ISOGEOMETRIC ANALYSIS: APPROXIMATION, STABILITY AND ERROR ESTIMATES FOR h-REFINED MESHES

Abstract

We begin the mathematical study of Isogeometric Analysis based on NURBS (nonuniform rational B-splines). Isogeometric Analysis is a generalization of classical Finite Element Analysis (FEA) which possesses improved properties. For example, NURBS are capable of more precise geometric representation of complex objects and, in particular, can exactly represent many commonly engineered shapes, such as cylinders, spheres and tori. Isogeometric Analysis also simplifies mesh refinement because the geometry is fixed at the coarsest level of refinement and is unchanged throughout the refinement process. This eliminates geometrical errors and the necessity of linking the refinement procedure to a CAD representation of the geometry, as in classical FEA. In this work we study approximation and stability properties in the context of h-refinement. We develop approximation estimates based on a new Bramble-Hilbert lemma in so-called "bent" Sobolev spaces appropriate for NURBS approximations and establish inverse estimates similar to those for finite elements. We apply the theoretical results to several cases of interest including elasticity, isotropic incompressible elasticity and Stokes flow, and advection-diffusion, and perform numerical tests which corroborate the mathematical results. We also perform numerical calculations that involve hypotheses outside our theory and these suggest that there are many other interesting mathematical properties of Isogeometric Analysis yet to be proved.

Overlapping Schwarz Methods for Isogeometric Analysis

“…As observed, for instance, in [4], in order to obtain spaces with homogeneous Dirichlet boundary conditions it is sufficient to eliminate the first and last functions in each coordinate. We therefore introduce the spline space (for instance, in two dimensions) living in parameter space,…”

“…The functions w, F are continuous, piecewise regular, and fixed once and for all at the coarsest level of discretization. We make, moreover, the standard assumption (see [4]) that the inverse mapping F −1 is well defined and with bounded derivatives. As a consequence, we have that w, F, F −1 are (componentwise) in the W 1,∞ Sobolev space.…”

“…As a consequence, we have that w, F, F −1 are (componentwise) in the W 1,∞ Sobolev space. Therefore, first a standard change of variables (see, for instance, [4]) from Ω to Ω, then a Poincaré inequality on Ω give…”

“…For instance, spaces of global C k regularity are easily built, thus allowing for fewer degrees of freedom, better performance in case of vibrations, easier approximation of higher order problems, and other advantages. IGA methodologies have been summarized in the recent book [18] and studied in, e.g., [2,4,9,10,19,23,28,29,11,5,8]. IGA methods are having a growing impact on fields as diverse as fluid dynamics [6,7,40,15,26], structural mechanics [3,1,12,20,30,39], and electromagnetics [17,16].…”

Computational cost estimates for parallel shared memory isogeometric multi-frontal solvers

“…Introduction Classical higher order finite element methods (FEM) [17,18] maintain only C 0 -continuity at the element interfaces, while isogeometric analysis (IGA) utilizes B-splines as basis functions, and thus, it delivers C k global continuity [14]. The higher continuity obtained across elements allows IGA to attain optimal convergence rates for any polynomial order, while using fewer degrees of freedom [3,1]. Nevertheless, this reduced count in the number of degrees of freedom may not immediately correlate with a computational cost reduction, since solution time per degree of freedom augments as the continuity is increased [10,13].…”