PRECISE MID-NODE LUMPED MASS MATRICES FOR 3D 20-NODE HEXAHEDRAL AND 10-NODE TETRAHEDRAL FINITE ELEMENTS

The three-dimensional (3D) 20-node hexahedral and 10-node tetrahedral elements have been widely used in structural analysis, while the row-sum mass lumping for such elements leads to certain negative entries for the lumped mass matrices, which cause considerable difficulty for dynamic finite element analysis. Meanwhile, despite that the diagonal scaling mass lumping technique, namely, the HRZ method, ensures a non-negative lumped mass formulation, a theoretical accuracy investigation of this frequently employed approach is still missed in 3D scenarios. In this work, a generalized lumped mass matrix template with specifically devised adjustable parameters is introduced for 3D 20-node hexahedral elements to assess the frequency accuracy of different mass lumping methods, which include the HRZ lumped mass matrix formulation as a specific circumstance. Subsequently, a frequency accuracy measure is rationally derived for this generalized lumped mass matrix template. With the aid of the frequency error measure, it is found that the HRZ lumped mass matrix formulation for 3D 20-node hexahedral elements does not give the optimal frequency accuracy. On the other hand, a precise mid-node lumped mass matrix formulation is attained by optimizing the frequency accuracy with respect to the adjustable parameters. A straightforward extension of the proposed formulation immediately yields an accurate mid-node lumped mass matrix formulation for 10-node tetrahedral elements. Furthermore, since the corner nodes of the mid-node lumped mass matrices for 20-node hexahedral and 10-node tetrahedral elements have zero diagonal mass entries, a standard static condensation operation on the discrete finite element equations enables a computationally efficient reduced order model that only contains the mid-node degrees of freedom for subsequent dynamic analysis. The frequency computation and transient analysis results consistently buttress that compared to the conventional HRZ lumped mass matrices, the proposed mid-node lumped mass matrices for 3D 20-node hexahedral and 10-node tetrahedral elements exhibit salient advantages regarding both accuracy and efficiency.