From 827d19f70023c4c945db3f346fe26a82d6b7ae32 Mon Sep 17 00:00:00 2001 From: wolf Date: Mon, 23 May 2005 23:03:34 +0000 Subject: [PATCH] Mention a few more topics. git-svn-id: https://svn.dealii.org/trunk@10711 0785d39b-7218-0410-832d-ea1e28bc413d --- .../step-18.data/intro.tex | 27 ++++++++++++++++++- 1 file changed, 26 insertions(+), 1 deletion(-) diff --git a/deal.II/doc/tutorial/chapter-2.step-by-step/step-18.data/intro.tex b/deal.II/doc/tutorial/chapter-2.step-by-step/step-18.data/intro.tex index b7f0bef08d..5afaebcfb6 100644 --- a/deal.II/doc/tutorial/chapter-2.step-by-step/step-18.data/intro.tex +++ b/deal.II/doc/tutorial/chapter-2.step-by-step/step-18.data/intro.tex @@ -378,6 +378,32 @@ elastoplasticity: I. Mixed displacement-pressure formulation'' pp. 3559--3586, 2004). +\paragraph*{Stabilization issues.} The formulation we have chosen, i.e. using +piecewise (bi-, tri-)linear elements for all components of the displacement +vector, and treating the stress as a variable dependent on the displacement is +appropriate for most materials. However, this so-called displacement-based +formulation becomes unstable and exhibits spurious modes for incompressible or +nearly-incompressible materials. While fluids are usually not elastic (in most +cases, the stress depends on velocity gradients, not displacement gradients, +although there are exceptions such as electro-rheologic fluids), there are a +few solids that are nearly incompressible, for example rubber. Another case is +that many plasticity models ultimately let the material become incompressible, +although this is outside the scope of the present program. + +Incompressibility is characterized by Poisson's ratio +\begin{gather*} + \nu = \frac{\lambda}{2(\lambda+\mu)}, +\end{gather*} +where $\lambda,\mu$ are the Lam'e constants of the material. +Physical constraints indicate that $-1\le \nu\le \tfrac 12$. If $\nu$ +approaches $\tfrac 12$, then the material becomes incompressible. In that +case, pure displacement-based formulations are no longer appropriate for the +solution of such problems, and stabilization techniques have to be employed +for a stable and accurate solution. The book and paper cited above give +indications as to how to do this, but there is also a large volume of +literature on this subject. + + \paragraph*{Refinement during timesteps.} In the present form, the program only refines the initial mesh a number of times, but then never again. For any kind of realistic simulation, one would want to extend this so that the mesh @@ -395,7 +421,6 @@ communication of these data elements is therefore necessary, making the entire process a little more unpleasant. -Pressure stabilization Make sure that cells are always well-formed -- 2.39.5