  Currently, components of consistent mass matrix are computed using various
numerical integration schemes, each one alters in number of integration (Gauss)
points, requires different amount of computations and possess different level
of accuracy. We discuss the closed-form mass matrix based on analytical
integration. Curved-sided and straight-sided elements are considered. For a
straight-sided element we derive an exact analytical easy to implement
consistent mass matrix. For a curved-sided element an exact analytical mass
matrix is derived, however it is rather lengthy, hence approximations are
proposed. Three systematic approximations to the metric (jacobian determinant)
are suggested; constant metric (CM), linearly varying metric (LM) and quadratic
metric (QM). CM requires evaluation of the metric at the centroid, LM requires
metric evaluations at the four corner nodes and QM uses metric values at all
the ten nodes. Analytical integration together with approximated metric models
yields closed-form semi-analytical mass matrices. The accuracy of the schemes
is studied numerically using randomly generated coarse mesh. Our findings
reveal significant superiority in accuracy and computations over equivalent
schemes. An important implication of this study is that based on the results,
it is superior to use our CM, LM and QM semi-analytical mass matrices over mass
matrices based on numerical integration schemes which involve four, five and
fifteen point Gauss quadrature. For a straight-sided element, CM, LM and QM
admit an exact consistent mass matrix.

Numerical Analysis