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Research Papers: Design Automation

Optimal Design of Nonlinear Multimaterial Structures for Crashworthiness Using Cluster Analysis

[+] Author and Article Information
Kai Liu

School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47907
e-mail: liu915@purdue.edu

Duane Detwiler

Honda R&D Americas, Inc.,
Raymond, OH 43067
e-mail: ddetwiler@oh.hra.com

Andres Tovar

Associate Professor
Department of Mechanical Engineering,
Indiana University-Purdue
University Indianapolis,
Indianapolis, IN 46202
e-mail: tovara@iupui.edu

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received September 12, 2016; final manuscript received August 2, 2017; published online August 30, 2017. Assoc. Editor: Nam H. Kim.

J. Mech. Des 139(10), 101401 (Aug 30, 2017) (11 pages) Paper No: MD-16-1637; doi: 10.1115/1.4037620 History: Received September 12, 2016; Revised August 02, 2017

This study presents an efficient multimaterial design optimization algorithm that is suitable for nonlinear structures. The proposed algorithm consists of three steps: conceptual design generation, clustering, and metamodel-based global optimization. The conceptual design is generated using a structural optimization algorithm for linear models or a heuristic design algorithm for nonlinear models. Then, the conceptual design is clustered into a predefined number of clusters (materials) using a machine learning algorithm. Finally, the global optimization problem aims to find the optimal material parameters of the clustered design using metamodels. The metamodels are built using sampling and cross-validation and sequentially updated using an expected improvement function until convergence. The proposed methodology is demonstrated using examples from multiple physics and compared with traditional multimaterial topology optimization (MTOP) method. The proposed approach is applied to a nonlinear, multi-objective design problems for crashworthiness.

