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

不适合的有限元方法允许模拟物理系统，这些物理系统即使不是不可能用需要贴体网格的经典有限元方法来模拟也是困难的。例如，不适合的有限元方法可以直接应用于显示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在人体心脏四个主要腔室之间流过心脏瓣膜。不适合的有限元方法还简化了计算设计优化技术的构建，该技术基于规定的性能指标优化工程系统的几何和材料布局。然而，即使对于领域专家来说，计算机实现不适合的有限元方法仍然是一项具有挑战性和耗时的任务。该项目的总体目标是构建一个新的软件库Ehume(高阶非拟合有限元方法提取)，以便能够使用经典有限元程序进行非拟合有限元分析。挖掘遗骸将使一大批科学家和工程师能够在自己的工作中使用不合适的有限元方法，使他们能够进行生物医学、材料科学和地球物理模拟，而使用经典有限元方法实现这些模拟的成本太高或太不稳定。挖掘还将提高学术界、国家实验室和工业界几乎每天进行的设计优化的保真度。非贴合有限元方法通过放宽有限元近似空间定义在贴体网格上的要求，简化了复杂和/或变形区域几何上的偏微分方程(PDE)的有限元求解。早期的非拟合有限元方法表现出低阶收敛速度，但最近的进展导致了高阶方法的出现。高阶非拟合有限元方法成功的关键是对截断单元(即区域边界剖分的非拟合单元)进行精确的数值积分。Exhume使用抽取的概念，以典型有限元程序中已实现的基本运算、积分网格和以正则形函数表示不适合的有限元基函数的抽取算子来表示截断单元上的数值积分。Exhume在特定有限元代码的范围之外生成积分网格和提取操作符，因此它可以与现有代码配对，而几乎不需要执行工作。该项目的一个关键目标是通过将其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挖掘。与软件开发平行的一项工作是探索与1)提取过程中的近似和2)切割单元的新的数值求积方案相关的精度与效率的权衡。Ehume的技术影响的广度是由几个因素保证的：1)几乎所有的科学和工程学科中普遍存在的偏微分方程组，2)库与现有有限元程序的互操作性，以及3)Ehume+FENICS示范例子的通用性质，它可以应用于任意的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对严格的科学原理的注意力。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.