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Abstract When the nonlinear transformation is used to evaluate three-dimensional (3D) weakly singular integral, the singularities of integrand can be eliminated by the Jacobian of the transformation. However, if the integration element has poor shape, such as including large angle or large ratio of edge length, the near singularities still exist in the weakly singular boundary integral, which can reduce the computational accuracy of weakly singular integral, and even lead to wrong results. Therefore, a new method for removing near singularity of weak singular boundary integral based on (α, β) transformation and distance transformation is proposed in this paper to accurately compute the 3D weakly singular integral. Firstly, the (α, β) transformation is employed to eliminate the singularity in the α direction. The (α, β) transformation is similar to the polar transformation and the singularity in the α direction can be removed by the Jacobian of the (α, β) transformation. Furthermore, the (α, β) transformation can separate the near singularity in the β direction. Then, the distance transformation is constructed according to the form of integral function in the β direction. The near singularity in the β direction can be eliminated by the Jacobian of the newly constructed distance transformation. Finally, several numerical examples of weakly singular integral with large angle and large ratio of edge length are given. The numerical results show that the relative error obtained by the proposed method can be less than 10-13 even for angle = 1700 or ratio of edge length = 100. These demonstrate that accurate evaluation of weakly singular integral with different shapes can be obtained using the (α, β) transformation combined with the distance transformation in the β direction. The last numerical example shows that the thin-walled structure problem can be analyzed by the presented method with high accuracy and computational efficiency.
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Received: 28 May 2020
Published: 14 April 2021
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2021, Vol. 42 
