First-order equivalent static loads for dynamic response structural optimization

TL;DR

The paper tackles the challenge of dynamic-response optimization by improving the classic ESL approach. It introduces First-order Equivalent Static Loads (F-ESL), which embeds first-order design sensitivity into the static loads to align the static subproblems with the original dynamic problem, thereby satisfying first-order optimality conditions at the solution. Through three reproducible numerical examples, F-ESL achieves the same optimal designs as direct dynamic optimization but with significantly fewer time-history analyses, confirming both accuracy and computational efficiency. The work provides a practical, robust framework for transient dynamic topology optimization, preserving ESL's ease of implementation while delivering guaranteed first-order consistency with the original problem.

Abstract

A novel first-order equivalent static loads approach for optimization of structural dynamic response, F-ESL, is presented and compared to the basic equivalent static load formulation, ESL. F-ESL simplifies dynamic optimization problems by converting them into a series of static optimization sub-problems. The ESL algorithm in its original formulation does not have a guaranteed capability of reaching, or recognizing, final designs that satisfy necessary first-order optimality conditions. F-ESL addresses this limitation by including first-order terms directly into the equivalent static load definition. This new mathematical information guides the optimization algorithm more effectively toward solutions that satisfy both feasibility and optimality conditions. Using reproducible numerical examples, we show that F-ESL overcomes the known limitations of the original ESL, often with few outer function evaluations and fast convergence. At the same time, F-ESL maintains ESL simplicity, robustness, and ease of implementation, providing practitioners with an effective tool for structural dynamic optimization problems.

Figures (15)

• Figure 1: Example of Sec. \ref{['subsec:studycasedes']}. A two-bar structure with one degree of freedom subjected to a vertical sinusoidal force.
• Figure 2: Illustration of the numerical example of Sec. \ref{['subsec:studycasedes']}. The figure shows the design domain, the contour of the objective function of the original dynamic problem, and the optimization paths across the iterations for ESL, F-ESL and the solution of the reference (original) problem. The area in gray defines the feasible design domain, enclosed by the box constraints and the volume constraints, marked by black and red dotted lines, respectively.
• Figure 3: Numerical example of Sec. \ref{['subsec:studycasedes']}. Comparison of the displacements sensitivities with respect to the design variables $x_1$ and $x_2$, of the original dynamic-response optimization problem, the ESL and the F-ESL formulations, at the final solution of the F-ESL, which is the optimal solution of the dynamic-response optimization problem \ref{['eq:dynoptform_example1']}.
• Figure 4: Numerical example of Sec. \ref{['subsec:studycasedes']}. Comparison of the displacements sensitivities with respect to the design variables $x_1$ and $x_2$, of the original dynamic-response optimization problem, the ESL and the F-ESL formulations, at the final solution of the ESL, which is not the optimal solution of the dynamic-response optimization problem \ref{['eq:dynoptform_example1']}.
• Figure 5: Numerical example of Sec. \ref{['subsec:studycasedes']}. Gradient of the equivalent static loads (Eq. \ref{['eq:nablafESL']}) at the optimal solution $\mathbf{x}^*_{opt}$ of the dynamic problem \ref{['eq:dynoptform_example1']}.