Eigenvalue Problems

7. Eigenvalue Problems#

Modal solutions form the basis of much of the analysis performed in Sierra/SD. It is essential that we understand the accuracy of the solution computed eigenvalue pairs may have errors for a variety of reasons, the most common is that the linear solvers all have tolerances, and errors in these solutions feed directly into errors in eigenpairs. It is well-known that errors in eigenvectors are typically significantly larger than errors in eigenvalues. If the relative error in an eigenvalue is \(\varepsilon\), the relative error in the eigenvector is of the order of \(\sqrt{\varepsilon}\).

At the conclusion of a modal analysis, the Sierra/SD application reports the eigenvalues and associated error estimates. An example of this output is provided below. The first column of data is the eigenvalue, which is related to the frequency \(f_j\) by,

\[ \lambda_j = (2\pi f_j)^2. \]

The second column is an estimate of the error bound on the eigenvalue, \(\epsilon_j=|(K-\lambda_j M)\phi_j|_2\). Generally, except for zero energy modes, the error bound should be tiny relative to \(\lambda\).

 Ritz values (Real, Imag) and direct residuals  
 ---------------------------------------------  
                  Col   1           Col   2  
  Row     1:     -2.16338D-06     7.34617D-07  
  Row     2:      2.07696D+07     2.25677D-06  
  Row     3:      2.07858D+07     8.73909D-07  
  Row     4:      3.56376D+08     1.48725D-06  
  Row     5:      4.84777D+08     1.69662D-06  
  Row     6:      4.84906D+08     5.01020D-06  
  Row     7:      9.59039D+08     6.06316D-06  
  Row     8:      1.11917D+09     1.22741D-05  
  Row     9:      1.11917D+09     3.30643D-06

7.1. Geometric Rigid Body Modes#

An example input and mesh (shown below) of the use of the Geometric Rigid Body Modes capability are provided. This example assumes that the reader is familiar with the parameter num_rigid_mode, and has read the GeometricRigidBodyModes Solution Case section of the User’s manual.

Eigenvalue Problems

Contents

7. Eigenvalue Problems#

7.1. Geometric Rigid Body Modes#