A Formulation of Structural Design Optimization Problems for Quantum Annealing

TL;DR

The paper addresses structural design optimization for quantum annealing by embedding the governing equations directly into a single optimization objective. It formulates a total complementary energy $\Pi^{*}=U^{*}+W^{*}$ and, through a force-based representation and binary/qubit encoding, maps the nested min($\alpha$,$\sigma$) problem to a QUBO suitable for QA hardware, demonstrated on a 1D rod under self-weight. Key contributions include a unified analysis-design formulation, a binary-qubit encoding of cross-sectional decisions, and hardware-aware strategies (e.g., penalty tuning and post-processing) enabling small-scale QA solutions without iterative classical solvers. The results show good agreement with analytical solutions and establish feasibility on current hardware for simple problems, providing a framework and roadmap for scaling QA-based structural optimization as hardware advances.

Abstract

We present a novel formulation of structural design optimization problems specifically tailored to be solved by quantum annealing (QA). Structural design optimization aims to find the best, i.e., material-efficient yet high-performance, configuration of a structure. To this end, computational optimization strategies can be employed, where a recently evolving strategy based on quantum mechanical effects is QA. This approach requires the optimization problem to be present, e.g., as a quadratic unconstrained binary optimization (QUBO) model. Thus, we develop a novel formulation of the optimization problem. The latter typically involves an analysis model for the component. Here, we use energy minimization principles that govern the behavior of structures under applied loads. This allows us to state the optimization problem as one overall minimization problem. Next, we map this to a QUBO problem that can be immediately solved by QA. We validate the proposed approach using a size optimization problem of a compound rod under self-weight loading. To this end, we develop strategies to account for the limitations of currently available hardware and find that the presented formulation is suitable for solving structural design optimization problems through QA and, for small-scale problems, already works on today's hardware.

Figures (7)

• Figure 1: A generic elastic body $\Omega$ under external loading with prescribed surface traction $\hat{t}_i$ and displacement $\hat{u}_i$ on the boundary portions $\Gamma^{\sigma}$ and $\Gamma^{u}$, respectively, and body force density $f_i$.
• Figure 2: Generic setup for a rod under self-weight loading that is composed of multiple elements $e$.
• Figure 3: Structural Analysis Problem: Setup and pattern.
• Figure 4: Structural Analysis Problem: solution for the force function $F(x)$ obtained by compared to the analytical solution $F^*(x)$.
• Figure 5: Structural Design Optimization Problem: Setup for composed rod with variable cross sections.