# Three-Dimensional Elastodynamic Analysis Employing Partially Discontinuous Boundary Elements
###### Abstract  
Compared with continuous elements, discontinuous elements advance in processing the discontinuity of physical variables at corner points and discretized models with complex boundaries. However, the computational accuracy of discontinuous elements is sensitive to the positions of element nodes. To reduce the side effect of the node position on the results, this paper proposes employing partially discontinuous elements to compute the time-domain boundary integral equation of 3D elastodynamics. Using the partially discontinuous element, the nodes located at the corner points will be shrunk into the element, whereas the nodes at the non-corner points remain unchanged. As such, a discrete model that is continuous on surfaces and discontinuous between adjacent surfaces can be generated. First, we present a numerical integration scheme of the partially discontinuous element. For the singular integral, an improved element subdivision method is proposed to reduce the side effect of the time step on the integral accuracy. Then, the effectiveness of the proposed method is verified by two numerical examples. Meanwhile, we study the influence of the positions of the nodes on the stability and accuracy of the computation results by cases. Finally, the recommended value range of the inward shrink ratio of the element nodes is provided. boundary element method; elastodynamics; partially discontinuous element; singular integral; stability +
Footnote †: journal:  
0000-0001-9991-4099  
the element node is determined by the Gauss-Legendre integration point. The research of [PERSON] [17] and [PERSON] [18] reached similar conclusions to [PERSON] and [PERSON] [16]. [PERSON] and [PERSON] [19] also discussed the node position problem. They claimed that the nodes should be distributed as uniformly as possible in the discontinuous element. A similar conclusion can be found in [15]. Although the discontinuous element advances in processing the discontinuity of physical variables at corner points, the completely discontinuous element will significantly increase the number of nodes in the computation model, which increases the computational cost. Additionally, the computational accuracy of the completely discontinuous element is sensitive to the positions of element nodes, making the numerical results unstable. A better way to deal with the corner problem is to use partially discontinuous elements [20], in which only the nodes at the corner points are moved into the element, and the nodes at the non-corner points remain unchanged. Therefore, a discrete model that is continuous on surfaces and discontinuous between adjacent surfaces can be generated. [PERSON] and [PERSON] [21] studied the influence of the position of the element node in partially discontinuous elements on computational accuracy based on the Laplace equation. Their results demonstrated that the computational accuracy is insensitive to the position of the element node when the distance moved inward is equal to [0.05, 0.3] times the edge length of the element. A similar conclusion can be found in [22], that is, the computational accuracy of the partially discontinuous element is less sensitive to the position of the element node. Moreover, the partially discontinuous element will not add too many nodes to the computation model, and the computational cost is comparable to that of the continuous element. Therefore, partially discontinuous boundary elements are employed to compute the 3D transient elastodynamic problems in this paper. The BEM for solving elastodynamic problems [23] mainly includes the time-domain BEM, the frequency-domain BEM, and the double reciprocity BEM. For transient problems, the time-domain BEM is the most natural, direct, and applicable method. However, compared with the other two methods, the numerical stability of the time-domain BEM is poor. The computation results are easily affected by discrete parameters such as the time step and the element size [24; 25; 26].