NewmarkBeta

Background

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

\[ V_{n+1} = \frac{2}{\Delta t} \left( U_{n+1} - U_n \right) - V_n \\ A_{n+1} = \frac{4}{\Delta t^2} \left( U_{n+1} - U_n \right) - \frac{4}{\Delta t} V_n - A_n \]

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

\[ \textbf{K}_{\textrm{eff}} = \textbf{K}^{(i-1)}_{n+1} + \frac{4}{\Delta t^2} \textbf{M} + \frac{2}{\Delta t} \textbf{C} \\ \textrm{F}_{\textrm{eff}} = \textrm{R}_{n+1}^{(i)} - \textrm{F}_{n+1}^{(i-1)} + \textbf{M} \left(\frac{4}{\Delta t}V_n + A_n - \frac{4}{\Delta t^2} \Delta U \right) + \textbf{C} \left(V_n - \frac{2}{\Delta t} \Delta U\right) \]

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:

• Bathe, K. Jurgen, "Finite Element Procedures", Chapter 6: pages 780-782. Prentice-Hall, 1996.
• J. P. Conte and P. K. Vijalapura and M. Meghella, "Consistent Finite-Element Response Sensitivity Analysis", Journal of Engineering Mechanics, volume 129 (12), 1380-1393, 2003.

Pre-Analysis

The python Pre-Analysis in the 01-Pre_Process/Method/Attach.py file provides with an interface to create a ExtendedNewmarkBeta 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 NEWMARK
    • '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 NEWMARK integrator can be defined using the python interface as follows:
  SVL.addIntegrator(tag=1, attributes={'name': 'Newmark', 'dt': 0.00100})

  Application
  Please refer to F03-DY_Lin_2DPointLoad_J2PStrain_Quad4.py file located at 03-Validations/01-Debugging/ to see an example on how to define a NEWMARK 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/03-Newmark/ExtendedNewmarkBeta.cpp file provides the class implementation. A ExtendedNewmarkBeta 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": "NEWMARK",
                      "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 NewmarkBeta formulation here presented correspond to the constant acceleration.

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