QuasiStatic

Background

The QuasiStatic keyword creates an integrator class that is used for StaticAnalysis. This integrator assumes that velocity and acceleration are \(V_{n+1} = 0, \; A_{n+1} = 0\), therefore, the effective stiffness matrix and force vector become,

\[ \textbf{K}_{\textrm{eff}} = \textbf{K}^{(i-1)} \nonumber \\ \textrm{F}_{\textrm{eff}} = \textrm{R}_{\textrm{n}}^{(i)} - \textrm{F}_{\textrm{int}}^{(i-1)} \nonumber \]

where \(\textbf{K}_{\textrm{eff}}\) and \(\textrm{F}_{\textrm{eff}}\) are used to compute \(\delta U^{(i)} = \textbf{K}_{\textrm{eff}}^{-1} \textrm{F}_{\textrm{eff}}\) and \(\Delta U = \displaystyle{\sum_i \delta U^{(i)}}\) used in the Algorithm.

REFERENCE:

• Bathe, K. Jurgen, "Finite Element Procedures", Chapter 8. Prentice-Hall, 1996.

Pre-Analysis

The python Pre-Analysis in the 01-Pre_Process/Method/Attach.py file provides with an interface to create a QuasiStatic integrator. We use the addIntegrator() as follows:

• addIntegrator(tag, attributes):

  • tag : The identifier of this integrator, i.e., tag > -1
  • attributes : Specific properties for the created integrator, for example
    • 'name' : The integrator's name, in this case STATIC
    • 'dt' : The integrator's time step
    • 'ktol' : Tolerance to set a stiffness matrix value kij = 0.0
    • 'mtol' : Tolerance to set a mass matrix value mij = 0.0
    • 'ftol' : Tolerance to set a force vector value fj = 0.0

  Example

  A STATIC integrator can be defined using the python interface as follows:
  SVL.addIntegrator(tag=2, attributes={'name': 'Static'})

  Application
  Please refer to C08-ST_kin_2DCantilever_Elastic_Truss2.py file located at 03-Validations/01-Debugging/ to see an example on how to define a STATIC integrator using the addAlgorithm function.

On the contrary, the 01-Pre_Process/Method/Remove.py file provides with an interface to depopulate the Entities dictionary. For example, to remove an already define Integrator, use:

• delIntegrator(tag):
  • tag : The identifier of the integrator to be removed, i.e., tag > -1

Run-Analysis

The C++ Run-Analysis in the 02-Run_Process/10-Integrators/01-QuasiStatic/QuasiStatic.cpp file provides the class implementation. A QuasiStatic is created using the built-in json parse-structure provided in the Driver.hpp and is defined inside the "Simulations" json field indicating its "Tag" as follows,

• {
      "Simulations": {
          "Tag": {
              "combo": int,
              "attributes": {
                  "integrator": {
                      "name": "STATIC",
                      "ktol": double,
                      "mtol": double,
                      "ftol": double,
                      "dt": double,
                  }
              }
          }
      }
  }
  

  Variable	Description
  combo	The combination to which this Integrator will be defined.
  mTol	The specified tolerance for setting \(M_{ij} \sim 0.0\).
  kTol	The specified tolerance for setting \(K_{ij} \sim 0.0\).
  fTol	The specified tolerance for setting \(F_{j} \sim 0.0\).
  dt	The time step, for this case \(\Delta_t = 0.0\).

  
  

  Attention

  The default tolerance for \(M_{ij}\), \(K_{ij}\), and \(F_{j}\) is \(10^{-15}\).

  The value dt is not used in static analysis.