Reduced integral order 3D scalar wave integral equation Derivation and BEM approach
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The Boundary Element Method (BEM) is a numerical method to solve partial differential equations (PDEs), which is derived from the integral equation (IE) that can be developed from certain PDEs. Among IEs, the 3D transient wave integral equation has a very special property which makes it distinguished from other integral equations; Dirac-delta and its derivative δ feet appear in the fundamental-solution (or kernel-function). These δ and δ feet generalized functions have continuity C -2 and C -3 , respectively, and become a major hurdle for BEM implementation, because many numerical methods including BEM are based on the idea of continuity. More specifically, the integrands (kernel - shape function products) in the 3D transient wave IE become discontinuous (C -2 and C -3 ) and make numerical integration difficult. There are several existing approaches to overcome the δ difficulty, but none use the character of the Dirac-delta to cancel the integral. In this dissertation, a new method called the "Reduced order wave integral equation (Reduced IE)" is developed to deal with the difficulty in the 3D transient wave problem. In this approach, the sifting properties of δ and δ feet are used to cancel an integration. As a result, smooth integrands are derived and the integral orders are reduced by one. Smooth integrands result in the more efficient and accurate numerical integration. In addition, there is no more coupling between the space-element size and time-step size. Non-zero initial condition (IC) can be considered also. Furthermore, space integrals need to be performed once, not per time-step. All of this reduces dramatically the computational requirement. As a result, the computation order for both time and space are reduced by 1 and one obtains an O ( M N 2 ) method, where M is the number of time steps and N is the number of spatial nodes on the boundary of the problem domain. A numerical approach to deal with the reduced IE is also suggested, and a simple example is considered to demonstrate the validity of the new BEM approach.