Quanling Deng: Tuesday 16 May at 3 pm, A415 Inria Paris.
The isogeometric analysis (IgA) is a powerful numerical tool that unifies the finite element analysis (FEA) and computer-aided design (CAD). Under the framework of FEA, IgA uses as basis functions those employed in CAD, which are capable of exactly represent various complex geometries. These basis functions are called the B-Splines or more generally the Non-Uniform Rational B-Splines (NURBS) and they lead to an approximation which may have global continuity of order up to $p-1$, where $p$ is the order of the underlying polynomial, which in return delivers more robustness and higher accuracy than that of finite elements.
We apply IgA to wave propagation and structural vibration problems to study their dispersion and spectrum properties. The dispersion and spectrum analysis are unified in the form of a Taylor expansion for eigenvalue errors. By blending optimally two standard Gaussian quadrature schemes for the integrals corresponding to the stiffness and mass, the dispersion error of IgA is minimized. The blending schemes yield two extra orders of convergence (superconvergence) in the eigenvalue errors, while the eigenfunction errors are of optimal convergence order. To analyze the eigenvalue and eigenfunction errors, the Pythagorean eigenvalue theorem (Strang and Fix, 1973) is generalized to establish an equality among the eigenvalue, eigenfunction (in L2 and energy norms), and quadrature errors.