−Table of Contents
Explicit dynamic integration schemes
Description
The equilibrium equation between internal forces FintFint, inertial forces MaMa (where MM is the diagonalized mass matrix and aa the acceleration) and external forces FextFext :
Ma+Fint=FextMa+Fint=Fext
Central difference method
Relations between displacements xx, velocities vv and accelerations aa are:
v(tn+1/2)=v(tn−1/2)+(tn+1−tn)a(tn)v(tn+1/2)=v(tn−1/2)+(tn+1−tn)a(tn)
x(tn+1)=x(tn)+(tn+1−tn)v(tn+1/2)x(tn+1)=x(tn)+(tn+1−tn)v(tn+1/2)
The equilibrium equation becomes :
a(tn+1)=(Fext(tn+1)−Fint(tn+1))/Ma(tn+1)=(Fext(tn+1)−Fint(tn+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−αM)a(tn+1)+αMa(tn)=Fext(tn)−Fint(tn)M(1−αM)a(tn+1)+αMa(tn)=Fext(tn)−Fint(tn)M
Relations between displacements xx, velocities vv and accelerations aa are:
x(tn+1)=x(tn)+(tn+1−tn)v(tn)+(tn+1−tn)2((0.5−β)a(tn)+βa(tn+1))x(tn+1)=x(tn)+(tn+1−tn)v(tn)+(tn+1−tn)2((0.5−β)a(tn)+βa(tn+1)) v(tn+1)=v(tn)+(tn+1−tn)(1−γ)a(tn)+γa(tn+1)v(tn+1)=v(tn)+(tn+1−tn)(1−γ)a(tn)+γa(tn+1)
Specific values leading to an optimal numerical dissipation are given as function of the spectral radius ρβρβ (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):
αM=(2ρβ−1)/(1+ρβ)αM=(2ρβ−1)/(1+ρβ)
γ=3/2−αMγ=3/2−αM
β=5−3ρβ(1+ρβ)2(2−ρβ)β=5−3ρβ(1+ρβ)2(2−ρβ)
Conditionally stable.
Tchamwa Scheme
Explicit algorithm where numerical dissipation is monitored by the parameter ϕϕ.
Equilibrium computed with
a(tn+1)=Fext(tn+1)−Fint(tn+1)Ma(tn+1)=Fext(tn+1)−Fint(tn+1)M
Relations between displacements xx, velocities vv and accelerations aa are:
x(tn+1)=x(tn)+(tn+1−tn)v(tn)+ϕ(tn+1−tn)2a(tn)x(tn+1)=x(tn)+(tn+1−tn)v(tn)+ϕ(tn+1−tn)2a(tn)
v(tn+1)=v(tn)+(tn+1−tn)a(tn)v(tn+1)=v(tn)+(tn+1−tn)a(tn)
Stability guaranteed for ϕ≥1ϕ≥1 and high frequencies killed over a single time step for \phi = 2$. the scheme is of :
- second order for ϕ=1 (no numerical dissipation)
- first order for ϕ>1 (numerical dissipation)
Relation between ϕ and spectral radius for the bifurcation ρβ (user parameter MDR_ECHR
) is:
- ϕ=2(1−ρ1/2β)(1−ρβ) if ρβ<1
- ϕ=1 if ρβ=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 = TchamwaExplicitTimeIntegration(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