Generalized Alpha Method: Difference between revisions

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This command is used to construct a Generalized <math>\alpha</math> integration object.  This is an implicit method that like the HHT method allows for high frequency energy dissipation and second order accuracy, i.e. <math>\Delta t^2</math>. Depending on choices of input parameters, the method can be unconditionally stable.
This command is used to construct a Generalized <math>\alpha</math> integration object.  This is an implicit method that like the HHT method allows for high frequency energy dissipation and second order accuracy, i.e. <math>\Delta t^2</math>. Depending on choices of input parameters, the method can be unconditionally stable.


{|  
{|  
| style="background:yellow; color:black; width:800px" | '''integrator HHT $alphaM $alphaF <$gamma $beta> '''
| style="background:yellow; color:black; width:800px" | '''integrator GeneralizedAlpha $alphaM $alphaF <$gamma $beta> '''
|}
|}


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integrator GeneralizedAlpha 1.0 0.5
integrator GeneralizedAlpha 1.0 0.8




NOTES:
NOTES:
#<math> \gamma_F</math> and <math>\gamma_M</math> are defined differently that in the paper, we use <math>\gamma_F = (1-\gamma_f)</math> and <math>\gamma_M=(1-\gamma_m)</math> where <math>\gamma_f</math> and <math>\gamma_m</math> are those used in the paper.
#<math> \alpha_F</math> and <math>\alpha_M</math> are defined differently that in the paper, we use <math>\alpha_F = (1-\alpha_f)</math> and <math>\alpha_M=(1-\gamma_m)</math> where <math>\alpha_f</math> and <math>\alpha_m</math> are those used in the paper.
# Like Newmark and all the implicit schemes, the unconditional stability of this method applies to linear problems. There are no results showing stability of this method over the wide range of nonlinear problems that potentially exist. Experience indicates that the time step for implicit schemes in nonlinear situations can be much greater than those for explicit schemes.
# Like Newmark and all the implicit schemes, the unconditional stability of this method applies to linear problems. There are no results showing stability of this method over the wide range of nonlinear problems that potentially exist. Experience indicates that the time step for implicit schemes in nonlinear situations can be much greater than those for explicit schemes.
# <math> \alpha_M = 1.0, \alpha_F = 1.0</math> produces the Newmark Method.
# <math> \alpha_M = 1.0, \alpha_F = 1.0</math> produces the Newmark Method.
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The defaults are:
The defaults are:


<math> \gamma = \tfrac{1}{2} + \gamma_M - \gamma_F</math>
:<math> \gamma = \tfrac{1}{2} + \gamma_M - \gamma_F</math>


and
and


<math> \beta = \tfrac{1}{4}(1 + \gamma_M - \gamma_F)^2</math>
:<math> \beta = \tfrac{1}{4}(1 + \gamma_M - \gamma_F)^2</math>


REFERENCES
REFERENCES
J. Chung, G.M.Hubert.  "A Time Integration Algorithm for Structural Dynamics with Improved Numerical Dissipation: The Generalized-<math>\alpha</math> Method" ASME Journal of Applied Mechanics, 60, 371:375, 1993.
J. Chung, G.M.Hubert.  "A Time Integration Algorithm for Structural Dynamics with Improved Numerical Dissipation: The Generalized-<math>\alpha</math> Method" ASME Journal of Applied Mechanics, 60, 371:375, 1993.
note: the definition of <math>\alpha</math> is different from the original paper and in the book by Hughes.


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The Generalized <math>\alpha</math> method (sometimes called the <math> \alpha</math> method) is a one step implicit method for solving the transient problem which attempts to increase the amount of numerical damping present without degrading the order of accuracy. In the HHT method, the same Newmark approximations are used:
The Generalized <math>\alpha</math> method (sometimes called the <math> \alpha</math> method) is a one step implicit method for solving the transient problem which attempts to increase the amount of numerical damping present without degrading the order of accuracy. In the HHT method, the same Newmark approximations are used:


<math> U_{t+\Delta t} = U_t + \Delta t \dot U_t + [(0.5 - \beta) \Delta t^2] \ddot U_t + [\beta \Delta t^2]  \ddot U_{t+\Delta t}</math>
:<math> U_{t+\Delta t} = U_t + \Delta t \dot U_t + [(0.5 - \beta) \Delta t^2] \ddot U_t + [\beta \Delta t^2]  \ddot U_{t+\Delta t}</math>


