A quantum annealing-sequential quadratic programming assisted finite element simulation for non-linear and history-dependent mechanical problems

TL;DR

This paper introduces a novel hybrid classical-quantum framework using quantum annealing to efficiently solve complex non-linear, history-dependent finite element problems in mechanics, leveraging quantum speedups for quadratic minimizations.

Contribution

It develops a quantum annealing-assisted sequential quadratic programming method for finite element simulations of non-linear mechanics, a novel integration of quantum computing into this domain.

Findings

Successfully applied to 1D and 2D elasto-plastic benchmarks.

Demonstrated feasibility of quantum-assisted finite element simulations.

Outlined pathways for future quantum-enhanced computational mechanics.

Abstract

We propose a framework to solve non-linear and history-dependent mechanical problems based on a hybrid classical computer -- quantum annealer approach. Quantum Computers are anticipated to solve particular operations exponentially faster. The available possible operations are however not as versatile as with a classical computer. However, quantum annealers (QAs) are well suited to evaluate the minimum state of a Hamiltonian quadratic potential. Therefore, we reformulate the elasto-plastic finite element problem as a double-minimisation process framed at the structural scale using the variational updates formulation. In order to comply with the expected quadratic nature of the Hamiltonian, the resulting non-linear minimisation problems are iteratively solved with the suggested Quantum Annealing-assisted Sequential Quadratic Programming (QA-SQP): a sequence of minimising quadratic problems is performed by approximating the objective function by a quadratic Taylor's series. Each quadratic minimisation problem of continuous variables is then transformed into a binary quadratic problem. This binary quadratic minimisation problem can be solved on quantum annealing hardware such as the D-Wave system. The applicability of the proposed framework is demonstrated with one- and two-dimensional elasto-plastic numerical benchmarks. The current work provides a pathway of performing general non-linear finite element simulations assisted by quantum computing.