# Tutorial 6

<br> <h1>The Boundary Element Method and CMISS</h1>

<p>We will now investigate the Boundary Element Method (BEM) for the solution of partial differential equations. This is discussed in Chapters 3 of the <a href="/documentation/course_notes/fembem_notes/" target="new">FEM/BEM Notes</a>.</p>

<h1>Simple BEM Problem</h1>

<p>The first problem we will consider is that of solving Laplace's equation inside a circle. Run through <a href="http://cmiss.bioeng.auckland.ac.nz/development/examples/3/32/321/index.html" target="new">example_321</a>.

You will need to: <ul>

<li> Define a <b>BOUNDARY ELEMENT</b> basis function. If you get the error "<b>>>ERROR: >>Not a boundary element basis</b>" then you have not defined a correct basis function. <b>Note:</b> the second question you will be asked is not the default answer of 1 (the default answer will define a Finite Element basis function. <li> Define the elements <li> Define the boundary conditions <li> Define the solution methods <li> After the problem has solved you can draw the solution field on the mesh using the command "fem draw field"

</ul>

<center> <img src="321.jpg" width="200"/>    <img src="321_field.jpg" width="200"/> </center>

<h1>Comparison of FEM and BEM</h1>

<p>We will now compare the BEM to that of the familiar Finite Element Method (FEM).</p>

- <ul>
- <li> The problem we will analyse is that of finding the steady state temperature distribution inside an annulus of inner radius 1 and outer radius 2. <li> The governing equation is Laplace's equation and the boundary conditions are a prescribed temperature distribution on the inside of the annulus, and a no-flux condition on the outside of the annulus. <li> The problem will be solved first using the FEM with 16 2D bilinear elements, and then using the BEM with 32 1D linear elements (16 on each circle). <li> In both cases, the number of degrees-of-freedom for the problem are the same (i.e. 32).

</ul>

<p>First run through the FEM simulation using <a href="http://cmiss.bioeng.auckland.ac.nz/development/examples/3/31/312/index.html" target="new">example_312</a>. Then run through <a href="http://cmiss.bioeng.auckland.ac.nz/development/examples/3/32/324/index.html" target="new">example_324</a>, the BEM simulation.</p>

<center> <img src="312.jpg" width="200"/>     <img src="324.jpg" width="200"/> </center>

<p>Compare the values of temperature obtained on the outer circle and the values of flux obtained on the inner circle. How are the values of flux on the inner circle from the BEM related to those obtained using the FEM?</p>

<h1>Cavity Problem</h1>

<p>We will now investigate the problem of finding the steady state water distribution arising from a water filled cylindrical cavity buried deep underground in a porous medium. This is a classic example were the BEM is more appropriate for the problem. If the FEM were to be used, the entire solution domain (i.e. all of underground) would have to be modelled and meshed. Using the BEM, only the boundary of the domain (i.e. the boundary of the underground cavity) needs to be discretised.</p>

<p>The governing equation for this problem is the modified Helmholtz equation, i.e. del-squared *H* = *s*^2 *H*, where *s* is the non-dimensionalised radius of the cavity, and *H* is the dependent variable related to the saturation of the soil.</p>

<p>In this case, *H* = *t* exp {*s* *y*} where *t* is the saturation of the soil (i.e. *t* = 0 when the soil is dry and *t* = 1 when the soil is fully saturated with water), and *y* is the vertical distance (measured downwards i.e. in the direction of gravity) from the center of the circular cavity. The boundary conditions are thus *t* = 1 around the cavity (or rather the equivalent in terms of the solution variable *H* = exp {*s* *y*} ) and *t* = *H* = 0 at infinity.</p>

<p>Run through <a href="http://cmiss.bioeng.auckland.ac.nz/development/examples/3/32/325/index.html" target="new">example_325</a> to solve this problem.</p>

<center> <img src="325.jpg" width="200"/> </center>

<hr> <p><b><a href="http://cmiss.bioeng.auckland.ac.nz/cgi-bin/lab-question-form.cgi?stage=1&laboratory=6" target="new">Tutorial 6 Quiz</a></b> | Tutorial 5</p> <hr>