−Table of Contents
Explicit dynamic integration schemes
Description
The equilibrium equation between internal forces Fint, inertial forces Ma (where M is the diagonalized mass matrix and a the acceleration) and external forces Fext :
Ma+Fint=Fext
Central difference method
Relations between displacements x, velocities v and accelerations a are:
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)
The equilibrium equation becomes :
a(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
Relations between displacements x, velocities v and accelerations a are:
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)
Specific values leading to an optimal numerical dissipation are given as function of the spectral radius ρβ (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)
γ=3/2−αM
β=5−3ρβ(1+ρβ)2(2−ρβ)
Conditionally stable.
Tchamwa Scheme
Explicit algorithm where numerical dissipation is monitored by the parameter ϕ.
Equilibrium computed with
a(tn+1)=Fext(tn+1)−Fint(tn+1)M
Relations between displacements x, velocities v and accelerations a are:
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)
Stability guaranteed for ϕ≥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