We compare isogeometric collocation with isogeometric Galerkin and standard C0 finite element methods with respect to the cost of forming the matrix and residual vector, the cost of direct and iterative solvers, the accuracy versus degrees of freedom and the accuracy versus computing time. On this basis, we show that isogeometric collocation has the potential to increase the computational efficiency of isogeometric analysis and to outperform both isogeometric Galerkin and standard C0 finite element methods, when a specified level of accuracy is to be achieved with minimum computational cost. We then explore an adaptive isogeometric collocation method that is based on local hierarchical refinement of NURBS basis functions and collocation points derived from the corresponding multi-level Greville abscissae. We introduce the concept of weighted collocation that can be consistently developed from the weighted residual form and the two-scale relation of B-splines. Using weighted collocation in the transition regions between hierarchical levels, we are able to reliably handle coincident collocation points that naturally occur for multi-level Greville abscissae. The resulting method combines the favorable properties of isogeometric collocation and hierarchical refinement in terms of computational efficiency, local adaptivity, robustness and straightforward implementation, which we illustrate by numerical examples in one, two and three dimensions.
|Original language||English (US)|
|Number of pages||63|
|Journal||Computer Methods in Applied Mechanics and Engineering|
|State||Published - Dec 1 2013|
Bibliographical noteFunding Information:
D. Schillinger was supported by the German National Science Foundation (Deutsche Forschungsgemeinschaft DFG) under grants SCHI 1249/1-1 and SCHI 1249/1-2. J.A. Evans, M.A. Scott, and T.J.R. Hughes were supported by grants from the Office of Naval Research (N00014-08-1-0992), the National Science Foundation (CMMI-01101007), and SINTEF (UTA10-000374), with the University of Texas at Austin. A. Reali was supported by the European Research Council through the FP7 Ideas Starting Grant n. 259229 ISOBIO, and by the Italian MIUR through the FIRB “Futuro in Ricerca” Grant n. RBFR08CZ0S.
- Hierarchical refinement of NURBS
- Isogeometric analysis
- Isogeometric collocation methods
- Local adaptivity
- Reduced quadrature
- Weighted collocation