3.2.9. eigen Command

This command is used to perform the eigenvalue analysis.

eigen <type> <$solver> $numEigenvalues

Argument	Type	Description
$numEigenvalues	integer	number of eigenvalues required.
$solver	string	optional string detailing type of solver: -genBandArpack (default), -fullGenLapack, -symmBandLapack.
$type	string	optional string indicating type of eigenvalue problem to solve: ‘general’ (default) or ‘standard’

Returns

A list containing the eigenvalues

Note

The eigenvectors are stored at the nodes and can be printed out using a Node Recorder, the nodeEigenvector command, or the Print command.
The default eigensolver is able to solve only for N-1 eigenvalues, where N is the number of inertial DOFs. When running into this limitation the -fullGenLapack solver can be used instead of the default Arpack solver.
The -symmBandLapack option works only standard eigenvalue analysis of the stiffness matrix, i.e., K*x = lam*x

Warning

The default eigen solver utilizes ARPACK, an iterative eigenvalue solver. Like other iterative methods, ARPACK can struggle to accurately compute eigenvectors when the model contains repeated (degenerate) eigenvalues or clusters of very closely spaced eigenvalues. The issue arises from difficulties in forming an orthonormal basis for the eigenspace numerically.

In such cases, ARPACK may fail to find all of the requested eigenvalues, or it may return correct eigenvalues but incorrect eigenvectors. When modal damping is used, inaccurate eigenvectors may cause issues with the damping forces calculated. If you are unsure if this is an issue with your model, switch the solver used and check the reslts again – this check works as the eigenvectors will vary greatly with the chosen solver when this issue exists.

Fortunately, small adjustments to the model can prevent this numerical issue. In actuality, as material properties (e.g., Young’s modulus) or nodal masses are rarely identical and by introducing slight variations in your model, ARPACK is more likely to form the basis correctly.

As an illustration of the problem, consider the following script (provided by mhscott):

model Basic -ndm 1 -ndf 1
node 0 0; fix 0 1
node 1 0; mass 1 1.0
node 2 0; mass 2 1.0
node 3 0; mass 3 1.0
node 4 0; mass 4 1.0

set k 610
uniaxialMaterial Elastic 1 $k

element zeroLength 1 0 1 -mat 1 -dir 1
element zeroLength 2 1 2 -mat 1 -dir 1
element zeroLength 3 0 3 -mat 1 -dir 1
element zeroLength 4 3 4 -mat 1 -dir 1

systetm ProfileSPD
numberer RCM

set nModes 2
set eigV [eigen $nModes]
puts "Eigenvalues: $eigV"

# print eigenvectors
for {set i 1} {$i <= 4} {incr i 1}  {
   puts "$i [nodeEigenvector $i 1] [nodeEigenvector $i 2]"
}

will produce:

Eigenvalues:  232.99926686256409880116   232.99926686256412722287
  0.50235827201282656773    0.15501409223134476889
  0.81283275864641846287    0.25081806996552691302
  0.15501409223134465787   -0.50235827201282667875
  0.25081806996552685751   -0.81283275864641857389

wheras if I change the mass at node 4 using:

node 4 0; mass 4 1.0000000001

I would get:

Eigenvalues:  232.99926679816474006657   232.99926686256424090971
 -0.00000004574590262152    0.52573111211913170493
 -0.00000007401842538890    0.85065080835203654708
  0.52573111211146139610    0.00000004574590273254
  0.85065080819431726500    0.00000007401842544441

For this in an academic setting dealing with small problems, the -fullGenLapack option will provide correct eigenvalues and eigenvectors to the original problem:

will produce:

Eigenvalues:   232.99926684570411339337   232.99926686256412722287
  0.00000000000000000000   -0.52573111211913370333
  0.00000000000000000000   -0.85065080835203987775
  0.52573111209361389484    0.00000000000000000000
  0.85065080832527950605   -0.00000000000000000000

3.2.9.1. Theory

A generalized eigenvalue problem for two symmetric matrices $K$ and $M$ of size $n \times n$ is given by:

\[\left (K - \lambda M \right ) \Phi = 0\]

where:

$K$ is the stiffness matrix

$M$ is the mass matrix

$\lambda$ is the eigenvalue

and $\Phi$ is the associated eigenvector

Example

The following example shows how to use the eigen command to obtain a list of eigenvalues.

Tcl Code

# obtain 10 eigenvalues using the default solver (-genBandArpack)
set eigenvalues [eigen 10]

# or, obtain 10 eigenvalues explicitly specifying the solver
set eigenvalues [eigen -fullGenLapack 10]

# obtain 10 eigenvalues of the stiffness matrix
set eigenvalues [eigen standard -symmBandLapack 10]

Python Code

# obtain 10 eigenvalues using the default solver (-genBandArpack)
eigenvalues = eigen(10)

# or, obtain 10 eigenvalues explicitly specifying the solver
eigenvalues = eigen('-fullGenLapack', 10)

# obtain 10 eigenvalues of the stiffness matrix
eigenvalues = eigen('standard','-symmBandLapack',10)

Code Developed by: fmk