  This paper marks the debut of a Galerkin isogeometric method for solving a
Fredholm integral eigenvalue problem, enabling random field discretization by
means of the Karhunen-Loeve expansion. The method involves a Galerkin
projection onto a finite-dimensional subspace of a Hilbert space, basis splines
(B-splines) and non-uniform rational B-splines (NURBS) spanning the subspace,
and standard methods of eigensolutions. Compared with the existing Galerkin
methods, such as the finite-element and mesh-free methods, the NURBS-based
isogeometric method upholds exact geometrical representation of the physical or
computational domain and exploits regularity of basis functions delivering
globally smooth eigensolutions. Therefore, the introduction of the isogeometric
method for random field discretization is not only new; it also offers a few
computational advantages over existing methods. In the big picture, the use of
NURBS for random field discretization enriches the isogeometric paradigm. As a
result, an uncertainty quantification pipeline of the future can be envisioned
where geometric modeling, stress analysis, and stochastic simulation are all
integrated using the same building blocks of NURBS. Three numerical examples,
including a three-dimensional random field discretization problem, illustrate
the accuracy and convergence properties of the isogeometric method for
obtaining eigensolutions.
