On the significance of basis interpolation for accurate lumped mass isogeometric formulation

Although the consistent mass isogeometric formulation enjoys superior frequency accuracy compared with the conventional finite element method, the lumped mass isogeometric formulation performs very poorly and only has a 2nd order accuracy regardless of the basis degree. In fact, an increase of basis degree may adversely decrease the frequency accuracy for the lumped mass isogeometric analysis of structural vibrations. In this work, it is found that the non-interpolatory property of basis functions attributes to this unfavorable performance of lumped mass isogeometric formulation and the significance of basis interpolation for accurate lumped mass formulation is disclosed. Meanwhile, it is revealed that the continuity rather than positivity of basis functions controls the frequency accuracy in isogeometric structural vibration analysis. Consequently, a set of interpolatory transformed basis functions are introduced to construct accurate isogeometric lumped mass matrices. These transformed basis functions inherit the smoothness and geometry exactness of the original isogeometric basis functions. It turns out that the lumped mass isogeometric formulation based upon the interpolatory transformed basis functions not only improves the frequency accuracy, but also simultaneously eliminates the spurious modes produced by the standard lumped mass isogeometric formulation. Numerical results demonstrate that the proposed lumped mass isogeometric formulation with interpolatory transformed basis functions yields much more favorable frequency accuracy in comparison with the standard lumped mass isogeometric approach.