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

Abstract

a b s t r a c tThis 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.

Topology optimization considering stress, fatigue and load uncertainties

“…Worst-case compliance problems have previously been solved using an eigenvalue formulation, [9,16,63]. If f 0 = 0 in (19), a similar formulation may be obtained from (20) where the equality follows from the Rayleigh-Ritz theorem ([29, Theorem 4.2.2]) and the eigenvalues λ i (H (x)) , i = 1, .…”

Game theory approach to robust topology optimization with uncertain loading

“…Appendix A). Choosing the compliance as the objective function and a suitable loading parametrization one can retrieve the generalized eigenvalue problems (Brittain et al 2012;Cherkaev and Cherkaev 2008;Takezawa et al 2011;Kobelev 1993) or semi-definite programming problems Thore et al 2015) mentioned above. When applicable these formulations may be more efficient, but compared to the proposed game theory framework they are very limited in that they only apply to zero-sum games with certain choices of functions and parametrizations.…”

“…These problems may, assuming small deformations and using certain load parametrizations, be cast as generalized eigenvalue problems (Brittain et al 2012;Cherkaev and Cherkaev 2008;Takezawa et al 2011) or as semidefinite programming problems ; Thore et al 2015). In this paper however we propose a much more general game theoretic framework for TO under loaduncertainty including a wide range of different objective functions and constraints.…”

Robust topology optimization of thin plate structure under concentrated load with uncertain load position

“…One of the authors of this paper [Kogiso et al, 2008] applied robust topology optimization to a compliant mechanism design problem considering load direction uncertainty. Takezawa et al [Takezawa et al, 2011] proposed a topology optimization for a design problem formulated to minimize robust compliance defined as the maximum compliance induced by the worst load case of an uncertain load set. Chen et al [Chen et al, 2010] applied the random field process to evaluate a reduced set of random variables.…”