Exponential convergence of mixed hp-DGFEM for Stokes flow in polygons

Summary.: We analyze mixed hp-discontinuous Galerkin finite element methods (DGFEM) for Stokes flow in polygonal domains. In conjunction with geometrically refined quadrilateral meshes and linearly increasing approximation orders, we prove that the hp-DGFEM leads to exponential rates of convergence for piecewise analytic solutions exhibiting singularities near corner

The MINI mixed finite element for the Stokes problem: An experimental investigation

Super-convergence of order 1.5 in pressure and velocity has been experimentally investigated for the two-dimensional Stokes problem discretised with the MINI mixed finite element. Even though the classic mixed finite element theory for the MINI element guarantees linear convergence for the total error, recent theoretical results indicate that super-convergence of order 1.5 in pressure and of the linear part of the computed velocity to the piecewise linear nodal interpolation of the exact velocity is in fact possible with structured, three-directional triangular meshes. The numerical experiments presented here suggest a more general validity of super-convergence of order 1.5, possibly to automatically generated and unstructured triangulations. In addition, the approximating properties of the complete computed velocity have been compared with the approximating properties of the piecewise-linear part of the computed velocity, finding that the former is generally closer to the exact velocity, whereas the latter conserves mass better

Hybrid finite difference/finite element immersed boundary method

The immersed boundary method is an approach to fluid-structure interaction that uses a Lagrangian description of the structural deformations, stresses, and forces along with an Eulerian description of the momentum, viscosity, and incompressibility of the fluid-structure system. The original immersed boundary methods described immersed elastic structures using systems of flexible fibers, and even now, most immersed boundary methods still require Lagrangian meshes that are finer than the Eulerian grid. This work introduces a coupling scheme for the immersed boundary method to link the Lagrangian and Eulerian variables that facilitates independent spatial discretizations for the structure and background grid. This approach employs a finite element discretization of the structure while retaining a finite difference scheme for the Eulerian variables. We apply this method to benchmark problems involving elastic, rigid, and actively contracting structures, including an idealized model of the left ventricle of the heart. Our tests include cases in which, for a fixed Eulerian grid spacing, coarser Lagrangian structural meshes yield discretization errors that are as much as several orders of magnitude smaller than errors obtained using finer structural meshes. The Lagrangian-Eulerian coupling approach developed in this work enables the effective use of these coarse structural meshes with the immersed boundary method. This work also contrasts two different weak forms of the equations, one of which is demonstrated to be more effective for the coarse structural discretizations facilitated by our coupling approach

Development of an integrated BEM approach for hot fluid structure interaction

A comprehensive boundary element method is presented for transient thermoelastic analysis of hot section Earth-to-Orbit engine components. This time-domain formulation requires discretization of only the surface of the component, and thus provides an attractive alternative to finite element analysis for this class of problems. In addition, steep thermal gradients, which often occur near the surface, can be captured more readily since with a boundary element approach there are no shape functions to constrain the solution in the direction normal to the surface. For example, the circular disc analysis indicates the high level of accuracy that can be obtained. In fact, on the basis of reduced modeling effort and improved accuracy, it appears that the present boundary element method should be the preferred approach for general problems of transient thermoelasticity

An Hp-Adaptive Finite Element Procedure For Fluid-Structure Interaction In Fully Eulerian Framework

This thesis attempts to implement a fully automatic hp-adaptive finite element procedure for fluid-structure interaction (FSI) problems in two dimensions. This work hypotesizes the efficacy of Fully Eulerian framework of FSI in hp-adaptivity on an a posteriori error estimator and adaptation for minimization of error in energy norm. Automatic mesh adaptation over triangular elements is handled by red-green-blue (RGB) refinement method. An effective mesh adaptivity to avoid excessive growth of unknowns is also addressed. Since the hp-method uses high order polynomials as approximation functions, the resulting system matrices are less sparse leading to the notion of FSI computation with parallelism. The parallel hp-adaptive computation is assessed with the conventional uniform and h refinement on a number of benchmark test cases. Subsequently, the efficacy of the fully Eulerian framework is compared to the well known Arbitrary Lagrangian Framework( ALE) for two different material models, namely, the St. Venant Kirchoff and the Neo-Hookean models. It was found that the fully Eulerian framework provides accurate FSI predictions for large deformation without need of frequent remeshing. The hp-adaptive method was also found to be a viable approach in obtaining accurate solutions without much compromise in computer memory and time. Furthermore, the integration of parallelism is successful in reducing the computation time by up to two orders of magnitude relative to the serial solver. For the comparisons between the ALE and the fully Eulerian frameworks, the computed solutions in all test cases are observed to be in agreement with each other