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"cell_type": "markdown",
"id": "sorted-rugby",
"metadata": {},
"source": [
"# Weak imposition of Dirichlet conditions for the Poisson problem\n",
"Author: Jørgen S. Dokken\n",
"\n",
"In this section, we will go through how to solve the Poisson problem from the [Fundamentals](./fundamentals_code.ipynb) tutorial using Nitsche's method {cite}`Nitsche1971`.\n",
"The idea of weak imposition is that we add additional terms to the variational formulation to impose the boundary condition, instead of modifying the matrix system using strong imposition (lifting).\n",
"\n",
"We start by importing the required modules and creating the mesh and function space for our solution"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "imposed-christopher",
"metadata": {},
"outputs": [],
"source": [
"from dolfinx import fem, mesh, plot, default_scalar_type\n",
"from dolfinx.fem.petsc import LinearProblem\n",
"import numpy\n",
"from mpi4py import MPI\n",
"from ufl import (Circumradius, FacetNormal, SpatialCoordinate, TrialFunction, TestFunction,\n",
" div, dx, ds, grad, inner)\n",
"\n",
"N = 8\n",
"domain = mesh.create_unit_square(MPI.COMM_WORLD, N, N)\n",
"V = fem.functionspace(domain, (\"Lagrange\", 1))"
]
},
{
"cell_type": "markdown",
"id": "surrounded-carrier",
"metadata": {},
"source": [
"Next, we create a function containing the exact solution (which will also be used in the Dirichlet boundary condition) and the corresponding source function for the right hand side. Note that we use `ufl.SpatialCoordinate` to define the exact solution, which in turn is interpolated into `uD` and used to create the source function `f`."
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "quality-andorra",
"metadata": {},
"outputs": [],
"source": [
"uD = fem.Function(V)\n",
"x = SpatialCoordinate(domain)\n",
"u_ex = 1 + x[0]**2 + 2 * x[1]**2\n",
"uD.interpolate(fem.Expression(u_ex, V.element.interpolation_points()))\n",
"f = -div(grad(u_ex))"
]
},
{
"cell_type": "markdown",
"id": "distinguished-obligation",
"metadata": {},
"source": [
"As opposed to the first tutorial, we now have to have another look at the variational form.\n",
"We start by integrating the problem by parts, to obtain\n",
"\\begin{align}\n",
" \\int_{\\Omega} \\nabla u \\cdot \\nabla v~\\mathrm{d}x - \\int_{\\partial\\Omega}\\nabla u \\cdot n v~\\mathrm{d}s = \\int_{\\Omega} f v~\\mathrm{d}x.\n",
"\\end{align}\n",
"As we are not using strong enforcement, we do not set the trace of the test function to $0$ on the outer boundary.\n",
"Instead, we add the following two terms to the variational formulation\n",
"\\begin{align}\n",
" -\\int_{\\partial\\Omega} \\nabla v \\cdot n (u-u_D)~\\mathrm{d}s + \\frac{\\alpha}{h} \\int_{\\partial\\Omega} (u-u_D)v~\\mathrm{d}s.\n",
"\\end{align}\n",
"where the first term enforces symmetry to the bilinear form, while the latter term enforces coercivity.\n",
"$u_D$ is the known Dirichlet condition, and $h$ is the diameter of the circumscribed sphere of the mesh element.\n",
"We create bilinear and linear form, $a$ and $L$\n",
"\\begin{align}\n",
" a(u, v) &= \\int_{\\Omega} \\nabla u \\cdot \\nabla v~\\mathrm{d}x + \\int_{\\partial\\Omega}-(n \\cdot\\nabla u) v - (n \\cdot \\nabla v) u + \\frac{\\alpha}{h} uv~\\mathrm{d}s,\\\\\n",
" L(v) &= \\int_{\\Omega} fv~\\mathrm{d}x + \\int_{\\partial\\Omega} -(n \\cdot \\nabla v) u_D + \\frac{\\alpha}{h} u_Dv~\\mathrm{d}s\n",
"\\end{align}"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "4b7634d9",
"metadata": {},
"outputs": [],
"source": [
"u = TrialFunction(V)\n",
"v = TestFunction(V)\n",
"n = FacetNormal(domain)\n",
