非结构化改进差分法在波动方程中的应用

Application of improved unstructured differential method in wave-equation

在计算声学领域中,有限差分法是一种比较传统的数值模拟方法。差分法简单易行而且效果突出,然而此方法只能在结构网格中使用,很难计算几何边界较为复杂的区域。基于高斯定理构造单元梯度的方式,在二维空间上,提出了一种改进的有限差分法(Improved Finite Difference Method,IFDM)。IFDM可以使用任意的三角形和四边形等单元处理二阶偏微分方程,并将计算域拓展为任意几何形状。通过编程,计算了固体和流体声传播的波动方程,并将IFDM的计算结果与差分的计算结果进行了对比,验证了此方法的可行性与稳定性。由于IFDM基于梯度重构思想,所以此方法很容易推广到三维空间。

In the field of computational acoustics,Finite Difference Method（FDM） is a traditional numerical simulation method.Difference method is simple and effective,but this method can only be used in structured grids,and it is diffi-cult to calculate in the area with more complex geometry boundary.Based on using Gauss theorem to construct unit gradient,in two-dimensional space,an Improved Finite Difference Method（IFDM） is put forward.IFDM can use any triangle and quadrilateral element to deal with two order partial differential equation,which can expands the com-puta-tional domain to arbitrary geometry.Through programming,the solid and fluid acoustic wave equations are computed,and the contrast between the results of IFDM and FDM verifies the feasibility and stability of IFDM.Because IFDM is based on the idea of reconstructing gradient,this method could be easily extended to three-dimensional space.