Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement.

*(English)* Zbl 1151.74419
Summary: 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 advectionâ€“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.

##### MSC:

74S05 | Finite element methods in solid mechanics |

74S30 | Other numerical methods in solid mechanics |

76M10 | Finite element methods (fluid mechanics) |

76M25 | Other numerical methods (fluid mechanics) |

65N50 | Mesh generation and refinement (BVP of PDE) |