., Veldhuizen, D., and Lamont, G., 2002, Evolutionary Algorithms for Solving Multi-Objective Problems, Kluwer Academic Publishers, MA.
Deb, K., Pratab, A., Agrawal, S., and Meyarivan, T., 2002, “A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II,” IEEE Trans. Evol. Comput., 6(2), pp. 182–197. [CrossRef]
Sigmund, O., 2001, “A 99 Line Topology Optimization Code Written in Matlab,” Struct. Multidiscip. Optim., 21, pp. 120–127. [CrossRef]
Michalewiz, Z., and Fogel, D. B., 2000, How to Solve it: Modern Heuristics, Springer-Verlag, Berlin Heidelberg, New York.
Xu, S., and Deng, X., 2004, “An Evaluation of Simplified Finite Element Models for Spot-Welded Joints,” Finite Elem. Anal. Design, 40, pp. 1175–1194. [CrossRef]
Zitler, E., and Thiele, L., 1999, “Multi-Objective Evolutionary Algorithms: A Comparative Case Study and the Strength Pareto Approach,” IEEE Trans. Evol. Comput., 3(4), pp. 257–271. [CrossRef]
Hamza, K., Aly, M., and Hegazi, H., 2013, “An Explicit Level-Set Approach for Structural Topology Optimization,” ASME DETC 2013/DAC-12155.


Grahic Jump Location
Fig. 3

Repairing disconnected structures: (a) detection of disconnected material islands, (b) some-to-some shortest path identification, (c) addition of material along the shortest path

Grahic Jump Location
Fig. 4

Repairing joint elements: (a) detection of cut edges within one component, (b) connecting the cut edges

Grahic Jump Location
Fig. 5

Illustration of convex hull and bounding box enclosing a component

Grahic Jump Location
Fig. 6

Spider-web diagram illustration of a Pareto-optimal design and its equivalent monolithic structure

Grahic Jump Location
Fig. 7

Illustration of geometric crossover

Grahic Jump Location
Fig. 8

Localized joint morphing mutation operator: (a) complementary joints, (b) single-cell island removal, and (c) joint propagation

Grahic Jump Location
Fig. 9

Twelve-cell example design domain and boundary conditions

Grahic Jump Location
Fig. 10

Select Pareto-optimal designs for the twelve-cell example

Grahic Jump Location
Fig. 11

Short cantilever example design domain, loading and boundary conditions

Grahic Jump Location
Fig. 12

Monolithic designs in Φ for two-stage approaches in short cantilever example

Grahic Jump Location
Fig. 13

Normalized Pareto-plots for GA1, GA2, GA3, and GA4

Grahic Jump Location
Fig. 14

Normalized Pareto-plots for GA3, GA4, GA5, and GA6

Grahic Jump Location
Fig. 15

Normalized Pareto-plots for GA3, GA4, GA7, and GA8

Grahic Jump Location
Fig. 16

Normalized Pareto-plots for GA5, GA6, GA9, and GA10

Grahic Jump Location
Fig. 17

Structural details of select designs in short cantilever example

Grahic Jump Location
Fig. 18

Spider-web diagram illustration of objective values for select designs in short cantilever example

Grahic Jump Location
Fig. 19

Design domain and boundary conditions for simplified vehicle floor model

Grahic Jump Location
Fig. 20

Monolithic designs in Φ for vehicle floor study

Grahic Jump Location
Fig. 21

Normalized Pareto-plots for vehicle floor study

Grahic Jump Location
Fig. 22

Structural details of select designs in vehicle floor study

Grahic Jump Location
Fig. 23

Spider-web diagram illustration of objective values for select designs in vehicle floor study



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In