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References

Bendsøe, M. P. , 1995, Optimization of Structural Topology, Shape and Material, Springer, New York. [CrossRef]
Allaire, G. , 2001, Shape Optimization by the Homogenization Method, Springer, New York.
Sigmund, O. , and Torquato, S. , 1997, “ Design of Materials With Extreme Thermal Expansion Using a Three-Phase Topology Optimization Method,” J. Mech. Phys. Solids, 45(6), pp. 1037–1067. [CrossRef]
Sigmund, O. , and Torquato, S. , 1999, “ Design of Smart Composite Materials Using Topology Optimization,” Smart Mater. Struct., 8(3), pp. 365–379. [CrossRef]
Gibiansky, L. V. , and Sigmund, O. , 2000, “ Multiphase Composites With Extremal Bulk Modulus,” J. Mech. Phys. Solids, 48(3), pp. 461–498. [CrossRef]
Bendsøe, M. P. , 1989, “ Optimal Shape Design as a Material Distribution Problem,” Struct. Multidiscip. Optim., 1(4), pp. 193–202. [CrossRef]
Mlejnek, H. , 1992, “ Some Aspects of the Genesis of Structures,” Struct. Optim., 5(1–2), pp. 64–69. [CrossRef]
Zhou, M. , and Rozvany, G. , 1991, “ The COC Algorithm—Part II: Topological, Geometrical and Generalized Shape Optimization,” Comput. Methods Appl. Mech. Eng., 89(1–3), pp. 309–336. [CrossRef]
Bendsøe, M. P. , and Sigmund, O. , 1999, “ Material Interpolations in Topology Optimization,” Arch. Appl. Mech., 69, pp. 635–654. [CrossRef]
Gao, T. , and Zhang, W. , 2011, “ A Mass Constraint Formulation for Structural Topology Optimization With Multiphase Materials,” Int. J. Numer. Methods Eng., 88(8), pp. 774–796. [CrossRef]
Cui, M. T. , and Chen, H. F. , 2014, “ An Improved Alternating Active-Phase Algorithm for Multi-Material Topology Optimization Problems,” Appl. Mech. Mater., 635–637, pp. 105–111. [CrossRef]
Osher, S. , and Santosa, F. , 2001, “ Level Set Methods for Optimization Problem Involving Geometry and Constraints—I: Frequencies of a Two-Density Inhomogeneous Drum,” J. Comput. Phys., 171(1), pp. 272–288. [CrossRef]
Wang, M. Y. , Wang, X. , and Guo, D. , 2003, “ A Level Set Method for Structural Topology Optimization,” Comput. Methods Appl. Mech. Eng., 192(1–2), pp. 227–246. [CrossRef]
Allaire, G. , Jouve, F. , and Toader, A.-M. , 2004, “ Structural Optimization Using Sensitivity Analysis and a Level-Set Method,” J. Comput. Phys., 194(1), pp. 363–393. [CrossRef]
Allaire, G. , and Castro, C. , 2002, “ Optimization of Nuclear Fuel Reloading by the Homogenization Method,” Struct. Multidiscip. Optim., 24(1), pp. 11–22. [CrossRef]
Mei, Y. , and Wang, X. , 2004, “ A Level Set Method for Structural Topology Optimization and Its Applications,” Adv. Eng. Software, 35(7), pp. 415–441. [CrossRef]
Wang, M. Y. , and Wang, X. , 2004, ““ Color” Level Sets: A Multi-Phase Method for Structural Topology Optimization With Multiple Materials,” Comput. Methods Appl. Mech. Eng., 193(6–8), pp. 469–496. [CrossRef]
Wang, M. Y. , and Wang, X. , 2005, “ A Level-Set Based Variational Method for Design and Optimization of Heterogeneous Objects,” CAD Comput. Aided Des., 37(3), pp. 321–337. [CrossRef]
Dombre, E. , Allaire, G. , Pantz, O. , and Schmitt, D. , 2012, “ Shape Optimization of a Sodium Fast Reactor Core,” CEMRACS, Marseille, France, July 18–Aug. 26, pp. 319–334.
Wei, P. , and Wang, M. Y. , 2009, “ Piecewise Constant Level Set Method for Structural Topology Optimization,” Int. J. Numer. Methods Eng., 78(4), pp. 379–402. [CrossRef]
Luo, Z. , Tong, L. , Luo, J. , Wei, P. , and Wang, M. Y. , 2009, “ Design of Piezoelectric Actuators Using a Multiphase Level Set Method of Piecewise Constants,” J. Comput. Phys., 228(7), pp. 2643–2659. [CrossRef]
Hamza, K. , Aly, M. , and Hegazi, H. , 2013, “ A Kriging-Interpolated Level-Set Approach for Structural Topology Optimization,” ASME J. Mech. Des., 136(1), p. 011008. [CrossRef]
Guirguis, D. , Hamza, K. , Aly, M. , Hegazi, H. , and Saitou, K. , 2015, “ Multi-Objective Topology Optimization of Multi-Component Continuum Structures Via a Kriging-Interpolated Level Set Approach,” Struct. Multidiscip. Optim., 51(3), pp. 733–748. [CrossRef]
