Variational formulations and numerical analysis of some problems in small strain elastoplasticity

Bibliography: pages 316-322.In this thesis we study the mathematical structure and numerical approximation of two boundary-value problems in small strain elastoplasticity. The first problem, which we call the incremental holonomic problem, is based on a consistent incremental holonomic constitutive law, which in turn derives from the notion of extremal paths in stress and strain space as originally proposed by PONTER & MARTIN (1972); the second problem which we study is the classical rate problem. We show that both problems can be formulated as variational inequalities, with internal variables being included explicitly in the formulation. Corresponding minimisation problems follow naturally from standard results in convex analysis

On the accuracy of asymptotic approximations for longitudinal deformation of a thin plate

Este trabajo presenta un análisis crítico de los argumentos que se aducen contra el sistema público de pensiones. Muestra que la defensa del sistema privado a partir de las virtudes y fortalezas de su solvencia y de su capacidad para hacer frente de manera automática a los procesos de envejecimiento de la sociedad son simples falacias. Demuestra que el sistema pay-as-you-go y el de capitalización individual son equivalentes en el sentido de que enfrentan los mismos problemas en el largo plazo. Por último, con ayuda de proyecciones razonables de algunos indicadores convencionales se muestra que los gritos de alarma de los economistas de Fedesarrollo y del BBVA no tienen sólidos fundamentos

Stabilized continuous and discontinuous Galerkin techniques for Darcy flow

We design stabilized methods based on the variational multiscale decomposition of Darcy's problem. A model for the subscales is designed by using a heuristic Fourier analysis. This model involves a characteristic length scale, that can go from the element size to the diameter of the domain, leading to stabilized methods with different stability and convergence properties. These stabilized methods mimic different possible functional settings of the continuous problem. The optimal method depends on the velocity and pressure approximation order. They also involve a subgrid projector that can be either the identity (when applied to finite element residuals) or can have an image orthogonal to the finite element space. In particular, we have designed a new stabilized method that allows the use of piecewise constant pressures. We consider a general setting in which velocity and pressure can be approximated by either continuous or discontinuous approximations. All these methods have been analyzed, proving stability and convergence results. In some cases, duality arguments have been used to obtain error bounds in the L2-norm

Complex flow and transport phenomena in porous media

This thesis analyzes partial differential equations related to the coupled surface and subsurface flows and develops efficient high order discontinuous Galerkin (DG) methods to solve them numerically. Specifically, the coupling of the Navier-Stokes and the Darcy's equations, which is encountered in the environmental problem of groundwater contamination through lakes and rivers, is considered. Predicting accurately the transport of contaminants by this coupled flow is of great importance for the remediation strategies. The first part of this thesis analyzes a weak formulation of the time-dependent Navier-Stokes equation coupled with the Darcy's equation through the Beavers-Joseph-Saffman condition. The analysis changes depending on whether the inertial forces are included in the interface conditions or not. The inclusion of the inertial forces (Model I) remedies the difficulty in the analysis caused by the nonlinear convection term; however, it does not reflect the physical interactions on the interface correctly. Hence, I also analyze the weak problem by omitting these forces (Model II) which complicates the analysis and necessitates an extra small data condition. For Model I, a fully discrete scheme based on the DG method and the Crank-Nicolson method is introduced. The convergence of the scheme is proven with optimal error estimates. The second part couples the surface flow and a convection-diffusion type transport with miscible displacement in the subsurface. Initially, I consider the coupled stationary Stokes and Darcy's equations for the flow and establish the existence of a weak solution. Next, imposing additional assumptions on the data, I extend the result to the nonlinear case where the surface flow is given by the Navier-Stokes equation. The analysis also applies to the particular case where the flow is loosely coupled to the transport, that is, the velocity field obtained from the flow is an input for the transport equation. The flow is discretized by combinations of the continuous finite element method and the DG method whereas the discretization of the transport is done by a combined DG and backward Euler methods. The scheme yields optimal error estimates and its robustness for fractured porous media is shown by a numerical example

Finite element schemes for elliptic boundary value problems with rough coefficients

