Novel higher order mass matrices for isogeometric structural vibration analysis

A set of novel higher order mass matrices are presented for isogeometric analysis of structural vibrations using NURBS. The proposed method for the construction of higher order mass matrices contains two essential steps. Firstly based upon the standard consistent mass matrix a special reduced bandwidth mass matrix is designed. This reduced bandwidth mass matrix meets the requirement of mass conservation while simultaneously preserves the same order of frequency accuracy as the corresponding consistent mass matrix. Subsequently a mixed mass matrix is formulated through a linear combination of the reduced bandwidth mass matrix and the consistent mass matrix. The desired higher order mass matrix is then deduced from the mixed mass matrix by optimizing the linear combination parameter to achieve the most favorable order of accuracy. Both quadratic and cubic formulations are discussed in detail and it is shown that with regard to the vibration frequency, the proposed higher order mass matrices have 6th and 8th orders of accuracy in contrast to the 4th and 6th orders of accuracy associated with the quadratic and cubic consistent mass matrices. A generalization to two dimensional higher order mass matrix is further realized by the tensor product operation on the one dimensional reduced bandwidth and consistent mass matrices. A series of benchmark examples congruously demonstrate that the proposed higher order mass matrices are capable of achieving the theoretically derived optimal accuracy orders for structural vibration analysis.