An aspect ratio dependent lumped mass formulation for serendipity finite elements with severe side-length discrepancy

The frequency solutions of finite elements may significantly deteriorate as the mesh aspect ratios become large, which implies a severe element side-length discrepancy. In this work, an aspect ratio dependent lumped mass (ARLM) formulation is proposed for serendipity elements, i.e., the two-dimensional eight-node and three dimensional twenty-node quadratic elements for linear problems. In particular, a generalized parametric lumped mass matrix template taking into account the mesh aspect ratios is introduced to examine the frequency accuracy of serendipity elements. This generalized lumped mass matrix template completely meets the mass conservation and non-negativity requirements. Subsequently, analytical frequency error estimates are developed for serendipity elements, which clearly illustrate the relationship between the frequency accuracy and element aspect ratios. Accordingly, optimal mass parameters are obtained as the functions of element aspect ratios through solving a constrained optimization problem for frequency accuracy. It turns out that the resulting aspect ratio dependent lumped mass matrices yield much more accurate frequency solutions, in comparison to the diagonal scaling lumped mass (HRZ) matrices and the mid-node lumped mass (MNLM) matrices without consideration of the element aspect ratios, especially for finite element discretizations with severe element side-length discrepancy. The superior accuracy and robustness of the proposed ARLM over HRZ and MNLM are consistently demonstrated by numerical examples.