An interpolatory basis lumped mass isogeometric formulation with rigorous assessment of frequency accuracy for Kirchhoff plates

A noticeable drawback associated with the isogeometric free vibration analysis of Kirchhoff plates with lumped mass formulation is that its frequency accuracy is strongly limited to 2nd order, no matter what degrees of basis functions are used. This issue is resolved herein by a lumped mass isogeometric formulation using a set of interpolatory basis functions. These interpolatory basis functions are constructed via transforming the standard isogeometric basis functions with respect to the Greville nodes. A direct consequence of the basis interpolation property is that the resulting lumped mass matrices can be realized with a nodal integration technique, and the corresponding nodal quadrature rules are then developed for cubic and quartic basis functions. Furthermore, based upon the transformation relationship between the standard and interpolatory basis functions, a frequency equivalence is rationally established between the standard and transformed isogeometric formulations, which enables a rigorous analytical study of the frequency accuracy for the proposed approach. In particular, the theoretical frequency error estimates are obtained for both cubic and quartic basis functions, which clearly illustrate the frequency accuracy superiority of the proposed method over the standard lumped mass isogeometric formulation for Kirchhoff plates. The theoretical observations are simultaneously demonstrated by numerical results.