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

非拟合有限元方法允许模拟物理系统，这些物理系统很难（如果不是不可能的话）用经典的有限元方法模拟，需要体拟合网格。例如，非拟合有限元方法可以直接应用于模拟显示域拓扑变化的物理系统，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建，该技术可以根据规定的性能指标优化工程系统的几何形状和材料布局。然而，计算机实现非拟合有限元方法仍然是一项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（高阶未拟合有限元方法提取），以便使用经典有限元代码进行未拟合有限元分析。EXHUME将使大量科学家和工程师能够在他们自己的工作中使用非拟合有限元方法，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太不稳定，无法使用经典的有限元方法来实现。EXHUME还将提高学术界、国家实验室和工业中几乎每天进行的设计优化的保真度。非拟合有限元方法简化了复杂和/或变形域几何上偏微分方程的有限元解，它放宽了在单元满足形状约束和连通性约束的体拟合网格上定义有限元近似空间的要求。早期的非拟合有限元方法表现出低阶收敛率，但最近的进展导致了高阶方法。高阶非拟合有限元方法成功的关键因素是对切割单元（即由区域边界切割的非拟合单元）进行精确的数值积分。EXHUME使用提取的概念，根据典型有限元代码中已经实现的基本操作来表示切割单元的数值积分，一个积分网格，以及用规范形状函数表示非拟合有限元基函数的提取算子。EXHUME在特定的有限元代码范围之外生成集成网格和提取运算符，因此它可以与现有的代码配对，几乎没有实现工作。该项目的一个关键目标是通过将EXHUME与现有的研究代码和流行的FEniCS工具链连接起来进行有限元分析，从而演示EXHUME。与软件开发并行的一项工作探索了准确性与效率之间的权衡，这与1)提取过程中的近似和2)切割细胞的新型数值正交方案有关。EXHUME技术影响的广度是由几个因素保证的：1)pde在几乎所有科学和工程学科中无处不在，2)图书馆与现有有限元代码的互操作性，以及3)EXHUME+FEniCS演示示例的通用性质，可以应用于任意pde系统。通过简化基于pde的计算模型的设置，EXHUME+FEniCS使课堂演示能够模拟复杂的物理场景，而不会让数值方法的技术细节分散正在教授的科学原理。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.