TY - JOUR

T1 - Topology optimization for worst load conditions based on the eigenvalue analysis of an aggregated linear system

AU - Takezawa, Akihiro

AU - Nii, Satoru

AU - Kitamura, Mitsuru

AU - Kogiso, Nozomu

N1 - Funding Information:
The authors thank Prof. Makoto Ohsaki at Hiroshima University for his valuable comments and advice on robust optimization and sensitivity analysis for repeated eigenvalue problems. We are also deeply grateful to Prof. Krister Svanberg at Albanova University Center for providing the source code for the Method of Moving Asymptotes (MMA). This research was partially supported by Grant-in-Aid for Scientific Research (21360432).

PY - 2011/6/15

Y1 - 2011/6/15

N2 - This paper proposes a topology optimization for a linear elasticity design problem subjected to an uncertain load. The design problem is formulated to minimize a robust compliance that is defined as the maximum compliance induced by the worst load case of an uncertain load set. Since the robust compliance can be formulated as the scalar product of the uncertain input load and output displacement vectors, the idea of "aggregation" used in the field of control is introduced to assess the value of the robust compliance. The aggregation solution technique provides the direct relationship between the uncertain input load and output displacement, as a small linear system composed of these vectors and the reduced size of a symmetric matrix, in the context of a discretized linear elasticity problem, using the finite element method. Introducing the constraint that the Euclidean norm of the uncertain load set is fixed, the robust compliance minimization problem is formulated as the minimization of the maximum eigenvalue of the aggregated symmetric matrix according to the Rayleigh-Ritz theorem for symmetric matrices. Moreover, the worst load case is easily established as the eigenvector corresponding to the maximum eigenvalue of the matrix. The proposed structural optimization method is implemented using topology optimization and the method of moving asymptotes (MMA). The numerical examples provided illustrate mechanically reasonable structures and establish the worst load cases corresponding to these optimal structures.

AB - This paper proposes a topology optimization for a linear elasticity design problem subjected to an uncertain load. The design problem is formulated to minimize a robust compliance that is defined as the maximum compliance induced by the worst load case of an uncertain load set. Since the robust compliance can be formulated as the scalar product of the uncertain input load and output displacement vectors, the idea of "aggregation" used in the field of control is introduced to assess the value of the robust compliance. The aggregation solution technique provides the direct relationship between the uncertain input load and output displacement, as a small linear system composed of these vectors and the reduced size of a symmetric matrix, in the context of a discretized linear elasticity problem, using the finite element method. Introducing the constraint that the Euclidean norm of the uncertain load set is fixed, the robust compliance minimization problem is formulated as the minimization of the maximum eigenvalue of the aggregated symmetric matrix according to the Rayleigh-Ritz theorem for symmetric matrices. Moreover, the worst load case is easily established as the eigenvector corresponding to the maximum eigenvalue of the matrix. The proposed structural optimization method is implemented using topology optimization and the method of moving asymptotes (MMA). The numerical examples provided illustrate mechanically reasonable structures and establish the worst load cases corresponding to these optimal structures.

KW - Eigenvalue analysis

KW - Finite element method

KW - Robust design

KW - Sensitivity analysis

KW - Topology optimization

KW - Worst case design

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U2 - 10.1016/j.cma.2011.03.008

DO - 10.1016/j.cma.2011.03.008

M3 - Article

AN - SCOPUS:79955395461

VL - 200

SP - 2268

EP - 2281

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0374-2830

IS - 25-28

ER -