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.

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Grahic Jump Location
Fig. 1

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

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Fig. 2

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

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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.

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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

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Fig. 5

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

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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.

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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.

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Fig. 8

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

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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.

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Fig. 10

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



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