The Discontinuous Petrov Galerkin Method with Optimal Test Functions for Compressible Flows and Ductile-to-Brittle Phase Transitions

具有最佳测试函数的不连续 Petrov Galerkin 方法用于可压缩流动和延性到脆性相变

基本信息

批准号：
1819101
负责人：
Leszek Demkowicz
金额：
$ 25万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-15 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1819101&HistoricalAwards=false
关键词：
Discontinuous Petrov Galerkin Method Optimal

项目摘要

The project aims at a further development of the Discontinuous Petrov Galerkin (DPG) Finite Element (FE) method with optimal test functions introduced by Jay Gopalakrishnan and Leszek Demkowicz in 2009. The DPG methodology represents a breakthrough in the Finite Element simulations of challenging Engineering and Science processes. The proposed research directions are: the solution of 2D and 3D compressible flow problems, the modeling of ductile to brittle phase transitions using Phase Fields theories. The method will have a lasting impact on the construction of software for difficult applications requiring high accuracy. Application areas targeted in this project include aerospace (transonic flows) and high energy density electrical motors (insulation failure), building on collaborations with Boeing and the Navy.The DPG method minimizes residuals in the dual norm corresponding to a specified test norm. Computation of the residual requires inversion of the Riesz operator in the test space. With the use of broken test spaces and localizable test norms, the inversion can be done element-wise using standard Galerkin and "enriched" spaces. With the error of inverting the Riesz operator controlled locally, i.e. on the element level, the method automatically guarantees discrete stability for any well-posed linear problem in a Hilbert setting, in the sense of the classic theory of closed operators. The methodology leads to uniform stability for singular perturbation problems and, being a minimization method, does not suffer from any preasymptotic instabilities. The residual is computed rather than estimated and provides a basis for automatic adaptivity. Optimality in the L2-norms does not preclude the Gibbs phenomenon and this project aims at extending the DPG technology to Banach spaces. The first focus area deals with a difficult classical subject, namely compressible Navier-Stokes equations with applications to flow around a three-dimensional wing model and its simplified model, the full potential equation. The second line of research aims at modeling ductile-to-brittle phase transitions in polymers using Phase Field theories, with applications to understanding damage and crack initiation in polymer insulation. In both application areas above, solutions experience strong boundary or/and internal layers. The proposed work includes both analysis and software development in a joint computational effort with Cracow University of Technology, Boeing, and collaborators at the University of Texas at Austin.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.

该项目旨在进一步开发不连续的Petrov Galerkin（DPG）有限元（FE）方法，该方法具有Jay Gopalakrishnan和Leszek Demkowicz在2009年引入的最佳测试功能。DPG方法论代表了挑战工程和科学过程的有限元素模拟中的突破性。拟议的研究方向是：2D和3D可压缩流问题的解决方案，使用相位场理论将延性到脆性相变的建模。该方法将对需要高准确性的困难应用程序的软件构建产生持久影响。该项目中针对的应用领域包括航空航天（跨气流）和高能电动机（绝缘故障），基于与波音和海军的合作建立。DPG方法最大程度地减少了对应于指定测试规范的双重规范中的残差。残差的计算需要在测试空间中的Riesz运算符反转。通过使用破裂的测试空间和可本质的测试规范，可以使用标准的盖尔金和“富集”空间来完成元素的反转。由于在元素级别上进行了RIESZ操作员的颠倒错误，该方法可以自动保证Hilbert环境中任何良好的线性问题的离散稳定性，从封闭的操作员的经典理论的意义上讲。该方法导致奇异扰动问题的稳定性统一，并且作为一种最小化方法，并未遭受任何抑制性不稳定性的影响。残差是计算而不是估计的，并为自动适应性提供了基础。 L2-Norms中的最佳性并不排除Gibbs现象，该项目旨在将DPG技术扩展到Banach空间。第一个焦点区域涉及一个困难的经典主题，即可压缩的Navier-Stokes方程，其应用程序可以围绕三维机翼模型流动及其简化模型，即完整的电势方程。第二种研究旨在使用相位场理论对聚合物的延性到脆性相变建模，并应用于理解聚合物绝缘层中的损伤和裂纹启动。在上述两个应用领域，解决方案都具有强大的边界或/和内部层。拟议的工作包括与克拉科夫技术大学，波音大学的联合计算工作中的分析和软件开发，以及德克萨斯大学奥斯汀分校的合作者。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的影响审查标准来评估通过评估来支持的。