A Mid-Node Mass Lumping Scheme for Accurate Structural Vibration Analysis with Serendipity Finite Elements

Abstract

A mid-node mass lumping scheme is proposed to formulate the lumped mass matrices of serendipity elements for accurate structural vibration analysis. Since the row-sum technique leads to unacceptable negative lumped mass components for serendipity elements, the diagonal scaling HRZ method is frequently employed to construct lumped mass matrices of serendipity elements. In this work, through introducing a lumped mass matrix template that includes the HRZ lumped mass matrix as a special case, an analytical frequency accuracy measure is rationally derived with particular reference to the classical eight-node serendipity element. The theoretical results clearly reveal that the standard HRZ mass matrix actually does not offer the optimal frequency accuracy in accordance with the given lumped mass matrix template. On the other hand, by employing the nature of non-negative shape functions associated with the mid-nodes of serendipity elements, a mid-node lumped mass matrix (MNLM) formulation is introduced for the mass lumping of serendipity elements without corner nodal mass components, which essentially corresponds to the optimal frequency accuracy in the context of the given lumped mass matrix template. Both theoretical and numerical results demonstrate that MNLM yields better frequency accuracy than the standard HRZ lumped mass matrix formulation for structural vibration analysis.