Imposing Dirichlet boundary conditions with point collocation method in isogeometric analysis

Contrasting to the conventional finite element method, isogeometric analysis uses the exact geometric representations for the modeling and numerical simulations by NURBS basis functions. It eliminates the geometric approximation errors during the mesh discretization, and the high-order conforming elements can be conveniently constructed. Due to the lack of the interpolation properties for the NURBS basis functions, it's difficult to impose the Dirichlet boundary condition directly. In order to solve this problem, the point collocation method is proposed to impose the boundary condition basing on the spline approximation theory. It approximates the constraints by introducing an array of distributed points along with boundary edges. The coefficient matrix of the resulting boundary systems will be singular or ill-condition while the unsuitable collocation schemes are adopted. Therefore, the distribution criterions of points are discussed in detail, and two robust collocation schemes that are Greville abscissa and Chebyshev sites are proposed. Meanwhile, the collocation method is extended to the least-squares forms. A simple and effective reduction approach of the degree of freedom is also devised. The numerical examples demonstrate the feasibility and effectiveness of the proposed methods. The results also show that almost all schemes remain the same magnitude of the convergence rate and the key issue of the collocation method is to ensure the numerical stability. © 2012 Journal of Mechanical Engineering.