A numerical method for determining electromagnetic fields in cracked metals
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A NUMERICAL METHOD FOR DETERMINING ELECTROMAGNETIC FIELDS IN CRACKED METALS Z. T. Nazarchuk, Ya. P. Kulynych, and Ya. V. Datsko
UDC 537.874
We improve the numerical algorithm of the solution of a three-dimensional hypersingular integral equation of first kind based on the collocation method. The efficiency of the developed approach is shown. We also give some numerical results and their interpretation.
For detecting undersurface cracks in metal structures, it is customary to use methods based on the analysis of interaction between eddy currents induced in a conductor and a defect [1]. If the crack opening is small, one usually applies for this purpose the model of a crack in the form of a cut, where a certain distribution of electrical dipoles is assigned in such a way that their electromagnetic field is equivalent to the field of the crack [2]. The determination of this distribution is reduced to the solution of a two-dimensional integral equation of first kind, where the order of singularity in its kernel is greater than the dimensionality of the domain of integration (a hypersingular integral equation) [2, 3]. Similar hypersingular integral equations are used in the solution of different applied problems, e.g., of the theory of elasticity for cracked bodies [4, 5], the mechanics of brittle fracture [6 – 9], thermoelasticity [10], aerodynamics [11], electromagnetic wave diffraction on conducting screens [12], etc. Despite numerous publications devoted to this research area, the general theory of the solution of such equations is far from being completed, and, hence, the creation of efficient numerical methods for their solution is topical. In particular, numerical methods based on the preliminary regularization of the hypersingular components of integral equations are developing intensely [4, 5, 10]. For regularization, the order of singularity is often decreased with the help of integration by parts or expansion of the desired function into a Taylor series in the neighborhood of the pole. In such a way, the initial integral equation is reduced to its approximate, already regular integro-differential analog. The subsequent stages of this algorithm are connected with the division of the crack domain into boundary elements, the approximation of the desired function and its derivatives on them, and, finally, the collocation construction of a finite-dimensional discrete analog of the initial equation in the form of a linear system of equations. However, such an algorithm has the following shortcomings: the fullness and dissymmetry of the matrix of this system and the loss of accuracy of the solution near the crack contour. This complicates substantially its application in three-dimensional problems. However, the described algorithm enables one to construct efficiently a numerical solution if the desired function in the near-contour zone has a root asymptotics [4]. In recent years, researchers have actively developed direct numerical methods for the solution of hypersingular integral equations based on special form
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