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

合作研究：Elements：EXHUME：高阶未拟合有限元方法的提取

• 批准号: 2104106
• 负责人: John Evans
• 金额: $ 30.03万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-06-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104106&HistoricalAwards=false
• 关键词: Collaborative; Research; Elements; EXHUME; Extraction

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.

未拟合的有限元方法允许对物理系统进行模拟，而使用需要贴体网格的经典有限元方法来模拟这些物理系统即使不是不可能，也是很困难的。例如，不适合的有限元方法可以直接应用于模拟表现出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心脏四个主要腔室之间的心脏瓣膜。人心。不拟合的有限元方法还简化了计算设计优化技术的构建，这些技术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，即使对于领域专家来说，不拟合的有限元方法的计算机实现仍然是一项具有挑战性且耗时的任务。该项目的总体目标是构建一个新颖的软件库 EXHUME（高阶不拟合有限元方法提取），以便能够使用经典有限元代码进行不拟合有限元分析。 EXHUME 将使大量科学家和工程师能够在自己的工作中采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，而这些模拟由于成本太高或太不稳定而无法使用经典有限元方法实现。 EXHUME 还将提高学术界、国家实验室和工业界几乎每天进行的设计优化的保真度。不拟合的有限元方法通过以下方式简化了复杂和/或变形域几何形状上的 PDE（偏微分方程）的有限元求解：放宽了在贴体网格上定义有限元近似空间的要求，该网格的元素满足限制性形状和连通性约束。早期未拟合的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。高阶未拟合有限元方法成功的关键因素是对切割单元（即由域边界切割的未拟合元素）进行精确数值积分。 EXHUME 使用提取的概念，根据已在典型有限元代码中实现的基本运算、积分网格和根据规范形状函数表示未拟合的有限元基函数的提取算子来表达切割单元上的数值积分。 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 工具链进行有限元分析来演示 EXHUME。与软件开发并行的一项工作探索了与 1) 提取过程中进行的近似和 2) 用于切割单元的新颖数值求积方案相关的精度与效率权衡。 EXHUME 技术影响的广度由以下几个因素保证：1) 偏微分方程在几乎所有科学和工程学科中普遍存在，2) 库与现有有限元代码的互操作性，以及 3) EXHUME+FEniCS 演示的通用性质例如，它可以应用于任意偏微分方程组。通过简化基于偏微分方程的计算模型的设置，EXHUME+FEniCS 可以在课堂上进行演示，模拟复杂的物理场景，同时又不会让数值方法的技术细节分散对所教授的科学原理的注意力。该奖项反映了 NSF 的法定使命，并被认为值得支持通过使用基金会的智力优点和更广泛的影响审查标准进行评估。

项目成果

Interpolation-based immersed finite element and isogeometric analysis

基于插值的浸入式有限元和等几何分析

• DOI: 10.1016/j.cma.2023.115890
• 发表时间: 2023-02
• 期刊: Computer Methods in Applied Mechanics and Engineering
• 影响因子: 7.2
• 作者: Fromm, Jennifer E.;Wunsch, Nils;Xiang, Ru;Zhao, Han;Maute, Kurt;Evans, John A.;Kamensky, David
• 通讯作者: Kamensky, David

Extended isogeometric analysis of multi-material and multi-physics problems using hierarchical B-splines

使用分层 B 样条对多材料和多物理问题进行扩展等几何分析

• DOI: 10.1007/s00466-023-02306-x
• 发表时间: 2023-06
• 期刊: Computational Mechanics
• 影响因子: 4.1
• 作者: Schmidt, Mathias;Noël, Lise;Doble, Keenan;Evans, John A.;Maute, Kurt
• 通讯作者: Maute, Kurt