Explicit Methods For Numerical Simulation Of Wave Motion In The Interior Of Computational Region

How to simulate the wave motion in the interior of a complicated system of large scale accurately and effectively is a very important research subject for developing and improving techniques for numerical simulation of near-field wave motion. Traditionally, the explicit method with small computational cost is employed for the numerical simulation of wave motion within the computational region. However, the existing time-space decoupling explicit finite element method is only second order accurate. The low order of accuracy not only affects precision of the numerical simulation, but also restricts improving efficiency of the computation. Accordingly, this paper is devoted to explore stable explicit methods with high order of accuracy for the numerical simulation of wave motion within the computational region. The paper further develops the existing time-space decoupling explicit numerical simulation technique, and proposes a new highly accurate stable explicit method.First, a new explicit method of numerical simulation is proposed based on the exact solution of wave equations. The method shares similarity with the lumped mass FEM, which is suitable for dealing with irregular grids and the recursion formulas derived are explicit and decoupling both in time and space. But, the recursion formulas developed by the method are 2M-order (M is the positive integer) accurate both in time and space. In this paper, the feasibility of the method is demonstrated via 1-D model: the recursion formulas at nodal points of an irregular grid are constructed, meanwhile, the stability and accuracy of the recursion formulas for a uniform grid are discussed in detail; according, an approach is proposed to construct the stable formulas which are of 2M-order of accuracy both in time and space with M being a positive integer and illustrated by constructing formulas of the second order (M=1) and the fourth order (M=2).Second, the method is generalized to the multi-dimensional cases and the corresponding recursion formulas for an irregular grid in 2-D and 3-D are constructed respectively. For a uniform quadrate grid, the approach used to construct the stable formulas of 2M-order of accuracy is deliberated in detail, and the stable formulas of the second order (M=1) and the fourth order (M=2) are presented; for a 3-D cubic grid, the recursion formulas are discussed briefly.Third, the stability and accuracy of the method are demonstrated by a series of 1-D and 2-D numerical tests, and the value of high order formulas for the improvement of calculation efficiency is pointed out. Last, starting from the space decoupling FEM ordinary differential equation system (the equations of motion of the structure), a group of explicit time integration schemes are derived via the Lagrange polynomial interpolation and integration by parts, which are of high order of accuracy; furthermore, the expanding form of the schemes are given for any mathematical- physical model; the stability of the schemes are investigated preliminarily by a simple linear time-invariant system.