Isogeometric analysis and numerical modeling of the fine scales within the variational multiscale method

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Title: Isogeometric analysis and numerical modeling of the fine scales within the variational multiscale method
Author: Cottrell, John Austin, 1980-
Abstract: This work discusses isogeometric analysis as a promising alternative to standard finite element analysis . Isogeometric analysis has emerged from the idea that the act of modeling a geometry exactly at the coarsest levels of discretization greatly simplifies the refinement process by obviating the need for a link to an external representation of that geometry . The NURBS based implementation of the method is described in detail with particular emphasis given to the numerous refinement possibilities , including the use of functions of higher -continuity and a new technique for local refinement . Examples are shown that highlight each of the major features of the technology : geometric flexibility , functions of high continuity , and local refinement . New numerical approaches are introduced for modeling the fine scales within the variational multiscale method . First , a general framework is presented for seeking solutions to differential equations in a way that approximates optimality in certain norms . More importantly , it makes possible for the first time the approximation of the fine -scale Green's functions arising in the formulation , leading to a better understanding of machinery of the variational multiscale method and opening new avenues for research in the field . Second , a simplified version of the approach , dubbed the "parameter -free variational multiscale method ," is proposed that constitutes an efficient stabilized method , grounded in the variational multiscale framework , that is free of the ad hoc stabilization parameter selection that has plagued classical stabilized methods . Examples demonstrate the efficacy of the method for both linear and nonlinear equations .
URI: http : / /hdl .handle .net /2152 /3190
Date: 2008-08-28

Citation

Isogeometric analysis and numerical modeling of the fine scales within the variational multiscale method. Doctoral dissertation, The University of Texas at Austin. Available electronically from http : / /hdl .handle .net /2152 /3190 .

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