A study on singular boundary integrals and stability of 3D time domain boundary element method

Abstract

The paper presents a novel numerical scheme to effectively solve the weak, strong, and hyper-singular time-space integrals in the 3D time domain boundary element method (BEM). To deal with the singularities of P- or S-wavefront elements, the element subdivision method is used to identify non-zero valued sub-elements on the wavefronts, and the non-zero valued sub-elements containing source points are solved by the power series expansion method, which achieves the effect of obtaining satisfactory accuracy with fewer Gaussian points. In the present scheme, the non-uniform rational B-Spline (NURBS) basis functions are used to determine the physical coordinates of each sub-element with very high precision. Numerical results show that the proposed scheme can effectively implement a high-precision singular time-space integral solution under a unified framework. In addition, this paper uses the time-weighting method to improve the time integration algorithm to solve the instability problem. The validity and accuracy of the algorithm are verified by dynamics examples of the finite elastic body and infinite elastic body, and the effects of different weight points and integration schemes in the time-weighting method on the stability are discussed. The effectiveness of the proposed algorithm for large-scale elastodynamics problems is also examined using the 27 spherical cavities problem.