Eigenfrequency
The Eigenfrequency () study and study step are used to compute eigenmodes and eigenfrequencies of a linear or linearized model.
For example, in electromagnetics, the eigenfrequencies correspond to the resonant frequencies and the eigenmodes correspond to the normalized electromagnetic field at the eigenfrequencies. In solid mechanics, the eigenfrequencies correspond to the natural frequencies of vibrations and the eigenmodes correspond to the normalized deformed shapes at the eigenfrequencies. In acoustics, the eigenfrequencies correspond to the resonant frequencies and the eigenmodes correspond to the normalized acoustic field at the eigenfrequencies.
Selecting an Eigenfrequency study gives a solver with an Eigenvalue Solver. Use this study to solve an eigenvalue problem for a set of eigenmodes and associated eigenfrequencies. Also see The Relationship Between Study Steps and Solver Configurations.
The Physics and Variables Selection, Values of Dependent Variables, Mesh Selection, Adaptation and Error Estimates, and Geometric Entity Selection for Adaptation sections and the Include geometric nonlinearity check box are described in Common Study Step Settings. There is also detailed information in the Physics and Variables Selection and Values of Dependent Variables sections.
Study Settings
From the Eigenfrequency solver list, choose ARPACK (the default), FEAST, or LAPACK (filled matrix):
The ARPACK algorithm is based on an algorithmic variant of an Arnoldi process When the matrix A is symmetric it reduces to a variant of the Lanczos process. For its settings, see Study Settings for ARPACK below.
The FEAST algorithm uses an inverse residual iteration algorithm and seeks to accelerate the convergence of the subspace eigenfrequency problem. It can be effective for finding clustered eigenfrequencies by using an ellipse to search within. For its settings, see Study Settings for FEAST.
The LAPACK algorithm is useful for finding all eigenvalues for a filled matrix. This option is only applicable for small eigenfrequency problems. For its settings, see Study Settings for LAPACK (Filled Matrix).
For more information about these eigenfrequency solvers, see The Eigenvalue Solver Algorithms.
Study Settings for ARPACK
From the Eigenfrequency search method list, select a search method:
•
 Around shift (the default), to specify some search criteria manually. See Manual Eigenfrequency Search Settings (Around Shift) below.
•
 Rectangular, to define a rectangular eigenfrequency search region in a complex plane. See Rectangular Eigenfrequency Search Region Settings below and The Eigenvalue Solver Algorithms.
Manual Eigenfrequency Search Settings (Around Shift)
By default, the physics interfaces suggest a suitable number of eigenfrequencies to search for. To specify the number of eigenfrequencies, select the check box in front of the Desired number of eigenfrequencies field to specify the number of eigenfrequencies you want the solver to return (default: 6).
In a 3D model, the first six eigenfrequencies are typically zero and correspond to the rigid-body modes of a 3D geometry. You may therefore need to specify a larger number of eigenfrequencies. The largest number of eigenfrequencies that the solver can compute is equal to the number of unconstrained degrees of freedom minus two.
From the Unit list, choose a suitable unit (default: Hz).
By default, the physics interfaces suggest a suitable value around which to search for eigenvalues. To specify the value to search for eigenvalues around (shift), select the check box in front of the Search for eigenvalues around field; you can then specify a value (as an eigenfrequency) around which the eigenvalue solver should look for solutions to the eigenvalue equation (default: 0).
Use the Eigenfrequency search method around shift list to control how the eigenvalue solver searches for eigenfrequencies around the specified shift value:
•
Select Closest in absolute value (the default value) to search for eigenfrequencies that are closest to the shift value when measuring the distance as an absolute value.
•
Select Larger real part to search for eigenfrequencies with a larger real part than the shift value.
•
Select Smaller real part to search for eigenfrequencies with a smaller real part than the shift value.
•
Select Larger imaginary part to search for eigenfrequencies with a larger imaginary part than the shift value.
•
Select Smaller imaginary part to search for eigenfrequencies with a smaller imaginary part than the shift value.
When using a search for eigenfrequencies, the nonsymmetric eigenvalue solver is always used, even when the problem to solve is symmetric.
Rectangular Eigenfrequency Search Region Settings
Use the Approximate number of eigenfrequencies field to specify the approximate number of eigenfrequencies you want the solver to return (default: 20). The value of the Approximate number of eigenfrequencies will affect the Dimension of Krylov space used by the algorithm; see the Advanced section of the Eigenvalue Solver. It means that increasing the value of the Approximate number of eigenfrequencies will increase the memory requirement and the computational time. If the solver indicates that the value of the Approximate number of eigenfrequencies is smaller than the actual number of eigenfrequencies in the given region, it will perform a search for more eigenfrequencies, which increases the computational time; see The Eigenvalue Region Algorithm. Within limits it is often more efficient to provide a too large value of Approximate number of eigenfrequencies than a too small.
In the Maximum number of eigenfrequencies field, you can specify a maximum number of eigenfrequencies to limit the eigenvalue solver’s search for additional eigenfrequencies (default: 200).
Under Rectangle search region, you define a unit (default: Hz) and the size of the search region for eigenfrequencies as a rectangle in the complex plane by specifying the Smallest real part (eigenfrequency), Largest real part (eigenfrequency), Smallest imaginary part (eigenfrequency), and Largest imaginary part (eigenfrequency) in the respective text fields. The search region also works as an interval method if the Smallest imaginary part and Largest imaginary part are equal; the eigenvalue solver then only considers the real axis and vice versa.
