File fleigxr.rdm

An alter to allow a symmetric formulation for Complex Eigenvalues, fleigx.v707



1. Background

The original delivery of fluid-structure analysis used an unsymmetric
formulation.  An option was developed later to allow a symmetric
formulation for frequency response analysis, and is the default option for
Version 70.5.  It provides a more economical, reliable solution.  This
option was not enabled for complex eigenvalue analysis.  The alter
described here allows use of the symmetric formulation for complex
eigensolutions in SOLs 107 and 110.

2. Theory.  

In the unsymmetric formulation, without damping, the problem solved is of
the form 

	[p**2*M + K]*phi = 0,

where M and K both contain the area factor matrix in opposite off-diagonal
partitions.  (See the MSC/NASTRAN Reference Manual for details).  The
solution methods tend to find one frequency per mode, with small damping
coefficients that are numerical noise.

For the symmetric formulation, the area factor matrices are placed in the
damping (B) matrix, in opposite corners, in a manner that provides
symmetric matrices in the equation

	[p**2*M + p*B + K]*phi=0

The program tends to find the roots as complex pairs, where the frequency
component matches that of the unsymmetric formulation, and the damping
component is numerical noise.  When these roots are squared their value (as
printed out in the (real) and (imag) columns of the eigenvalue output) is
complex conjugates, as theory would predict.  Both the unsymmetric and
symmetric solutions are valid.  They differ only because they use a
different theory, but the answers from one are derivable from the answers
from the other formulation.

The mathematics that allows the symmetric formulation replaces the fluid
DOFs in the displacement vector phi with velocity variables.  This causes
the eigenvectors computed by the two methods to appear to be different, but
if the velocity components are returned to the dimensions of displacement
by dividing them by 1/p, the eigenvectors are also equivalent.  This
problem does not occur in frequency response because excitation of a fluid
at zero frequency has no physical meaning.  The fluid displacements are
returned in displacement units by dividing the computed response by
i*omega, where omega is the excitation frequency. 

Since fluid structure eigensolutions often have zero frequency solutions,
dividing the velocities by a number approaching zero leads to numerical
difficulties.  It is proposed to deliver this capability in Version 71
using velocity terms in the output of the fluid variables, and describe
this in the user documentation.  Client input on this topic is requested.

3. Input

Use the alter provided above. 

4. Output

The eigenvalues will come out as complex conjugate pairs.  The units of
measure on structural DOFs are in displacement units, while the fluid DOFs
are in velocity units.

5. Limitations

No new limitations are imposed by the alter.