Vector potential formulation for the three-dimensional finite element analysis of nonlinear electromechanical problems

Ferroelectroelastic materials exhibit nonlinear behavior when they are subjected to high electromechanical loadings. Using the standard formulation with the scalar potential as electric nodal variable in the nonlinear finite element analysis can lead to a low convergence of the iteration procedures. Therefore the formulation with a vector potential as electric nodal variable is developed, which ensures a positive definite stiffness matrix. Solutions of boundary value problems using the scalar potential formulation lie on a saddle point in the space of the nodal degrees of freedom, whereas solutions for the vector potential formulation are in a minimum. Unfortunately, the latter solutions involving the "curl-curl" operator are non-unique in the three-dimensional case. A Coulomb gauge condition imposed on the electric vector potential improves the convergence behavior of nonlinear problems, and in combination with appropriate boundary conditions, it can enforce unique vector potential solutions. A penalized version of the weak vector potential formulation with the Coulomb gauge is proposed and tested on some numerical examples in ferroelectricity.