# Metafor

ULiege - Aerospace & Mechanical Engineering

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# 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 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 1):

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

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 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

### New Metafor Version > 2422

#### Centered Difference

ti = CentralDifferenceTimeIntegration(metafor)
metafor.setTimeIntegration(ti)

#### Chung Hulbert

ti = ChExplicitTimeIntegration(metafor)
ti.setRhoB(_rhoB)
metafor.setTimeIntegration(ti)

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

#### Tchamwa

ti = TchamwaTimeIntegration(metafor)
ti.setRhoB(_rhoB)
metafor.setTimeIntegration(ti)

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

Other parameters : see Quasi-static integration schemes

1)
see real parameters