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The dynamic behaviour of magneto-mechanical sensors and actuators can be completely described by Maxwell's and Navier-Lamé's partial differential equations (PDEs) with appropriate coupling terms reflecting the interactions of these fields and with the corresponding initial, boundary and interface conditions. Neglecting the displacement currents, which can be done for the classes of problems considered in this paper, and introducing the vector potential for the magnetic field, we arrive at a system of degenerate parabolic PDEs for the vector potential coupled with the hyperbolic PDEs for the displacements.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 arising at each step of the time integration procedure. We use geometrical multigrid solvers which are different for both parts. These multigrid solvers enable us to solve quite efficiently not only academic test problems, but also transient 3D technical magneto-mechanical systems of high complexity such as solenoid valves and electro-magnetic-acoustic transducers. The results of the computer simulation are in very good agreement with the experimental data.

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The focus of this paper is on the efficient numerical computation of 3D electromagnetic field problems by using the finite element (FE) and multigrid (MG) methods. The magnetic vector potential is used as the field variable and the discretization is performed by Lagrange (nodal) as well as Ne´de´lec (edge) finite elements. The resulting system of equations is solved by applying a preconditioned conjugate gradient (PCG) method with an adapted algebraic multigrid (AMG) as well as an appropriate geometric MG preconditioner.

Due to the high geometric complexity and the interaction of different physical field types, like mechanical displacement, acoustic pressure, and magnetic induction, the design of electromagnetic transducers is a challenge for the developers. Since the fabrication of prototypes and experimental-based design is a lengthy and costly process, needs for appropriate numerical simulation tools arise. In this paper, a simulation program is presented which is especially adapted to multifield problems like the one described above. First, the description of the underlying physical fields (acoustic, mechanical, and magnetic) with partial differential equations (PDE) and their coupling is reported. The solution of these PDEs is based on either finite-element methods (FEM), boundary-element methods (BEM), or coupled FE-BE methods, depending on the considered problem type. However, these numerical techniques yield to long computer time, especially in the 3-D case. Therefore, a multigrid solver has been developed which enables considerably faster solutions of the presented numerical tasks. Finally, two application examples, the calculation of an EMAT (electromechanical acoustic transducer) in transmitting and receiving mode and the prediction of the sound field of a vibrating machine part, show the reliability of the simulations.