Research Papers: Design Automation

Topology Optimization of Structures Made of Discrete Geometric Components With Different Materials

[+] Author and Article Information
Hesaneh Kazemi

Department of Mechanical Engineering,
University of Connecticut,
191 Auditorium Road, U-3139,
Storrs, CT 06269
e-mail: hesaneh.kazemi@uconn.edu

Ashkan Vaziri

Department of Mechanical and
Industrial Engineering,
Northeastern University,
254 Richards Hall,
360 Huntington Ave.,
Boston, MA 02115
e-mail: vaziri@coe.neu.edu

Julián A. Norato

Department of Mechanical Engineering,
University of Connecticut,
191 Auditorium Road, U-3139,
Storrs, CT 06269
e-mail: norato@engr.uconn.edu

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received December 11, 2017; final manuscript received June 18, 2018; published online September 7, 2018. Assoc. Editor: Andres Tovar.

J. Mech. Des 140(11), 111401 (Sep 07, 2018) (11 pages) Paper No: MD-17-1817; doi: 10.1115/1.4040624 History: Received December 11, 2017; Revised June 18, 2018

We present a new method for the simultaneous topology optimization and material selection of structures made by the union of discrete geometric components, where each component is made of one of multiple available materials. Our approach is based on the geometry projection method, whereby an analytical description of the geometric components is smoothly mapped onto a density field on a fixed analysis grid. In addition to the parameters that dictate the dimensions, position, and orientation of the component, a size variable per available material is ascribed to each component. A size variable value of unity indicates that the component is made of the corresponding material. Moreover, all size variables can be zero, signifying the component is entirely removed from the design. We penalize intermediate values of the size variables via an aggregate constraint in the optimization. We also introduce a mutual material exclusion constraint that ensures that at most one material has a unity size variable in each geometric component. In addition to these constraints, we propose a novel aggregation scheme to perform the union of geometric components with dissimilar materials. These ingredients facilitate treatment of the multi-material case. Our formulation can be readily extended to any number of materials. We demonstrate our method with several numerical examples.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


