A rigorous and unified mass lumping scheme for higher-order elements

In dynamic analysis with explicit time integration schemes, a lumped mass matrix (LMM) is preferable, because LMM can avoid solving the large scale simultaneous algebraic equations. Mathematically rigorous mass lumping schemes, such as the mass lumping by nodal quadrature and the row-sum technique, are applicable to only linear or bilinear elements. For higher-order elements, such as 8-node serendipity elements, the diagonal scaling procedure is the only lumping method that can be recommended to generate positive definite diagonal element mass matrices. Unfortunately, there is no mathematical theory in support of this approach. This study proposes a general mass lumping scheme applicable to higher order elements, where the virtual work of initial force is integrated over the problem domain that is viewed as the manifold covered by the finite element patches. By a series of numerical experiments, both free and forced vibration problems, it is suggested that even in the implicit time integration scheme the consistent mass matrix (CMM) can be superseded by the proposed LMM. Furthermore, the proposed LMM has much stronger adaptability to distorted meshes.