| 100 |
Neumann boundary conditions describe the radiation or flux normal to the boundary surface. This aptly describes our insulation conditions as we do not want to exert a constant temperature as with the heat source. However, we do want to prevent any loss of energy from the system. These natural boundary conditions can be described by specifying a radiation condition which prescribes the normal component of the flux $\kappa T\hackscore{,i}$ to be proportional |
Neumann boundary conditions describe the radiation or flux normal to the boundary surface. This aptly describes our insulation conditions as we do not want to exert a constant temperature as with the heat source. However, we do want to prevent any loss of energy from the system. These natural boundary conditions can be described by specifying a radiation condition which prescribes the normal component of the flux $\kappa T\hackscore{,i}$ to be proportional |
| 101 |
to the difference of the current temperature to the surrounding temperature $T\hackscore{ref}$; in general terms this is; |
to the difference of the current temperature to the surrounding temperature $T\hackscore{ref}$; in general terms this is; |
| 102 |
\begin{equation} |
\begin{equation} |
| 103 |
\kappa T\hackscore{,i} n\hackscore i = \eta (T\hackscore{ref}-T) |
\kappa T\hackscore{,i} \hat{n}\hackscore i = \eta (T\hackscore{ref}-T) |
| 104 |
\label{eqn:hdbc} |
\label{eqn:hdbc} |
| 105 |
\end{equation} |
\end{equation} |
| 106 |
and simplified to our one dimensional model we have; |
and simplified to our one dimensional model we have; |
| 107 |
\begin{equation} |
\begin{equation} |
| 108 |
\kappa \frac{\partial T}{\partial dx} n\hackscore x = \eta (T\hackscore{ref}-T) |
\kappa \frac{\partial T}{\partial dx} \hat{n}\hackscore x = \eta (T\hackscore{ref}-T) |
| 109 |
\end{equation} |
\end{equation} |
| 110 |
where $\eta$ is a given material coefficient depending on the material of the block and the surrounding medium and $n\hackscore i$ is the $i$-th component of the outer normal field \index{outer normal field} at the surface of the domain. These two conditions form a boundary value problem that has to be solved for each time step. Due to the perfect insulation in our model we can set $\eta = 0$ which results in zero flux - no energy in or out - we do not need to worry about the Neumann terms of the general form for this example. |
where $\eta$ is a given material coefficient depending on the material of the block and the surrounding medium and $\hat{n}\hackscore i$ is the $i$-th component of the outer normal field \index{outer normal field} at the surface of the domain. These two conditions form a boundary value problem that has to be solved for each time step. Due to the perfect insulation in our model we can set $\eta = 0$ which results in zero flux - no energy in or out - we do not need to worry about the Neumann terms of the general form for this example. |
| 111 |
|
|
| 112 |
\subsection{A \textit{1D} Clarification} |
\subsection{A \textit{1D} Clarification} |
| 113 |
It is necessary for clarification that we revisit the general PDE from \refeq{eqn:commonform nabla} under the light of a two dimensional domain. \ESCRIPT is inherently designed to solve problems that are greater than one dimension and so \ref{eqn:commonform nabla} needs to be read as a higher dimensional problem. In the case of two spatial dimensions the \textit{Nabla operator} has in fact two components $\nabla = (\frac{\partial}{\partial x}, \frac{\partial}{\partial y})$. In full, \ref{eqn:commonform nabla} assuming a constant coefficient $A$, takes the form; |
It is necessary for clarification that we revisit the general PDE from \refeq{eqn:commonform nabla} under the light of a two dimensional domain. \ESCRIPT is inherently designed to solve problems that are greater than one dimension and so \ref{eqn:commonform nabla} needs to be read as a higher dimensional problem. In the case of two spatial dimensions the \textit{Nabla operator} has in fact two components $\nabla = (\frac{\partial}{\partial x}, \frac{\partial}{\partial y})$. In full, \ref{eqn:commonform nabla} assuming a constant coefficient $A$, takes the form; |
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