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]
Bendsøe, M. P. , 1989, “Optimal Shape Design as a Material Distribution Problem,” Struct. Multidiscip. Optim., 1(4), pp. 193–202. [CrossRef]
Rozvany, G. , and Zhou, M. , 1991, “The COC Algorithm, Part I: Cross-Section Optimization or Sizing,” Comput. Methods Appl. Mech. Eng., 89(1–3), pp. 281–308. [CrossRef]
Gibiansky, L. V. , and Sigmund, O. , 2000, “Multiphase Composites With Extremal Bulk Modulus,” J. Mech. Phys. Solids, 48(3), pp. 461–498. [CrossRef]
Carbonari, R. C. , Silva, E. C. , and Nishiwaki, S. , 2007, “Optimum Placement of Piezoelectric Material in Piezoactuator Design,” Smart Mater. Struct., 16(1), p. 207. [CrossRef]
Luo, Z. , Gao, W. , and Song, C. , 2010, “Design of Multi-Phase Piezoelectric Actuators,” J. Intell. Mater. Syst. Struct., 21(18), pp. 1851–1865. [CrossRef]
Kang, Z. , Wang, R. , and Tong, L. , 2011, “Combined Optimization of Bi-Material Structural Layout and Voltage Distribution for in-Plane Piezoelectric Actuation,” Comput. Methods Appl. Mech. Eng., 200(13–16), pp. 1467–1478. [CrossRef]
Luo, Y. , and Kang, Z. , 2013, “Layout Design of Reinforced Concrete Structures Using Two-Material Topology Optimization With Drucker–Prager Yield Constraints,” Struct. Multidiscip. Optim., 47(1), pp. 95–110. [CrossRef]
Sigmund, O. , 2001, “Design of Multiphysics Actuators Using Topology Optimization–Part II: Two-Material Structures,” Comput. Methods Appl. Mech. Eng., 190(49–50), pp. 6605–6627. [CrossRef]
Yin, L. , and Ananthasuresh, G. , 2001, “Topology Optimization of Compliant Mechanisms With Multiple Materials Using a Peak Function Material Interpolation Scheme,” Struct. Multidiscip. Optim., 23(1), pp. 49–62. [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]
Lund, E. , 2009, “Buckling Topology Optimization of Laminated Multi-Material Composite Shell Structures,” Compos. Struct., 91(2), pp. 158–167. [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]
Hvejsel, C. F. , and Lund, E. , 2011, “Material Interpolation Schemes for Unified Topology and Multi-Material Optimization,” Struct. Multidiscip. Optim., 43(6), pp. 811–825. [CrossRef]
Buehler, M. J. , Bettig, B. , and Parker, G. G. , 2004, “Topology Optimization of Smart Structures Using a Homogenization Approach,” J. Intell. Mater. Syst. Struct., 15(8), pp. 655–667. [CrossRef]
Mirzendehdel, A. M. , and Suresh, K. , 2015, “A Pareto-Optimal Approach to Multimaterial Topology Optimization,” ASME J. Mech. Des., 137(10), p. 101701. [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. , Chen, S. , Wang, X. , and Mei, Y. , 2005, “Design of Multimaterial Compliant Mechanisms Using Level-Set Methods,” ASME J. Mech. Des., 127(5), pp. 941–956. [CrossRef]
Guo, X. , Zhang, W. , and Zhong, W. , 2014, “Stress-Related Topology Optimization of Continuum Structures Involving Multi-Phase Materials,” Comput. Methods Appl. Mech. Eng., 268, pp. 632–655. [CrossRef]
Wang, M. Y. , and Wang, X. , 2005, “A Level-Set Based Variational Method for Design and Optimization of Heterogeneous Objects,” Comput.-Aided Des., 37(3), pp. 321–337. [CrossRef]
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]
Wang, Y. , Luo, Z. , Kang, Z. , and Zhang, N. , 2015, “A Multi-Material Level Set-Based Topology and Shape Optimization Method,” Comput. Methods Appl. Mech. Eng., 283, pp. 1570–1586. [CrossRef]
Zhou, S. , and Wang, M. Y. , 2007, “Multimaterial Structural Topology Optimization With a Generalized Cahn–Hilliard Model of Multiphase Transition,” Struct. Multidiscip. Optim., 33(2), pp. 89–111. [CrossRef]
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]
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]
Wallin, M. , Ivarsson, N. , and Ristinmaa, M. , 2015, “Large Strain Phase-Field-Based Multi-Material Topology Optimization,” Int. J. Numer. Methods Eng., 104(9), pp. 887–904. [CrossRef]
Bell, B. , Norato, J. , and Tortorelli, D. , 2012, “A Geometry Projection Method for Continuum-Based Topology Optimization of Structures,” AIAA Paper No. AIAA 2012-5485..
Norato, J. , Bell, B. , and Tortorelli, D. , 2015, “A Geometry Projection Method for Continuum-Based Topology Optimization With Discrete Elements,” Comput. Methods Appl. Mech. Eng., 293, pp. 306–327. [CrossRef]
Zhang, S. , Norato, J. A. , Gain, A. L. , and Lyu, N. , 2016, “A Geometry Projection Method for the Topology Optimization of Plate Structures,” Struct. Multidiscip. Optim., 54(5), pp. 1173–1190. [CrossRef]
Guo, X. , Zhang, W. , and Zhong, W. , 2014, “Doing Topology Optimization Explicitly and Geometrically—a New Moving Morphable Components Based Framework,” ASME J. Appl. Mech., 81(8), p. 081009. [CrossRef]
Deng, J. , and Chen, W. , 2016, “Design for Structural Flexibility Using Connected Morphable Components Based Topology Optimization,” Sci. China Technol. Sci., 59(6), pp. 839–851. [CrossRef]
Watts, S. , and Tortorelli, D. A. , 2017, “A Geometric Projection Method for Designing Three-Dimensional Open Lattices With Inverse Homogenization,” Int. J. Numer. Methods Eng., 112 (11), pp. 1564–1588. [CrossRef]
Zhang, W. , Song, J. , Zhou, J. , Du, Z. , Zhu, Y. , Sun, Z. , and Guo, X. , “Topology Optimization With Multiple Materials Via Moving Morphable Component (MMC) Method,” Int. J. Numer. Methods Eng., 113(11), 1653–1675.
Faure, A. , Michailidis, G. , Parry, G. , Vermaak, N. , and Estevez, R. , 2017, “Design of Thermoelastic Multi-Material Structures With Graded Interfaces Using Topology Optimization,” Struct. Multidiscip. Optim., 56(4), pp. 823–837. [CrossRef]
Kreisselmeier, G. , 1979, “Systematic Control Design by Optimizing a Vector Performance Index,” IFAC Symposium, Zürich, Switzerland, Aug. 29–31, pp. 113–117.
Le, C. , Norato, J. , Bruns, T. , Ha, C. , and Tortorelli, D. , 2010, “Stress-Based Topology Optimization for Continua,” Struct. Multidiscip. Optim., 41(4), pp. 605–620. [CrossRef]
Svanberg, K. , 2002, “A Class of Globally Convergent Optimization Methods Based on Conservative Convex Separable Approximations,” SIAM J. Optim., 12(2), pp. 555–573. [CrossRef]
Svanberg, K. , 2007, “MMA and GCMMA, Versions September 2007,” Optimization and Systems Theory, KTH, Stockholm, Sweden, accessed July 12, 2018, https://people.kth.se/~krille/mmagcmma.pdf
Michell, A. G. M. , 1904, “LVIII. the Limits of Economy of Material in Frame-Structures,” Philos. Mag. J. Sci., 8(47), pp. 589–597. [CrossRef]
Bangerth, W. , Hartmann, R. , and Kanschat, G. , 2007, “Deal.II—A General Purpose Object Oriented Finite Element Library,” ACM Trans. Math. Software, 33(4), pp. 24/1–24/27. [CrossRef]
Bangerth, W. , Davydov, D. , Heister, T. , Heltai, L. , Kanschat, G. , Kronbichler, M. , Maier, M. , Turcksin, B. , and Wells, D. , “The Deal.II Library, Version 8.4,” J. Numer. Math., 24(3), 135–141.


