Introduction

The 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 boundary element method for linear water waves, standard boundary element method, in which the fundamental solution to 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

Two-Dimensional Equations

We begin with the Standard Linear Wave Scattering Problem in Finite Depth,

We then apply Green's second identity to a finite region U containing the wetted body surface [math]\displaystyle{ \partial\Omega }[/math], the free surface [math]\displaystyle{ z=0 }[/math], the bottom surface [math]\displaystyle{ z=-h }[/math] and two vertical boundaries. This gives us

We then substitute

which is the free space Green function which satisfies

where [math]\displaystyle{ \delta }[/math] is the Dirac delta function. This leads to

where [math]\displaystyle{ G_{n}\left( \mathbf{x},\mathbf{x}^{\prime }\right) }[/math] is the outward normal derivative of [math]\displaystyle{ G }[/math] (with respect to the [math]\displaystyle{ \mathbf{x}^{\prime } }[/math] coordinate). This gives us the following equation relating the potential and its outward normal derivative on the boundary [math]\displaystyle{ \partial U }[/math]

We solve this equation by a modified constant panel method which reduces it to the following matrix equation

In this equation [math]\displaystyle{ \vec{\phi}\mathcal{\ } }[/math]and [math]\displaystyle{ \vec{\phi} _{n} }[/math] are vectors which approximate the potential and its normal derivative around the boundary [math]\displaystyle{ \partial \Omega }[/math], and [math]\displaystyle{ \mathbf{G} }[/math] and [math]\displaystyle{ \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]\displaystyle{ \vec{\phi}_{n}, }[/math] and the potential, [math]\displaystyle{ \vec{\phi}, }[/math] are related by the conditions on the boundary [math]\displaystyle{ \partial U }[/math] given by [math]\displaystyle{ L }[/math].

Numerical Calculation of [math]\displaystyle{ \mathbf{G} }[/math] and [math]\displaystyle{ \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]\displaystyle{ \phi , }[/math] or its outward normal derivative, [math]\displaystyle{ \phi _{n}, }[/math] are assumed to be constant. The free-space Green's function, [math]\displaystyle{ G, }[/math] and its normal derivative, [math]\displaystyle{ G_{n} }[/math] are more rapidly varying and have a singularity at [math]\displaystyle{ \mathbf{x}=\mathbf{x}^{\prime } }[/math]. For this reason, over each panel, while [math]\displaystyle{ \phi }[/math] and [math]\displaystyle{ \phi _{n} }[/math] are assumed constant, [math]\displaystyle{ G }[/math] and [math]\displaystyle{ G_{n} }[/math] are integrated exactly. For example, we use the following approximation to calculate the integral of [math]\displaystyle{ G }[/math] and [math]\displaystyle{ \phi }[/math] over a single panel

where [math]\displaystyle{ \mathbf{x}_{i} }[/math] is the midpoint of the panel and [math]\displaystyle{ h }[/math] is the panel length. The integral on the right hand side, because of the simple structure of [math]\displaystyle{ G }[/math], can be calculated exactly.

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

Matlab Code

A program to calculate the matrics [math]\displaystyle{ G }[/math] and [math]\displaystyle{ G_{n} }[/math] can be found here bem_constant_panel.m