Eigenvalue Solver
Use the Eigenvalue Solver () to find the solution to linear or linearized eigenvalue problems (also called eigenfrequency problems). This solver is automatically used when a Eigenvalue or Eigenfrequency study is added to the model.
Also see The Eigenvalue Solver Algorithm.
General
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, which is always available).
Use the Desired number of eigenvalues field to specify the number of eigenvalues and eigenvectors (default: 6) that the solver should compute and store in the output.
The number in the Relative tolerance field (default 1.0*10−6) controls the relative error in the computed eigenvalues.
Use the Search for eigenvalues around field to search for eigenvalues close to the specified real or complex scalar. This value is often called the shift. Select the Enter transformed value check box to enter the value to search around as the transformed value (an eigenfrequency, for example) instead of the corresponding eigenvalue.
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.
For other settings, see the Eigenvalue or Eigenfrequency study settings. When the eigenvalue search settings are defined by the study step, these settings, including the ones above except the Relative tolerance, are not available.
Values of Linearization Point
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:
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Initial expression (the default) to use the expressions specified on the Initial Values nodes under a specific physics interface as a linearization point.
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Solution to use a solution as linearization point.
Use the Solution list to specify which solution to use if Prescribed by is set to Solution:
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Select Zero (the default) to use a linearization point that is identically equal to zero.
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Select any other available solution to use it as linearization point.
Select the Store linearization point and deviation in output check box to store the linearization point and the deviation from that linearization instead of the total solution.
If the eigenvalue itself appears nonlinearly, the solver reduces the problem to a quadratic approximation around an eigenvalue linearization point. Use the settings under Value of eigenvalue linearization point to specify such a scalar. Select the Transform point check box to transform the linearization point value using the selected eigenvalue transformation. Specify the value of the linearization point in the Point field (default value: 0).
Output
Select an option from the Scaling of eigenvectors list to specify the scaling method used to normalize the eigenvectors. Select:
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RMS to use root mean square normalization. This RMS scaling of the eigenvector u is such that .
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Maximum to use maximum norm normalization. The degree of freedom with the largest absolute value will be assigned the value specified in the Maximum absolute value field (default: 1), so that . You can change the maximum absolute value to, for example, keep the eigenmodes small in value. The physics interfaces for structural mechanics set this value to a small number during the solver generation.
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Mass matrix to scale the eigenvectors such that the modal masses become unity.
Advanced
The eigenvalue solver is an iterative algorithm. Use the Maximum number of eigenvalue iterations field to limit the number of iterations (default: 300).
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.
Constants
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). Some examples of when it can be useful to define constants for a solver:
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When doing more advanced solver sequences, where constants need to be changed in between calls (for example, in for-loops).
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When you do not want to change the global definition of a parameter or when you cannot or do not want to use a Parametric Solver feature.
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When you want to define auxiliary parameters that are part of the equations like CFLCMP or niterCMP and where the solver does not define these parameters.
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.
Log
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.
Conical Quantum Dot: Application Library path COMSOL_Multiphysics/Equation_Based/conical_quantum_dot