A variational quantum algorithm for nonlinear finite element analysis of hyperelastic materials

This manuscript explores a variational quantum formulation for nonlinear elasticity problems arising from hyperelastic material models. The approach leverages the potential energy structure of hyperelasticity and employs a hybrid quantum classical framework in which the energy functional is evaluated using parameterized quantum circuits and optimized through classical routines. To enable a hybrid (classical quantum implementation), polynomial approximations of the nonlinear terms in strain energy density are introduced, yielding a representation compatible with variational quantum algorithms. The methodology is demonstrated on a special case of the NeoHookean material model in a one dimensional setting using finite element discretizations with first and second order shape functions and nonhomogeneous boundary conditions. Numerical experiments investigate the influence of the polynomial approximation order on the accuracy and efficiency of the proposed approach, illustrating its feasibility for near-term quantum devices.