Worst-case topology optimization of self-weight loaded structures using semi-definite programming

Abstract

Worst-case topology optimization of self-weight loaded structures using semi-definite programmingErik Holmberg · Carl-Johan Thore · Anders KlarbringAbstract The paper concerns worst-case compliance optimization by finding the structural topology with minimum compliance for the loading due to the worst possible acceleration of the structure and attached non-structural masses. A main novelty of the paper is that it is shown how this min-max problem can be formulated as a non-linear semi-definite programming (SDP) problem involving a small-size constraint matrix and how this problem is solved numerically. Our SDP formulation is an extension of an eigenvalue problem seen previously in the literature; however, multiple eigenvalues naturally arise which makes the eigenvalue problem non-smooth, whereas the SDP problem presented in this paper provides a computationally tractable problem. Optimized designs, where the uncertain loading is due to acceleration of applied masses and the weight of the structure itself, are shown in two and three dimensions and we show that these designs satisfy optimality conditions that are also presented.

Game formulations for structural optimization under uncertainty

“…The load vector is given by where r satisfies ‖ r ‖ ≤ 1, in which ‖·‖ is the Euclidean norm, and Q is a given matrix. This is an example of the so‐called ellipsoidal model used by many researchers to model load uncertainty . The compliance is now given by Since c is convex as a function of r , the maximizers are found among the extreme points of the set .…”

Game theory approach to robust topology optimization with uncertain loading

“…Due to the penalization in (5) the intermediate design variable values in this transition give a non-physical stiffness, which makes them undesirable. Therefore, in order to obtain a more "black-and-white" design without the intermediate design variable values, we remove the filter after convergence of Algorithm 1 and continue with a TO using only those ρ i (x) that are close to a boundary as design variables, while those that are at the lower or upper bound and surrounded by elements with the same value are fixed to their current value (see Holmberg et al (2015) for additional details). As the design changes are typically quite small we have chosen not to update the loads after the filter has been removed, and we have found (3k)-(3q) to be very close to satisfied anyway, see Figs.…”

“…This type of loading has been used in Holmberg et al (2015) where the components of r were used to vary the loading. In this paper we choose to parameterize r by the angles θ = (θ, φ) T in a spherical coordinate system (Fig.…”

“…But for some problems the worst-case loading is not unique; consequently, an optimized design will not be optimal for any of the loads in the set if these are applied as unique load cases. This observation is related to the multiple eigenvalue problem discussed in Holmberg et al (2015). Uncertainty, not only concerning loading, but also elastic moduli and material defects such as cracks was considered by Banichuk and Neittaanmäki (2007).…”

Topology optimization of continuum structures with stress constraints and uncertainties in loading

“…A review of the literature reveals many papers dealing with continuum topology optimization formulations to solve stochastic problems and find optimal structures in the presence of uncertainties. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] Two main techniques are often used to address design optimization problems under uncertainties: reliability-based design optimization and robust design optimization. 18 In the reliability-based design optimization approach, uncertainties affect only the design constraints, which are written in such a way as to warrant a minimal level of safety for the optimal design.…”