Validation and further development of the perfectly matched layer technique for the numerical treatment of elastodynamic boundary value problems

分析弹性介质中的波的传播即使在简单的系统中也是一项困难的任务。对于某些边值问题有解析解，但对于土-结构动力相互作用问题的计算，一般采用数值方法，所有数值方法的基础是土和结构的离散化。因此，选择合适的边界条件是获得真实的数值计算结果的关键，因为它们必须能够模拟波在人工边界处无限传播而没有反射。到目前为止，有两种方法来模拟波在无限介质中的传播，边界元法和一种形式的局部透射边界。一种相对较新的方法是完美匹配层（PML）方法，它在电磁场的计算中很常见。它结合了上述两种技术的优点，在弹性动力学中使用PML时，将在人工边界上附加一个材料层，其目的是模拟几乎完美的波传播到无穷远。但由于材料并不完全基于物理原理，因此不可能对PML公式的数值质量做出机械合理的陈述。另外，对于PML公式的收敛性，如果它是在简单的边值问题之外，则缺乏理论上的研究。因此，所选PML公式的质量无法预测，这就是为什么该技术在弹性动力学领域几乎未被使用的原因。由于PML的这些特性，统计方法的使用是显而易见的。因此，在提交的研究项目中，敏感性分析方法被应用于定性和定量地识别影响因素，这些因素对用PML解决的边值问题的数值解有影响。通过这种方法，可以弥补PML的人工性质与其物理微分方程之间的差距，从而为处理复杂的土-结构动力相互作用问题提供一种工具。这一新开发的程序将在实际问题上得到验证。

The analysis of wave propagation in elastic media is even in simple systems a difficult task. For some boundary value problems there are analytical solutions, but generally for the computation of dynamic soil-structure-interaction problems the application of numerical methods is standard.The base for all numerical methods is the discretization of soil and structure. Thereby the choice of appropriate boundary conditions is the key to obtain realistic results of the numerical computation because they have to be able to simulate the propagation of waves to infinite without reflections at the artificial boundary. So far, there are two techniques to simulate the wave propagation in infinite media, the boundary-element-method and a form of local transmitting boundaries. A relatively new approach is the perfectly matched layer (PML) method, which is common in the computation of electromagnetic fields. It combines the advantages of the two techniques mentioned above.When using PML in elastodynamics, a material layer will be attached to the artificial boundary, which is designed to simulate almost perfect wave propagation to infinity. But because the material is not entirely based on physical principles, it is not possible to make mechanically reasonable statements to the numerical quality of a PML formulation. Also, there is a lack of mathematical-theoretical research to the convergence of PML formulations if it is taken beyond simple boundary value problems. Hence, the quality of a chosen PML formulation cannot be forecasted which is why the technique is almost unused in elastodynamic fields.Because of these specific properties of PML, the usage of statistical methods is obvious. Thus, in the submitted research project sensitivity analysis methods are applied to qualitatively and quantitatively identify the factors of influence, which have an impact on the numerical solutions of boundary value problems solved with PML. By this way, the gap between the mathematically-artificially properties and the underlying physical differential equation of the PML can be closed, to create a tool, to handle complex dynamic soil-structure-interaction-problems. This newly developed procedure will be validated on practical questions.