Small update the Theory Manual related to Neumann boundary condition

Chris Smartt
1 parent 036ba562
Showing 2 changed files with 2 additions and 2 deletions Show diff stats
DOCUMENTATION/SACAMOS_TheoryManual.pdf
DOCUMENTATION/THEORY_MANUAL/Tex/Laplace_Solver_Documentation.tex
@@ -123,7 +123,7 @@ The Neumann boundary condition is expressed as
 \begin{equation} 
 \frac{\partial \phi(r) }{\partial r}=0
 \end{equation}
-This boundary condition enforces a condition that the normal derivative of the potential is zero and hence the charge on the boundary is zero everywhere. The zero charge propertiy is required by the capacitance matrix calculation and using the Neumann boundary condition leads to convergence of the capacitance matrix elements to the correct values as the boundary distance and the mesh density are increased.
+This boundary condition enforces a condition that the normal derivative of the potential is zero and hence the charge inside the boundary is zero (by application of the integral form of Gauss' law). The zero charge propertiy is required by the capacitance matrix calculation and using the Neumann boundary condition leads to convergence of the capacitance matrix elements to the correct values as the boundary distance and the mesh density are increased.
  
 \subsubsection{Asymptotic boundary conditions}
  
@@ -148,7 +148,7 @@ Expression  immediately implies that the potential for $r \gg r&#39;$, with $r&#39;$ the
 \begin{equation}\frac{\partial }{{\partial r}}\phi \left( {r,\theta } \right) = \frac{1}{{r\ln \left( r \right)}}\phi \left( {r,\theta } \right)\end{equation}
 \\
  
-It has been observed that the use of the asymptotic boundary condition described here can have an effect on the convergence of the capacitance matrix calculation due to a net charge on the outer boundary which is not taken into account in the capacitance matrix calculation described in section \ref{LC_energy}. Currently it is not known whether a 'generalised capacitance matrix' type solution can be used along with this boundary condition as the outer boundary is not an equipotential (as it is in the Dirichelet case).
+It has been observed that the use of the asymptotic boundary condition described here can have an effect on the convergence of the capacitance matrix calculation due to the total charge within the boundary not being null, which impacts the capacitance matrix calculation described in section \ref{LC_energy}. Currently it is not known whether a 'generalised capacitance matrix' type solution can be used along with this boundary condition as the outer boundary is not an equipotential (as it is in the Dirichelet case).
 As a consequence of this the Neumann boundary condition is recommended and used in the SACAMOS software as the default open boundary condition.