From bf4f738f60207ee2cddbb0d770d7fc577757787e Mon Sep 17 00:00:00 2001 From: wolf Date: Fri, 15 Apr 2005 02:45:15 +0000 Subject: [PATCH] Something on the model. git-svn-id: https://svn.dealii.org/trunk@10505 0785d39b-7218-0410-832d-ea1e28bc413d --- .../step-18.data/intro.tex | 107 ++++++++++++++++-- 1 file changed, 95 insertions(+), 12 deletions(-) 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 2dadd395cb..b4b04dba55 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 @@ -1,6 +1,7 @@ \documentclass{article} \usepackage{amsmath} - +\renewcommand{\vec}[1]{\mathbf{#1}} +\renewcommand{\div}{\mathrm{div}\ } \begin{document} This tutorial program is another one in the series on the elasticity problem @@ -8,23 +9,105 @@ that we have already started with step-8 and step-17. It extends it into two different directions: first, it solves the quasistatic but time dependent elasticity problem for large deformations with a Lagrangian mesh movement approach. Secondly, it shows some more techniques for solving such problems -using parallel processing with PETSc's linear algebra; however, as before in -step-17, the program runs just as fine on a single sequential machine as long -as you have PETSc installed. In addition to this, we show how to work around -the main bottleneck of step-17, namely there that we only generated graphical -output from one process, and that this scaled very badly with larger numbers -of processes and on large problems. Finally, a good number of assorted -improvements and techniques are demonstrated that have not been shown yet in -previous programs. +using parallel processing with PETSc's linear algebra. In addition to this, we +show how to work around the main bottleneck of step-17, namely there that we +only generated graphical output from one process, and that this scaled very +badly with larger numbers of processes and on large problems. Finally, a good +number of assorted improvements and techniques are demonstrated that have not +been shown yet in previous programs. + +As before in step-17, the program runs just as fine on a single sequential +machine as long as you have PETSc installed. Information on how to tell +deal.II about a PETSc installation on your system can be found in the deal.II +README file, which is linked to from the main documentation page +\texttt{doc/index.html} in your installation of deal.II, or on the deal.II +webpage \texttt{http://www.dealii.org/}. \subsection*{Quasistatic elastic deformation} -In general, elastic deformation is described by the elastic wave equation +In general, small elastic deformations are described by the elastic wave +equation \begin{gather*} - \rho + \rho \frac{\partial^2 \vec u}{\partial t^2} + + c \frac{\partial \vec u}{\partial t} + - \div ( C \varepsilon(\vec u)) = \vec f + \qquad + \text{in $\Omega$}, \end{gather*} - +where $\vec u=\vec u (\vec x,t)$ is the deformation of the body, $\rho$ and +$c$ the density and attenuation coefficient, and $\vec f$ external forces. In +addition, initial conditions +\begin{align*} + \vec u(\cdot, 0) = \vec u_0(\cdot) + \qquad + \text{on $\Omega$}, +\end{align*} +and Dirichlet (displacement) or Neumann (force) boundary conditions need +to be specified for a unique solution: +\begin{align*} + \vec u(\vec x,t) &= d(\vec x,t) + \qquad + &&\text{on $\Gamma_D\subset\partial\Omega$}, + \\ + \vec n \ C \varepsilon(\vec u(\vec x,t)) &= b(\vec x,t) + \qquad + &&\text{on $\Gamma_N=\partial\Omega\backslash\Gamma_D$}. +\end{align*} +In above formulation, $\varepsilon(\vec u)= \tfrac 12 (\nabla \vec u + \nabla +\vec u^T)$ is the symmetric gradient of the displacement, also called the +\textit{strain}. $C$ is a tensor of rank 4, called the \textit{stress-strain + tensor} that contains knowledge of the elastic strength of the material. We +will comment on the roles of the strain and stress tensors more below. For the +moment it suffices to say that we interpret the term $\div ( C +\varepsilon(\vec u))$ as the vector with components $\tfrac \partial{\partial + x_j} C_{ijkl} \varepsilon(\vec u)_{kl}$, where summation over indices +$j,k,l$ is implied. + +The quasistatic limit of this equation is motivated as follows: each small +perturbation of the body, for example by changes in boundary condition or the +forcing function, will result in a corresponding change in the configuration +of the body. In general, this will be in the form of waves radiating away from +the location of the disturbance. Due to the presence of the damping term, +these waves will be attenuated on a time scale of, say, $\tau$. Now, assume +that all changes in external configuration happen on times scales that are +much larger than $\tau$. In that case, the dynamic nature of the change is +unimportant: we can consider the body to always be in static equilibrium, +i.e. we can assume that at all times the body satisfies +\begin{align*} + - \div ( C \varepsilon(\vec u)) &= \vec f + &&\text{in $\Omega$}, + \\ + \vec u(\vec x,t) &= d(\vec x,t) + \qquad + &&\text{on $\Gamma_D$}, + \\ + \vec n \ C \varepsilon(\vec u(\vec x,t)) &= b(\vec x,t) + \qquad + &&\text{on $\Gamma_N$}. +\end{align*} +Note that the differential equation does not contain any time derivatives any +more -- all time dependence is introduced through boundary conditions and a +possibly time-varying force function $\vec f(\vec x,t)$. + +While these equations are sufficient to describe small deformations, computing +large deformations is a little more complicated. To do so, let us first +introduce a stress variable $\sigma$, and write the differential equations in +terms of the stress: +\begin{align*} + - \div \sigma &= \vec f + &&\text{in $\Omega(t)$}, + \\ + \vec u(\vec x,t) &= d(\vec x,t) + \qquad + &&\text{on $\Gamma_D\subset\partial\Omega(t)$}, + \\ + \vec n \ C \varepsilon(\vec u(\vec x,t)) &= b(\vec x,t) + \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. \subsection*{Parallel graphical output} -- 2.39.5