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"Steady state heat equation\n",
"======\n",
"\n",
"This notebook will setup and solve the steady state heat equation:\n",
"\n",
"\\\\[\n",
"\\nabla(k\\nabla)T = h\n",
"\\\\]\n",
"\n",
"where $k$ is the diffusivity, T the temperature field and $h$ the source term.\n",
"This model is solved in a rectangular domain with Dirichlet boundary conditions on the top and bottom walls.\n",
"\n",
"**Keywords:** initial conditions, boundary conditions, heat equation\n",
"\n",
"\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import underworld as uw\n",
"import underworld.visualisation as vis"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"# Set some python variables for model parameters\n",
"boxHeight = 1.0\n",
"boxLength = 2.0\n",
"resx = 16\n",
"resy = 8\n",
"\n",
"# create mesh and variables\n",
"mesh = uw.mesh.FeMesh_Cartesian( elementType = (\"Q1/dQ0\"), \n",
" elementRes = (resx, resy), \n",
" minCoord = (0., 0.), \n",
" maxCoord = (boxLength, boxHeight))"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"# Create mesh variables for the temperature field & initialise. \n",
"temperatureField = mesh.add_variable( nodeDofCount=1 )\n",
"temperatureField.data[:] = 0."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"# Using the `specialSets` of the mesh we designate the vertices to \n",
"# build the `DirichletCondition` object - Refer to the `Systems` section \n",
"# of the user guide.\n",
"botWalls = mesh.specialSets[\"Bottom_VertexSet\"]\n",
"topWalls = mesh.specialSets[ \"Top_VertexSet\"]\n",
"bcWalls = botWalls + topWalls\n",
"tempBC = uw.conditions.DirichletCondition( variable=temperatureField, indexSetsPerDof=(bcWalls,) )"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"# set bottom wall temperature bc\n",
"temperatureField.data[botWalls] = 1.0\n",
"# set top wall temperature bc\n",
"temperatureField.data[topWalls] = 0.0"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"scrolled": false
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"source": [
"# visualisation of temperature field & mesh\n",
"fig = vis.Figure(figsize=(800,400))\n",
"fig.append( vis.objects.Mesh(mesh) )\n",
"fig.append( vis.objects.Surface( mesh, temperatureField, colours=\"blue white red\" ) )\n",
"fig.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Heat Equation System setup & solve\n",
"-----\n",
"\n",
"Temperature field, diffusivity and boundary conditions are passed to the SteadyStateHeat system function."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"heatequation = uw.systems.SteadyStateHeat(temperatureField = temperatureField, \n",
" fn_diffusivity = 1.0, \n",
" conditions = tempBC)\n",
"\n",
"# get the default heat equation solver\n",
"heatsolver = uw.systems.Solver(heatequation)\n",
"# solve\n",
"heatsolver.solve()"
]
},
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"cell_type": "code",
"execution_count": 8,
"metadata": {},
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"source": [
"# visualisation of temperature field & mesh\n",
"fig.show()"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [],
"source": [
"# As a test, ensure we reproduce the required average temperature\n",
"import numpy as np\n",
"tottemp = uw.utils.Integral(temperatureField, mesh)\n",
"avtemp = tottemp.evaluate()[0] / (boxHeight*boxLength)\n",
"if not np.isclose(avtemp,0.5):\n",
" raise RuntimeError(\"Incorrect average temperature produced by model. \")"
]
},
{
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"execution_count": null,
"metadata": {},
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}
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