CIMMS Seminar: Isogeometric Analysis:  CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement

Dr. Thomas J. R. Hughes
Professor of Aerospace Engineering and Engineering Mechanics
Computational and Applied Mathematics Chair III
Institute for Computational Engineering and Sciences (ICES)
The University of Texas at Austin

The concept of Isogeometric Analysis is proposed.  Basis functions generated from NURBS (Non-Uniform Rational B-Splines) are employed to  construct an exact geometric model.  For purposes of analysis, the basis is  refined and/or its order elevated without changing the geometry or its parameterization.  Analogues of finite element h- and p- refinement schemes  are presented and a new, more efficient, higher-order concept,  k-refinement, is introduced.  Refinements are easily implemented and exact geometry is maintained at all levels without the necessity of subsequent communication with a CAD (Computer Aided Design) description.  In the  context of structural mechanics, it is established that the basis functions  are complete with respect to affine transformations, meaning that all rigid  body motions and constant strain states are exactly represented.  Standard   patch tests are likewise satisfied.  Numerical examples exhibit optimal rates of convergence for linear elasticity problems and convergence to thin  elastic shell solutions.  A k-refinement strategy is shown to converge toward monotone solutions for advection-diffusion processes with sharp internal and boundary layers, a very surprising result.  It is argued that Isogeometric Analysis is a viable alternative to standard, polynomial-based, finite element analysis and possesses several advantages.