Collaborative Research: Elements: EXHUME: Extraction for High-Order Unfitted Finite Element Methods

非拟合有限元方法允许模拟的物理系统是困难的，如果不是不可能使用经典的有限元方法，需要贴体网格模拟。例如，非拟合有限元方法可以直接应用于表现出域拓扑变化的物理系统的模拟，例如血细胞通过人体毛细血管的运动或血液流过人体心脏的四个主要腔室之间的心脏瓣膜。非拟合有限元方法还简化了计算设计优化技术的构造，该计算设计优化技术基于规定的性能度量来优化工程系统的几何形状和材料布局。然而，计算机实现的非拟合有限元方法仍然是一个具有挑战性和耗时的任务，即使是领域专家。该项目的总体目标是构建一个新的软件库，EXHUME（提取高阶不拟合有限元方法），使经典的有限元代码不拟合有限元分析的使用。EXHUME将使大量的科学家和工程师能够在自己的工作中使用不适合的有限元方法，使他们能够进行生物医学，材料科学和地球物理模拟，这些模拟过于昂贵或过于不稳定，无法使用经典有限元方法实现。EXHUME还将提高学术界、国家实验室、非拟合有限元方法简化了偏微分方程的有限元解（偏微分方程）复杂和/或变形域几何形状的放松的要求，有限元近似空间被定义在一个机构-拟合网格，其元素满足限制性形状和连接约束。早期的非拟合有限元方法表现出低阶收敛速度，但最近的进展导致高阶方法。高阶非拟合有限元法成功的关键因素是在切割单元（即被域边界切割的非拟合单元）上的精确数值积分。EXHUME使用提取的概念来表达数值积分切割细胞的基本操作已经实现了典型的有限元代码，一个集成网格，和提取运营商表示不适合的有限元基函数的规范形状函数。EXHUME在特定有限元代码的范围之外生成积分网格和提取算子，因此它可以与现有代码配对，只需很少的实现工作。该项目的一个关键目标是通过将EXHUME连接到现有的研究代码和流行的FEniCS有限元分析工具链来演示EXHUME。与软件开发并行的努力探索了与1）提取期间进行的近似和2）切割细胞的新数值求积方案相关的精度与效率的权衡。EXHUME技术影响的广度由以下几个因素保证：1）PDE在几乎所有科学和工程学科中的普遍存在，2）库与现有有限元代码的互操作性，3）EXHUME + FEniCS演示示例的通用性，可应用于任意PDE系统。通过简化基于偏微分方程的计算模型的设置，EXHUME + FEniCS能够在课堂上模拟复杂的物理场景，而不会让数值方法的技术细节分散对科学原理的讲授。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Unfitted finite element methods allow for the simulation of physical systems that are difficult if not impossible to simulate using classical finite element methods requiring body-fitted meshes. For instance, unfitted finite element methods can be directly applied to the simulation of physical systems exhibiting a change of domain topology, such as the movement of blood cells through a human capillary or the flow of blood past the heart valves between the four main chambers of the human heart. Unfitted finite element methods also streamline the construction of computational design optimization technologies that optimize the geometry and material layout of an engineered system based on prescribed performance metrics. However, the computer implementation of an unfitted finite element method remains a challenging and time-consuming task even for domain experts. The overarching objective of this project is to construct a novel software library, EXHUME (EXtraction for High-order Unfitted finite element MEthods), to enable the use of classical finite element codes for unfitted finite element analysis. EXHUME will empower a large community of scientists and engineers to employ unfitted finite element methods in their own work, allowing them to carry out biomedical, materials science, and geophysical simulations that have been too expensive or too unstable to realize using classical finite element methods. EXHUME will also improve the fidelity of design optimizations being performed in academia, national laboratories, and industry on a near daily basis.Unfitted finite element methods simplify the finite element solution of PDEs (Partial Differential Equations) on complex and/or deforming domain geometries by relaxing the requirement that the finite element approximation space be defined on a body-fitted mesh whose elements satisfy restrictive shape and connectivity constraints. Early unfitted finite element methods exhibited low-order convergence rates, but recent progress has led to high-order methods. The key ingredient to success of a high-order unfitted finite element method is accurate numerical integration over cut cells (i.e, unfitted elements cut by domain boundaries). EXHUME uses the concept of extraction to express numerical integration over cut cells in terms of basic operations already implemented in typical finite element codes, an integration mesh, and extraction operators expressing unfitted finite element basis functions in terms of canonical shape functions. EXHUME generates integration meshes and extraction operators outside of the confines of a particular finite element code so it may be paired with existing codes with little implementation effort. A key goal of the project is demonstration of EXHUME by connecting it to existing research codes and the popular FEniCS toolchain for finite element analysis. An effort parallel to software development explores accuracy versus efficiency trade-offs associated with 1) approximations made during extraction and 2) novel numerical quadrature schemes for cut cells. The breadth of EXHUME's technical impact is ensured by several factors: 1) the ubiquity of PDEs across nearly all disciplines of science and engineering, 2) the library's interoperability with existing finite element codes, and 3) the generic nature of the EXHUME+FEniCS demonstrative example, which can be applied to arbitrary systems of PDEs. By simplifying the setup of PDE-based computational models, EXHUME+FEniCS enables classroom demonstrations simulating complicated physical scenarios without letting the technical details of numerical methods distract from the scientific principles being taught.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.