An isogeometric collocation method for efficient random field discretization

This paper presents an isogeometric collocation method for a computationally expedient random field discretization by means of the Karhunen-Loeve expansion. The method involves a collocation projection onto a finite-dimensional subspace of continuous functions over a bounded domain, basis splines (B-splines) and nonuniform rational B-splines (NURBS) spanning the subspace, and standard methods of eigensolutions. Similar to the existing Galerkin isogeometric method, the isogeometric collocation method preserves an exact geometrical representation of many commonly used physical or computational domains and exploits the regularity of isogeometric basis functions delivering globally smooth eigensolutions. However, in the collocation method, the construction of the system matrices for a d-dimensional eigenvalue problem asks for at most d-dimensional domain integrations, as compared with 2d-dimensional integrations required in the Galerkin method. Therefore, the introduction of the collocation method for random field discretization offers a huge computational advantage over the existing Galerkin method. Three numerical examples, including a three-dimensional random field discretization problem, illustrate the accuracy and convergence properties of the collocation method for obtaining eigensolutions.