From: bangerth Date: Thu, 22 Mar 2007 20:45:08 +0000 (+0000) Subject: Fix a few issues found by Yulia. X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=9d860d412071cbb69b9b963df6b52b0f963886ce;p=dealii-svn.git Fix a few issues found by Yulia. git-svn-id: https://svn.dealii.org/trunk@14595 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-24/doc/intro.dox b/deal.II/examples/step-24/doc/intro.dox index 441c30ac71..7253135cc6 100644 --- a/deal.II/examples/step-24/doc/intro.dox +++ b/deal.II/examples/step-24/doc/intro.dox @@ -45,27 +45,38 @@ pressure by accelleration: \rho \frac{\partial^2}{\partial t^2}u(t,\mathbf r) = -\nabla p(t,\mathbf r). @f] -Furthermore, it expands based on changes in temperature: +Furthermore, it contracts due to excess pressure and expands based on changes in temperature: @f[ -\nabla \cdot u(t,\mathbf r) = -\frac{p(t,\mathbf r)}{\rho c_0^2}+\beta T(t,\mathbf r) +\nabla \cdot u(t,\mathbf r) = -\frac{p(t,\mathbf r)}{\rho c_0^2}+\beta T(t,\mathbf r) . @f] +Here, $\beta$ is a thermoexpansion coefficient. -If we combine these equations and assume that heating only happens on a time +Let us now make the assumption that heating only happens on a time scale much shorter than wave propagation through tissue (i.e. the temporal length of the microwave pulse that heats the tissue is much shorter than the -time it takes a wave to cross the domain), then we can rewrite the above -equations as follows: +time it takes a wave to cross the domain). In that case, the heating +rate $H(t,\mathbf r)$ can be written as $H(t,\mathbf r) = a(\mathbf +r)\delta(t)$ (where $a(\mathbf r)$ is a map of absorption strengths for +microwave energy), which together with the first equation above will yield +an instantaneous jump in the temperature $T(\mathbf r)$ at time $t=0$. +Using this assumption, and taking all equations together, we can +rewrite and combine the above as follows: @f[ -\Delta p-\frac{1}{c_0^2} \frac{\partial^2 p}{\partial^2 t} = \lambda \delta(t)a(\mathbf r) +\Delta p-\frac{1}{c_0^2} \frac{\partial^2 p}{\partial^2 t} = \lambda +a(\mathbf r)\frac{d\delta(t)}{dt} @f] where $\lambda = - \frac{\beta}{C_p}$. This corresponds to a wave equation with initial conditions as follows: @f{eqnarray*} \Delta \bar{p}- \frac{1}{c_0^2} \frac{\partial^2 \bar{p}}{\partial^2 t} & = & f(t,\mathbf r) \\ -\bar{p}(0,\mathbf r) &=&\lambda a(\mathbf r) = b(\mathbf r) +\bar{p}(0,\mathbf r) &=&\lambda a(\mathbf r) = b(\mathbf r) \\ +\frac{\partial\bar{p}(0,\mathbf r)}{\partial t} &=& 0. @f} -In the inverse problem, it is this right hand side $\lambda a(\mathbf r)$ that +(With $f=0$, though we usually keep it around to derive formulas that +are valid even for the case that $f$ was non-zero.) + +In the inverse problem, it is the initial condition $b(\mathbf r) = \lambda a(\mathbf r)$ that one would like to recover, since it is a map of absorption strengths for microwave energy, and therefore presumably an indicator to discern healthy from diseased tissue. @@ -112,7 +123,7 @@ two seperate equations: with initial conditions: @f{eqnarray*} \bar{p}(0,\mathbf r) & = & b(r) \\ -v(0,\mathbf r)=\bar{p}_t(0,\mathbf r) & = & 1 +v(0,\mathbf r)=\bar{p}_t(0,\mathbf r) & = & 0 @f} The semi-discretized, weak version of this model, using the general $\theta$ scheme @@ -123,21 +134,21 @@ introduced in @ref step_23 "step-23" is then: -\left(\nabla((\theta\bar{p}^n+(1-\theta)\bar{p}^{n-1})),\nabla\phi\right)_\Omega- \frac{1}{c_0}\left(\frac{\bar{p}^n-\bar{p}^{n-1}}{k},\phi\right)_{\partial\Omega} - \frac{1}{c_0^2}\left(\frac{v^n-v^{n-1}}{k},\phi\right)_\Omega & = -& \theta f^{n}+(1-\theta)f^{n-1}, +& \left(\theta f^{n}+(1-\theta)f^{n-1}, \phi\right)_\Omega, @f} where $\phi$ is an arbitrary test function, and where we have used the absorbing boundary condition to integrate by parts: -absoring boundary conditions are incorporated into the weak form by using +absorbing boundary conditions are incorporated into the weak form by using @f[ \int_\Omega\varphi \, \Delta p\; dx = -\int_\Omega\nabla \varphi \cdot \nabla p dx + -\int_{\partial\Omega}\varphi \frac{\partial p}{\partial t}ds. +\int_{\partial\Omega}\varphi \frac{\partial p}{\partial {\mathbf n}}ds. @f] From this we obtain the discrete model by introducing a finite number of shape functions, and get @f{eqnarray*} -M\bar{p}^{n}-k \theta M v^{n-1} & = & M\bar{p}^{n-1}+k (1-\theta)Mv^{n-1},\\ +M\bar{p}^{n}-k \theta M v^n & = & M\bar{p}^{n-1}+k (1-\theta)Mv^{n-1},\\ (-c_0^2k \theta A-c_0 B)\bar{p}^n-Mv^{n} & = & (c_0^2k(1-\theta)A-c_0B)\bar{p}^{n-1}-Mv^{n-1}+c_0^2k(\theta F^{n}+(1-\theta)F^{n-1}).