非结构化改进差分法在波动方程中的应用
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摘要
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.
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.
引文
[1](美)Xiao Lin. 固体应力波的数值解法[M]. 北京: 国防工业版社, 2008.
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[10]彭健新. 时域有限差分法及其在室内声场模拟中的应用[J]. 声学技术, 2009, 26(1): 53‐57. PENG Jianxin. Finite difference time domain method and its appli‐cation in room acoustical simulation[J]. Technical Acoustics, 2009, 26(1): 53‐57.
[2]李太宝. 计算声学[M]. 北京: 科学出版社, 2006.
[3]侯鹏. 改进的间断Galerkin法在激波管中的应用[J]. 哈尔滨工程大学学报, 2010, 31(2): 188‐194.
[4]刑丽. 地震声波数值模拟中的吸收边界条件[M]. 上海第二工业大学学报, 2006, 23(4): 273‐278.
[5]王礼立. 应力波基础[M]. 北京: 国防工业出版社. 2005.
[6]张文生. 科学计算中的偏微分方程有限差分方法[M]. 北京: 高等教育出版社. 2006.
[7]杨莹. 二维地震波场有限差分法数值模拟研究[D]. 中国地质大学硕士学位论文. 2009.
[8]Fang Q Hu. An Analysis of the Discontinuous Galerkin Method for wave propagation problems[J]. Journal of Computational Physics, 2001, 151: 921‐946.
[9]李树榜, 李书光. 固体中圆柱形孔脉冲超声波散射的有限元模拟[J]. 声学技术, 2007, 26(3): 417‐421. LI Shubang, LI Shuguang. Finite element modeling of pulsed ultra‐sonic waves scattered from cylindrical hole in solid[J]. Technical Acoustics, 2007, 26(3): 417‐421.
[10]彭健新. 时域有限差分法及其在室内声场模拟中的应用[J]. 声学技术, 2009, 26(1): 53‐57. PENG Jianxin. Finite difference time domain method and its appli‐cation in room acoustical simulation[J]. Technical Acoustics, 2009, 26(1): 53‐57.
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