Hi,
I am trying to use the generalized alpha method in this website, Time-integration of elastodynamics equation — Numerical tours of continuum mechanics using FEniCS master documentation
In my system, the displacement u^{n+1} is the only unknown, after getting the displacement I will get the acceleration and velocity. If my system doesn’t have the body force, which is the F in that website, what is the difference between generalized alpha method and Newmark method.
Hi,
the difference between the generalized alpha and Newmark method is that in the Newmark method, the evolution equation is written exactly at time n+1 (for all terms). In the generalized-alpha method the evolution equation is written at time n+1-\alpha_f, except for the inertial term which is written at time n+1-\alpha_m. Hence, both methods coincide only when \alpha_f=\alpha_m=0. This has nothing to do with the presence or not of an external force. But honestly, the Newmark method is already
fine in most applications.
Thank you so much bleyerj, I have another question, if I follow the idea of this website, [Time-integration of elastodynamics equation — Numerical tours of continuum mechanics using FEniCS master documentation ], to be more specific, this part, (Time-integration of elastodynamics equation — Numerical tours of continuum mechanics using FEniCS master documentation)
In the generalized \alpha-method, one makes the choice of satisfying the evolution equation at an intermediate time step rather at the end of the time step. This is for enhanced accuracy and/or stability reasons. This is inherent to the choice of the method. If you choose to satisfy your evolution equation at the end of the time step then you are using the Newmark method.
I see, so this website is actually using Newmark? So the results are oscillatory?
No it is using the \alpha-method. See the first line of this section where the evolution equation is clearly written at an intermediate time step
So the idea is to use the u^{n+1} and u^{n} to denote the intermediate time step via the formula
X_{n+1-\alpha}=(1-\alpha)X_{n+1}+\alpha X_n
so we can get u^{n+1}, and from u^{n+1}, using
to get a^{n+1} and v^{n+1}
So still the u^{n+1} is the only unknown in the evolution equation, but with some \alpha?
Am I right?
Yes that’s the idea.
From the original paper, I found there are still timestep_alpha, like
this is only used for evaluating the external force vector. It is considered in the demo, see this line:
# Forces are evaluated at t_{n+1-alpha_f}=t_{n+1}-alpha_f*dt
p.t = t-float(alpha_f*dt)
