# Find this integral which containing normal derivative

I have a mesh of many triangles. Now, I only want to consider two triangles of this mesh. Three nodes (vertices) of the first triangle are $A(\frac{1}{4},0), B(\frac{1}{4},\frac{1}{4}), C(\frac{1}{2},\frac{1}{4})$ and the second one are $B(\frac{1}{4},\frac{1}{4}), C(\frac{1}{2},\frac{1}{4}), D(\frac{1}{2},\frac{1}{2})$.

I also have two basic functions $\phi_B,\phi_C$ defined on this mesh.

On $\Delta ABC$, I have

\begin{align*} \phi_B &= -4x+4y+1 \qquad (\phi_B \text{ =1 on node B and =0 on other nodes of ABC})\\ \phi_C &= 4x-1 \qquad (\phi_C \text{ =1 on node C and =0 on other nodes of ABC}) \end{align*}

On $\Delta BCD$, I have

\begin{align*} \phi_B &= -4x+2 \qquad (\phi_B \text{ =1 on node B and =0 on other nodes of BCD})\\ \phi_C &= 4x-4y \qquad (\phi_C \text{ =1 on node B and =0 on other nodes of BCD}) \end{align*}

There is one "interface" $x=0.3$ that cuts the two triangles. I call the intersection-segment $\Gamma$. Now, I want to compute the following integral on $\Gamma$

\begin{align*} -\frac{1}{2}\int_{\Gamma}{\phi_B\kappa_1(\nabla\phi_C\cdot\mathbf{n})} \end{align*}

where, $\kappa_1=0.1$ and $\mathbf{n}$ is normal vector.

• When you say 'interface' $x=0.3$, do you mean the line $x=0.3$? Also, can you please mention what you've tried as of now and where you are getting stuck? Feb 17, 2015 at 14:40
• @PrajaktaBedekar : Sory, I have just edited the expression inside the integral. What I have tried is that : we have $\Gamma: x=0.3$, it implies that $\nabla\phi_C\cdot\mathbf{n} = \partial_x\phi_C$ and $\int_{\Gamma}=\int_{\Gamma_{ABC}}+\int_{\Gamma_{BCD}}$. Substitute the expression of $\phi_B$ and $\phi_C$ on 2 triangles, after some computations, I have got the value $-0.024$. Feb 17, 2015 at 17:12