Hoang,
I have some doubts over the contact algorithm used in step-41. Maybe someone
can help me to clarify:
+ In step-41, instead of solving for delta_U and delta_Lambda, we solve for
U_k+1 by using a finite difference equation. Is it some form of explicit
contact algorithm?
I'm not familiar with the literature on contact problems, so don't know the
specific terms. You can relate the approach taken in step-41 to what one can
do when using a the Newton method to minimize an energy functional. There, you
can solve for the update dx^(k+1) in step k+1 via
H^k dx^(k+1) = -J^k
(H^k=Hessian, J^k=Jacobian of the energy functional, both linearized around
the current iterate x^k), but you can also add H^k x^k to both sides to obtain
the equation
H^k (x^k+dx^(k+1)) = H^k x^k - J^k
which you can rewrite in terms of the *next* iterate
H^k x^(k+1) = H^k x^k - J^k
In other words, you can use Newton's method to either solve for the
*increment* dx^(x+1) or for the *next iterate* x^(x+1). It's still the same
method.
It is also worth noting that the gap function is not
linearized. In this example, the gap G is constant function by the way,
however considering possible sliding of the node, the gap still need to be
linearized if using the implicit contact algorithm.
I don't quite understand. The gap function is a linear function in this
example -- it's just the distance between the obstacle and the membrane.
+ The convergence of the contact algorithm is in 18 Newton steps, comparing to
typically 7-8 steps for an implicit contact algorithm. Although it's hard to
compare here because the primal-dual active set is used. But if implicit
contact is used, will the convergence be improved?
I don't know. I think details such as the number of Newton steps required
critically depend on so many factors (the obstacle function, the energy
functional, the exact choice of penalty factors, ...) that it's difficult to
make such comparisons. One would have to try this out by implementing an
alternative method. If you wanted to try this, we'd be very happy to take your
program and incorporate it into the tutorial or the code gallery!
Best
W.
--
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Wolfgang Bangerth email: bange...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/
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