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XLatexIt! run report...
better with LatexIt, sorry.*** Found _expression_ $$dt_e,dt_v$$
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*** Found _expression_ $$dt_e$$
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*** Found _expression_ $$dt_v<2b/a$$
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*** Found _expression_ $$\dot{X} = (1 - dt.a/b)\dot{X}$$
*** Found _expression_ $$\dot{X}_{t+dt} = \dot{X}_{t}-dt.a/b.\dot{X}_{t}$$
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*** Found _expression_ $$a\ddot{X}+b\dot{X}+X=0$$
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A lazy time-step determination :I recall the problem we solve is: -- _______________ Bruno Chareyre Associate Professor ENSE³ - Grenoble INP Lab. 3SR BP 53 - 38041, Grenoble cedex 9 - France Tél : +33 4 56 52 86 21 Fax : +33 4 76 82 70 43 ________________ |
1/ If the viscous effects "b" are high compared to elastic effects, then
it means b>>1 and the "X" term can be neglected.
In that case a first order(*) explicit scheme will give
or
which obviously converges only if 
.
In the absence of a better derivation, this condition should at least be
verified for all visco-simulation.
(*) If the current implementation uses previous velocity to determine
the viscous force, then it is first order scheme. Is it the case?
2/ If viscous effects are small in comparison with stiffness (highly
"under-relaxed", if you like), then the elastic times-step 
given by GSTStepper is a good enough approximate.
3/ If nothing can be neglected, then there a critical timestep smaller
than the two timesteps defined above, this is the interesting but
difficult part.
It could be workarounded, maybe, by defining a sufficiently small safety
factor, like dt=0.5*min(
). It would need numerical
experiments to see if this method applies.
I examined the true visco-elastic problem today and found a big equation
giving a conditon on dt, but I couldn't solve it by hand yet... :-(
Bruno
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