Hilber-Hughes-Taylor Method: Difference between revisions

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This command is used to construct a Hilber-Hughes-Taylor (HHT) integration object.   
{{CommandManualMenu}}
 
This command is used to construct a Hilber-Hughes-Taylor (HHT) integration object.  This is an implicit method that allows for energy dissipation and second order accuracy (which is not possible with the regular Newmark method). Depending on choices of input parameters, the method can be unconditionally stable.


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


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{|
{|
|  style="width:150px" | '''$alpha ''' || <math>\alpha</math> factor
|  style="width:150px" | '''$alpha ''' || <math>\alpha</math> factor
|-
|'''$gamma''' || <math>\gamma</math> factor
|-
|'''$beta''' || <math>\beta</math> factor
|}
|}


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integrator HHT 0.5
integrator HHT 0.9




NOTES:
NOTES:
# If the accelerations are chosen as the unknowns and <math>\beta</math> is chosen as 0, the formulation results in the fast but conditionally stable explicit Central Difference method. Otherwise the method is implicit and requires an iterative solution process.
<math>\alpha</math> is defined differently that in the paper, we use <math>\alpha = (\alpha_{HHT} - 1})</math> where <math>\alpha_{HHT}</math> is that used in the paper.
# Two common sets of choices are
# 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.
## Average Acceleration Method (<math>\gamma=\tfrac{1}{2}, \beta = \tfrac{1}{4}</math>)
# <math> \alpha = 1.0</math> corresponds to the Newmark method.
## Constant Acceleration Method (<math>\gamma=\tfrac{1}{2}, \beta = \tfrac{1}{6}</math>)
# <math> \alpha</math> should be between 0.67 and 1.0. The smaller the <math>\alpha</math> the greater the numerical damping.
# <math>\gamma</math> and <math>\beta</math> are optional. The default values ensure the method is second order accurate and unconditionally stable when <math>\alpha</math> is <math>\tfrac{2}{3} <= \alpha <= 1.0</math>. The defaults are:
 
:<math> \beta = \frac{(2 - \alpha)^2}{4}</math>
 
and
 
:<math> \gamma = \frac{3}{2} - \alpha</math>


REFERENCES
REFERENCES


Hilber, H.M, Hughes,T.J.R and Talor, R.L.  "Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics" Earthquake Engineering and Structural Dynamics, 5:282-292, 1977.
Hilber, H.M, Hughes,T.J.R and Talor, R.L.  "Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics" Earthquake Engineering and Structural Dynamics, 5:282-292, 1977.


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The HHT 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 the same Newmark approximations are used
The HHT 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 equation of motion is used:
 
<math>M \ddot U_{t + \Delta t} + R(U_{t + \alpha, \dot U_{t+\alpha}) = F_{t+\Delta t} </math>
 
 
<math> M \ddot U_{t + \Delta t} + + (1 + \alpha) C \dot U_{t + \Delta t} - |alpha C \dot U_t + R(U_{t + \Delta t}) = F_{t + \Delta t} </math>                             
 
Using the Taylor series approximation of <math>U_{t+\Delta t}</math> and <math>\dot U_{t+\Delta t}</math>:
 
<math> U_{t+\Delta t} = U_t + \Delta t \dot U_t + \tfrac{\Delta t^2}{2} \ddot U_t + \tfrac{\Delta t^3}{6} \dot \ddot U_t + \cdots </math>
 
<math> \dot U_{t+\Delta t} = \dot U_t + \Delta t \ddot U_t + \tfrac{\Delta t^2}{2} \dot \ddot U_t +  \cdots </math>
 
Newton truncated these using the following:
 
<math> U_{t+\Delta t} = u_t + \Delta t \dot U_t + \tfrac{\Delta t^2}{2} \ddot U +  \beta {\Delta t^3} \dot \ddot U </math>
 
<math> \dot U_{t + \Delta t} = \dot U_t + \Delta t \ddot U_t + \gamma \Delta t^2 \dot \ddot U </math>
 
in which he assumed linear acceleration within a time step, i.e.
 
<math> \dot \ddot U = \frac{{\ddot U_{t+\Delta t}} -  \ddot U_t}{\Delta t} </math>
 
which results in the following expressions:
 
<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>
 
The variables <math>\beta</math> and <math>\gamma</math> are numerical parameters that control both the stability of the method and the amount of numerical damping introduced into the system by the method. For <math>\gamma=\tfrac{1}{2}</math> there is no numerical damping; for <math>\gamma>=\tfrac{1}{2}</math> numerical damping is introduced. Two well known and commonly used cases are:
# Average Acceleration Method (<math>\gamma=\tfrac{1}{2}, \beta = \tfrac{1}{4}</math>)
# Constant Acceleration Method (<math>\gamma=\tfrac{1}{2}, \beta = \tfrac{1}{6}</math>)
 
 
The advancement of the nonlinear solution from one step to the next requires setting initial conditions at the first trial step<math> U_{t+\Delta t}^0, \dot U_{t+\Delta t}^0, \ddot U_{t+\Delta t}^0</math> linearization of the equation, and solution of the linearized equations and updating. When the displacements are the unknowns this results in the following:
 
1) Initial Conditions at a time step: For an implicit solution we choose:
 
<math>U_{t+\Delta t}^0 = U_t</math>
 
The values for the velocity and accelerations are determined using the Newmark formulas.
 
