Eigenvalue
The Eigenvalue () study and study step are used to compute the eigenvalues and eigenmodes of a linear or linearized model in a generic eigenvalue formulation where the eigenvalues are not necessarily frequencies. The Eigenvalue study gives you full control of the eigenvalue formulation, in contrast to the eigenfrequency study that is adapted for specific physics interfaces. The Eigenvalue study is typically used for equation-based modeling.
Selecting an Eigenvalue study gives a solver configuration with an Eigenvalue Solver.
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 Eigenvalue solver list, choose ARPACK (the default) or FEAST.
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 eigenvalue problem. For its settings, see Study Settings for FEAST. For more information about these eigenvalue solvers, see The Eigenvalue Solver Algorithms.
Study Settings for ARPACK
From the Eigenvalue search method list, select a search method:
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 Manual (the default), to specify some search criteria manually. See Manual Eigenvalue Search Settings below.
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 Region, to define an eigenvalue search region in a complex plane. See Manual Eigenvalue Search Settings below and The Eigenvalue Solver Algorithms.
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 All (filled matrix) to find all eigenvalues for a filled matrix. This option is only applicable for small eigenvalue problems. You can then specify a Maximum matrix size (default: 2000).
Manual Eigenvalue Search Settings
By default, the physics interfaces suggest a suitable number of eigenvalues to search for. To specify the number of eigenvalues, select the check box in front of the Desired number of eigenvalues field to specify the number of eigenvalues you want the solver to return (default: 6).
In a 3D model, the first six eigenvalues are typically zero and correspond to the rigid-body modes of a 3D geometry. You may therefore need to specify a larger number of eigenvalues. The largest number of eigenvalues that the solver can compute is equal to the number of unconstrained degrees of freedom minus two.
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 or expression around which the eigenvalue solver should look for solutions to the eigenvalue equation (default: 0).
Use the Eigenvalue search method around shift list to control how the eigenvalue solver searches for eigenvalues around the specified shift value:
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Select Closest in absolute value (the default value) to search for eigenvalues that are closest to the shift value when measuring the distance as an absolute value.
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Select Larger real part to search for eigenvalues with a larger real part than the shift value.
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Select Smaller real part to search for eigenvalues with a smaller real part than the shift value.
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Select Larger imaginary part to search for eigenvalues with a larger imaginary part than the shift value.
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Select Smaller imaginary part to search for eigenvalues with a smaller imaginary part than the shift value.
When using a search for eigenvalues, the nonsymmetric eigenvalue solver is always used, even when the problem to solve is symmetric.
Eigenvalue Search Region Settings
Use the Approximate number of eigenvalues field to specify the approximate number of eigenvalues you want the solver to return (default: 20). The value of the Approximate number of eigenvalues 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 eigenvalues will increase the memory requirement and the computational time. If the solver indicates that the value of the Approximate number of eigenvalues is smaller than the actual number of eigenvalues in the given region, it will perform a search for more eigenvalues, 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 eigenvalues than a too small.
In the Maximum number of eigenvalues field, you can specify a maximum number of eigenvalues to limit the eigenvalue solver’s search for additional eigenvalues (default: 200).
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.
Under Search region, you define a unit (defaults: rad/s) and the size of the search region for eigenvalues as a rectangle in the complex plane by specifying the Smallest real part, Largest real part, Smallest imaginary part, and Largest imaginary part 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 eigenvalues (up to twice the desired number). These are eigenvalues that the eigenvalue solver finds without additional computational effort.
Symmetry Settings
Eigenvalue computations can be performed with a nonsymmetric solver or, if applicable, a real symmetric solver. From the Use real symmetric eigenvalue solver list, choose Automatic (the default) or Off. For the Automatic option there is the option to select the Real symmetric eigenvalue solver consistency check check box. This check increases the computational time and memory requirements but provides a rigorous check of the applicability of the real symmetric solver.
Study Settings for FEAST
From the Eigenvalue search contour list, select a search contour:
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 Whole contour (the default), to define an eigenvalue search contour in a complex plane. See Whole Search Contour Settings settings below.
