Transient dynamic response topology optimization methods always face the challenge of high computational costs due to the need for repeated time-domain discrete structural response calculations. To address this issue, the Equivalent Static Loads Method (ESLM) calculates structural responses with Equivalent Static Loads (ESLs), and solves a sequence of static response optimization problems to approximate the original dynamic problem. However, it has been noted that ESLM may not always identify a Karush-Kuhn-Tucker (KKT) point, and ESLM may produce nonnegligible errors due to using the static response sensitivity to approximate the dynamic response sensitivity, unless the dynamic characteristics of problems are weak enough. In this paper, we proposed a dynamic topology optimization method with the approximate dynamic response sensitivity by the Adjoint Variable Method (AVM) using ESLs, to replace the static response sensitivity used in ESLM. After conducting the dynamic structural analysis, the approximate sensitivity is equivalent to the dynamic sensitivity obtained by AVM. Nevertheless, the errors between them may occur and increase with increasing iterations of the static structural analysis, as the differential relationships between displacement, velocity and acceleration are relaxed. Thus, the similarity assessment criteria using necessary and sufficient conditions were proposed, which control the errors of sensitivities within an acceptable range and simplify the double-loop algorithm of ESLM into a single-loop. And the approximate structural responses can also be calculated in parallel using ESLs. Several numerical examples are presented to demonstrate the effectiveness and efficiency of the proposed method. It is shown that the proposed method exhibits similar capabilities to AVM in achieving optimized objectives and convergence rates. The proposed method also benefits from the efficiency of parallel computing, and the numerical example demonstrates that the overall optimization process can achieve a maximum speedup of up to around 4, with a structural response speedup of 12.7. The formula for estimating the upper limit of the overall speedup based on the structural response speedup is provided at the end.
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