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

• Applicant: Dr.-Ing. Sascha Eisenträger
• Subject Area: Mechanics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 497531141

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.

• DFG Programme: Research Grants