Non-uniqueness and symmetry of optimal topology of a shell for minimum compliance

  • Authors:

  • Ryo Watada;Makoto Ohsaki;Yoshihiro Kanno

  • Affiliations:

  • Department of Architecture and Architectural Engineering, Kyoto University, Kyoto, Japan;Department of Architecture and Architectural Engineering, Kyoto University, Kyoto, Japan;Department of Mathematical Informatics, University of Tokyo, Tokyo, Japan 113-8656

  • Venue:
  • Structural and Multidisciplinary Optimization
  • Year:
  • 2011

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Abstract

Uniqueness and symmetry of solution are investigated for topology optimization of a symmetric continuum structure subjected to symmetrically distributed loads. The structure is discretized into finite elements, and the compliance is minimized under constraint on the structural volume. The design variables are the densities of materials of elements, and intermediate densities are penalized to prevent convergence to a gray solution. A path of solution satisfying conditions for local optimality is traced using the continuation method with respect to the penalization parameter. It is shown that the rate form of the solution path can be formulated from the optimality conditions, and the uniqueness and bifurcation of the path are related to eigenvalues and eigenvectors of the Jacobian of the governing equations. This way, local uniqueness and symmetry breaking process of the solution are rigorously investigated through the bifurcation of a solution path.