Postdoc: Parallel Adaptive Partition of Unity Methods

该博士后将与伦斯勒科学计算研究中心（SCOREC）的研究人员合作，开发并行自适应单元划分方法。这些方法代表了求解偏微分方程的一类新的离散化技术，它不需要传统的不相交、不重叠的单元格，如典型的有限元、有限差分或有限体积方法中使用的网格。然而，它们确实需要对区域进行离散化，从而使一组适当的分割细胞与自身和区域边界相互作用。构造分区的关键是确定分区细胞如何与自身和区域边界相互作用。这些方法的最终有效性将取决于分区的有效构造和控制。基于可扩展并行算法在一般三维域上的自动自适应有限元分析开发的专业知识，提议的研究计划将考虑自动构建和自适应解决方案的替代方法，并行地，根据实体模型定义的几何域的统一离散化分割。具体的发展领域包括：(i)检查在并行计算机上使用分布式八叉树结构来定义八叉树域上的基本分区的替代方法，（ii）在统一单元分区上执行所需集成的有效技术，特别关注集成那些相交域边界的单元，以及（iii）开发并行的自适应统一解决方案分区方法

The postdoctoral associate will work with researchers in Rensselaer's Scientific Computation Research Center (SCOREC) on the development of parallel adaptive partition of unity methods. These methods represent a new class of discretization techniques for solving partial differential equations, which do not require the traditional grids of disjoint, non-overlapping cells, as used in typical finite element, finite difference or finite volume methods. However, they do require a discretization of the domain from which an appropriate set of partition cells interact with themselves and the domain boundary. Key to the construction of the partitions is determining how the partition cells interact with themselves and the domain boundary. The ultimate effectiveness of these methods will depend on efficient construction and control of the partitions. Building on expertise on the development of scalable parallel algorithms for automated adaptive finite element analysis over general three-dimensional domains, the proposed research program will consider alternative methodologies for the automatic construction and adaptive solution, in parallel, of partition of unity discretizations for geometric domains defined in terms of solid models. Specific areas of development include: (i) examination of alternative methods to use distributed octree structures on parallel computers to define basic partitions over the domain of the octree, (ii) effective techniques to perform the required integrations over partition of unity cells, with particular concern for integrating those cells that intersect boundaries of the domain, and (iii) develop parallel adaptive partition of unity solution methodologies