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# Efficient methods for solving boundary integral equation in diffusive scalar problem and eddy current nondestructive evaluation

Yang, Ming (2010) Efficient methods for solving boundary integral equation in diffusive scalar problem and eddy current nondestructive evaluation. PhD thesis, Iowa State University.

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## Abstract

Eddy current nondestructive evaluation (NDE) of airframe structures involves the detection of electromagnetic field irregularities due to non-conducting inhomogeneities in an electrically conducting material. Usually, the eddy current NDE problem can be formulated by the boundary integral equations (BIE) and discretized into matrix equations by the method of moments (MoM) or the boundary element method (BEM). The fast multipole method (FMM) is a well-established and effective method for accelerating numerical solutions of the matrix equations. Accelerated by the FMM, the BIE method can now solve large-scale electromagnetic wave propagation and diffusion problems. The traditional BIE method requires O(N^2) operations to compute the system of equations and another O(N^3) operations to solve the system using direct solvers, with N being the number of unknowns; in contrast, the BIE method accelerated by the two-level FMM can potentially reduce the operations and memory requirement to O(N^3/2). Moreover, several approaches have been proposed for the field calculation in the presence of flaws in three dimensional NDE; however, seldom work has been done in applying efficient methods to seek rapid solution in eddy current NDE simulation. As elaborated in the dissertation, we introduce a fast multipole BIE method for two-dimensional diffusive scalar problem and an efficient BIE method for three-dimensional eddy current NDE. Firstly, we work with the two-dimensional Helmholtz equation with a complex wave number for non-trivial boundary geometry. We describe the FMM acceleration procedure of the BIE method and its features briefly, explaining that the FMM is not only efficient in meshing complicated geometries, accurate for solving singular fields or fields in finite domains, but also practical and often superior to other methods in solving large-scale problems. Subsequently, computational tests of the numerical FMM solutions against the conventional BIE results and their complexity are presented. Secondly, for the eddy current NDE, a BIE method in three dimensions has been demonstrated. The eddy current problem is formulated by the BIE and discretized into matrix equations by the method of moments (MoM) or the boundary element method (BEM). In our implementation of the Stratton-Chu formulation for the conductive medium, the equivalent electric and magnetic surface currents are expanded in terms of Rao-Wilton-Glisson (RWG) vector basis function while the normal component of magnetic field is expanded in terms of the pulse basis function. Also, a low frequency approximation is applied in the external medium, that is, free space in our case. Computational tests are presented to demonstrate the accuracy and capability of the three-dimensional BIE method with a complex wave number for arbitrarily shaped objects described by a number of triangular patches. The results of this research set the stage for the efficient BIE method to be applied in more practical eddy current NDE simulation and be embedded with the FMM in the future.

EPrint Type: Thesis (PhD) Electrical Engineering > ELECTROMAGNETICS & NONDESTRUCTIVE EVALUATION > Computational Electromagnetics 602 Identification Number UNSPECIFIED Ming Yang 01 December 2010

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