MPM Solver (Base)

Overview

The solver is responsible for handling the simulation physics and solving the problem. We should remark the following points, that apply to all Kratos Applications:

• solvers are always implemented at the Python level;
• all solvers extend the base PythonSolver class, which can be found in this script. This base solver does not include any physics and its purpose is to define the solver class API.

Each Kratos Multiphysics application may implement multiple solvers, each tailored to different types of problems or simulation strategies. For example, the MPMApplication implements the following solvers:

• MPMImplicitDyanamicSolver - solves time-dependent problems by means of an implicit time scheme;
• MPMExplicitSolver - solves time-dependent problems using an explicit time scheme;
• MPMQuasiStaticSolver - solves quasi-static problems;
• MPMStaticSolver - solves static problems.

In Kratos, it is common practice to define a base solver for a specific physics-based application and then derive specialized solvers from it. For instance, all MPM solvers listed before extend the base MPMSolver, which manages common aspects of MPM-based simulations. The MPMSolver itself extendes the base PythonSolver class.

The solver is also responsible for creating and managing the solution strategy (e.g., the Newton-Raphson algorithm for non-linear problems) and the linear solver (if required) to solve the system of equations.

The solver is initialized through a custom AnalysisStage class, which uses (part of) the parameters from the solver_settings section of the ProjectParameters.json input file (introduced here). The solver_settings also contain other parameters used by the solver to define the linear solver and the solution strategy.

Once initialised, the solver can be accessed from the AnalysisStage class by calling the _GetSolver() method.

Parameters & Defaults

The most important parameters contained in the solver_settings section of the ProjectParameters.json input file and required for the initialisation and configuration of MPM solvers extending the MPMSolver class are the following.

{
    "solver_settings"  : {
        "model_part_name"                 : "MPM_Material",
        "echo_level"                      : 0,
        "domain_size"                     : 2,
        "solver_type"                     : "Dynamic",
        "time_stepping"                   : {
            "time_step" : 0.01
        },
        "grid_model_import_settings"      : {
            "input_type"     : "mdpa",
            "input_filename" : "file_name_Grid"
        },
        "model_import_settings"           : {
            "input_type"     : "mdpa",
            "input_filename" : "file_name_Body"
        },
        "material_import_settings"        : {
            "materials_filename" : "ParticleMaterials.json"
        },
        "analysis_type"                   : "non_linear",
        "convergence_criterion"           : "residual_criterion",
        "displacement_relative_tolerance" : 0.0001,
        "displacement_absolute_tolerance" : 1e-9,
        "residual_relative_tolerance"     : 0.0001,
        "residual_absolute_tolerance"     : 1e-9,
        "max_iteration"                   : 10,
        "pressure_dofs"                   : false,
        "linear_solver_settings"          : {
            "solver_type" : "LinearSolversApplication.sparse_lu"
        },
        "auxiliary_variables_list"        : ["NORMAL","IS_STRUCTURE"]
    }
}

• model_part_name: string with the name to be assigned to the material point model part.
• echo_level: an integer controlling the verbosity level of the simulation output. Higher values result in more messages printed to the standard output.
• domain_size: size of the domain in which the problem is set; admissible values are 2 (two-dimensional problems) and 3 (three-dimensional problems).
• solver_type: string used to initialize the appropriate MPM solver class depending on the type of problem (dynamic, quasi-static, static) to be solved. Admissible values are:
  • "dynamic": initializes either the MPMImplicitDyanamicSolver or the MPMExplicitSolver, depending on whether the additional parameter time_integration_method is set to "implicit" or "explicit", respectively;
  • "quasi-static": initializes the MPMQuasiStaticSolver;
  • "static": initializes the MPMStaticSolver.

  The following table summarizes the available solvers and the parameters used to initialise them

• time_stepping: object defining the time step (if required).
• grid_model_import_settings: object containing the following name-value pairs:
  • input_filename: name (without extension) of the file defining the background grid
  • input_type: input file type.
• model_import_settings: object containing the following name-value pairs:
  • input_filename: name (without extension) of the file defining the mesh discretising the body to be simulated by means of the MPM
  • input_type: input file type.
• material_import_settings: name of the json file (typically ParticleMaterials.json) containing the information about the physical properties of the materials.
• analysis_type: admissible values are "linear" and "non_linear". If set to "non_linear", a linearization procedure based on the Newton-Raphson strategy is used. In this case, the following additional parameters defining the convergence criterion of the Newton-Raphson algorithm must be defined.
  • convergence_criterion: type of convergence criterion to be used; available options are "displacement_criterion" and "residual_criterion".
  • displacement_relative_tolerance: relative tolerance value for displacement-based convergence criterion.
  • displacement_absolute_tolerance: absolute tolerance value for displacement-based convergence criterion.
  • residual_relative_tolerance: relative tolerance value for residual-based convergence criterion.
  • residual_absolute_tolerance: absolute tolerance value for residual-based convergence criterion.
  • max_iteration: maximum number of non-linear iterations.
• pressure_dofs: boolean variable identifying the type of MPM formulation to be used (displacement based if false, mixed displacement-pressure if true).
• linear_solver_settings: object defining the type of solver to be used for solving the linear system.
• auxiliary_variables_list: list of strings, each one representing an additional variable required in the simulation process.

Source Code

MPMApplication/python_scripts/mpm_solver.py