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 Half contour (Hermitian problem), to define an eigenvalue search contour by half of the whole contour. See Half Search Contour Settings below.
Whole Search Contour Settings
You define the Whole contour by specifying the Unit (default: deg/s), Center of the ellipse contour, Horizontal radius of the ellipse contour, Vertical/horizontal axis ratio of ellipse contour (%) and Rotation angle of the ellipse contour in the respective text fields. It is important that the horizontal radius that you specify in the Horizontal radius of the ellipse contour field is large enough to enclose the eigenvalues of interest. In the Vertical/horizontal axis ratio of ellipse contour (%) field, specify the ratio of the vertical radius of the ellipse over its horizontal radius, assuming that the horizontal radius is 100. In the Rotation angle of the ellipse contour field, specify the rotation angle in degree from the vertical axis in the range of −180 degrees to 180 degrees.
From the Number of eigenvalues list, select the method for evaluating the number of eigenvalues inside the eigenvalue search contour:
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 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).
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 Manual to specify the number of eigenvalues inside the eigenvalue search contour manually in the Approximate number of eigenvalues field (default: 6).
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 Stochastic estimation only to use only stochastic estimation to evaluate the number of eigenvalues.
In the Size of initial search subspace for estimation field (default: 6), specify the initial guess of the search subspace dimension, which can be interpreted as an initial guess for the number of eigenvalues inside the contour. This setting is only available when using a stochastic estimation.
From the Integration type for estimation list, select the type for integration:
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 Automatic (the default) to choose the integration type automatically depending on eigenvalue solver. It means the Gauss type for real symmetric or Hermitian eigenvalue solver and the Trapezoidal type for other types of solvers.
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 Gauss to use Gauss integration.
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 Trapezoidal to use trapezoidal integration.
From the Number of integration points for estimation list, select the number of points for integration:
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 Automatic (the default) to define the number of points for integration automatically. It is 3 for real symmetric or Hermitian eigenvalue solvers and 6 for other types of solvers.
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 Manual to specify the number of integration points for estimation manually in the Number of integration points field.
Eigenvalue computations can be performed with a nonsymmetric solver or, if applicable, a real symmetric solver. From the Use real symmetric or Hermitian eigenvalue solver list, choose Automatic (the default) or Off. For the Automatic option there is the option to select the Real symmetric or Hermitian eigenvalue solver consistency check check box. This check increases the computational time and memory requirements but provides a rigorous check of the applicability of the real symmetric solver.
There is also and 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.
Half Search Contour Settings
You define the Half contour (Hermitian problem) by specifying the Unit, Lower bound of search interval, Upper bound of search interval, and Vertical/horizontal axis ratio of ellipse contour (%) in the respective text fields.
From the Number of eigenvalues list, select the method for evaluating the number of eigenvalues inside the eigenvalue search contour:
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 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).
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 Manual to specify the number of eigenvalues inside the eigenvalue search contour manually in the Approximate number of eigenvalues field (default: 6).
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 Stochastic estimation only to use only stochastic estimation to evaluate the number of eigenvalues.
From the Integration type for estimation list, select the type for integration:
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 Automatic (the default) to choose the integration type automatically depending on eigenvalue solver. It means the Gauss type for real symmetric or Hermitian eigenvalue solver and the Trapezoidal type for other types of solvers.
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 Gauss to use Gauss integration.
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 Trapezoidal to use trapezoidal integration.
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 Zolotarev to use Zolotarev integration.
From the Number of integration points for estimation list, select the number of points for integration:
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 Automatic (the default) to define the number of points for integration automatically. It is 3 for real symmetric or Hermitian eigenvalues.
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 Manual to specify the number of integration points for estimation manually in the Number of integration points field.
In the Size of initial search subspace for estimation field (default: 6), specify the initial guess of the search subspace dimension, which can be interpreted as an initial guess for the number of eigenvalues inside the half contour. This setting is only available when using a stochastic estimation.
If required, there is an option to select the Real symmetric or Hermitian eigenvalue solver consistency check check box. There is also 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.
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.