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Example of interface structure

If you are going to replace the built-in geometry pre-processor by your own one, it has to provide the interface structure IX. Probably the best way to explain how it has to look like is a concrete example.

Light blue lines indicate the grid lines and the domain $\Omega^*$ is in this case a unit square with $n_x=n_y=11$ and $h_x=h_y=0.1$. On the black marked part of the boundary, Dirichlet boundary condition is given, the green one is the Neumann boundary.

Blue filled circles mark x-type intersection points, red filled circles are intersection points of y-type. Blue crosses mark the anchor points for x-intersections, red empty circles mark the anchor points for y-intersections.

Interface structure has following fields:

• coord: coordinates of the intersection points
• type: has values `x' or `y' corresponding to x and y-type intersections
• anch: indices of the anchor point. To a given intersection point $(x_\alpha, y_\alpha)$ the anchor is defined as follows: It is a point $(x_i,y_j)\in \Omega^-$ such that
  • for x-intersections: $y_j=y_\alpha$ and $\vert x_i-x_\alpha\vert<h_h$
  • for y-intersections: $x_i=x_\alpha$ and $\vert y_j-y_\alpha\vert<h_y$
• n: components of the normalised normal vector at each of the intersection points.
• t: components of the tangential vector at each of the intersection points. The pair $(n,t)$ has to form a right hand system. In two dimensions, $t=(-n_2,n_1)$.

Ordering of the intersection points
Currently, the code is constructed in such way that IX has to consist of two parts, IX_D (contains points along the Dirichlet boundary) and IX_N (contains the intersection points along the Neumann boundary). It is,

   IX$$= \begin{pmatrix} \mbox{\textcolor{mygreen}{\texttt{IX\_D}}} \\ \mbox{\textcolor{mygreen}{\texttt{IX\_N}}} \end{pmatrix}$$

Each of structures IX_D and IX_N has to be ordered in such way that x-type intersections come first and then come y-intersections.

In our example this would look as follows:

Nr	Nr	Nr
(IX)	(IX_D)	(IX_N)	coord	type	anch	n	t
1.	1.	-	0.275, 0.400	`x'	4, 5	(-1, -1)$/\sqrt{2}$	(1, -1)$/\sqrt{2}$
2.	2.	-	0.450, 0.300	`x'	6, 4	(-1, 0)	(0, -1)
3.	3.	-	0.875, 0.300	`x'	9, 4	(1, 0)	(0, 1)
4.	4.	-	0.850, 0.400	`x'	9, 5	(1, 1)$/\sqrt{2}$	(-1, 1)$/\sqrt{2}$
5.	5.	-	0.775, 0.500	`x'	8, 6	(1, 0)	(0, 1)
6.	6.	-	0.300, 0.375	`y'	4, 5	(-1, -1)$/\sqrt{2}$	(1, -1)$/\sqrt{2}$
7.	7.	-	0.400, 0.350	`y'	5, 5	(0, -1)	(1, 0)
8.	8.	-	0.500, 0.225	`y'	6, 4	(0, -1)	(1, 0)
9.	9.	-	0.600, 0.225	`y'	7, 4	(0, -1)	(1, 0)
10.	10.	-	0.700, 0.225	`y'	8, 4	(0, -1)	(1, 0)
11.	11.	-	0.800, 0.225	`y'	9, 4	(0, -1)	(1, 0)
12.	12.	-	0.800, 0.450	`y'	9, 5	(1, 1)$/\sqrt{2}$	(-1, 1)$/\sqrt{2}$
13.	13.	-	0.700, 0.525	`y'	8, 6	(0, 1)	(-1, 0)
14.	14.	-	0.600, 0.550	`y'	7, 6	(1, 1)$/\sqrt{2}$	(-1, 1)$/\sqrt{2}$
15.	-	1.	0.175, 0.500	`x'	3, 6	(-1, -1)$/\sqrt{2}$	(1, -1)$/\sqrt{2}$
16.	-	2.	0.125, 0.600	`x'	3, 7	(0, -1)	(1, 0)
17.	-	3.	0.125, 0.700	`x'	3, 8	(0, -1)	(1, 0)
18.	-	4.	0.125, 0.800	`x'	3, 9	(0, -1)	(1, 0)
19.	-	5.	0.525, 0.800	`x'	6, 9	(1, 1)$/\sqrt{2}$	(-1, 1)$/\sqrt{2}$
20.	-	6.	0.575, 0.700	`x'	6, 8	(1, 0)	(0, 1)
21.	-	7.	0.575, 0.600	`x'	6, 7	(1, 0)	(0, 1)
22.	-	8.	0.200, 0.475	`y'	3, 6	(-1, -1)$/\sqrt{2}$	(1, -1)$/\sqrt{2}$
23.	-	9.	0.200, 0.850	`y'	3, 9	(0, 1)	(-1, 0)
24.	-	10.	0.300, 0.850	`y'	4, 9	(0, 1)	(-1, 0)
25.	-	11.	0.400, 0.850	`y'	5, 9	(0, 1)	(-1, 0)
26.	-	12.	0.500, 0.825	`y'	6, 9	(1, 1)$/\sqrt{2}$	(-1, 1)$/\sqrt{2}$