Yoshimura, M. , Shimoyama, K. , Misaka, T. , and Obayashi, S. , 2017, “ Topology Optimization of Fluid Problems Using Genetic Algorithm Assisted by the Kriging Model,” Int. J. Numer. Methods Eng., 109(4), pp. 514–532. [CrossRef]
Chopp, D. , 1993, “ Computing Minimal Surface Via Level Set Curvature Flow,” J. Comput. Phys., 106(1), pp. 77–91. [CrossRef]
Sussman, M. , Smereka, P. , and Osher, S. , 1994, “ A Level Set Approach for Computing Solutions to Incompressible Two-Phase flow,” J. Comput. Phys., 114(1), pp. 146–159. [CrossRef]
Sethian, J. , 1999, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Science, Cambridge University Press, Cambridge, UK.
Osher, S. , and Fedkiw, R. , 2002, Level Set Methods and Dynamic Implicit Surfaces, Springer, New York.
Bourdin, B. , and Chambolle, A. , 2003, “ Design-Dependent Loads in Topology Optimization,” ESAIM: Control Optim. Calculus Var., 9, pp. 19–48. [CrossRef]
Wang, M. Y. , and Zhou, S. , 2005, “ Synthesis of Shape and Topology of Multi-Material Structures With a Phase-Field Method,” J. Comput. Aided Mater. Des., 11(2–3), pp. 117–138.
Bourdin, B. , and Chambolle, A. , 2006, “ The Phase-Field Method in Optimal Design,” IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials, Dordrecht, The Netherlands, pp. 207–215. [CrossRef]
Takezawa, A. , Nishiwaki, S. , and Kitamura, M. , 2010, “ Shape and Topology Optimization Based on the Phase Field Method and Sensitivity Analysis,” J. Comput. Phys., 229(7), pp. 2697–2718. [CrossRef]
Blank, L. , Garcke, H. , Sarbu, L. , and Styles, V. , 2012, “ Primal-Dual Active Set Methods for Allen-Cahn Variational Inequalities With Nonlocal Constraints,” Numer. Methods Partial Differ. Equations, 29(3), pp. 999–1030.
Tavakoli, R. , 2014, “ Multimaterial Topology Optimization by Volume Constrained Allen-Cahn system and Regularized Projected Steepest Descent Method,” Comput. Methods Appl. Mech. Eng., 276, pp. 534–565. [CrossRef]
Huang, X. , and Xie, Y. M. , 2009, “ Bi-Directional Evolutionary Topology Optimization of Continuum Structures With One or Multiple Materials,” Comput. Mech., 43(3), pp. 393–401. [CrossRef]
Huang, X. , and Xie, M. , 2010, Evolutionary Topology Optimization of Continuum Structures: Methods and Applications, Wiley, Chichester, UK. [CrossRef]
Lund, E. , and Stegmann, J. , 2005, “ On Structural Optimization of Composite Shell Structures Using a Discrete Constitutive Parametrization,” Wind Energy, 8(1), pp. 109–124. [CrossRef]
Stegmann, J. , and Lund, E. , 2005, “ Discrete Material Optimization of General Composite Shell Structures,” Int. J. Numer. Methods Eng., 62(14), pp. 2009–2027. [CrossRef]
Tovar, A. , Patel, N. M. , Niebur, G. L. , Sen, M. , and Renaud, J. E. , 2006, “ Topology Optimization Using a Hybrid Cellular Automation Method With Local Control Rules,” ASME J. Mech. Des., 128(6), pp. 1205–1216. [CrossRef]
Tovar, A. , Patel, N. M. , Kaushik, A. K. , and Renaud, J. E. , 2007, “ Optimality Conditions of the Hybrid Cellular Automata for Structural Optimization,” AIAA J., 45(3), pp. 673–683. [CrossRef]
Goetz, J. , Tan, H. , Renaud, J. , and Tovar, A. , 2012, “ Two-Material Optimization of Plate Armour for Blast Mitigation Using Hybrid Cellular Automata,” Eng. Optim., 44(8), pp. 985–1005. [CrossRef]
Holmberg, E. , Torstenfelt, B. , and Klarbring, A. , 2013, “ Stress Constrained Topology Optimization,” Struct. Multidiscip. Optim., 48(1), pp. 33–47. [CrossRef]
Ishikawa, T. , Nakayama, K. , Kurita, N. , and Dawson, F. P. , 2014, “ Optimization of Rotor Topology in PM Synchronous Motors by Genetic Algorithm Considering Cluster of Materials and Cleaning Procedure,” IEEE Trans. Magn., 50(2), pp. 637–640. [CrossRef]
Ishikawa, T. , Mizuno, S. , and Krita, N. , 2017, “ Topology Optimization Method for Asymmetrical Rotor Using Cluster and Cleaning Procedure,” IEEE Trans. Magn., 53(6), pp. 1–4.
