Zero and negative masses in finite element vibration and transient analysis

Abstract Mass matrix lumping by quadrature is considered. Accuracy requirements seem to dictate the use of zero or negative masses for multidimensional higher-order elements. It is shown that the zero and/or negative masses do not destroy the essential algebraic properties of the discrete eigenproblem, in spite of the negative or infinite eigenvalues which may result. Explicit transient methods require positive-definite lumping which, for some elements, may only be achieved by sacrificing accuracy to avoid the negative or zero masses that would render the lumping indefinite. An implicit-explicit time integration method based on quadratic triangles with optimal lumping is devised, analyzed, and tested. It treats the nodes with nonzero masses explicitly and the nodes with zero masses implicitly. Analysis and numerical tests show that this formulation is optimally accurate and less costly than a similar method with nonzero masses, based on optimally lumped biquadratic rectangles. The method is also found to be substantially more accurate than the fully explicit method based on lumping the triangular elements in an ad hoc fashion to retain nonzero masses.