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Research Papers

Stiffness Design of Continuum Structures by a Bionics Topology Optimization Method

[+] Author and Article Information
Kun Cai, Hong-wu Zhang

State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, P.R.C.

Biao-song Chen1

State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, P.R.C.chenbs@dlut.edu.cn

Jiao Shi

College of Water Resources and Architectural Engineering, Northwest A&F University, Yangling 712100, P.R.C.

1

Corresponding author.

J. Appl. Mech 75(5), 051006 (Jul 15, 2008) (11 pages) doi:10.1115/1.2936929 History: Received May 24, 2007; Revised March 27, 2008; Published July 15, 2008

A heuristic approach is presented to solve continuum topology optimization problems with specified constraints, e.g., structural volume constraint and/or displacement constraint(s). The essentials of the present approach are summarized as follows. First, the structure is regarded as a piece of bone and the topology optimization process is viewed as bone remodeling process. Second, a second-rank positive and definite fabric tensor is introduced to express the microstructure and anisotropy of a material point in the design domain. The eigenpairs of the fabric tensor are the design variables of the material point. Third, Wolff’s law, which states that bone microstructure and local stiffness tend to align with the stress principal directions to adapt to its mechanical environment, is used to renew the eigenvectors of the fabric tensor. To update the eigenvalues, an interval of reference strain, which is similar to the concept of dead zone in bone remodeling theory, is suggested. The idea is that, when any one of the absolute values of the principal strains of a material point is out of the current reference interval, the fabric tensor will be changed. On the contrary, if all of the absolute values of the principal strains are in the current reference interval, the fabric tensor remains constant and the material point is in a state of remodeling equilibrium. Finally, the update rule of the reference strain interval is established. When the length of the interval equals zero, the strain energy density in the final structure distributes uniformly. Simultaneously, the volume and the displacement field of the final structure are determined uniquely. Therefore, the update of the reference interval depends on the ratio(s) between the current constraint value(s) and their critical value(s). Parameters, e.g., finite element mesh the initial material and the increments of the eigenvalues of fabric tensors, are studied to reveal their influences on the convergent behavior. Numerical results demonstrate the validity of the method developed.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

(a) Design domain and (b) optimal topology obtained by SIMP method

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Figure 2

Iteration histories of the supremums of the intervals of the reference strain of the structure with different initial designs: (a) 1–40 steps and (b) 1–80 steps

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Figure 3

Iteration histories of the supremums of reference intervals for different grow speed pairs: (a) g1=0.08 and (b) g2=0.08

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Figure 4

Iteration histories of the supremums of reference strain intervals when the growth speed pairs are different

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Figure 5

Initial design domain

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Figure 6

Iteration histories of the supremums of the intervals of the reference strain and the volume ratios of structure with different mesh schemes: (a) results of Case A, (b) results of Case B, and (c) results of Case C

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Figure 7

Optimal material distributions of structure with different FE mesh cases: (a) results of structure with Mesh Case A after 64 iterations, (b) results of structure with Mesh Case B after 84 iterations, and (c) results of structure with Mesh Case C after 100 iterations

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Figure 8

Initial design domain

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Figure 9

Iteration histories of the supremum of the interval of the reference strain and the structural volume ratio

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Figure 10

The final material distribution of structure

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Figure 11

Initial design domain

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Figure 12

Optimal topologies of structure in different cases: (a) Case 1 and (b) Case 2

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Figure 13

Iteration histories of the volume ratio of the structure, y-deflection of Point A, and the supremum of the reference strain interval

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Figure 14

Iteration histories of the y-deflections of Points A and B and the supremum of the reference strain interval

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