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Research Papers

Topology Optimization for Structures With Nonlinear Behavior Using the Equivalent Static Loads Method

[+] Author and Article Information
Hyun-Ah Lee

Department of Mechanical Engineering,  Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, Republic of Koreafannail@hanyang.ac.kr

Gyung-Jin Park

Department of Mechanical Engineering,  Hanyang University, 1271 Sa 3-dong, Sangnok-gu, Ansan City, Gyeonggi-do 426-791, Republic of Koreagjpark@hanyang.ac.kr

J. Mech. Des 134(3), 031004 (Feb 18, 2012) (14 pages) doi:10.1115/1.4005600 History: Received March 21, 2011; Revised November 17, 2011; Published February 17, 2012; Online February 18, 2012

Many structures in the real world show nonlinear responses. The nonlinearity may be due to some reasons, such as nonlinear material (material nonlinearity), large deformation of the structures (geometric nonlinearity), or contact between the parts (contact nonlinearity). Conventional optimization algorithms considering the nonlinearities are fairly difficult and expensive because many nonlinear analyses are required. It is quite difficult to perform topology optimization considering nonlinear static behavior because of the many design variables. In the current element density based topology optimization considering nonlinear behavior, low-density finite elements cause serious numerical problems due to excessive mesh distortion. Updating the material of the finite elements based on the density is considerably complicated because of the relationship between the element density and structural material. The equivalent static loads method for nonlinear static response structural optimization (ESLSO) has been proposed for size and shape optimization. The equivalent static loads (ESLs) are defined as the linear static load sets which generate the same displacement field from nonlinear static analysis. In this research, a new algorithm is proposed for topology optimization considering all kinds of nonlinearities by modifying the existing ESLSO. The new ESLSO can overcome the difficulties which may occur in topology optimization with nonlinear static behavior. A nonlinear static response optimization problem is converted to cyclic use of linear static response optimization with ESLs. Therefore, the new ESLSO can generate results of nonlinear static response topology optimization by using well established nonlinear static analysis and linear static response topology optimization methods. Four structural examples are demonstrated using the finite element method. Different kinds of nonlinearities are involved in each example.

Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

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Schematic process between the analysis domain and the design domain

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Generation of ESLs

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Topology optimization process using ESLs

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Problem description of a cantilever plate

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Bilinear elastoplastic strain–stress curve for a cantilever plate

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Results of topology optimization considering geometric nonlinearity using ESLs for a cantilever plate

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The histories of the objective function value of a geometric nonlinear topology optimization problem under the applied force of 144 kN

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Example of unstable elements under large deformation

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Result of topology optimization considering material nonlinearity using ESLs for a cantilever plate

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Problem description of a long slender beam

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Bilinear elastoplastic strain–stress curve for a long slender beam

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Results of topology optimization considering nonlinearities using ESLs for a long slender beam

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The history of the results of topology optimization considering geometric nonlinearity for a long slender beam

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The histories of the objective function value of topology optimization considering geometric nonlinearity for a long slender beam

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Problem description of two solid structures

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Bilinear elastoplastic strain–stress curve for two solid structures

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Results of topology optimization considering contact nonlinearity for two solid structures

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Results of topology optimization considering contact and material nonlinearity for two solid structures

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Results of topology optimization considering contact and geometric nonlinearity for two solid structures

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Results of topology optimization considering contact, material, and geometric nonlinearity for two solid structures

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Schematic view of a spacer grid set

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Finite element model of the spacer grid spring

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The relation between the stress and the strain of a spacer grid spring

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Results of nonlinear response topology optimization of a spacer grid spring

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Bilinear elastoplastic material

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Contact state between two bodies

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Nonlinear load–displacement curve with the linear and nonlinear topology optimization results for a cantilever plate

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