Discontinuous Petrov Galerkin (DPG) Method with Optimal Test Functions. Space-Time Formulations and Elements of Irregular Shapes

该项目侧重于开发一种用计算机模拟复杂物理和工程过程的新技术-不连续彼得罗夫伽辽金（DPG）方法。DPG方法有可能改变美国国家实验室和工业界现有的有限元软件。在不破坏现有代码逻辑的情况下，该方法要求利用新兴的多核体系结构对局部操作（在元素级别）进行彻底检查。该方法将对需要高精度仿真的应用中出现的难题的软件开发产生持久的影响。这项工作的应用领域包括航空航天工程（跨声速和超声速流动）、地质力学和生物工程（植入物、软组织成像）。在应用方面，该项目建立在与桑迪亚和阿贡国家实验室、本古里安大学和波音公司的合作基础上。项目重点研究两个方向：(a) DPG时空元素，(b)不规则形状元素的使用。DPG方法最大限度地减少与预先指定的测试范数对应的对偶范数的残差。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，即在元水平上，该方法自动保证了经典闭算子理论意义上的任意适定问题的离散稳定性。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。第一个重点领域处理对流主导问题的时空要素，以及在二维空间中可压缩和不可压缩Navier-Stokes （NS）方程的应用。第二步研究了该方法对单元形状的敏感性，以期使用任意多面体形状的单元。项目的这一部分将在三维空间的线性弹性和超弹性的背景下进行，并应用于地质力学和生物工程。建议的工作包括分析和半专业软件开发。该项目建立在现有代码和与阿贡国家实验室和桑迪亚国家实验室联合计算的基础上。

The project focuses on the development of a new technique for simulating complicated physical and engineering processes with computers - the Discontinuous Petrov Galerkin (DPG) method. The DPG methodology has the potential to transform existing Finite Element software at US National Labs and industry. Without undermining the logic of existing codes, the method asks for a complete overhaul of local operations (on the element level) utilizing emerging multicore architectures. The method will have a lasting impact on software development for difficult problems arising in applications which require high accuracy simulations. Application areas targeted in this work include aerospace engineering (transonic and supersonic flows), geomechanics and bioengineering (implants, imaging of soft tissues). On the application side, the project builds on collaboration with Sandia and Argonne National Laboratories, Ben Gurion University and Boeing.The project focuses on two research directions: (a) DPG space time elements, and (b) use of elements with irregular shapes. The DPG method minimizes residuals in the dual norm corresponding to a prespecified 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 problem in the sense of 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. The first focus area deals with space-time elements for convection-dominated problems with applications to compressible and incompressible Navier-Stokes (NS) equations in two space dimensions. The second line of research investigates the sensitivity of the method to element shapes with a view to using elements of arbitrary polyhedral shapes. This part of the project will be done in the context of linear elasticity and hyperelasticity in three space dimensions with applications to geomechanics and bioengineering. The proposed work includes both analysis and semi-professional software development. The project builds on existing codes and joint computational effort with Argonne and Sandia National Laboratories.