214 research outputs found
Weak Galerkin finite element methods for elasticity and coupled flow problems
Includes bibliographical references.2020 Summer.We present novel stabilizer-free weak Galerkin finite element methods for linear elasticity and coupled Stokes-Darcy flow with a comprehensive treatment of theoretical results and the numerical methods for each. Weak Galerkin finite element methods take a discontinuous approximation space and bind degrees of freedom together through the discrete weak gradient, which involves solving a small symmetric positive-definite linear system on every element of the mesh. We introduce notation and analysis using a general framework that highlights properties that unify many existing weak Galerkin methods. This framework makes analysis for the methods much more straightforward. The method for linear elasticity on quadrilateral and hexahedral meshes uses piecewise constant vectors to approximate the displacement on each cell, and it uses the Raviart-Thomas space for the discrete weak gradient. We use the Schur complement to simplify the solution of the global linear system and increase computational efficiency further. We prove first-order convergence in the L2 norm, verify our analysis with numerical experiments, and compare to another weak Galerkin approach for this problem. The method for coupled Stokes-Darcy flow uses an extensible multinumerics approach on quadrilateral meshes. The Darcy flow discretization uses a weak Galerkin finite element method with piecewise constants approximating pressure and the Arbogast-Correa space for the weak gradient. The Stokes domain discretization uses the classical Bernardi-Raugel pair. We prove first-order convergence in the energy norm and verify our analysis with numerical experiments. All algorithms implemented in this dissertation are publicly available as part of James Liu's DarcyLite and Darcy+ packages and as part of the deal.II library
Two-field finite element solver for linear poroelasticity, A
Includes bibliographical references.2020 Summer.Poroelasticity models the interaction between an elastic porous medium and the fluid flowing in it. It has wide applications in biomechanics, geophysics, and soil mechanics. Due to difficulties of deriving analytical solutions for the poroelasticity equation system, finite element methods are powerful tools for obtaining numerical solutions. In this dissertation, we develop a two-field finite element solver for poroelasticity. The Darcy flow is discretized by a lowest order weak Galerkin (WG) finite element method for fluid pressure. The linear elasticity is discretized by enriched Lagrangian () elements for solid displacement. First order backward Euler time discretization is implemented to solve the coupled time-dependent system on quadrilateral meshes. This poroelasticity solver has some attractive features. There is no stabilization added to the system and it is free of Poisson locking and pressure oscillations. Poroelasticity locking is avoided through an appropriate coupling of finite element spaces for the displacement and pressure. In the equation governing the flow in pores, the dilation is calculated by taking the average over the element so that the dilation and the pressure are both approximated by constants. A rigorous error estimate is presented to show that our method has optimal convergence rates for the displacement and the fluid flow. Numerical experiments are presented to illustrate theoretical results. The implementation of this poroelasticity solver in deal.II couples the Darcy solver and the linear elasticity solver. We present the implementation of the Darcy solver and review the linear elasticity solver. Possible directions for future work are discussed
Non-negative mixed finite element formulations for a tensorial diffusion equation
We consider the tensorial diffusion equation, and address the discrete
maximum-minimum principle of mixed finite element formulations. In particular,
we address non-negative solutions (which is a special case of the
maximum-minimum principle) of mixed finite element formulations. The discrete
maximum-minimum principle is the discrete version of the maximum-minimum
principle.
In this paper we present two non-negative mixed finite element formulations
for tensorial diffusion equations based on constrained optimization techniques
(in particular, quadratic programming). These proposed mixed formulations
produce non-negative numerical solutions on arbitrary meshes for low-order
(i.e., linear, bilinear and trilinear) finite elements. The first formulation
is based on the Raviart-Thomas spaces, and is obtained by adding a non-negative
constraint to the variational statement of the Raviart-Thomas formulation. The
second non-negative formulation based on the variational multiscale
formulation.
For the former formulation we comment on the affect of adding the
non-negative constraint on the local mass balance property of the
Raviart-Thomas formulation. We also study the performance of the active set
strategy for solving the resulting constrained optimization problems. The
overall performance of the proposed formulation is illustrated on three
canonical test problems.Comment: 40 pages using amsart style file, and 15 figure
Enforcing the non-negativity constraint and maximum principles for diffusion with decay on general computational grids
In this paper, we consider anisotropic diffusion with decay, and the
diffusivity coefficient to be a second-order symmetric and positive definite
tensor. It is well-known that this particular equation is a second-order
elliptic equation, and satisfies a maximum principle under certain regularity
assumptions. However, the finite element implementation of the classical
Galerkin formulation for both anisotropic and isotropic diffusion with decay
does not respect the maximum principle.
We first show that the numerical accuracy of the classical Galerkin
formulation deteriorates dramatically with increase in the decay coefficient
for isotropic medium and violates the discrete maximum principle. However, in
the case of isotropic medium, the extent of violation decreases with mesh
refinement. We then show that, in the case of anisotropic medium, the classical
Galerkin formulation for anisotropic diffusion with decay violates the discrete
maximum principle even at lower values of decay coefficient and does not vanish
with mesh refinement. We then present a methodology for enforcing maximum
principles under the classical Galerkin formulation for anisotropic diffusion
with decay on general computational grids using optimization techniques.
Representative numerical results (which take into account anisotropy and
heterogeneity) are presented to illustrate the performance of the proposed
formulation
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