Processing math: 100%

Metafor

ULiege - Aerospace & Mechanical Engineering

User Tools

Site Tools


doc:user:integration:scheme:dynexpl

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

(1αM)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+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)

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=(2ρβ1)/(1+ρβ)
γ=3/2αM
β=53ρβ(1+ρβ)2(2ρβ)

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

Stability guaranteed for ϕ1 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: 2022/12/21 11:35 by boman

Donate Powered by PHP Valid HTML5 Valid CSS Driven by DokuWiki