CentralDifference

Background

The CentralDifference class creates an explicit second order integrator that is used in a DynamicAnalysis. This integrator assumes that velocity and acceleration can be approximated as,

\[ V_{n+1} = \frac{1}{2 \,\Delta t} \left( U_{n+1} - U_{n-1} \right)\,, \\ A_{n+1} = \frac{1}{\Delta t^2} \left( U_{n+1} - 2 \, U_n + U_{n-1} \right) \,, \]

thus, the effective stiffness matrix and force vector are computed as,

\[ \textbf{K}_{\textrm{eff}} = \frac{1}{\Delta t^2} \textbf{M} + \frac{1}{2 \Delta t} \textbf{C} \\ \textrm{F}_{\textrm{eff}} = \rm{R}_n - \rm{F}_n + \left(\frac{1}{\Delta t} \textbf{M} - \frac{1}{2 \Delta t} \textbf{C} \right) \left( U_n - U_{n-1}\right) \]

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:

• Finite Element Procedures, Bathe, K.J., Chapter 9: pages 770-774. 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 CentralDifference 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 CENTRALDIFFERENCE
    • '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 CENTRALDIFFERENCE integrator can be defined using the python interface as follows:
  SVL.addIntegrator(tag=1, attributes={'name': 'CentralDifference', 'dt': 0.00100})
  

  Application
  Please refer to A06-DY_Lin_1DPointMass_Elastic_ZeroLength_SupportMotion.py file located at 03-Validations/01-Debugging/ to see an example on how to define a CENTRALDIFFERENCE 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/02-CentralDifference/CentralDifference.cpp file provides the class implementation. A CentralDifference 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": "CENTRALDIFFERENCE",
                      "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 the analysis.

  
  

  Attention

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