Aulig, N. , and Olhofer, M. , 2016, “ State-Based Representation for Structural Topology Optimization and Application to Crashworthiness,” IEEE Congress on Evolutionary Computation (CEC), Vancouver, BC, Canada, July 24–29, pp. 1642–1649.
Liu, K. , Tovar, A. , and Detwiler, D. , 2014, “ Thin-Walled Component Design Optimization for Crashworthiness Using Principles of Compliant Mechanism Synthesis and Kriging Sequential Approximation,” Engineering Optimization, CRC Press, Boca Raton, FL, pp. 775–780. [CrossRef]
Hashin, Z. , and Shtrikman, S. , 1963, “ A Variational Approach to the Elastic Behavior of Multiphase Minerals,” J. Mech. Phys. Solids, 11(2), pp. 127–140. [CrossRef]
Wang, Y.-J. , Zhang, J.-S. , and Zhang, G.-Y. , 2007, “ A Dynamic Clustering Based Differential Evolution Algorithm for Global Optimization,” Eur. J. Oper. Res., 183(1), pp. 56–73. [CrossRef]
Xu, H. , Chuang, C.-H. , and Yang, R.-J. , 2015, “ A Data Mining-Based Strategy for Direct Multidisciplinary Optimization,” SAE Int. J. Mater. Manuf., 8(2), pp. 357–363. [CrossRef]
MacQueen, J. B. , 1967, “ Some Methods for Classification and Analysis of Multivariate Observations,” Fifth Berkeley Symposium on Mathematical Statistics and Probability, June 21–July 18 and Dec. 27–Jan. 7, Berkeley, CA, pp. 281–297. https://projecteuclid.org/euclid.bsmsp/1200512992
Liu, K. , Tovar, A. , Nutwell, E. , and Detwiler, D. , 2015, “ Thin-Walled Compliant Mechanism Component Design Assisted by Machine Learning and Multiple Surrogates,” SAE Paper No. 2015-01-1369.
MacKay, D. , 2003, Information Theory, Inference, and Learning Algorithms, Cambridge University Press, Cambridge, UK.
Bandi, P. , Schmiedeler, J. P. , and Tovar, A. , 2013, “ Design of Crashworthy Structures With Controlled Energy Absorption in the Hybrid Cellular Automaton Framework,” ASME J. Mech. Des., 135(9), p. 091002. [CrossRef]
Lophaven, S. N. , Nielsen, H. B. , and Sondergaard, J. , 2002, “ ‘Dace’—A ‘Matlab’ Kriging Toolbox,” Technical University of Denmark, Lyngby, Denmark, Technical Report No. IMM-TR-2002-12 http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.17.3530&rep=rep1&type=pdf.
Myers, R. , and Montgomery, D. , 1995, Response Surface Methodology: Process and Product Optimization Using Designed Experiments, Wiley, New York.
Owen, A. B. , 1994, “ Controlling Correlations in Latin Hypercube Samples,” J. Am. Stat. Assoc., 89(428), pp. 1517–1522. [CrossRef]
Johnson, M. , Moore, L. , and Ylvisaker, D. , 1990, “ Minimax and Maximin Distance Designs,” J. Stat. Plann. Inference, 26(2), pp. 131–148. [CrossRef]
Jones, D. R. , Schonlau, M. , and Welch, W. J. , 1998, “ Efficient Global Optimization of Expensive Black-Box Functions,” J. Global Optim., 13(4), pp. 455–492. [CrossRef]
Viana, F. A. C. , Haftka, R. T. , and Steffen, V. , 2009, “ Multiple Surrogates: How Cross-Validation Errors Can Help Us to Obtain the Best Predictor,” Struct. Multidiscip. Optim., 39(4), pp. 439–457. [CrossRef]
Forrester, A. I. J. , Sóbester, A. , and Keane, A. J. , 2008, Engineering Design Via Surrogate Modelling: A Practical Guide, Wiley, Chichester, UK.
Tavakoli, R. , and Mohseni, S. M. , 2014, “ Alternating Active-Phase Algorithm for Multimaterial Topology Optimization Problems: A 115-line MATLAB implementation,” Struct. Multidiscip. Optim., 49(4), pp. 621–642. [CrossRef]
Liu, K. , and Tovar, A. , 2014, “ An Efficient 3D Topology Optimization Code Written in Matlab,” Struct. Multidiscip. Optim., 50(6), pp. 1175–1196. [CrossRef]
Tovar, A. , and Khandelwal, K. , 2013, “ Topology Optimization for Minimum Compliance Using a Control Strategy,” Eng. Struct., 48, pp. 674–682. [CrossRef]
Bandi, P. , Detwiler, D. , Schmiedeler, J. P. , and Tovar, A. , 2015, “ Design of Progressively Folding Thin-Walled Tubular Components Using Compliant Mechanism Synthesis,” Thin-Walled Struct., 95, pp. 208–220. [CrossRef]
Saxena, A. , and Ananthasuresh, G. , 2000, “ On an Optimal Property of Compliant Topologies,” Struct. Multidiscip. Optim., 19(1), pp. 36–49. [CrossRef]

