Ultra-Accurate Isogeometric Analysis Of Structural Vibrations

The isogeometric analysis employs the non-uniform rational B-splines as the basis functions simultaneously for geometric description and finite element analysis. The method is capable of eliminating the geometric discretization errors and the effect of negative elements for the mass matrix, as provides a very favorable way for structural vibration analysis. In this thesis an ultra-accurate isogeometric analysis of structural vibrations is proposed. The essential component for the proposed method is to construct novel higher order mass matrices for the vibration analysis. More specifically, for the1D rod problems,2D membrane problems and thin plate problems, the higher order mass matrices are formulated using a new two-step method. 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 also 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. Nonetheless, for the Euler beam problems, it turns out that the desired higher order mass matrices can be directly established through optimally reducing the bandwidth of consistent mass matrix.For the1D rod vibration problems, an elevation of two orders of accuracy for the vibration frequencies is observed for the proposed ultra-accurate isogeometric method compared with the standard isogeometric analysis with the consistent mass matrices. It is shown that for1D rod problems, with regard to the vibration frequency the proposed higher order mass matrices have6th and8th orders of accuracy in contrast to the4th and6th orders of accuracy associated with the quadratic and cubic consistent mass matrices. For Euler beam problems, the proposed higher order mass matrices have4th and6th orders of accuracy, while they are2th and4th orders of accuracy for the quadratic and cubic consistent mass matrix formulations. A generalization to2D higher order mass matrices are further realized by the tensor product operation on the one dimensional reduced bandwidth and consistent mass matrices. It is proved that the2D higher order mass matrices also enable an ultra-accurate analysis for membranes and thin plates by improving two orders of frequency accuracy in comparison with the vibration analysis with 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.