Selective and reduced numerical integrations for NURBS-based isogeometric analysis

We propose a new approach to construct selective and reduced integration rules for isogeometric analysis based on NURBS elements. The notion of an approximation space that approximates the target space is introduced. We explore the use of various approximation spaces associated with optimal patch-wise numerical quadratures that exactly integrate the polynomials in approximation spaces with the minimum number of quadrature points. Patch rules exploit the higher continuity of spline basis functions. The tendency of smooth spline functions to exhibit numerical locking in nearly-incompressible problems when using a full Gauss-Legendre quadrature is alleviated with selective or reduced integration. Stability and accuracy of the schemes are examined analyzing the discrete spectrum in a generalized eigenvalue problem. We propose a local algorithm, which is robust and computationally efficient, to compute element-by-element the quadrature points and weights in patch rules. The performance of the methods is assessed on several numerical examples in two-dimensional elasticity and Reissner-Mindlin shell structures.