# Tutorial 3

<br> <h1>CMISS and the Finite Element Method for solving partial differential equations</h1>

<p>Given a finite element mesh, this tutorial demonstrates how to solve partial differential equations in order to model various physical situations.</p>

<h1>Simulating steady-state heat flow in a plate</h1>

<p>We now move on to a real-life application of the finite element method to calculate the steady-state temperature distribution and heat flow across a metal plate. This is discussed in Chapter 2 of the <a href="/documentation/course_notes/fembem_notes/" target="new">FEM/BEM Notes</a>.

<p>Run <a href="http://cmiss.bioeng.auckland.ac.nz/development/examples/3/31/311/index.html" target="new">example_311</a>. This example uses the 'fem_mesh.ip****' files that you created in example_111. (If you do not have these files you will have to re-run example_111 - refer to Tutorial 1).</p>

<h1>Analysis of stability for different time integration schemes</h1>

<p>Here we investigate the effects of varying length and time steps on the stability of time integration schemes. This is discussed in Chapter 4 of the <a href="/documentation/course_notes/fembem_notes/" target="new">FEM/BEM Notes</a>.</p>

<p>The problem is a one-dimensional diffusion equation on a domain of length 1. The initial conditions in the domain are u(x,0)=0 and the boundary conditions are u(0,t)=0 and u(1,t)=1. Consider the problem with delta-x=0.25. This is covered in <a href="http://cmiss.bioeng.auckland.ac.nz/development/examples/3/33/332/3321/index.html" target="new">example_3321</a>. Note that there are 'wiggles' or oscillatory noise in the nodal time history.</p>

<p>Now re-run this example with a time step of delta-t=0.05. Notice that the nodal time history is now smooth and gets the analytic steady state solution of u(x,t)=x.</p>

<p>Consider now the case with delta-x=0.1. This is covered in <a href="http://cmiss.bioeng.auckland.ac.nz/development/examples/3/33/332/3322/index.html" target="new">example_3322</a>. Notice that the nodal history has oscillatory noise even though the time step delta-t=0.05 was acceptable for example_3321. This time step is not satisfactory for the model with this many degrees-of-freedom (nodal parameters). Rerun the example with a time step of delta-t=0.01 and see what happens.</p>

<h1>Heat flow through a cast iron furnace wall</h1>

<p>Here we investigate the heat flow through a cast iron furnace wall. The cast iron furnace wall is 1 cm thick and is initially at 0 degrees Celsius. At time zero the furnace is switched on and the temperature at one side of the wall held at 1000 degrees Celsius whilst the other side is held at zero degrees Celsius. This problem is covered in <a href="http://cmiss.bioeng.auckland.ac.nz/development/examples/3/33/333/index.html" target="new">example_333</a>.</p>

<h1>Contamination of Lake Pacific</h1>

<p>You have been asked as an engineering consultant to investigate the long term effects of an environmental disaster. The area concerned is Lake Pacific (a square lake, somewhere near here) which has a river flowing into it at the bottom left of the lake and an island in the centre. Some terrible government has been testing nuclear weapons on the island and cracks have formed through which radioactive material is being released into the lake. The river that flows into the lake (along with prevailing winds) causes a natural current in the lake, which results in the water flowing to the North and to the East. The current is twice as strong in the North direction as in the East direction. You decide that this process is governed by the advection-diffusion equation and that you will use CMISS to perform the analysis. Run through <a href="http://cmiss.bioeng.auckland.ac.nz/development/examples/3/33/334/index.html" target="new">example_334</a>.</p>

<hr> <p><b><a href="http://cmiss.bioeng.auckland.ac.nz/cgi-bin/lab-question-form.cgi?stage=1&laboratory=3" target="new">Tutorial 3 Quiz</a></b> | Tutorial 2 | Tutorial 4</p> <hr>