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| Evaluating nearly singular integrals accurately and efficiently based on sphere subdivision method |
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Abstract The boundary element method (BEM) has been widely used for solving engineering and scientific problems. Compared with the finite element method (FEM), the BEM is more attractive for its dimension reduction feature and higher accuracy. Accurate and efficient evaluation of nearly singular integrals is of crucial importance for successful implementation of the BEM. The nearly singular integrals in BEM have been studied for a long time, and many methods have been proposed. However, none of them can evaluate these integrals accurately and efficiently. In this paper, a method based on sphere subdivision technique is proposed for evaluating nearly singular integrals. With the method, the nearly singular integrals can be evaluated accurately and efficiently for cases of arbitrary type fundamental solution, arbitrary shape of element and arbitrary location of source point. In the proposed method, the minimum and maximum distances between the source point and the integration element are firstly computed, which determine the beginning and ending of sphere radius. Then triangular and quadrilateral sub-elements can be obtained by subdividing the integration element through a sequence of spheres with exponential increasing radius. Finally the obtained sub-elements are turned into arc-shape ones, i.e. the triangular and quadrilateral sub-elements are changed to flabellate and annular sub-elements, respectively. The sphere subdivision is performed in 3D Cartesian coordinate system, thus the proposed method is suitable for any elements. In addition, fundamental solution is a function of the distance between the source point and the field point, so in the same level of accuracy, the number of Gaussian point can greatly decrease in circular direction for evaluating nearly singular integrals on sub-elements. Because of exponential growth of sphere radius, the integration can be of the same level of accuracy in the radial direction. The numerical examples have demonstrated that the proposed method has much better stability and accuracy than conventional methods.
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Received: 15 December 2017
Published: 28 October 2018
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2018, Vol. 39 
