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doc:user:integration:scheme:dynexpl

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(tn1/2)+(tn+1tn)a(tn)v(tn+1/2)=v(tn1/2)+(tn+1tn)a(tn)
x(tn+1)=x(tn)+(tn+1tn)v(tn+1/2)x(tn+1)=x(tn)+(tn+1tn)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 :

(1αM)a(tn+1)+αMa(tn)=Fext(tn)Fint(tn)M(1αM)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+1tn)v(tn)+(tn+1tn)2((0.5β)a(tn)+βa(tn+1))x(tn+1)=x(tn)+(tn+1tn)v(tn)+(tn+1tn)2((0.5β)a(tn)+βa(tn+1)) v(tn+1)=v(tn)+(tn+1tn)(1γ)a(tn)+γa(tn+1)v(tn+1)=v(tn)+(tn+1tn)(1γ)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=(2ρβ1)/(1+ρβ)αM=(2ρβ1)/(1+ρβ)
γ=3/2αMγ=3/2αM
β=53ρβ(1+ρβ)2(2ρβ)β=53ρβ(1+ρβ)2(2ρβ)

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+1tn)v(tn)+ϕ(tn+1tn)2a(tn)x(tn+1)=x(tn)+(tn+1tn)v(tn)+ϕ(tn+1tn)2a(tn)
v(tn+1)=v(tn)+(tn+1tn)a(tn)v(tn+1)=v(tn)+(tn+1tn)a(tn)

Stability guaranteed for ϕ1ϕ1 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(1ρ1/2β)(1ρβ) 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

(see Global Parameters [REMOVED])

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

1)
see real parameters
doc/user/integration/scheme/dynexpl.txt · Last modified: by boman

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