COMSOL 6.4 - Mode Following

The Mode Following algorithm considers the eigenvalue λp+1 for the current parameter value to be the continuation of the eigenvalue λp for the previous parameter value if the corresponding left and right eigenmodes fulfill a specific pseudo-orthogonal condition, known as the Modal Assurance Criterion. More specifically, for a symmetric eigenvalue problem with no second order coefficients, that is, E = 0, and a positive definite mass matrix D, the eigenvectors are orthogonal; namely

if i ≠ j and

if i = j. This result is usually referred to as the spectral theorem. It is possible to extend this result also for the case with second order coefficients and show that

if i ≠ j and

if i = j. As such, the left and right eigenvectors are pseudo-orthogonal with respect to the pseudoscalar product defined by the matrix D − (λi + λj)E. The mode following functionality uses this result to determine if a specific eigenpair is a continuation of another eigenpair by checking if the pseudo-orthogonality is almost fulfilled. More precisely, the eigenvalue λp+1 and its corresponding eigenvector, among the found eigenvalues, that obtains the maximum value for the pseudoscalar product defined above compared to the eigenvalue λp and its corresponding eigenvector, will be the continuation of the eigenvalue λp.

Select the checkbox to keep track of any extra eigenvalues that appear after the first parameter value. Then enter the number of expected extra modes with one of the methods below:

•

From the list, choose (the default) and then enter an allocation factor in the field (default: 1). For example, if 20% extra modes are expected, the allocation factor should be set to 1.2. If set to 1 and new eigenvalues appear, they will not be stored.

The checkbox is selected by default, which allows the algorithm to attempt to follow the previous solutions. It also attempts to detect if, among the converged eigenvalues, some eigenvalues are not the continuation of the previous solutions.

When the checkbox is selected, you can specify the following settings:

•

Enter a threshold value in the field (default: 0.01). This is the tolerance factor to detect if an eigenpair is new. The software checks if the pseudoscalar product for a specific eigenpair is equal to the previous parameter up to the specified threshold. This indicates that the current eigenpair is pseudo-orthogonal with respect to all the previous eigenpairs and therefore is a new solution.

•

Select in the list to detect if a given eigenpair is new by checking if the pseudoscalar product is small in general, as opposed to checking for new eigenpairs in relative terms as above. A comparison between the pseudoscalar product of the potential new eigenpair, the maximum value of the pseudoscalar product, and the other eigenpairs is performed. Select (the default) to disable the or to get a warning when a new eigenpair is detected.