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

即使在简单的系统中，分析弹性介质中的波传播也是一项艰巨的任务。对于一些边值问题，一般都有解析解，但对于土-结构-动力相互作用问题的计算，一般采用的是数值方法，所有数值方法的基础都是对土和结构的离散。因此，选择合适的边界条件是获得实际数值计算结果的关键，因为它们必须能够模拟波在无反射人工边界上的传播。到目前为止，用于模拟无限大介质中波传播的技术主要有两种：边界单元法和局部传输边界形式。一种相对较新的方法是完全匹配层(PML)方法，它是电磁场计算中常用的方法。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层材料层，以模拟近乎完美的波在无限远的传播。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上合理的陈述。此外，如果超越简单的边值问题，对PML公式的收敛也缺乏数学理论研究。因此，所选择的PML公式的质量是无法预测的，这就是为什么该技术在弹性动力学领域几乎没有使用。因此，在提交的研究项目中，应用灵敏度分析方法定性和定量地识别影响用PML求解边值问题的数值解的影响因素。通过这种方式，可以弥合PML的数学人工特性和基本物理微分方程之间的差距，从而创建一种工具，来处理复杂的土-结构-相互作用-问题。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.