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.We consider the task of computing reliable numerical approximations of the solutions of elliptic equations and systems where the coefficients vary discontinuously, rapidly, and by large orders of magnitude. Such problems, which occur in diffusion and in linear elastic deformation of composite materials, have solutions with low regularity with the result that reliable numerical approximations can be found only in approximating spaces, invariably with high dimension, that can accurately represent the large and rapid changes occurring in the solution. The use of the Galerkin approach with such high dimensional approximating spaces often leads to very large scale discrete problems which at best can only be solved using efficient solvers. However, even then, their scale is sometimes so large that the Galerkin approach becomes impractical and alternative methods of approximation must be sought. In this thesis we adopt two approaches. We propose a new asymptotic method of approximation for problems of diffusion in materials with periodic structure. This approach uses Fourier series expansions and enables one to perform all computations on a periodic cell; this overcomes the difficulty caused by the rapid variation of the coefficients. In the one dimensional case we have constructed problems with discontinuous coefficients and computed the analytical expressions for their solutions and the proposed asymptotic approximations. The rates at which the given asymptotic approximations converge, as the period of the material decreases, are obtained through extensive computational tests which show that these rates are fundamentally dependent on the level of regularity of the right hand sides of the equations. In the two dimensional case we show how one can use the Galerkin method to approximate the solutions of the problems associated with the periodic cell. We construct problems with discontinuous coefficients and perform extensive computational tests which show that the asymptotic properties of the approximations are identical to those observed in the one dimensional case. However, the computational results show that the application of the Galerkin method of approximation introduces a discretization error which can obscure the precise asymptotic rate of convergence for low regularity right hand sides. For problems of two dimensional linear elasticity we are forced to consider an alternative approach. We use domain decomposition techniques that interface the subdomains with conjugate gradient methods and obtain algorithms which can be efficiently implemented on computers with parallel architectures. We construct the balancing preconditioner, M,, and show that it has the optimal conditioning property k(Mh(^-1)Sh) = 0 is a constant which is independent of the magnitude of the material discontinuities, H is the maximum subdomain diameter, and h is the maximum finite element diameter. These properties of the preconditioning operator Mh allow one to use the computational power of a parallel computer to overcome the difficulties caused by the changing form of the solution of the problem. We have implemented this approach for a variety of problems of planar linear elasticity and, using different domain decompositions, approximating spaces, and materials, find that the algorithm is robust and scales with the dimension of the approximating space and the number of subdomains according to the condition number bound above and is unaffected by material discontinuities. In this we have proposed and implemented new inner product expressions which we use to modify the bilinear forms associated with problems over subdomains that have pure traction boundary conditions.This work is funded by the Engineering and Physical Sciences Research Council

A finite element analysis of crack propagation problems with applications to seismology

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Quasi-optimal nonconforming methods for symmetric elliptic problems. III-discontinuous Galerkin and other interior penalty methods

We devise new variants of the following nonconforming finite element methods: discontinuous Galerkin methods of fixed arbitrary order for the Poisson problem, the Crouzeix-Raviart interior penalty method for linear elasticity, and the quadratic C0 interior penalty method for the biharmonic problem. Each variant differs from the original method only in the discretization of the right-hand side. Before applying the load functional, a linear operator transforms nonconforming discrete test functions into conforming functions such that stability and consistency are improved. The new variants are thus quasi-optimal with respect to an extension of the energy norm. Furthermore, their quasi-optimality constants are uniformly bounded for shape regular meshes and tend to 1 as the penalty parameter increases

Stochastic homogenization of rate-independent systems

We study the stochastic and periodic homogenization 1-homogeneous convex functionals. We proof some convergence results with respect to stochastic twoscale convergence, which are related to classical Gamma-convergence results. The main result is a general lim inf-estimate for a sequence of 1-homogeneous functionals and a two-scale stability result for sequences of convex sets. We apply our results to the homogenization of rate-independent systems with 1-homogeneous dissipation potentials and quadratic energies. In these applications, both the energy and the dissipation potential have an underlying stochastic microscopic structure. We study the particular homogenization problems of Prandlt-Reuss plasticity, Coulomb friction on a macroscopic surface and Coulomb friction on microscopic fissures

Stochastic homogenization of rate-independent systems

We study the stochastic and periodic homogenization 1-homogeneous convex functionals. We proof some convergence results with respect to stochastic two-scale convergence, which are related to classical Gamma-convergence results. The main result is a general liminf-estimate for a sequence of 1-homogeneous functionals and a two-scale stability result for sequences of convex sets. We apply our results to the homogenization of rateindependent systems with 1-homogeneous dissipation potentials and quadratic energies. In these applications, both the energy and the dissipation potential have an underlying stochastic microscopic structure. We study the particular homogenization problems of Prandlt-Reuss plasticity, Coulomb friction on a macroscopic surface and Coulomb friction on microscopic fissure

Mathematical and computational aspects of the enhanced strain finite element method

Bibliography: pages 102-107.This thesis deals with further investigations of the enhanced strain finite element method, with particular attention given to the analysis of the method for isoparametric elements. It is shown that the results established earlier by B D Reddy and J C Simo for affine-equivalent meshes carry over to the case of isoparameric elements. That is, the method is stable and convergent provided that a set of three conditions are met, and convergence is at the same rate as in the standard method. The three conditions differ in some respects, though, from their counterparts for the affine case. A procedure for recovering the stress is shown to lead to an approximate stress which converges at the optimal rate to the actual stress. The concept of the equivalent parallelogram associated with a quadrilateral is introduced. The quadrilateral may be regarded as a perturbation of this parallelogram, which is most conveniently described by making use of properties of the isoparametric map which defines the quadrilateral. The equivalent parallelogram generates a natural means of defining a regular family of quadrilaterals; this definition is used together with other properties to obtain in a relatively simple manner estimates, in appropriate seminorms or norms, of the isoparametric map and it's Jacobian, for use in the determination of finite element interpolation error estimates, with regard to computations, a new basis for enhanced strains is introduced, and various examples have been tested. The results obtained are compared with those obtained using other bases, and with those found from an assumed stress approach. Favourable comparisons are obtained in most cases, with the present basis exhibiting an improvement over existing bases. Convergence of the finite element results are verified; it is observed numerically that the improvement of results due to enhancement is as a result of a smaller constant appearing in the error estimates