Mass Matrix Types

Mass Matrix Types

The“Lumped without rotations”,“Lumped with rotations”and“Consistent”mass matrices of
dynamic analysis can be applied to a structure.
The“Lumped without rotations”and“Lumped with rotations”are the diagonal mass
matrices. These types of mass matrices require minimum computational effort.
The“Consistent”mass matrix appears when the user wishes to consider a system with the
distributed parameters. 

It is commonly believed that a consistent mass matrix describes
inertial properties of a structure more exactly than the lumped one. However, in most cases
the lumped mass matrix provides a good approximation, since it is obvious that the inertial
parameters can be presented less precisely than the stiffness ones. In fact, that kinetic energy
is described as displacements of a structure, but potential energy is expressed through spatial
derivative of displacements. It is a well-known fact that approximation error increases
considerably during each differentiation [4]. Thus, for continual objects (solid, shells, plates),
it is possible to approximate the mass parameters less precisely than the stiffness ones for the
same mesh.

Usually, Hermit polynomials are used as shape functions for bars. It is an exact solution for
most of the static problems and the dynamic problems when lumped mass matrix is
considered. However, exact solutions for dynamic problems of a bar with distributed masses
belong to the class of Krylov functions (it is a specific combination of hyperbolic and
trigonometric functions). It enables the stiffness parameters in such case to be presented
approximately when Hermit polynomials are used simultaneously with a consistent mass
matrix. (Let us take note that, in fact, it is not intended for implementing a different type of
shape functions for static and dynamic problems). Therefore, for most cases it is not a great
benefit to complicate the dynamic model by the use of distributed mass parameters, since the
approximate solution with consistent masses occurs instead of the exact solution for an
approximate model (lumped masses).
Moreover, usually own masses of bar structural elements (girders, columns, etc.) are
negligible compared to masses of walls and roof (dead load), which are taken into account
through the technique of conversion of dead loads to masses. Such non-structural masses
usually reduce the effects of distributed element masses.
All that was mentioned above leads to the following conclusion: for most practical cases the
lumped mass matrix ensures a sufficiently precise approximation of structure inertial
properties. It should be remembered that a consistent mass matrix requires considerable
computational efforts, if a large-scale problem is analyzed. It should be certain that
implementation of a consistent mass matrix will be justified before selection of such a type of
matrix for analysis.

It is assumed that the mass matrix must be“Consistent”, if the rigid links are used into
computation model.
If sparse direct solver or iterative solver is applied, element-by-element (EBE) technique is
used for computation of matrix-vector product. It means, that the consistent mass matrix can
never be assembled, however, all operations are performed only on the element level. For
skyline solver, a consistent mass matrix is assembled and stored in the same way as a stiffness
matrix. For small problems (at the most ~3000 equations) skyline technique is faster, although
it still drastically time-consuming when the size of a problem increases.