The accurate solution of wave propagations in general two- and three-dimensional solids is still difficult and frequently impossible to achieve with the current computational schemes and computers available. We present in this paper a solution scheme that has much promise for the accurate solution of wave propagations in general solids. The procedure is based on the use of “overlapping finite elements” and direct time integration. The overlapping finite elements are effective because the spatial dispersion error is relatively small and can be monotonically reduced using a finer mesh. Similarly, the time integration dispersion errors can also be reduced monotonically as the time step becomes smaller. Hence the key property of the solution scheme is that the total dispersion error in the simulation of multiple waves traveling through solids is monotonically reduced as the spatial discretization and time stepping become finer. We summarize the ingredients of the solution scheme and illustrate the characteristics in the solution of some wave propagation problems that are difficult to solve accurately. These solutions may be benchmark solutions to use in the evaluations of other computational schemes.

在一般的二维和三维固体中，对波传播的精确解仍然是困难的，并且使用现有的计算方案和计算机通常是无法实现的。我们在本文中提出了一种解决方案，该方案对于精确求解一般固体中的波传播具有很大的希望。该过程基于“重叠有限元”的使用和直接时间积分。重叠的有限元是有效的，因为空间色散误差相对较小，并且可以使用更精细的网格单调减小。类似地，随着时间步长变小，时间积分分散误差也可以单调减少。因此，该解决方案的关键特性是，随着空间离散化和时间步长的变细，模拟穿过固体的多波波时的总色散误差将单调减小。我们总结了解决方案的组成部分，并说明了一些难以精确解决的波传播问题在解决方案中的特点。这些解决方案可能是基准解决方案，可用于其他计算方案的评估。