How to use the FEAST Solver in Kratos?

The FEAST Solver is used for solving an eigenvalue problem. It is based on a new algorithm which deviates fundamentally from the Krylov subspace based techniques (Arnoldi and Lanczos algorithms), Davidson-Jacobi techniques or other traditional subspace iteration techniques

E. Polizzi, Density-Matrix-Based Algorithms for Solving Eigenvalue Problems, Phys. Rev. B. Vol. 79, 115112 (2009)

A documentation about how FEAST 4.0 works internally is available here.

Compilation of the FEAST Library

In order to compile the FEAST Library, the LinearSolversApplication with Intel MKL is required. The following lines must be added to the configure.sh file:

-DLINEAR_SOLVERS_APPLICATION=ON                                 \
-DUSE_EIGEN_MKL=ON                                              \
-DUSE_EIGEN_FEAST=ON                                            \

Once these lines are included, it is just needed the execution of the configure.sh file. However, only compilation on Linux is supported at the moment. For more information, please consult the LinearSolversApplication readme.

How to use FEAST 4.0

FEAST 4.0 can be used for solving an eigenvalue problem in a structural mechanics context by specifying eigen_value as solver_type in the problem parameters json. The default input parameters for solving a real-valued problem using FEAST are:

"eigensolver_settings":{
	    "solver_type" : "feast",
            "symmetric" : true,
            "number_of_eigenvalues" : 0,
            "e_min": 0.0,
            "e_max": 0.2,
            "search_lowest_eigenvalues" : false,
            "search_highest_eigenvalues" : false,
            "sort_eigenvalues" : false,
            "sort_order" : "sr",
            "subspace_size" : 0,
            "max_iteration" : 20,
            "tolerance" : 1e-12,
            "echo_level" : 0
	},

The settings controlled with these parameters are:

• solver_type: If FEAST is used. Another possibility is eigen_eigensystem. The following settings are only applicable, if feast is chosen.
• symmetric: True, if the input matrices are symmetric.
• number_of_eigenvalues: Specifies the number of eigenvalues to be found.
• e_min and e_max: Defines the search interval, where eigenvalues are found as e_min < λ < e_max. Note, that for strucural mechanics, the relation between eigenvalue λ and angular eigenfrequency ω is ω²=λ
• search_lowest_eigenvalues and search_highest_eigenvalues: Either search for the highest or lowest eigenvalues in the specified region. Only available for real, symmetric problems and in combination with a defined number_of_eigenvalues.
• sort_eigenvalues: True, if the resulting eigenvalues should be sorted-
• sort_order: Defines the sort order. Possible choices are sr (smallest real part), sm (smallest magnitude), lr (largest real part), lm (largest magnitude).
• subspace_size: Manually defines the size of the subspace used to find eigenvalues. The FEAST documentation recommends a size of 1.5 * numer of eigenvalues to be found. This value is overwritten, if numer_of_eigenvalues is specified.
• max_iteration: Maximum number of FEAST iterations.
• tolerance: Defines the FEAST solver tolerance.

Printing Results

In the StructuralMechanicsApplication there is a process for of postprocessing eigenvalues. It animates the results and writtes all EigenValues into one file for easier postprocessing.

PostprocessEigenvaluesProcess

This process is written in C++ but can be called by adding the corresponding python process to the list_other_processes in the ProjectParameters.json

    "list_other_processes" : [{
        "python_module"   : "postprocess_eigenvalues_process",
        "kratos_module"   : "KratosMultiphysics.StructuralMechanicsApplication",
        "help"                  : "This process postprocces the eigen values for GiD",
        "process_name"          : "PostProcessEigenvaluesProcess",
        "Parameters"            : { 
            "result_file_name" : "Structure",
            "result_file_format_use_ascii" : false,
            "computing_model_part_name"   : "computing_domain",
            "animation_steps"   :  20,
            "list_of_result_variables" : ["DISPLACEMENT"],
            "label_type" : "frequency"
        }
    }]

Certain Parameters can be chosen (but don't have to, the settings above are the defaults. If the defaults are to be used, simply omit the corresponding parameter):

• result_file_name : Name of the result file, will be followed by "_EigenResults"
• result_file_format_use_ascii : Usually the results are printed in binary format, this setting is only used for testing and debugging
• computing_model_part_name : Name of the Computing Model Part that is used as input for the solver
• animation_steps : Number of steps used to animate the eigenvectors
• list_of_result_variables : Variables that are used for the computation of the eigen-solution. These are used for the animation. In Structural Analysis this is the DISPLACEMENT
• label_type : Output the results in either angular_frequency [rad/s] or frequency [Hz]