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

该项目旨在进一步发展Jay Gopalakrishnan和Leszek Demkowicz于2009年提出的具有最优测试函数的不连续Petrov Galerkin （DPG）有限元（FE）方法。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，利用相场理论对韧脆性相变进行建模。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（跨音速流动）和高能量密度电动机（绝缘失效），并与波音公司和海军合作。DPG方法最大限度地减少了对偶模中对应于特定测试范数的残差。残差的计算需要对测试空间中的Riesz算子进行反演。通过使用破碎的测试空间和可定位的测试规范，可以使用标准Galerkin和“丰富”空间进行元素反演。由于Riesz算子的反演误差被局部控制，即在元素水平上，该方法在经典闭算子理论的意义上自动保证了Hilbert集合中任何适定线性问题的离散稳定性。该方法使奇异摄动问题具有一致稳定性，并且作为一种最小化方法，不存在任何前渐近不稳定性。残差是计算而不是估计的，为自动自适应提供了基础。l2 -范数中的最优性并不排除Gibbs现象，该项目旨在将DPG技术扩展到Banach空间。第一个重点领域涉及一个困难的经典主题，即可压缩的Navier-Stokes方程及其在三维机翼模型及其简化模型全势方程中的应用。研究的第二个方向是利用相场理论对聚合物从延性到脆性的相变进行建模，并应用于了解聚合物绝缘中的损伤和裂纹起裂。在上述两个应用领域中，解决方案都经历了强大的边界或/和内层。提议的工作包括与克拉科夫科技大学、波音公司和德克萨斯大学奥斯汀分校的合作者联合计算工作中的分析和软件开发。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。

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.