A differentiable structural analysis framework for high-performance design optimization

TL;DR

This paper bridges the gap between computational efficiency and the freedom of problem formulation through a differentiable analysis framework designed for general structural optimization through leveraging Automatic Differentiation to manage the complex computational graph of structural analysis programs, and implementing specific derivation rules for performance critical functions along this graph.

Abstract

Fast, gradient-based structural optimization has long been limited to a highly restricted subset of problems -- namely, density-based compliance minimization -- for which gradients can be analytically derived. For other objective functions, constraints, and design parameterizations, computing gradients has remained inaccessible, requiring the use of derivative-free algorithms that scale poorly with problem size. This has restricted the applicability of optimization to abstracted and academic problems, and has limited the uptake of these potentially impactful methods in practice. In this paper, we bridge the gap between computational efficiency and the freedom of problem formulation through a differentiable analysis framework designed for general structural optimization. We achieve this through leveraging Automatic Differentiation (AD) to manage the complex computational graph of structural analysis programs, and implementing specific derivation rules for performance critical functions along this graph. This paper provides a complete overview of gradient computation for arbitrary structural design objectives, identifies the barriers to their practical use, and derives key intermediate derivative operations that resolves these bottlenecks. Our framework is then tested against a series of structural design problems of increasing complexity: two highly constrained minimum volume problem, a multi-stage shape and section design problem, and an embodied carbon minimization problem. We benchmark our framework against other common optimization approaches, and show that our method outperforms others in terms of speed, stability, and solution quality.

Figures (13)

• Figure 1: The computation graph of a structural optimization problem for a seven-bar truss structure with an objective function, $\mathcal{L}(u,\dots)$. Top: The effect of a single element parameter (cross sectional area) follows a single path along the graph, resulting in a relatively straightforward derivative calculation. Bottom: A single spatial node parameter, however, affects multiple streams of intermediate computations; its derivative calculation reflects the sum of all these paths.
• Figure 2: Problem formulation and variable definition for a minimum volume optimization problem.
• Figure 3: Comparing the total optimization time with the final solution geometry and volumes. Red indicates a final solution that violates one or more constraints.
• Figure 4: Optimization history of the minimum volume design problem. Our method converges to a feasible solution in the shortest time with the fewest iterations.
• Figure 5: A cantilevered spatial truss roof structure consisting of $n_n = 145$ nodes and $n_e = 512$ steel elements.