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

TL;DR

This work addresses the computational challenge of solving non-linear, history-dependent mechanical problems by coupling classical finite element assembly with quantum annealing. It introduces a QA-SQP framework that recasts the FE double-minimisation over displacements and internal variables into sequences of quadratic problems, which are then binarised to fit QUBO form for execution on quantum annealers. The approach is demonstrated on one- and two-dimensional elasto-plastic benchmarks, showing high accuracy and the capacity to model history-dependent behavior while outlining hardware-related limitations. Overall, the paper outlines a viable path toward accelerating non-linear FE simulations through hybrid classical-quantum computation as quantum hardware continues to mature.

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.

Figures (12)

• Figure 1: Illustration of the minimisation of a single variable $v$ function considering $L=3$ qubits. At each iteration, the minimum between 8 discrete values is found and is used as the starting point for the next iteration.
• Figure 2: Uniaxial-strain tensile test with $b_0=100\,\text{MPa}\cdot\text{mm}^{-1}$: (a, c, e) convergence histories of 5 realisations for each value of $L$; and (b, d, f) displacement distribution over the bar predicted by the first realisation for each value of $L$. The default annealing time equal to $20 \mu s$ is considered.
• Figure 3: Uniaxial-strain tensile test with $b_0=100\,\text{MPa}\cdot\text{mm}^{-1}$: (a) effect of the annealing time (annealing_time) for number_reads=100; and (b) effect of the number of reads (number_reads) for annealing_time=20 $\mu$s on the convergence history.
• Figure 4: Uniaxial-strain tensile test with $b_0=100\,\text{MPa}\cdot\text{mm}^{-1}$: total QPU access time and number of iterations for each realisation in function of the number of qubits and number of reads (number_reads). The default annealing time equal to $20 \mu s$ is considered.
• Figure 5: Uniaxial-strain tensile test with $b_0=400\,\text{MPa}\cdot\text{mm}^{-1}$: (a) displacement distribution; and (b) equivalent plastic strain distribution over the bar.