期刊
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
卷 16, 期 7, 页码 1031-1090出版社
WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S0218202506001455
关键词
B-splines; NURBS; finite elements; approximation; error estimates; inverse estimates; stability; elliptic boundary value problems; elasticity; incompressibility; Stokes flow; advection-diffusion; h-refinement
We begin the mathematical study of Isogeometric Analysis based on NURBS (non-uniform 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.
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