Multigrid methods for magnetomechanical problems

The transient behavior of some magneto-mechanical system can be completely described by Maxwell's and Lame's partial differential equations (PDEs) with appropriate coupling terms reflecting the interactions of these fields and with the corresponding boundary and initial conditions. Neglecting the displacement currents and introducing the vector potential for the magnetic field, we arrive at a system of parabolic PDEs for the vector potential coupled with the hyperbolic PDEs for the displacements. In the stationary case, we have to deal with two coupled systems of elliptic PDEs. Usually the computational domain, the finite element discretization, the time integration, and the solver are different for the magnetic and mechanical parts. For instance, the vector potential is approximated by edge elements whereas the finite element discretization of the displacements is based on nodal elements on different meshes. The most time consuming modules in the solution procedure are the solvers for both the magnetical and the mechanical finite element equations arizing in each step of the time integration procedure. We present geometrical multigrid solvers that are different for both parts. Already in their non-parallelized versions, these multigrid solvers enable us to solve quite efficiently not only academic test problems (e.g. TEAM problems), but also transient 3D technical magneto-mechanical systems of high complexity such as magnetic valves and electro-magnetic-acoustic-transducers (EMATs). The magneto-mechanical multigrid solver was implemented within the framework of the adaptive multigrid package FEPP. In some practical application it would be preferable to use algebraic versions of these multigrid methods that were also developed within this project. The parallelization of the magnetic multigrid methods (Maxwell solvers) as well as of the mechanical multigrid methods (Lame solvers) is based on a unified data distribution concept borrowed from the domain decomposition methods. Refs. 3 (author)