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:

  1. Compute the consistent mass matrix in the usual manner.

  2. Compute:

    (12–9)

    where:

    n = number of degrees of freedom (DOFs) in the element

  3. Set

    (12–10)

    (12–11)

For higher order elements the procedure suggested by Hinton, et al.([46]), is used. The steps are:

  1. Compute the consistent mass matrix in the usual manner.

  2. Compute:

    (12–12)

    (12–13)

  3. Set:

    (12–14)

    (12–15)

Note that this method ensures that:

  1. The element mass is preserved

  2. 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. Special Handling of Rotational Degrees of Freedom

Diagonalization of a mass matrix containing rotational degrees of freedom may lead to the following complications:

  1. The lumped mass matrix may lose the property of frame invariance.

  2. The coupling between translational and rotational degrees of freedom, generally created by unbalanced laminate construction or sections with offsets, may be present in the lumped mass matrix.

Rotational degrees of freedom are excluded in the lumped mass matrix, except for the current technology elements, which include SHELL181, BEAM188, BEAM189, SHELL208, SHELL209, SHELL281, PIPE288, PIPE289, and ELBOW290. The following options are available for handling the rotational degrees of freedom in current technology elements (see the KeyElt argument of the LUMPM command):

  1. Direct diagonalization (see Diagonalization Procedure), in which rotational degrees of freedom are treated the same as translational degrees of freedom. However, any coupling between the translational and rotational degrees of freedom is excluded in the process.

  2. Translational mass only, in which the mass contributions from rotational degrees of freedom, including the coupling with the translational degrees of freedom, are entirely excluded.

  3. Frame invariant diagonalization, in which the lumped rotational mass components at each node are made identical to achieve frame invariance. The procedure is carried out only for BEAM188, BEAM189, PIPE288, and PIPE289 elements. The value of the lumped mass contribution Me(j,j) of rotational degree of freedom j is given as follows:

    (12–16)

    Where Me(i,i) is the lumped translational mass contribution of degree of freedom i from the same node, Iyy and Izz are the moments of inertia about the local element y and z axes, and A is the cross section area.

 

12.5.3. Limitations of Lumped Mass Matrices

Lumped mass matrices have the following limitations:

  1. 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, BEAM188, BEAM189, SHELL208, SHELL209, SHELL281, PIPE288, PIPE289, and ELBOW290 (see Special Handling of Rotational Degrees of Freedom for details).

  2. 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.

Note that coupling due to constraint equations is always included.