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   "cell_type": "markdown",
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   "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'`,...) assuming that the names of geometric dimensions are `'x'`, `'y'`, and `'z'`\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": [
    "## 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`. When the geometric dimensions are x, y, and z, the string can consist 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": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "mesh = df.Mesh(region=region, n=n, bc=\"xy\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 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": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<strong>Mesh</strong>\n",
       "<ul>\n",
       "  <li><strong>Region</strong>\n",
       "<ul>\n",
       "  <li>pmin = [0.0, 0.0, 0.0]</li>\n",
       "  <li>pmax = [1e-07, 5e-08, 2e-08]</li>\n",
       "  <li>dims = ['x', 'y', 'z']</li>\n",
       "  <li>units = ['m', 'm', 'm']</li>\n",
       "</ul></li>\n",
       "  <li>n = [20, 10, 4]</li>\n",
       "    <li>bc = neumann</li>\n",
       "  </ul>"
      ],
      "text/plain": [
       "Mesh(Region(pmin=[0.0, 0.0, 0.0], pmax=[1e-07, 5e-08, 2e-08], dims=['x', 'y', 'z'], units=['m', 'm', 'm']), n=[20, 10, 4], bc=neumann)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mesh = df.Mesh(region=region, n=n, bc=\"neumann\")\n",
    "mesh"
   ]
  }
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