The purpose of this paper is to address the formulation, implementation, and adaptation of closely coupled multi‐physics problems with h‐ and p‐adaptive finite element methods. A general formulation is chosen allowing for coupled problems of various types. Adaptation algorithms for h‐ and p‐refinement are given.
A generic system of second‐order differential equations is used, where the field of each individual problem is represented as an entry in the list of field variables. Specific problems are implemented by mapping material coefficients to the coefficients of the generic form. An example with four natures is investigated with close coupling between electric, mechanical and thermal fields. h‐ and p‐refinement using a single mesh is considered, where the element order may differ for individual fields.
In coupled problems, the error in each single field is affected by approximation properties of all other field quantities. In order to allow for optimal convergence order in the number of degrees of freedom, the error contributions of all fields have to be considered. Separate error estimation in each field is needed especially in h‐adaptation on a single mesh. Energy coupling coefficients were introduced to derive an adaptation criterion. Convergence analysis of h‐ and p‐adaptation proves the feasibility of the approach.
Piezopyroelectricity considers thermal effects in high‐frequency piezoelectric materials, which is a coupled problem of four natures. The paper introduces an adaptation criterion for such complicated coupled problems and proves feasibility.
