Development of high order fictitious domain methods for unstructured grids

随着快速高分辨率计算机断层扫描(CT)扫描仪的出现，微结构的三维信息变得容易获得。利用这些数据，发展了一种用于结构模拟的全自动数值方法。因此，可以在没有任何用户干预的情况下评估异质微结构对组件的全局行为的影响。所选择的方法是基于虚域的概念，特别是基于作者最近提出的有限单元法(FCM)或谱单元法(SCM)，目前的研究重点是将FCM和SCM推广到非结构网格。在这种情况下，众所周知，任意几何形状可以由四面体单元离散，并且强大的网格生成器可用于该单元类型。因此，第一步是开发一种基于四面体网格的FCM和SCM的新公式。未来这个框架中还包括棱柱体、金字塔和多面体单元。为了保持p-有限元中遇到的高收敛速度，实现并测试了不同的高阶节点形函数和分层形函数。在为所提议的方法建立了良好的理论基础之后，对其进行了详细的评估。研究了单元尺寸对微结构的典型长度尺度的影响以及合适的时间积分方法等开放问题。根据微观结构细节的大小，可能需要进行局部网格细化来求解解场。在动力分析方面，时间积分格式对解的准确性和效率负有责任。因此，针对不同方法对高阶虚拟域法的适用性进行测试是不可避免的。最后，通过两个不同的工程问题对所提出的方法进行了验证。第一个应用领域是铸造部件的机械评估。第二个应用涉及发动机封装所用材料的分析。在这两种情况下，数值模型都是基于真实部分的CT扫描。根据所用材料的微观结构，可能需要对四面体单元进行局部meh细化。对这些复杂的面向实践的问题的分析表明了该方法的性能及其对复杂工业问题的适用性。

With the advent of fast high-resolution computed tomography (CT) scanners three-dimensional information of the micro-structure is readily available. Using this data a fully automated numerical method for structural simulations is developed. Therefore, it is possible to assess the influence of a heterogeneous micro-structure on the global behavior of the component without any user intervention. The chosen approach is based on the fictitious domain concept and especially on the finite cell method (FCM) or the spectral cell method (SCM) as recently proposed by the author.The focus of the current research proposal is on the extension of the FCM and the SCM to unstructured grids. In this context, it is generally known that arbitrary geometries can be discretized by tetrahedral elements and that powerful mesh generators are available for this element type. Therefore, the first step is to develop a novel formulation of the FCM and the SCM based on tetrahedral grids. In the future also prismatic, pyramidal and polyhedral elements are included in this framework. To retain high rates of convergence as encountered in the p-FEM different high order nodal and hierarchical shape functions are implemented and tested. After developing a sound theoretical basis for the proposed approach it is evaluated in detail. Open questions including the influence of the size of the cells with respect to a typical length scale of the micro-structure and suitable time integration methods are investigated. Depending on the size of the micro-structural details a local mesh refinement might be needed to resolve the solution field. Regarding dynamical analyses, time integration schemes are responsible for the accuracy of the solution and its efficiency. Therefore, it is inevitable to test different methods in view of their suitability for high order fictitious domain methods.Finally, the proposed method is validated using two different engineering problems. The first area of application is the mechanical assessment of cast components. The second application deals with the analysis of materials used for encapsulations of engines. In both cases the numerical model is based on CT scans of the real part. Depending on the micro-structure of the used materials a local meh refinement of the tetrahedral cells might be required. The analysis of these complex practice-oriented problems demonstrates the performance of the method and its applicability to complex industrial problems.