The eigenvalue solver can in some cases return more than the desired number of eigenfrequencies (up to twice the desired number). These are eigenfrequencies that the eigenvalue solver finds without additional computational effort.
Symmetry and Consistency Settings
If you have chosen Rectangle from the Eigenfrequency search method list, the Perform consistency check check box is selected by default to increase confidence that the solver finds all eigenvalues in the search region. The work required for performing the consistency check constitutes a significant part of the total work of the eigenvalue computation.
Study Settings for FEAST
Using the FEAST eigenfrequency solver, you can define an ellipse to explicitly exclude the eigenfrequencies outside the ellipse. Choosing a suitable search region determines which eigenfrequencies that will be returned for results analysis.
Ellipse Search Region
You define the Ellipse by specifying the Unit (default: Hz), Center, Real semiaxis (eigenfrequency), Imaginary semiaxis (eigenfrequency), and Rotation angle in the respective text fields. It is important that the real semiaxis that you specify in the Real semiaxis field is large enough to enclose the eigenvalues of interest. In the Imaginary semiaxis field, you specify the imaginary semiaxis in a similar way. In the Rotation angle field, specify the rotation angle in degrees from the vertical axis in the range of −180 degrees to 180 degrees.
From the Number of eigenfrequencies list, select the method for evaluating the number of eigenvalues inside the eigenvalue search ellipse:
•
 Stochastic estimation (the default) to use stochastic estimation to evaluate the number of eigenvalues. After the stochastic estimation finishes, the eigenvalue solver automatically calculates the eigenvalues inside the eigenvalue search contour, using the number of eigenvalues calculated from stochastic estimation as the Size of initial search subspace for estimation (default: 6).
•
 Manual to specify the number of eigenvalues inside the eigenvalue search contour manually in the Approximate number of eigenvalues field (default: 6).
For an eigenfrequency problem for which you know how many eigenfrequencies there are, the solver can compute the eigenfrequencies directly. For that case, choose Manual from the Number of eigenfrequencies list and enter the desired number of eigenfrequencies in the Approximate number of eigenfrequencies field. If you do not know the number of eigenfrequencies in the defined region, there are two cases: You can just estimate the number of eigenfrequencies without computing them, or you can let the software estimate the number of eigenfrequencies and then compute them automatically afterward. The former is achieved by clicking the Stochastic Estimation button () at the top right of the Study Settings section. The latter is achieved by choosing Stochastic estimation from the Number of eigenvalues list, and then click Compute. Both cases require doing stochastic estimation, which needs the setting of a value in the Size of initial search subspace for estimation field (default: 6).
The Stochastic Estimation works as a Compute to Selected step. Because the Stochastic Estimation action does not solve the eigenvalue problem, there will not be any valid solution from this action. So, if the Stochastic Estimation action is done for a single step or the first step in a multistep study, there will not be any solution that can be used for postprocessing and results analysis. If the action is done for a step after the first step in a multistep study, the main solution of the sequence corresponds to the study step right before the eigenstep for which the estimation is done.
There is an option to select the Store linear system factorization check box. If selected, linear system factorizations are stored from the first FEAST iteration and reused in later iterations.
If the Study>Batch and Cluster check box is selected in the Show More Options dialog box, select the Distribute linear system solution check box to run the FEAST eigenvalue solver in parallel. See Running FEAST in a Parallel MPI Mode for more information.
Study Settings for LAPACK (Filled Matrix)
To specify a shift to use in the modes computation, select the Shift used in the modes computation check box and then enter a shift (in Hz) in the associated text field.
In the Maximum matrix size field, enter an upper limit on the matrix size (default: 2000).
From the Unit list, choose a suitable unit (default: Hz).
Values of Linearization Point
Use the settings in this section to specify a linearization point.
From the Settings list, choose Physics controlled (the default) to use linearization point settings controlled by the physics interfaces. Choose User defined to specify the linearization point using the Method list. Select:
•
 Initial expression to use the expressions specified on the Initial Values nodes under a specific physics interface as a linearization point.
•
 Solution to use a solution as a linearization point.
Use the Study list to specify which solution to use from the available studies. Select:
•
 Zero solution to use a linearization point that is identically equal to zero.
•
Any other available solution to use it as a linearization point. It can be the current solution in the sequence, or a solution from another sequence, or a solution that was stored with the Solution Store node. You select a stored solution by changing Use to the name of the stored solution. Choose a solution using the Selection list (see Values of Dependent Variables under Common Study Step Settings).
Study Extensions
This section contains some optional extensions of the study, such as auxiliary sweeps (see Common Study Step Settings). Adding an auxiliary parametric sweep adds an Eigenvalue Parametric attribute node to the Eigenvalue Solver.
Distribute Parametric Solver
If you are running an auxiliary sweep and want to distribute it by sending one parameter value to each compute node, select the Distribute parametric solver check box. To enable this option, click the Show More Options button () and select Batch and Cluster in the Show More Options dialog box.
Tuning Fork: Application Library path COMSOL_Multiphysics/Structural_Mechanics/tuning_fork.
For a model that uses a search region for the eigenfrequencies, with the Acoustics Module, see Helmholtz Resonator Analyzed with Different Frequency Domain Solvers: Application Library path Acoustics_Module/Tutorials,_Pressure_Acoustics/helmholtz_resonator_solvers.