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

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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.