From: wolf Date: Fri, 15 Apr 2005 17:09:53 +0000 (+0000) Subject: More text for the model. X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=29e911eec18100c521eba61a2ca98d1738b2b6e8;p=dealii-svn.git More text for the model. git-svn-id: https://svn.dealii.org/trunk@10507 0785d39b-7218-0410-832d-ea1e28bc413d --- 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 4fbd42b085..bcaf0400f7 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 @@ -26,6 +26,8 @@ webpage \texttt{http://www.dealii.org/}. \subsection*{Quasistatic elastic deformation} +\subsubsection*{Motivation of the model} + In general, small elastic deformations are described by the elastic wave equation \begin{gather*} @@ -106,9 +108,68 @@ terms of the stress: \qquad &&\text{on $\Gamma_N=\partial\Omega(t)\backslash\Gamma_D$}. \end{align*} -Note that these equations are posed on a domain $\Omega(t)$ that changes with -time. +Note that these equations are posed on a domain $\Omega(t)$ that +changes with time, with the boundary moving according to the +displacements $\vec u(\vec x,t)$ of the points on the boundary. To +complete this system, we have to specify the relationship between the +stress and the strain, as follows: +\begin{align*} + \dot\sigma = C \varepsilon (\dot{\vec u}), +\end{align*} +where a dot indicates a time derivative. + +\subsubsection*{Time discretization} + +Numerically, this system is solved as follows: first, we discretize +the time component using a backward Euler scheme. This leads to a +discrete equilibrium of force at time step $n$: +\begin{align*} + \div \sigma^n &= f^n, +\intertext{where} + \sigma^n &= \sigma^{n-1} + C \varepsilon (\Delta \vec u^n), +\end{align*} +and $\Delta \vec u^n$ the incremental displacement for time step +$n$. This way, if we want to solve for the displacement increment, we +have to solve the following system: +\begin{align*} + - \div C \varepsilon(\Delta\vec u^n) &= \vec f - \div \sigma^{n-1} + &&\text{in $\Omega(t_{n-1})$}, + \\ + \Delta \vec u^n(\vec x,t) &= d(\vec x,t_n) - d(\vec x,t_{n-1}) + \qquad + &&\text{on $\Gamma_D\subset\partial\Omega(t_{n-1})$}, + \\ + \vec n \ C \varepsilon(\Delta \vec u^n(\vec x,t)) &= b(\vec x,t_{1})-b(\vec x,t_{n-1}) + \qquad + &&\text{on $\Gamma_N=\partial\Omega(t_{n-1})\backslash\Gamma_D$}. +\end{align*} +This system at time step $n$, to be solved on the old domain +$\Omega(t_{n-1})$, has exactly the form of a stationary elastic +problem, and is therefore similar to what we have already implemented +in previous example programs. We will therefore not comment on the +space discretization beyond saying that we again use lowest order +continuous finite elements. + +There are differences, however: +\begin{enumerate} + \item We have to move the mesh after each time step, in order to be + able to solve the next time step on a new domain; + + \item We need to know $\sigma^{n-1}$ to compute the next incremental + displacement, i.e. we need to compute it at the end of the time step + to make sure it is available for the next time step. Essentially, + the stress variable is our window to the history of deformation of + the body. +\end{enumerate} +These two operations are done in the functions ``move\_mesh'' and +``update\_\-quadrature\_\-point\_history'' in the program. While moving +the mesh is only a technicality, updating the stress is a little more +complicated and will be discussed in the next section. + + +\subsubsection*{Updating the stress variable} +x \subsection*{Parallel graphical output}