{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Mesh boundary conditions\n",
"\n",
"In Ubermag, boundary conditions are set by passing `bc` argument to the mesh. The value of the `bc` argument is a string. The following boudary conditions (BC) are allowed:\n",
"\n",
"1. open BC (`bc=''`)\n",
"2. periodic BC (`bc='x'`, `bc='xy'`, `bc='xyz'`, `bc='y'`,...)\n",
"3. Neumann BC (`bc='neumann'`) - experimental\n",
"4. Dirichlet BC (`bc='dirichlet'`) - experimental\n",
"\n",
"To demonstrate boundary conditions, we are going to use the following mesh:"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import discretisedfield as df\n",
"\n",
"p1 = (0, 0, 0)\n",
"p2 = (100e-9, 50e-9, 20e-9)\n",
"n = (20, 10, 4)\n",
"\n",
"region = df.Region(p1=p1, p2=p2)\n",
"mesh = df.Mesh(region=region, n=n)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"By default, boudary conditions are open (empty string):"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"''"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mesh.bc"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can now ask the mesh for neigbouring cells. If we ask for the neighbours of a cell, which is not at the boundary, we expect to get 6 neighbouring cell indices."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[(1, 2, 2), (3, 2, 2), (2, 1, 2), (2, 3, 2), (2, 2, 1), (2, 2, 3)]"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mesh.neighbours((2, 2, 2))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"On the other hand, if we ask for neighbours of a cell which is at the corner of the sample:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[(1, 0, 0), (0, 1, 0), (0, 0, 1)]"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mesh.neighbours((0, 0, 0))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"we get only three neighbouring cell indices.\n",
"\n",
"## Periodic boundary conditions\n",
"\n",
"Now, let us define a mesh with periodic boundary conditions. The periodic boudary conditions are defined by passing a string to `bc`. String consists of characters `'x'`, `'y'`, and/or `'z'`, depending on the directions in which the mesh is periodic. For instance, if our mesh is periodic in x and y directions, we pass `bc='xy'`."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"mesh = df.Mesh(region=region, n=n, bc=\"xy\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can have a look at the neghbouring cells again. For a cell in the middle of the sample, there are 6 neighbours as expected:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[(1, 2, 2), (3, 2, 2), (2, 1, 2), (2, 3, 2), (2, 2, 1), (2, 2, 3)]"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mesh.neighbours((2, 2, 2))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"However, if we now ask for neighbours of a corner cell:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[(19, 0, 0), (1, 0, 0), (0, 19, 0), (0, 1, 0), (0, 0, 1)]"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mesh.neighbours((0, 0, 0))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"we get 5 neighbouring cells:\n",
"\n",
"- 2 in the x-direction `(19, 0, 0)` and `(1, 0, 0)`,\n",
"- 2 in the y-direction `(0, 19, 0)` and `(0, 1, 0)`, and\n",
"- 1 in the z-direction `(0, 0, 1)`,\n",
"\n",
"because our mesh is not periodic in the z-direction.\n",
"\n",
"## Experimental: Neumann and Dirichet boundary conditions\n",
"\n",
"Neumann and Dirichlet BC are defined by passing `bc='neumann'` or `bc='dirichet'`, respectively.\n",
"\n",
"**IMPORTANT:** At the moment, only Neumann BC with zero value are supported and defining BC in a more general way will be included in the future releases of `discretisedfield`.\n",
"\n",
"Here we just include an example of defining Neumann BC:\n",
"\n",
"$$\\frac{df}{d\\mathbf{n}} = 0$$"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"Mesh\n",
"\n",
" - Region\n",
"
\n",
" - p1 = (0, 0, 0)
\n",
" - p2 = (1e-07, 5e-08, 2e-08)
\n",
"
\n",
" - n = (20, 10, 4)
\n",
" - bc = neumann
\n",
" - attributes:\n",
"
\n",
" - unit: m
\n",
" - fourierspace: False
\n",
" - isplane: False
\n",
"
\n",
" \n",
"
"
],
"text/plain": [
"Mesh(Region(p1=(0, 0, 0), p2=(1e-07, 5e-08, 2e-08)), n=(20, 10, 4), bc=neumann, attributes: (unit: m, fourierspace: False, isplane: False))"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"mesh = df.Mesh(region=region, n=n, bc=\"neumann\")\n",
"mesh"
]
}
],
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