12.5. Lumped Matrices

Some of the elements allow their consistent mass or specific heat matrices to be reduced to diagonal matrices (accessed with the LUMPM,ON command). This is referred to as “lumping”.

12.5.1. Diagonalization Procedure

One of two procedures is used for the diagonalization, depending on the order of the element shape functions. The mass matrix is used as an example.

For lower order elements (linear or bilinear) the diagonalized matrix is computed by summing rows (or columns). The steps are:

Compute the consistent mass matrix in the usual manner.
Compute:
(12–9)
where:
n = number of degrees of freedom (DOFs) in the element
Set
(12–10)
(12–11)

For higher order elements the procedure suggested by Hinton, et al.([45]), is used. The steps are:

Compute the consistent mass matrix in the usual manner.
Compute:
(12–12)
(12–13)
Set:
(12–14)
(12–15)

Note that this method ensures that:

The element mass is preserved
The element mass matrix is positive definite

It may be observed that if the diagonalization is performed by simply summing rows or columns in higher order elements, the resulting element mass matrix is not always positive definite.

12.5.2. Limitations of Lumped Mass Matrices

Lumped mass matrices have the following limitations:

Elements containing both translational and rotational degrees of freedom will have mass contributions only for the translational degrees of freedom. Rotational degrees of freedom are included for:
- SHELL181, SHELL208, SHELL209, SHELL281, and ELBOW290 unless an unbalanced laminate construction is used.
- BEAM188, BEAM189, PIPE288, and PIPE289 if there are no offsets.
Lumping, by its very nature, eliminates the concept of mass coupling between degrees of freedom. Therefore, the following restrictions exist:
- Lumping is not allowed for FLUID29, FLUID30, or FLUID38 elements.
- Lumping is not allowed for the mass matrix option of MATRIX27 elements if it is defined with nonzero off-diagonal terms.
- The use of lumping with constraint equations may effectively cause the loss of some mass for analyses that involve a mass matrix. For example, in modal analyses this typically results in higher frequencies. This loss of mass comes about because of the generation of off-diagonal terms by the constraint equations, which then are ignored.
  The exceptions to this are substructuring generation passes with the sparse solver and the PCG Lanczos mode extraction method in modal analyses. These exceptions contain the off-diagonal terms when lumped mass is used with constraint equations. It is important to note however, that the assembled mass matrix in a jobname.FULL file generated by the PCG Lanczos mode extraction method will not contain the off-diagonal mass terms for this case.