Introduction to quantum computing and non-linear finite-element (re)formulation for quantum annealing

Quantum Computing is a change of paradigm since quantum computers have the potential to solve particular operations exponentially faster than classical computers. The available possible operations that can be performed on a quantum computer are not as versatile as with a classical computer. Quantum annealers, a family of quantum computers, are dedicated in evaluating the minimum state of a Hamiltonian quadratic potential. In order to take advantage of this feature, we develop a hybrid classical computer - quantum annealer approach [1] by reformulating elasto-plastic finite-elements as a double minimisation process using the variational updates formulation [2].

Since in all generalities, the potentials resulting from an elasto-plastic behavior are non-quadratic, a series of Hamiltonian quadratic potentials is obtained 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 that can be encoded on a quantum annealing hardware such as the D-Wave system.

REFERENCES
[1] V. D. Nguyen, F. Remacle, L. Noels. A quantum annealing-sequential quadratic programming assisted finite element simulation for non-linear and history-dependent mechanical problems. Submitted to European Journal of Mechanics – A/solids, http://dx.doi.org/10.48550/arXiv.2310.06911 .
[2] M. Ortiz, L. Stainier (1999), The variational formulation of viscoplastic constitutive updates, Computer methods in applied mechanics and engineering, 171, 419-444, doi: https://doi.org/10.1016/S0045-7825(98)00219-9