<math>\dot U_{t+\Delta t}^0 =  (1 - \frac{\gamma}{\beta}) \dot U_t + \Delta t(1 - \frac{\gamma}{2 \beta}) \ddot U_t</math>
 
<math>\ddot U_{t+\Delta t}^0 = - \frac{1}{\beta \Delta t} \dot U_t - (\frac{1}{2 \beta} -1) \ddot U_t </math>


2) Linearizing the Newmark formulas
:<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>dU_{t+\Delta t}^{i+1} = \beta \Delta t^2 d \ddot U_{t+\Delta t}^{i+1}</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>d \dot U_{t+\Delta t}^{i+1} = \gamma \Delta t \ddot U_{t+\Delta t}^{i+1}</math>
but the time-discrete momentum equation is modified:


giving the update formula when displacement increment is used as unknown as:
:<math>R_{t + \alpha \Delta t} =  F_{t+\Delta t}^{ext} - M \ddot U_{t + \Delta t} - C \dot U_{t+\alpha \Delta t} - F^{int}(U_{t + \alpha \Delta t})
</math>


<math>U_{t+\Delta t}^{i+1} = U_{t+\Delta t}^i + dU_{t+\Delta t}^{i+1}</math>
where the displacements and velocities at the intermediate point are given by:


<math>\dot U_{t+\Delta t}^{i+1} = \dot U_{t+\Delta t}^i + \frac{\gamma}{\beta \Delta t}dU_{t+\Delta t}^{i+1}</math>
:<math>U_{t+ \alpha \Delta t} = (1 - \alpha) U_t + \alpha U_{t + \Delta t}</math>


<math>\ddot U_{t+\Delta t}^{i+1} = \ddot U_{t+\Delta t}^i + \frac{1}{\beta \Delta t^2}dU_{t+\Delta t}^{i+1}</math>
:<math>\dot U_{t+\alpha \Delta t} = (1-\alpha) \dot U_t + \alpha \dot U_{t + \Delta t}</math>


The linearization of the momentum equation using the displacements as the unknowns leads to the following
Following the methods outlined for Newmarks method, loinearization of the nonlinear momentum equation results in the following linear equations:


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


and
and


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


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


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Code Developed by: <span style="color:blue"> fmk </span>
Code Developed by: <span style="color:blue"> fmk </span>

Latest revision as of 19:19, 11 June 2015



This command is used to construct a Hilber-Hughes-Taylor (HHT) integration object. This is an implicit method that allows for energy dissipation and second order accuracy (which is not possible with the regular Newmark method). Depending on choices of input parameters, the method can be unconditionally stable.

integrator HHT $alpha <$gamma $beta>
$alpha <math>\alpha</math> factor
$gamma <math>\gamma</math> factor
$beta <math>\beta</math> factor

EXAMPLE:


integrator HHT 0.9


NOTES: <math>\alpha</math> is defined differently that in the paper, we use <math>\alpha = (\alpha_{HHT} - 1})</math> where <math>\alpha_{HHT}</math> is that used in the paper.

  1. 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.
  2. <math> \alpha = 1.0</math> corresponds to the Newmark method.
  3. <math> \alpha</math> should be between 0.67 and 1.0. The smaller the <math>\alpha</math> the greater the numerical damping.
  4. <math>\gamma</math> and <math>\beta</math> are optional. The default values ensure the method is second order accurate and unconditionally stable when <math>\alpha</math> is <math>\tfrac{2}{3} <= \alpha <= 1.0</math>. The defaults are:
<math> \beta = \frac{(2 - \alpha)^2}{4}</math>

and

<math> \gamma = \frac{3}{2} - \alpha</math>

REFERENCES

Hilber, H.M, Hughes,T.J.R and Talor, R.L. "Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics" Earthquake Engineering and Structural Dynamics, 5:282-292, 1977.


THEORY:


The HHT 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 \Delta t} = F_{t+\Delta t}^{ext} - M \ddot U_{t + \Delta t} - C \dot U_{t+\alpha \Delta t} - F^{int}(U_{t + \alpha \Delta t})

</math>

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

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

Following the methods outlined for Newmarks method, loinearization 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 K_t + \frac{\alpha \gamma}{\beta \Delta t} C_t + \frac{1}{\beta \Delta t^2} M</math>

and

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


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

Code Developed by: fmk

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