Newmark Method
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This command is used to construct a Newmark integrator object.
integrator Newmark $gamma $beta |
$gamma | <math>\gamma</math> factor |
$beta | <math>\beta</math> factor |
EXAMPLE:
integrator Newmark 0.5 0.25
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.
- Two common sets of choices are
- Average Acceleration Method (<math>\gamma=\tfrac{1}{2}, \beta = \tfrac{1}{4}</math>)
- Linear Acceleration Method (<math>\gamma=\tfrac{1}{2}, \beta = \tfrac{1}{6}</math>)
- <math> \gamma > \tfrac{1}{2}</math> results in numerical damping proportional to <math> \gamma - \tfrac{1}{2}</math>
- The method is second order accurate if and only if <math>\gamma=\tfrac{1}{2}</math>
- The method is conditionally stable for <math> \beta >= \frac{\gamma}{2} >= \tfrac{1}{4}</math>
REFERENCES
Newmark, N.M. "A Method of Computation for Structural Dynamics" ASCE Journal of Engineering Mechanics Division, Vol 85. No EM3, 1959.
THEORY:
The Newmark method is a one step implicit method for solving the transient problem, represented by the residual for the momentum equation:
- <math> R_{t + \Delta t} = F_{t+\Delta t}^{ext} - M \ddot U_{t + \Delta t} - C \dot U_{t + \Delta t} + F(U_{t + \Delta t})^{int}</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 linearization of the Newmark equations gives:
- <math>dU_{t+\Delta t}^{i+1} = \beta \Delta t^2 d \ddot U_{t+\Delta t}^{i+1}</math>
- <math>d \dot U_{t+\Delta t}^{i+1} = \gamma \Delta t \ddot U_{t+\Delta t}^{i+1}</math>
which gives the update formula when displacement increment is used as unknown in the linearized system as:
- <math>U_{t+\Delta t}^{i+1} = U_{t+\Delta t}^i + dU_{t+\Delta t}^{i+1}</math>
- <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>\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>
The linearization of the momentum equation using the displacements as the unknowns leads to the following linear equation:
- <math> K_{t+\Delta t}^{*i} \Delta U_{t+\Delta t}^{i+1} = R_{t+\Delta t}^i</math>
where
- <math>K_{t+\Delta t}^{*i} = K_t + \frac{\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 + \Delta t}^{i-1})^{int} - C \dot U_{t+\Delta t}^{i-1} - M \ddot U_{t+ \Delta t}^{i-1}</math>
Code Developed by: fmk