Figures

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Fig. 1

LHSs with P = 5 and (a) high correlation and (b) low correlation achieved using the maximization of the minimum inter-site distances

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Fig. 2

Leave-one-out cross-validation at the second sampled point exemplified by fitting a function with a Kriging metamodel

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Fig. 3

Thermal compliance problem—design domain and Dirichlet boundary condition: constant surface temperature u = 0

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Fig. 4

Thermal compliance problem—conceptual design with f = 1.94 × 106

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Fig. 5

Thermal compliance problem—clustered design with f = 1.97 × 106

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Fig. 6

Thermal compliance problem—comparison of the predicted number of function evaluations

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Fig. 7

Mechanical compliance problem—design domain and boundary conditions for the three-dimensional beam in cantilever

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Fig. 8

Mechanical compliance problem—conceptual design with 8320 distinct density values and f = 2217

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Fig. 9

Mechanical compliance problem—clustered design with three clusters and f = 2432

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Fig. 10

Armor plate problem—finite element model

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Fig. 11

Armor plate problem—initial design (left) and impact simulation (right) with maximum penetration f1 = 12.05 mm and mass fraction f2 = 0.50

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Fig. 12

Armor plate problem—conceptual design (left) and impact simulation (right) with maximum penetration f1 = 9.33 mm and mass fraction f2 = 0.50

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Fig. 13

Armor plate problem—clustered designs with K=1,…,18

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Fig. 14

Armor plate problem—maximum penetration and mass fraction as functions of the number of clusters

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Fig. 15

Armor plate problem—Pareto front of the design optimization problem and the optimized clustered design (K = 4) with maximum penetration f1 = 9.46 mm and mass fraction f2 = 0.50

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Fig. 16

S-rail problem—geometry of the thin-walled S-rail (side view). The cross section is squared of dimensions H × H and thickness xe.

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Fig. 17

S-rail problem—locations of input and output ports for a thin-walled S-tube following the wavelength λ corresponding to the progressive buckling after an ideal axial crushing condition

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Fig. 18

S-rail problem—initial design represented by a uniform thickness distribution in the “unfolded” thin-walled structure (left). The initial design depicts Euler-type buckling (right). The corresponding crashworthiness indicators are SEA = 3.39 kJ/kg and PCF = 267 kN.

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Fig. 19

S-rail problem—conceptual design represented the thickness distribution in the “unfolded” thin-walled structure (left). The conceptual design depicts progressive folding (right).The corresponding crashworthiness indicators are SEA = 5.05 kJ/kg and PCF = 359 kN.

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Fig. 20

S-rail problem—clustered designs with K=1,…,12

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Fig. 21

S-rail problem—SEA and PCF values as a function of the number of clusters K

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Fig. 22

S-rail problem—Pareto fronts for K=1,…,4. Initial, conceptual, and clustered designs are dominated. Clustered designs have the following (−SEA, PCF) coordinates: for K = 1: (−3.82, 274), for K = 2: (−4.81, 349), for K = 3: (−3.98, 423), and for K = 4: (−3.90, 381).

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