Non-uniform rational Lagrange functions and its applications to isogeometric analysis of in-plane and flexural vibration of thin plates

New basis functions were developed for isogeometric analysis (IGA) to overcome the difficulties of IGA using NURBS (Non-Uniform Rational B-Splines) on coping with Dirichlet boundary conditions. The new basis functions were constructed through nesting rational local Lagrange interpolations like the T-spline and were evaluated in similar procedure as the finite difference method. Explicit expressions for the new basis functions were presented. Due to their equivalence to the NURBS, the new basis functions were named as non-uniform rational Lagrange (NURL) basis. IGA using NURL includes the finite element method as a special case but the geometry in IGA using NURL is exact. IGA using NURL can carry out p-refinement that has the nesting feature of the k-refinement of NURBS. Dirichlet boundary conditions can be directly imposed in IGA using NURL because the NURL are interpolation basis functions. A method of directly transforming the tensor product basis of triangular patches to area coordinates was presented and the singularity problem at the edge degenerated to a single point was solved. The methods developed in this work were applied to in-plane and flexural vibration of thin plates. Comparisons with available results in literatures showed the fast convergence and high accuracy of IGA using NURL and the transformation method for triangular patches. Introduction of a MATLAB toolbox of the NURL was appended.