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

自适应网格细化(AMR)策略在科学和工程中的许多分支中都很受关注。由于有可能产生局部细化的离散化，具有实际意义的大规模问题变得容易处理。然而，对AMR技术现状的回顾揭示了几个尚未得到充分解决的问题。首先，通常只有低阶有限元结合h-扩张被应用于商业代码中，从而导致代数型的次优收敛。此外，为了确保保角耦合，必须使用由单纯形元素和张量积元素组成的非结构离散化，或者必须限制元素的ansatz空间，从而降低方法的总体精度。其次，主要在研究环境中实现的高阶hp型加密策略受到限制，只能使用1-不规则网格来降低实现复杂性，或者基于分层形状函数，不允许构造对角质量矩阵，而对角线质量矩阵是(显式)瞬变分析所必需的。然而，这些方法至少能够提供指数级的收敛速度。为此，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，通过使用所谓的超限内插技术(Gordon-Coons内插)可以很容易地克服上述缺点，从而能够构造任意过渡元素。原则上，可以耦合不同类型、大小和多项式阶数的元素，而不会对可获得的精度造成任何影响。这种独特的多功能性为这种元素类型开辟了广泛的可能应用领域。此外，还开发了高精度的质量集总格式，以便于应用于暂态分析。所提出的新的元素家族构成了实施动态细化策略的基石，其中在动态分析的每个时间步中都执行细化和粗化步骤。因此，用最少的数值计算就可以达到最优的收敛速度。为了进一步提高框架的效率，所有算法都在一个高性能的计算框架中实现。在波传播分析领域，动态网格加密技术对于高效的数值模拟是必不可少的，通过不同的数值例子展示了新方法的特殊性质。为此，讨论了微结构材料中结构健康监测中的导波和地震波在大范围内的传播。

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.