<math> \dot U_{t+\Delta t} = \dot U_t + [(1-\gamma)\Delta t] \ddot U_t + [\gamma \Delta t ] \ddot U_{t+\Delta t} </math>
:<math> \dot U_{t+\Delta t} = \dot U_t + [(1-\gamma)\Delta t] \ddot U_t + [\gamma \Delta t ] \ddot U_{t+\Delta t} </math>


but the time-discrete momentum equation is modified:
but the time-discrete momentum equation is modified:


<math>R_{t + \alpha_M \Delta t} =  F_{t+\Delta t}^{ext} - M \ddot U_{t + \alpha_M \Delta t} - C \dot U_{t+\alpha_F \Delta t} - F^{int}(U_{t + \alpha_F \Delta t})  
:<math>R_{t + \alpha_M \Delta t} =  F_{t+\Delta t}^{ext} - M \ddot U_{t + \alpha_M \Delta t} - C \dot U_{t+\alpha_F \Delta t} - F^{int}(U_{t + \alpha_F \Delta t})  
</math>
</math>


where the displacements and velocities at the intermediate point are given by:
where the displacements and velocities at the intermediate point are given by:


<math>U_{t+ \alpha_F \Delta t} = (1 - \alpha_F) U_t + \alpha_F U_{t + \Delta t}</math>
:<math>U_{t+ \alpha_F \Delta t} = (1 - \alpha_F) U_t + \alpha_F U_{t + \Delta t}</math>


<math>\dot U_{t+\alpha_F \Delta t} = (1-\alpha_F) \dot U_t + \alpha_F \dot U_{t + \Delta t}</math>
:<math>\dot U_{t+\alpha_F \Delta t} = (1-\alpha_F) \dot U_t + \alpha_F \dot U_{t + \Delta t}</math>


<math>\ddot U_{t+\alpha_M \Delta t} = (1-\alpha_M) \ddot U_t + \alpha_M \ddot U_{t + \Delta t}</math>
:<math>\ddot U_{t+\alpha_M \Delta t} = (1-\alpha_M) \ddot U_t + \alpha_M \ddot U_{t + \Delta t}</math>


Following the methods outlined for Newmarks method, linearization of the nonlinear momentum equation results in the following linear equations:
Following the methods outlined for Newmarks method, linearization of the nonlinear momentum equation results in the following linear equations:


<math> K_{t+\Delta t}^{*i} d U_{t+\Delta t}^{i+1} = R_{t+\Delta t}^i</math>
:<math> K_{t+\Delta t}^{*i} d U_{t+\Delta t}^{i+1} = R_{t+\Delta t}^i</math>


where
where


<math>K_{t+\Delta t}^{*i} = \alpha K_t + \frac{\alpha \gamma}{\beta \Delta t} C_t + \frac{1}{\beta \Delta t^2} M</math>
:<math>K_{t+\Delta t}^{*i} = \alpha_F K_t + \frac{\alpha_F \gamma}{\beta \Delta t} C_t + \frac{\alpha_M}{\beta \Delta t^2} M</math>


and
and


<math> R_{t+\Delta t}^i = F_{t + \Delta t}^{ext} - F(U_{t + \alpha F \Delta t}^{i-1})^{int} - C \dot U_{t+\alpha F \Delta t}^{i-1} - M \ddot U_{t+ \alpha M \Delta t}^{i-1}</math>
:<math> R_{t+\Delta t}^i = F_{t + \Delta t}^{ext} - F(U_{t + \alpha F \Delta t}^{i-1})^{int} - C \dot U_{t+\alpha F \Delta t}^{i-1} - M \ddot U_{t+ \alpha M \Delta t}^{i-1}</math>





Latest revision as of 18:01, 4 April 2012




This command is used to construct a Generalized <math>\alpha</math> integration object. This is an implicit method that like the HHT method allows for high frequency energy dissipation and second order accuracy, i.e. <math>\Delta t^2</math>. Depending on choices of input parameters, the method can be unconditionally stable.

integrator GeneralizedAlpha $alphaM $alphaF <$gamma $beta>

$alphaM <math>\alpha_M</math> factor
$alphaF <math>\alpha_F</math> factor
$gamma <math>\gamma</math> factor
$beta <math>\beta</math> factor

EXAMPLE:


integrator GeneralizedAlpha 1.0 0.8


NOTES:

