Robust topology optimization of phononic crystals with random field uncertainty

Abstract

Summary The uncertain spatial variation of material properties can remarkably affect the band gap characteristics of phononic crystals (PnCs). It is necessary to consider this issue when designing and manufacturing PnC materials/structures. This paper investigates a robust topology optimization method for designing the microstructures of PnCs by considering random‐field material properties. Herein, the spatial distribution of the material properties is first represented by a random field and then discretized into uncorrelated stochastic variables with the expansion optimal linear estimation method; stochastic band gap analysis is then conducted with polynomial chaos expansion. Furthermore, a robust topology optimization formulation of PnCs is proposed on the basis of the relative elemental density, where a weighted objective function handles the compromise of the mean value and standard deviation of the PnC band gap. The band gap response is analyzed, employing the finite element method for each sample of polynomial chaos expansion. In this context, the sensitivities of the stochastic band gap behaviors to the design variables are also derived. Numerical examples demonstrate that the proposed method can generate meaningful optimal topologies of PnCs with a relatively large width and less sensitive band gap. Additionally, the effects of the weight factors in the objective function and the variation coefficient of material properties are discussed.

Robust topology optimization methodology for continuum structures under probabilistic and fuzzy uncertainties

“…Step (5): Update the fuzzy variables by Equation (21). Convert the standard fuzzy variables q (k, t) to fuzzy variables a (k, t) .…”

Robust topology optimization for heat conduction with polynomial chaos expansion

“…The new objective function F is a weighted sum of the expected value and the standard deviation of the norm of the temperature field. Note that the weighted sum of the expected value and the standard deviation is frequently employed as the objective function in robust optimization [15,53,69]. Some authors also employ the variance instead of the standard deviation [6,23].…”

“…optimization approaches have been successfully applied to topology optimization in the last years [18-21, 23, 25, 33, 35, 36, 38, 39, 45, 46, 53, 57, 59, 63, 69]. In particular, the works by Tootkaboni et al [59], Keshavarzzadeh et al [35], Martínez-Frutos et al [46] and Zhang et al [69] employed polynomial chaos expansion (PCE) in the context of topology optimization, but only covered applications from solid mechanics and optics.…”

Parametric topology optimization with multiresolution finite element models

“…This performance deficiency can be overcome by incorporating uncertainty analysis in the optimization process; in robust design optimization (RDO), [8][9][10][11][12][13][14] the performance variation is minimized, and in reliability-based design optimization (RBDO), [15][16][17] the failure probability is constrained. Computational complexity is the outstanding challenge in these approaches due to requiring a considerable number of expensive simulations to capture variations in parameter/stochastic space.…”