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

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.