Dynamic Mesh Refinement Exploiting High-Order Transfinite Elements for the Analysis of Wave Propagation Phenomena on High-Performance Clusters

自适应网格加密（AMR）策略在科学和工程的许多分支中是感兴趣的。由于生成局部细化离散化的可能性，实际相关的大规模问题变得易于处理。然而，对AMR最新技术水平的审查揭示了一些尚未充分解决的问题。首先，通常只有低阶有限元与h-扩展相结合，应用在商业代码中，导致代数型的次优收敛。此外，为了确保保形耦合，无论是非结构化离散组成的单纯形和张量积元素必须使用或的ananterior空间的元素必须受到约束，降低整体精度的方法。其次，高阶HP型细化策略，主要是在研究环境中实施，受到限制1-不规则网格，以降低实现的复杂性，或基于分层形状函数，不允许对角质量矩阵的建设是必不可少的（显式）瞬态分析。然而，这些方法至少能够提供指数收敛率。因此，提出了一种基于（节点）高阶四边形和六面体有限元类型的动态加密网格构造方法。幸运的是，上述缺点可以很容易地克服，通过采用所谓的超限插值技术（Gordon-Coons插值），使任意过渡元素的建设。原则上，可以耦合不同类型、大小和多项式阶数的元素，而不会对可达到的精度产生任何影响。这种独特的多功能性为这种元件类型开辟了各种可能的应用。此外，高精度的质量集总计划的开发，以方便应用程序的瞬态分析。所提出的新的家庭的元素构成的基石，实现一个动态的细化策略，细化和粗化步骤中执行的每个时间步的动态分析。因此，最佳的收敛速度，可以实现最小的数值工作。为了进一步提高所提出的框架的效率，所有算法都在高性能计算框架中实现。使用不同的数值例子，从波传播分析，其中动态网格细化技术是不可缺少的数值高效的模拟领域的新方法的特殊性能。为此，导波在微结构材料的结构健康监测和地震波传播的大区域的背景下进行了讨论。

Adaptive mesh refinement (AMR) strategies are of interest in many branches in science and engineering. Due to the possibility to generate locally refined discretizations, large-scale problems of practical relevance become tractable. However, a review of the state of the art in AMR reveals several issues that are not sufficiently resolved yet. First, often only low-order finite elements in combination with h-extensions are applied in commercial codes resulting in a sub-optimal convergence of algebraic type. Additionally, to ensure a conformal coupling, either unstructured discretizations consisting of both simplex and tensor-product elements have to be utilized or the ansatz space of the elements has to be constrained reducing the overall accuracy of the approach. Second, high-order hp-type refinement strategies, mainly implemented in a research environment, suffer from restrictions to 1-irregular meshes to decrease the implementational complexity or are based on hierarchic shape functions that do not permit the construction of diagonal mass matrices which is essential for (explicit) transient analyses. However, these methods are at least capable of delivering exponential rates of convergence. Therefore, a novel methodology to construct dynamically refined meshes based on (nodal) high-order quadrilateral and hexahedral finite element types is proposed. Fortunately, the aforementioned drawbacks can easily be overcome by employing the so-called transfinite interpolation technique (Gordon-Coons interpolation) enabling the construction of arbitrary transition elements. In principle, it is possible to couple elements of different types, sizes, and polynomial orders without any compromise on the attainable accuracy. This unique versatility opens up a wide variety of possible applications for this element type. Additionally, highly accurate mass lumping schemes are developed to facilitate the application to transient analyses. The proposed novel family of elements constitutes the cornerstone for implementing a dynamic refinement strategy, where both refinement and coarsening steps are executed in each time step of a dynamic analysis. Hence, optimal rates of convergence can be achieved with minimal numerical effort. To increase the efficiency of the proposed framework even further, all algorithms are implemented in a high-performance computing framework. The exceptional properties of the novel methodology are showcased using different numerical examples from the area of wave propagation analysis, where dynamic mesh refinement techniques are indispensable for a numerically efficient simulation. To this end, guided waves in the context of structural health monitoring in micro-structured materials and seismic wave propagation over large regions are discussed.