Use the Eigenvalue Solver () to find the solution to linear or linearized eigenvalue problems (also called eigenfrequency problems). This solver is automatically used when an Eigenvalue or Eigenfrequency study is added to the model.

Also see The Eigenvalue Solver Algorithms. The solver settings are split in the following sections

Use the Defined by study step list to specify if the settings are synchronized with the corresponding study step. Select User defined to specify the properties below (in addition to the relative tolerance and the options for computing and storing left eigenvalues, which are always available).

The settings here are similar to those for the Eigenvalue study step, with the following exceptions and options, when applicable:

From the Eigenvalue transformation list, select a transformation method for transforming the eigenvalues into another related quantity. The default is None, which keeps the original eigenvalues. Depending on the physics in the model, other transformations might also be available.

The number in the Relative tolerance field (default 1.0·10−6) controls the relative error in the computed eigenvalues.

From the Compute and store left eigenvectors list, choose Automatic (the default), On, or Off. The Automatic option typically is the same as Off, but, depending on the other settings, if needed, the left eigenvectors will be computed (and eventually stored). The eigenvectors referred to as just “eigenvectors” are right eigenvectors. A left eigenvector is defined as a row vector XL that satisfy XL A = λLXL . For example, for sensitivity and optimization, when the matrices are nonsymmetric, left eigenvectors will be computed. Also, left eigenvectors can be used for projecting equations in the modal solver.

Select the Enter transformed values checkbox to enter the values to search around as the transformed value (an eigenfrequency, for example) instead of the corresponding eigenvalue.

Under FEAST settings, the default setting for the Integration type for estimation list and the Number of integration points for estimation list is Automatic. When you choose Automatic in the Integration type for estimation list, it uses the Gauss type for real symmetric or Hermitian eigenvalue solvers and the trapezoidal method for other types of solver. Otherwise, choose Gauss or Trapezoidal to use only one of those methods. When you choose Automatic in Number of integration points for estimation, it is 6 for real symmetric or Hermitian eigenvalue solvers and 16 for other types of solvers.

Select the Distribute linear system solution checkbox to run the FEAST eigenvalue solver in parallel. See Running FEAST in a Parallel MPI Mode for more information.

When you have selected the A posteriori residual check checkbox, the solver a posteriori explicitly computes the relative residual of the eigenpair and checks that this quantity is smaller that 100*tol, where tol is the Relative tolerance. Notice that, in order to do this computation, the solver needs to store the K, D, and E matrices, and therefore, this option requires more memory. If any residual is larger than the tolerance, a Warning node appears with information about which eigenpairs it concerns. That information also appears in the solver log.

If the eigenvalue itself appears nonlinearly, and the solver is not the ARPACK nonlinear solver, the solver reduces the problem to a quadratic approximation around an eigenvalue linearization point. Under Eigenvalue linearization point, select the Transform eigenvalue linearization point checkbox to transform the linearization point value using the selected eigenvalue transformation. Specify the value of the linearization point in the Value of eigenvalue linearization point field (SI unit: rad/s).

If you use the ARPACK nonlinear solver, the solver approximates the nonlinearity in the eigenvalue with a polynomial and, if you have selected the Quasi-Newton refinement checkbox, it will activate extra quasi-Newton iterations on the returned solutions given by the ARPACK call. The Maximum number of iterations (default: 20) does what the name suggests. The Tolerance factor (default: 1) instead makes it possible to refine the solutions more accurately. More precisely, the tolerance for stopping the quasi-Newton refinement process is the product between the values for the Tolerance factor and the Relative tolerance.

For other settings, see the Eigenvalue or Eigenfrequency study settings. When the eigenvalue search settings are defined by the study step, the settings above that correspond to similar study settings are not available.

This section is only visible when the study contains a Stationary Then Eigenfrequency study step. It can be used to specify which dependent variables are computed with the Eigenvalue solver. The same dependent variable cannot appear in the equivalent section of the Stationary Solver, control variables excepted.

From the Linearity list, choose Linear (the default) or Linear perturbation.

Both for linear and nonlinear PDE problems, the eigenvalue problem is that of the linearization about a solution. Such a solution is specified with the Prescribed by list. Select:

Use the Solution list to specify which solution to use if Prescribed by is set to Solution:

Select the Store linearization point and deviation in output checkbox to store the linearization point and the deviation from that linearization instead of the total solution.

Select an option from the Scaling of eigenvectors list to specify the scaling method used to normalize the eigenvectors. Select:

Select User defined from the Defined by study step list in the General section to specify the settings below for sorting and filtering the eigenmodes. Those settings can be useful to, for example, filter out physically uninteresting solutions or to sort the solutions according to given criteria.

The following parts of this section are available when the Defined by study step list is set to User defined.

The eigenvalues can be excluded if there is a filter expression that they do not satisfy. In the table below, in the Filter expression (store if true or >0) column, add expressions for the filtering Those expressions can be functions of the eigenvalue lambda or eigenfrequency freq and can be logical expressions such as lambda>10. If desired, add some descriptive text in the Description column for the expressions.

The Store solutions list is always available: Choose All converged solutions (the default) or First Nth for the first Nth solutions. Then specify that number in the Maximum number of stored solutions field (default: 1000).

The eigenvalues can be sorted in Ascending (the default) or Descending order depending on the Ordering setting. When the Sorting method is Predefined, you can choose to Sort primarily based on the Real part, Imaginary part, Real part magnitude, Imaginary part magnitude, or Absolute value. The same settings are available for the Sort secondly option, which is used to resolve conflicts. The defaults for eigenvalues are Real part for Sort primarily and Imaginary part magnitude for Sort secondly. For eigenfrequencies, the defaults are Imaginary part for Sort primarily and Imaginary part magnitude for Sort secondly. Also, the Sort based on transformed eigenvalues checkbox is selected by default to take and eigenvalue transformation into account when sorting.

Alternatively, Manual can be chosen for the Sorting method and then an arbitrary number of (ordered) custom sorting priority expressions can be defined in the table that appears. In the Sorting priority expression column, add expressions for the sorting, in order of priority. Those expressions can be functions of the eigenvalue lambda or eigenfrequency freq. For example, you can specify an expression such as abs(freq-1) to sort according to the distance from a given shift (1 in this case). If desired, add some descriptive text in the Description column for the expressions.

The Conjugate-pair consecutive sort checkbox is selected by default to make sure that complex-conjugate eigenpairs appear one after the other, regardless of the sorting rules.

The eigenvalue solver is an iterative algorithm. Use the Maximum number of eigenvalue iterations field to limit the number of iterations (default: 300).

When you use the ARPACK solver, you can use the Dimension of Krylov space field to control the algorithm’s memory use. The default value of 0 means that the solver sets the dimension automatically to approximately twice the number specified in the Desired number of eigenvalues field in the General section.

In this section you can define constants that can be used as temporary constants in the solver. You can use the constants in the model or to define values for internal solver parameters. These constants overrule any previous definition (for example, from Global Definitions). The constant values are expressions and can, for example, include the range() operator, units, and global expressions. The constant name can be a new or an existing global parameter. The constant is temporary in the sense that it is only defined during the solver run. You cannot override parameters used in the following parts of the model:

Click the Add button () to add a constant and then define its name in the Constant name column and its value (a numerical value or parameter expression) in the Constant value column. By default, any defined parameters are first added as the constant names, but you can change the names to define other constants. Click Delete () to remove the selected constant from the list.

The Log section contains logs of the eigenvalue solver results and properties of the assembled system, including the solver iterations and the total solution time. This log is stored in the Model MPH-file.