Simulation of flow through a heterogeneous porous medium with fine-scale features can be computationally expensive if the flow is fully resolved. Coarsening the problem gives a faster approximation of the flow but loses some detail. We propose an algorithm that obtains the fully resolved approximation but only iterates on a sequence of coarsened problems. The sequence is chosen by optimizing the shapes of the coarse finite element basis functions. As a stand-alone method, the algorithm converges globally and monotonically with a quadratic asymptotic rate. Computational experience indicates the number of iterations needed is independent of the resolution and heterogeneity of the medium. However, an externally provided error estimate is required; the algorithm could be combined as an accelerator with another iterative algorithm. A single "inner" iteration of the other algorithm would yield an error estimate; following it with an "outer" iteration of our algorithm would give a viable method. / text

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Solution algorithms are developed for the nonlinear finite element equations, resulting from the discretization of governing incremental equations of equilibrium. The equations of equilibrium in the total Lagrangian formation are derived, based on continuum mechanics principles and by invoking the principle of virtual work. A general purpose computer program, modular in structure, and interactive in nature, is developed, based on the theoretical formulation presented. A brief review of some of the existing solution algorithms for nonlinear structural analysis using the finite element formulation is presented. Arc-length methods, which facilitate the tracing of the load-configuration path beyond limit points, are systematically examined. Specific emphasis is focussed on obtaining the complete structural response in an efficient and reliable way. The arc-length methods treat the load parameter as a variable in addition to the unknown displacements. A constraint equation forms the supplemental equation for the determination of the additional variable. Attention is directed to the constraint equation, comprising the displacements and loads. Realising that the loads and displacements have different magnitudes, several authors had addressed the issue of including a scaling parameter, however, inadequately. In this research, several well formulated automatic means of computing the scaling parameter are suggested. A constant scaling parameter, which remains unchanged throughout all the load steps, and a variable scaling parameter, which is recalculated at each load step are introduced. The latter parameter is related to the current stiffness parameter. In the present investigation, the spherical constraint, the normal plane constraint and the updated normal plane constraint equations are examined with the proposed scaling parameters. An automatic load step calculation, based on the arc-length is suggested. The load step is rendered adaptive by including the load term with the scaling parameter in the constraint equation. Examples from large displacement problems, that exhibit severe geometric nonlinearities, are included to evaluate the proposed algorithms and substantiate their efficacies in tracing the complete load-configuration path. The relative performances of the proposed algorithms reveal their cost-effectiveness.

published_or_final_version / Civil and Structural Engineering / Doctoral / Doctor of Philosophy

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