A robust topology optimization for enlarging working bandwidth of acoustic devices

Abstract

We propose a novel robust topology optimization for designing acoustic devices that are effective for broadband sound waves. Here, we define the objective function as the weighted sum of the acoustic response to an incident wave consisting of a single frequency and its standard deviation (SD) against the frequency perturbation. By approximating the SD, under the assumption that the incident frequency follows the normal distribution, with the high‐order Taylor expansion of the (conventional) objective function, we deal with significant frequency variations. To calculate such an approximation, we need to calculate the high‐order frequency derivatives of the state variable. Here, we define them by integral representations, which enables us to characterize them even when the state variable is defined in an unbounded domain as is often the case with wave scattering problems. We further show that, based on this definition, the high‐order derivatives can efficiently be computed by a combination of the boundary element method and automatic differentiation. We also present a derivation and calculation of the topological derivative for the newly defined objective function. We install all the proposed techniques into a topology optimization method based on the level‐set method to design wideband acoustic devices. The validity and effectiveness of the proposed topology optimization are confirmed through several numerical examples.

A topology optimisation of acoustic devices based on the frequency response estimation with the Padé approximation

“…It has not yet been proved rigorously but inferred numerically that Eq. ( 44) holds for arbitrary n [9].…”

“…The angular frequency derivatives u (n) := d n u/dω n are required to obtain the Padé approximation of the angular frequency response. We have already established a low-cost method to compute u (n) [9], which is briefly reviewed in this section. Equations ( 4)-( 6) with the angular frequency ω 0 are equivalent to the following boundary integral equation (BIE):…”

“…( 25) and ( 27) at ω = ω 0 , we can compute the boundary values of u(x; ω 0 ) and u (n) (x; ω 0 ). The forward mode automatic differentiation [9] is available to compute the higher-order angular frequency derivatives of W , u in and q in in Eq.…”

“…One possible approach to realise such a wide-band device is robust topology optimisation [8,9]. In the robust topology optimisation, the (angular) frequency is regarded as a stochastic variable following the normal distribution.…”

“…Since it is time-consuming to evaluate the statistic quantity by the Monte-Carlo method, the deviation is often approximated through the Taylor expansion of the original objective at the mean angular frequency. We have pointed out that the high-order approximation is sometimes necessary to achieve a wide-band topology optimisation in acoustics and developed a low-cost method to evaluate the angular frequency derivatives up to an arbitrary order which is required in such a high-order approximation [9].…”