====== Explicit dynamic integration schemes ======

===== Description =====

The equilibrium equation between internal forces $F^{int}$, inertial forces $Ma$ (where $M$ is the diagonalized mass matrix and $a$ the acceleration) and external forces $F^{ext}$ :

$$Ma+F^{int}=F^{ext}$$


==== Central difference method ====

Relations between displacements $x$, velocities $v$ and accelerations $a$ are:

$$v(t^{n+1/2}) = v(t^{n-1/2}) + (t^{n+1}-t^n) a(t^n) $$ \\
$$x(t^{n+1}) = x(t^n) + (t^{n+1}-t^n) v(t^{n+1/2}) $$

The equilibrium equation becomes :

$$a(t^{n+1}) = (F^{ext}(t^{n+1}) - F^{int}(t^{n+1}))/M $$

This scheme is conditionally stable (time step limited) and non dissipative.

==== Alpha-generalized scheme ====

Same relations as in the implicit [[dynimpl|alpha-generalized]] scheme, but with the parameter used to weight internal and external forces equal to 1, leading to :
 
$$(1-\alpha_M) a(t^{n+1}) + \alpha_M a(t^n) = \frac{F^{ext}(t^n) - F^{int}(t^n)}{M}$$

Relations between displacements $x$, velocities $v$ and accelerations $a$ are:

$$x(t^{n+1}) = x(t^n) + (t^{n+1}-t^n) v(t^n) + (t^{n+1}-t^n)^2 \left( (0.5-\beta)a(t^n) + \beta a(t^{n+1})\right) $$
$$v(t^{n+1}) = v(t^n) + (t^{n+1}-t^n) {(1-\gamma)a(t^n) + \gamma a(t^{n+1})} $$

Specific values leading to an optimal numerical dissipation are given as function of the spectral radius $\rho_\beta$ (''MDR_ECHR'') for a bifurcation frequency (a spectral radius equal to 1 leads to a conservative algorithm when a spectral radius lower than 1 leads to a dissipative one ((see real parameters)):

$$\alpha_M = (2\rho_\beta-1)/(1+\rho_\beta) $$\\
$$\gamma = 3/2 - \alpha_M $$\\
$$\beta = \frac{5-3\rho_\beta}{(1+\rho_\beta)^2 (2-\rho_\beta)}$$

Conditionally stable.

==== Tchamwa Scheme ====



Explicit algorithm where numerical dissipation is monitored by the parameter $\phi$. 

Equilibrium computed with

$$a(t^{n+1}) = \frac{F^{ext}(t^{n+1}) - F^{int}(t^{n+1})}{M}$$

Relations between displacements $x$, velocities $v$ and accelerations $a$ are:

$$x(t^{n+1}) = x(t^n) + (t^{n+1}-t^n) v(t^n) + \phi (t^{n+1}-t^n)^2 a(t^n) $$\\
$$v(t^{n+1}) = v(t^n) + (t^{n+1}-t^n) a(t^n) $$

Stability guaranteed for  $\phi \geq 1 $ and high frequencies killed over a single time step for \phi = 2$. the scheme is of : 
  * second order for $\phi = 1$ (no numerical dissipation) 
  * first order for $\phi > 1$ (numerical dissipation) 

Relation between $\phi$ and spectral radius for the bifurcation $\rho_\beta$ (user parameter ''MDR_ECHR'') is:
  * $$\phi = \frac{2(1- \rho_\beta^{1/2})}{(1-\rho_\beta)} \mbox{ if } \rho_\beta < 1 $$
  * $$\phi = 1 \mbox{ if } \rho_\beta = 1 $$

===== Input file =====

See [[dynimpl|dynamic implicit]] scheme for definition of density and initial velocities.

==== Old Metafor Version <= 2422 ====

=== Choosing the algorithm ===

^       Scheme         ^  ''MDE_NDYN''  ^  ''MDR_ECHR''  ^
| Certered difference  |       1        |                |
| Chung Hulbert        |       3        |       X        |
| Tchamwa              |       6        |       X        |

(see [[doc:user:integration:general:parameters]])

==== New Metafor Version > 2422 ====

=== Centered Difference ===

<code>
ti = CentralDifferenceTimeIntegration(metafor)
metafor.setTimeIntegration(ti)
</code>

=== Chung Hulbert ===

<code>
ti = ChExplicitTimeIntegration(metafor)
ti.setRhoB(_rhoB)
metafor.setTimeIntegration(ti)
</code>

The parameter ''_rhoB'' is the spectral radius at bifurcation point ([0, 1]). The default value is 0.8182.

=== Tchamwa === 

<code>
ti = TchamwaExplicitTimeIntegration(metafor)
ti.setRhoB(_rhoB)
metafor.setTimeIntegration(ti)
</code>

The parameter ''_rhoB'' is the spectral radius at bifurcation point ([0, 1]). The default value is 0.8182.

Other parameters : see [[quasistatique]]