Three-dimensional Fem-bem Computation of Electromagnetic Responses of Flaws

Electromagnetic (EM) nondestructive testing of a lossy dielectric material involves exciting the material with an external EM source, measuring responses on a measuring surface, and inferring the location, shapes and properties of possible flaws. But the interaction of electromagnetic fields with the material is a highly complex process. Three-dimensional computation is therefore required to accurately predict the overall interaction of EM fields with the material and flaws. The problem is unbounded, and strong inhomogeneity exists inside the material, due to the presence of flaws. The coupling of the finite element method (FEM) and the boundary element method (BEM) provides an efFicient tool for modeling this type of problems [1]. Two questions must be answered before a successful use of this method. i) The interior EM fields are governed by a weak integral form based on the curlcurl equation of a field variable, leaving the divergence free equation unspecified, and the interface and boundary conditions to be forced separately. Spurious solutions may occur, when the standard nodal based finite element method is applied. Therefore, special care should be taken to prevent the so-ca lied vector parasites [2]. ii) The exterior EM fields are governed by a surface integral representation, which takes care of the infinite extent of the exterior domain. This integral equation invloves tangential surface currents. The standard nodal based boundary element method does not provide a rigorous description of the tangential surface currents especially near corners and edges, because of the ambiguity in normal directions. Semi-discontinuous superparametric elements can be used [3]. Also, discretization of the integral equation is central to couple the source fields into the materials and flaws. The boundary element mesh should not be too coarse and should not violate the sampling rule, in order not to cause aliasing for a given frequency. As a consequence, much computation effort is invested in the boundary integral, if a planar, linear boundary element mesh is used to model curved surfaces. It is common sense that one should use curvilinear and higher order boundary elements, to achieve a comparable resolution with less degrees of freedom. In this paper, we provide answers to the above questions by implementing an "edge" element based FEM-BEM procedure. Some of the ideas used here were reported in [4]. The tangentially continuous, "edge" element based FEM model is found to be free of spurious solutions for closed cavity problems [5]. A special type of boundary element with vector shape functions is developed to accurately represent the tangential curents. In addition, curvilinear "edge" elements allow a good approximation to material boundaries and interfaces. This model is used to compute the electromagnetic response of arbitrarily shaped flaws buried in