View source for Category:Boundary Element Method

= Introduction =

The [http://en.wikipedia.org/wiki/Boundary_element_method boundary element method] is a method for solving 
Laplace's equation (and other related elliptic equation).
It is a very general and powerful method well suited to the linear wave scattering problem and forms the
basis for many commercial wave scattering codes. The are two 
[http://en.wikipedia.org/wiki/Boundary_element_method boundary element method] for linear water waves,
standard [http://en.wikipedia.org/wiki/Boundary_element_method boundary element method], 
in which the fundamental solution to 
[http://en.wikipedia.org/wiki/Laplace%27s_Equation Laplace's Equation] is used, which we
decribe here and the
[[Green Function Solution Method]], in which the 
the [[Free-Surface Green Function]]
is used so that the problem reduces to solving only on the wetted body surface.

The theory for solving a wave scattering problem is described in [[Boundary Element Method for a Rigid Body in Finite Depth]]

= Two-Dimensional Equations = 

We begin with the [[Standard Linear Wave Scattering Problem]]
in [[Finite Depth]],
<center><math>
\nabla^{2}\phi=0, \, -h<z<0,\,\,\,\mathbf{x}\notin \Omega
</math></center>
<center><math>
\frac{\partial\phi}{\partial z}=0, \, z=-h,
</math></center>
<center><math>
 \frac{\partial\phi}{\partial z} = \alpha \phi,\,z=0,\,\,\mathbf{x}\notin\Omega,
</math></center>
<center><math>
 \frac{\partial\phi}{\partial z} = L\phi, \, z\in\partial\Omega.
</math></center>

We then apply [http://en.wikipedia.org/w/index.php?title=Green%27s_identities&action=edit Green's second identity] 
to a finite region ''U'' containing the wetted
body surface <math>\partial\Omega</math>, the free surface <math>z=0</math>,
the bottom surface <math>z=-h</math> and two vertical boundaries. This gives us
<center><math> \int_U \left( G \nabla^2 \phi - \phi \nabla^2 G\right)\, dV = \oint_{\partial U} \left( G {\partial \phi \over \partial n} - \phi {\partial G \over \partial n}\right)\, dS 
</math></center>
We then substitute 
<center><math>
G\left( \mathbf{x},\mathbf{x}^{\prime }\right) =\frac{1}{2\pi }\ln \,\left| 
\mathbf{x}-\mathbf{x}^{\prime }\right| ,  
</math></center>
which is the free space Green function
which satisfies 
<center><math> 
\nabla^{2}G=\delta(\mathbf{x}-\mathbf{x}^{\prime }),
</math></center>
where <math>\delta</math> is the 
[http://en.wikipedia.org/wiki/Dirac_delta Dirac delta function]. This leads to 
<center><math>
\oint_{\partial \Omega }\left(
G_{n}\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi \left( \mathbf{x}
^{\prime }\right) -G\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi
_{n}\left( \mathbf{x}^{\prime }\right) \right) d\mathbf{x}^{\prime }
  =
\left(
\begin{matrix}
0, \,\,\,x\notin U \cup \partial U, \\
\phi(\mathbf{x})/2,\,\,\,\mathbf{x} \in \partial U, \\
\phi(\mathbf{x}),\,\,\,\mathbf{x} \in U,
\end{matrix}
\right.
</math></center>
where <math>G_{n}\left( \mathbf{x},\mathbf{x}^{\prime }\right) </math> is the outward
normal derivative of <math>G</math> (with respect to the <math>\mathbf{x}^{\prime }</math>
coordinate).
This gives us the following equation relating the potential
and its outward normal derivative on the boundary <math>\partial U </math> 
<center><math>
\left. \frac{1}{2}\phi \left( \mathbf{x}\right) \right|_{\mathbf{x}\in\partial U}
=\oint_{\partial \Omega }\left(
G_{n}\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi \left( \mathbf{x}
^{\prime }\right) -G\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi
_{n}\left( \mathbf{x}^{\prime }\right) \right) d\mathbf{x}^{\prime },\;\;
\mathbf{x}\in \partial \Omega .  
</math></center>


We solve this equation by a modified constant panel method
which reduces it to the following matrix equation 
<center><math>
\frac{1}{2}\vec{\phi}=\mathbf{G}_{n}\vec{\phi}-\mathbf{G}\vec{\phi}_{n}.
</math></center>
In this equation <math>\vec{\phi}\mathcal{\ }</math>and <math>\vec{\phi}
_{n}</math> are vectors which approximate the potential and its normal derivative
around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
are matrices corresponding to the Green function and the outward normal
derivative of the Green function respectively. 
The outward normal derivative of the potential, <math>\vec{\phi}_{n},</math> and the
potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
\partial U </math> given by <math>L</math>.

==Numerical Calculation of <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>==

We solve the boundary element equation numerically by
a modified constant panel method. In this method, the boundary is divided
into panels over which the potential, <math>\phi ,</math> or its outward normal
derivative, <math>\phi _{n},</math> are assumed to be constant. The free-space Green's
function, <math>G,</math> and its normal derivative, <math>G_{n}</math> are more rapidly varying
and have a singularity at <math>\mathbf{x}=\mathbf{x}^{\prime }</math>. For this
reason, over each panel, while <math>\phi </math> and <math>\phi _{n}</math> are assumed constant, 
<math>G</math> and <math>G_{n}</math> are integrated exactly. For example, we use the following
approximation to calculate the integral of <math>G</math> and <math>\phi </math> over a single
panel 
<center><math>
\int_{\mathbf{x}_{i}-h/2}^{\mathbf{x}_{i}+h/2}G\left( \mathbf{x},\mathbf{x}
^{\prime }\right) \phi \left( \mathbf{x}^{\prime }\right) d\mathbf{x}
^{\prime }\approx \phi \left( \mathbf{x}_{i}\right) \int_{\mathbf{x}
_{i}-h/2}^{\mathbf{x}_{i}+h/2}G\left( \mathbf{x},\mathbf{x}^{\prime }\right)
d\mathbf{x}^{\prime }, 
</math></center>
where <math>\mathbf{x}_{i}</math> is the midpoint of the panel and <math>h</math> is the panel
length. The integral on the right hand side,
because of the simple structure of <math>G</math>, can be calculated exactly.

It is well known that, for higher accuracy, a higher order boundary element method works
better. 

{{boundary element code}}

= Matlab Code =



[[Category:Pages with Matlab Code]]
[[Category:Numerical Methods]]