Grahic Jump Location
Fig. 1

Bar geometry (left) and sample window Bpr for projected density calculation (right)

Grahic Jump Location
Fig. 3

Compliance versus weight fraction for optimal two-bar cantilever beams. Red (×) indicates material 1 and green (○) indicates material 2. The circled numbers indicate the corresponding runs in Table 1.

Grahic Jump Location
Fig. 4

Effective density ρeffi for materials i = 1 (left) and i = 2 (right) for initial (top) and optimal (bottom) designs for run 4 in Table 1

Grahic Jump Location
Fig. 5

Design envelope, boundary conditions, and initial design for the MBB beam

Grahic Jump Location
Fig. 6

MBB optimal designs with floating bars. Red (×) indicates material 1 and green (○) indicates material 2: (a) wf*=0.1, C = 1.439938; (b) wf*=0.11, C = 1.483786; (c) wf*=0.12, C = 1.230245; (d) wf*=0.13, C = 1.329151; (e) wf*=0.14, C = 1.075210; (f) wf*=0.15, C = 1.079675; (g) wf*=0.16, C = 0.919734; (h) wf*=0.17, C = 0.899547; (i) wf*=0.18, C = 0.872629; and (j) wf*=0.19, C = 0.783196.

Grahic Jump Location
Fig. 2

Two-bar cantilever beam design envelope, boundary conditions, and initial design

Grahic Jump Location
Fig. 7

MBB optimal designs with connected bars. Red (×) indicates material 1 and green (○) indicates material 2: (a) wf*=0.1, C = 2.074616; (b) wf*=0.11, C = 1.533285; (c) wf*=0.12, C = 1.342276; (d) wf*=0.13, C = 1.165188; (e) wf*=0.14, C = 1.312084; (f) wf*=0.15, C = 1.038830; (g) wf*=0.16, C = 1.008228; (h) wf*=0.17, C = 1.005593; (i) wf*=0.18, C = 1.050758; and (j) wf*=0.19, C = 0.876429.

Grahic Jump Location
Fig. 8

Design envelope, boundary conditions, and initial design for Michell cantilever design

Grahic Jump Location
Fig. 9

Michell cantilever optimal designs. Red (×) indicates material (stiffest/heaviest), green (○) indicates material 2, blue (△) indicates material 3, and magenta (□) indicates material 4 (weakest/lightest): (a) one material, C = 0.233370; (b) two materials, C = 0.176354; (c) three materials, C = 0.193584; and (d) four materials, C = 0.192687.

Grahic Jump Location
Fig. 10

Design envelope, boundary conditions and initial design for 3D cantilever design



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