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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.

该项目旨在进一步发展不连续彼得罗夫伽辽金（DPG）有限元（FE）方法，并在2009年由Jay Gopalakrishnan和Leszek Demkowicz引入了最佳测试函数。DPG方法代表了具有挑战性的工程和科学过程的有限元模拟的突破。提出的研究方向是：二维和三维可压缩流动问题的求解，利用相场理论模拟韧脆相变。该方法将对需要高精度的困难应用程序的软件构建产生持久的影响。该项目的目标应用领域包括航空航天（跨音速流）和高能量密度电机（绝缘失效），该项目建立在与波音公司和海军的合作基础上。DPG方法最大限度地减少了与指定测试规范对应的对偶规范中的残差。剩余的计算需要在测试空间中反演Riesz算子。通过使用破碎的测试空间和可局部化的测试范数，反演可以使用标准Galerkin和“丰富”空间进行。通过局部控制Riesz算子的逆运算误差，即在元素级别上，该方法自动保证Hilbert环境中任何适定线性问题的离散稳定性，从经典的闭算子理论的意义上来说。该方法导致奇异摄动问题的一致稳定性，作为一种最小化方法，不遭受任何前渐近不稳定性。残差是计算的，而不是估计的，并提供了一个自动适应的基础。L2-范数的最优性并不排除吉布斯现象，该项目旨在将DPG技术扩展到Banach空间。第一个重点领域涉及一个困难的经典主题，即可压缩纳维尔-斯托克斯方程与应用程序的三维机翼模型及其简化模型，全势方程周围的流动。研究的第二条线旨在使用相场理论对聚合物中的韧脆相变进行建模，并应用于理解聚合物绝缘中的损伤和裂纹萌生。在上述两个应用领域中，解决方案都具有强大的边界层或/和内部层。该奖项反映了NSF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。