University of Strathclyde, Department of Electronic and Electrical Engineering, Centre for Ultrasonic Engineering, Royal College Building R3-35, 204 George Street, Glasgow G11XW, United Kingdom.
Ultrasonics. 2010 Mar;50(3):431-8. doi: 10.1016/j.ultras.2009.10.009. Epub 2009 Oct 20.
The stability of the finite-difference approximation of elastic wave propagation in orthotropic homogeneous media in the three-dimensional case is discussed. The model applies second- and fourth-order finite-difference approaches with staggered grid and stress-free boundary conditions in the space domain and second-order finite-difference approach in the time domain. The numerical integration of the wave equation by central differences is conditionally stable and the corresponding stability criterion for the time domain discretisation has been deduced as a function of the material properties and the geometrical discretization. The problem is discussed by applying the method of VonNeumann. Solutions and the calculation of the critical time steps is presented for orthotropic material in both the second- and fourth-order case. The criterion is verified for the special case of isotropy and results in the well-known formula from the literature. In the case of orthotropy the method was verified by long time simulations and by calculating the total energy of the system.
讨论了各向异性均匀介质中弹性波传播的有限差分近似在三维情况下的稳定性。该模型采用交错网格的二阶和四阶有限差分方法以及空间域中的无应力边界条件和时间域中的二阶有限差分方法。通过中心差分对波动方程进行数值积分是条件稳定的，并且已经推导出了相应的时间域离散化稳定性准则，它是材料性质和几何离散化的函数。通过冯·诺依曼方法讨论了这个问题。针对各向异性材料的二阶和四阶情况，给出了数值解和临界时间步长的计算。该准则在各向同性的特殊情况下得到验证，得出了文献中已知的公式。在各向异性的情况下，通过长时间模拟和计算系统的总能量来验证该方法。
