Browsing by Subject "Mixed finite element method"
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Item Mixed hp-adaptive finite element methods for elasticity and coupled problems(2010-08) Qiu, Weifeng, 1978-; Demkowicz, Leszek; Prudhomme, Serge M.In my dissertation, I developed mixed hp-finite element methods for linear elasticity with weakly imposed symmetry, which is based on Arnold-Falk-Winther's stable mixed finite elements. I have proved the h-stability of my method for meshes with arbitrary variable orders. In order to show the h-stability, I need an upper limit of the highest order of meshes, which can be an arbitrary nonnegative integer.Item Modelling and simulations of hydrogels with coupled solvent diffusion and large deformation(2014-12) Bouklas, Nikolaos; Huang, Rui, doctor of civil and environmental engineering; Landis, Chad M.Swelling of a polymer gel is a kinetic process coupling mass transport and mechanical deformation. A comparison between a nonlinear theory for polymer gels and the classical theory of linear poroelasticity is presented. It is shown that the two theories are consistent within the linear regime under the condition of a small perturbation from an isotropically swollen state of the gel. The relationships between the material properties in the linear theory and those in the nonlinear theory are established by a linearization procedure. Both linear and nonlinear solutions are presented for swelling kinetics of substrate-constrained and freestanding hydrogel layers. A new procedure is suggested to fit the experimental data with the nonlinear theory. A nonlinear, transient finite element formulation is presented for initial boundary value problems associated with swelling and deformation of hydrogels, based on nonlinear continuum theories for hydrogels with compressible and incompressible constituents. The incompressible instantaneous response of the aggregate imposes a constraint to the finite element discretization in order to satisfy the LBB condition for numerical stability of the mixed method. Three problems of practical interests are considered: constrained swelling, flat-punch indentation, and fracture of hydrogels. Constrained swelling may lead to instantaneous surface instability. Indentation relaxation of hydrogels is simulated beyond the linear regime under plane strain conditions, and is compared with two elastic limits for the instantaneous and equilibrium states. The effects of Poisson’s ratio and loading rate are discussed. On the study of hydrogel fracture, a method for calculating the transient energy release rate for crack growth in hydrogels, based on a modified path-independent J-integral, is presented. The transient energy release rate takes into account the energy dissipation due to diffusion. Numerical simulations are performed for a stationary center crack loaded in mode I, with both immersed and non-immersed chemical boundary conditions. Both sharp crack and blunted notch crack models are analyzed over a wide range of applied remote tensile strains. Comparisons to linear elastic fracture mechanics are presented. A critical condition is proposed for crack growth in hydrogels based on the transient energy release rate. The applicability of this growth condition for simulating concomitant crack propagation and solvent diffusion in hydrogels is discussed.Item Preconditioning for the mixed formulation of linear plane elasticity(Texas A&M University, 2005-11-01) Wang, YanqiuIn this dissertation, we study the mixed finite element method for the linear plane elasticity problem and iterative solvers for the resulting discrete system. We use the Arnold-Winther Element in the mixed finite element discretization. An overlapping Schwarz preconditioner and a multigrid preconditioner for the discrete system are developed and analyzed. We start by introducing the mixed formulation (stress-displacement formulation) for the linear plane elasticity problem and its discretization. A detailed analysis of the Arnold-Winther Element is given. The finite element discretization of the mixed formulation leads to a symmetric indefinite linear system. Next, we study efficient iterative solvers for the symmetric indefinite linear system which arises from the mixed finite element discretization of the linear plane elasticity problem. The preconditioned Minimum Residual Method is considered. It is shown that the problem of constructing a preconditioner for the indefinite linear system can be reduced to the problem of constructing a preconditioner for the H(div) problem in the Arnold-Winther finite element space. Our main work involves developing an overlapping Schwarz preconditioner and a multigrid preconditioner for the H(div) problem. We give condition number estimates for the preconditioned systems together with supporting numerical results.