Multi-axial perfectly matched layer (M-PML), known to have lost the perfect-matching property owing to multi-axial coordinate stretching, has been numerically validated to be long-time stable and it is thus used extensively in linear anisotropic wave simulation and in isotropic cases where the PML becomes unstable. We are concerned with the construction of the M-PML for anisotropic wave simulation based on a second order wave equation implemented with the displacement-based numerical method. We address the benefit of the incorrect chain rule, which is implicitly adopted in the previous derivation of the M-PML. We show that using the frequency-shifted stretching function improves the absorbing efficiency of the M-PML for near-grazing incident waves. Then, through multi-axial complex-coordinate stretching the second order anisotropic wave equation in a weak form, we derive a time-domain multi-axial unsplit frequency-shifted PML (M-UFSPML) using the frequency-shifted stretching function and the incorrect chain rule. A new approach is provided to reduce the number of memory variables needed for computing convolution terms in the M-UFSPML. The obtained M-UFSPML is well suited for implementation with a finite element or the spectral element method. By providing several typical examples, we numerically verify the accuracy and long-time stability of the implementation of our M-UFSPML by utilizing the Legendre spectral element method.

多轴完美匹配层（M-PML），已知由于多轴坐标拉伸而失去完美匹配特性，已被数值验证为长期稳定，因此广泛用于线性各向异性波模拟在 PML 变得不稳定的各向同性情况下。我们关注基于位移数值方法实现的二阶波动方程的各向异性波模拟 M-PML 的构造。我们解决了不正确的链式规则的好处，该规则在之前的 M-PML 推导中隐式采用。我们表明，使用频移拉伸函数可以提高 M-PML 对近掠入射波的吸收效率。然后，通过多轴复坐标拉伸弱形式的二阶各向异性波动方程，我们使用频移拉伸函数和不正确的链规则推导了时域多轴未分裂频移PML（M-UFSPML） . 提供了一种新方法来减少计算 M-UFSPML 中的卷积项所需的内存变量的数量。获得的 M-UFSPML 非常适合用有限元或谱元方法实现。通过提供几个典型示例，我们利用勒让德谱元方法数值验证了 M-UFSPML 实现的准确性和长期稳定性。提供了一种新方法来减少计算 M-UFSPML 中的卷积项所需的内存变量的数量。获得的 M-UFSPML 非常适合用有限元或谱元方法实现。通过提供几个典型示例，我们利用勒让德谱元方法数值验证了 M-UFSPML 实现的准确性和长期稳定性。提供了一种新方法来减少计算 M-UFSPML 中的卷积项所需的内存变量的数量。获得的 M-UFSPML 非常适合用有限元或谱元方法实现。通过提供几个典型示例，我们利用勒让德谱元方法数值验证了 M-UFSPML 实现的准确性和长期稳定性。