Jean-Paul Pelteret

Convergence study of the h-adaptive PUM and the hp-adaptive FEM applied to eigenvalue problems in quantum mechanics

In this paper the h-adaptive partition-of-unity method and the h- and hp-adaptive finite element method are applied to eigenvalue problems arising in quantum mechanics, namely, the Schrödinger equation with Coulomb and harmonic potentials, and the all-electron Kohn–Sham density functional theory. The partition-of-unity method is equipped with an a posteriori error estimator, thus enabling implementation of error-controlled adaptive mesh refinement strategies. To that end, local interpolation error estimates are derived for the partition-of-unity method enriched with a class of exponential functions. The efficiency of the h-adaptive partition-of-unity method is compared to the h- and hp-adaptive finite element method. The latter is implemented by adopting the analyticity estimate from Legendre coefficients. An extension of this approach to multiple solution vectors is proposed. Numerical results confirm the theoretically predicted convergence rates and remarkable accuracy of the h-adaptive partition-of-unity approach. Implementational details of the partition-of-unity method related to enforcing continuity with hanging nodes are discussed.

Numerical modelling of nonlinear thermo-electro-elasticity

This work presents the numerical modelling of nonlinear thermo-electro-elasticity in the context of electro-active polymers (EAPs). EAPs are characterised by their electro-mechanical coupling behaviour that converts electrical into mechanical energy. As polymeric materials in general are sensitive to the influence of temperature, thermal effects play an important role in the material behaviour of EAPs. Based on a thermo-electro-mechanically coupled constitutive framework presented in an earlier contribution, a variational formulation is developed and the finite-element method is employed to solve the nonlinear thermo-electro-mechanical coupling problem. The numerical implementation is studied by means of several examples.