Fast Finite Element Solution for a Long Rectangular Surface Coil Placed Above a Flawed Layered Half Space

To analyze the interaction between an eddy current coil and a flaw, one generally needs to solve a three-dimensional vector diffusion equation, which is an involved task. Whenever possible, means to use instead two- and one-dimensional models are sought. The electromagnetic field of an eddy current surface coil often can be calculated through the spatial frequency analysis. Transforming the diffusion equation and boundary conditions into the spatial frequency domain, solution can be calculated using inverse transforms when the response at any spatial frequency is determined. The behavior of a long rectangular surface coil placed perpendicularly above a long surface breaking crack in a conductive layered half space, has been analyzed with this method. The interaction of a plane wave of spatial frequency a with a long flaw in a conductive half space represents a 2D problem. Solving the transformed diffusion equation with the finite element method (FEM) for a numbers of spatial frequencies, and carrying out the inverse transform, a good agreement between theory and experiment was obtained. As has been shown elsewhere, even a single main spatial frequency gives a reasonably good approximation. The two-dimensional FEM solution is performed for the two-component vector magnetic potential and the scalar electric potential. The system of algebraic equations was obtained using Galerkin weak formulation for the transformed diffusion equation, and solved using various numeric methods. The technique was validated at 100 MHz using long rectangular surface coils a set of austenitic stainless .steel samples coated with 60 and 100 µm tin layers, containing 500 µm deep EDM notches

Mathematical models and finite element approaches for nanosized piezoelectric bodies with uncoulped and coupled surface effects

In this chapter the dynamic problems for piezoelectric nanosized bodies with account for coupled damping and surface effects are considered. For these problems we propose new mathematical model which generalizes the models of the elastic medium with damping in sense of the Rayleigh approach and with surface effects for the cases of piezoelectric materials. Our model of attenuation and surface effects has coupling properties between mechanical and electric fields, both for the damping terms and constitutive equations for piezoelectric materials on the surface. For solving the problems stated the finite element approximations are discussed. A set of effective finite element schemes is examined for finding numerical solutions of week statements for nonstationary problems, steady-state oscillation problems, modal problems and static problems within the framework of modelling of piezoelectric nanosized materials with damping and surface effects. For transient and harmonic problems, we demonstrate that the proposed models allow the use of the mode superposition method. In addition, we note that for transient and static problems we can use efficient finite element algorithms for solving the systems of linear algebraic equations with symmetric quasi-definite matrices both in the case of uncoupled surface effects and in the case of coupled surface effects