  1. <math> \alpha_F</math> and <math>\alpha_M</math> are defined differently that in the paper, we use <math>\alpha_F = (1-\alpha_f)</math> and <math>\alpha_M=(1-\gamma_m)</math> where <math>\alpha_f</math> and <math>\alpha_m</math> are those used in the paper.
  2. Like Newmark and all the implicit schemes, the unconditional stability of this method applies to linear problems. There are no results showing stability of this method over the wide range of nonlinear problems that potentially exist. Experience indicates that the time step for implicit schemes in nonlinear situations can be much greater than those for explicit schemes.
  3. <math> \alpha_M = 1.0, \alpha_F = 1.0</math> produces the Newmark Method.
  4. <math> \alpha_M = 1.0</math> corresponds to the HHT method.
  5. The method is second-order accurate provided <math>\gamma = \tfrac{1}{2} + \alpha_M - \alpha_F</math>
  6. The method is unconditionally stable provided <math>\alpha_M >= \alpha_F >= \tfrac{1}{2}, \beta>=\tfrac{1}{4} +\tfrac{1}{2}(\gamma_M - \gamma_F)</math>
  7. <math>\gamma</math> and <math>\beta</math> are optional. The default values ensure the method is unconditionally stable, second order accurate and high frequency dissipation is maximized.

The defaults are:

<math> \gamma = \tfrac{1}{2} + \gamma_M - \gamma_F</math>

and

<math> \beta = \tfrac{1}{4}(1 + \gamma_M - \gamma_F)^2</math>

REFERENCES

J. Chung, G.M.Hubert. "A Time Integration Algorithm for Structural Dynamics with Improved Numerical Dissipation: The Generalized-<math>\alpha</math> Method" ASME Journal of Applied Mechanics, 60, 371:375, 1993.



THEORY:


The Generalized <math>\alpha</math> method (sometimes called the <math> \alpha</math> method) is a one step implicit method for solving the transient problem which attempts to increase the amount of numerical damping present without degrading the order of accuracy. In the HHT method, the same Newmark approximations are used:

<math> U_{t+\Delta t} = U_t + \Delta t \dot U_t + [(0.5 - \beta) \Delta t^2] \ddot U_t + [\beta \Delta t^2] \ddot U_{t+\Delta t}</math>
<math> \dot U_{t+\Delta t} = \dot U_t + [(1-\gamma)\Delta t] \ddot U_t + [\gamma \Delta t ] \ddot U_{t+\Delta t} </math>

but the time-discrete momentum equation is modified:

<math>R_{t + \alpha_M \Delta t} = F_{t+\Delta t}^{ext} - M \ddot U_{t + \alpha_M \Delta t} - C \dot U_{t+\alpha_F \Delta t} - F^{int}(U_{t + \alpha_F \Delta t})

</math>

where the displacements and velocities at the intermediate point are given by:

<math>U_{t+ \alpha_F \Delta t} = (1 - \alpha_F) U_t + \alpha_F U_{t + \Delta t}</math>
<math>\dot U_{t+\alpha_F \Delta t} = (1-\alpha_F) \dot U_t + \alpha_F \dot U_{t + \Delta t}</math>
<math>\ddot U_{t+\alpha_M \Delta t} = (1-\alpha_M) \ddot U_t + \alpha_M \ddot U_{t + \Delta t}</math>

Following the methods outlined for Newmarks method, linearization of the nonlinear momentum equation results in the following linear equations:

<math> K_{t+\Delta t}^{*i} d U_{t+\Delta t}^{i+1} = R_{t+\Delta t}^i</math>

where

<math>K_{t+\Delta t}^{*i} = \alpha_F K_t + \frac{\alpha_F \gamma}{\beta \Delta t} C_t + \frac{\alpha_M}{\beta \Delta t^2} M</math>

and

<math> R_{t+\Delta t}^i = F_{t + \Delta t}^{ext} - F(U_{t + \alpha F \Delta t}^{i-1})^{int} - C \dot U_{t+\alpha F \Delta t}^{i-1} - M \ddot U_{t+ \alpha M \Delta t}^{i-1}</math>


The linear equations are used to solve for <math> U_{t+\alpha F \Delta t}, \dot U_{t + \alpha F \Delta t} \ddot U_{t+ \alpha M \Delta t}</math>. Once convergence has been achieved the displacements, velocities and accelerations at time <math>t + \Delta t</math> can be computed.


Code Developed by: fmk