"h = 2 * Circumradius(domain)\n",
"alpha = fem.Constant(domain, default_scalar_type(10))\n",
"a = inner(grad(u), grad(v)) * dx - inner(n, grad(u)) * v * ds\n",
"a += - inner(n, grad(v)) * u * ds + alpha / h * inner(u, v) * ds\n",
"L = inner(f, v) * dx \n",
"L += - inner(n, grad(v)) * uD * ds + alpha / h * inner(uD, v) * ds"
]
},
{
"cell_type": "markdown",
"id": "971dc391",
"metadata": {},
"source": [
"As we now have the variational form, we can solve the linear problem"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "6187fc89",
"metadata": {},
"outputs": [],
"source": [
"problem = LinearProblem(a, L)\n",
"uh = problem.solve()"
]
},
{
"cell_type": "markdown",
"id": "fac3387f",
"metadata": {},
"source": [
"We compute the error of the computation by comparing it to the analytical solution"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "4851b7a4",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"$L^2$-error: 1.59e-03\n"
]
}
],
"source": [
"error_form = fem.form(inner(uh-uD, uh-uD) * dx)\n",
"error_local = fem.assemble_scalar(error_form)\n",
"errorL2 = numpy.sqrt(domain.comm.allreduce(error_local, op=MPI.SUM))\n",
"if domain.comm.rank == 0:\n",
" print(fr\"$L^2$-error: {errorL2:.2e}\")"
]
},
{
"cell_type": "markdown",
"id": "c5c43d69",
"metadata": {},
"source": [
"We observe that the $L^2$-error is of the same magnitude as in the first tutorial.\n",
"As in the previous tutorial, we also compute the maximal error for all the degrees of freedom."
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "b3413383",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Error_max : 5.41e-03\n"
]
}
],
"source": [
"error_max = domain.comm.allreduce(numpy.max(numpy.abs(uD.x.array-uh.x.array)), op=MPI.MAX)\n",
"if domain.comm.rank == 0:\n",
" print(f\"Error_max : {error_max:.2e}\")"
]
},
{
"cell_type": "markdown",
"id": "f5ba8c79",
"metadata": {},
"source": [
"We observe that as we weakly impose the boundary condition, we no longer fullfill the equation to machine precision at the mesh vertices. We also plot the solution using `pyvista`"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "b12d508e",
"metadata": {},
"outputs": [
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" var message = result.value;\n",
" console.log(message)\n",
" var content = {data: message.data, comm_id};\n",
" var buffers = []\n",
" for (var buffer of message.buffers || []) {\n",
" buffers.push(new DataView(buffer))\n",
" }\n",
" var metadata = message.metadata || {};\n",
" var msg = {content, buffers, metadata}\n",
" msg_handler(msg);\n",
" return messages.next().then(processIteratorResult);\n",
" }\n",
" return messages.next().then(processIteratorResult);\n",
" })\n",
" }\n",
" }\n",
"\n",
" JupyterCommManager.prototype.get_client_comm = function(plot_id, comm_id, msg_handler) {\n",
" if (comm_id in window.PyViz.comms) {\n",
" return window.PyViz.comms[comm_id];\n",
" } else if (window.comm_manager || ((window.Jupyter !== undefined) && (Jupyter.notebook.kernel != null))) {\n",
" var comm_manager = window.comm_manager || Jupyter.notebook.kernel.comm_manager;\n",
" var comm = comm_manager.new_comm(comm_id, {}, {}, {}, comm_id);\n",
" if (msg_handler) {\n",
" comm.on_msg(msg_handler);\n",
" }\n",
" } else if ((plot_id in window.PyViz.kernels) && (window.PyViz.kernels[plot_id])) {\n",
" var comm = window.PyViz.kernels[plot_id].connectToComm(comm_id);\n",
" comm.open();\n",
" if (msg_handler) {\n",
" comm.onMsg = msg_handler;\n",
" }\n",
" } else if (typeof google != 'undefined' && google.colab.kernel != null) {\n",
" var comm_promise = google.colab.kernel.comms.open(comm_id)\n",
" comm_promise.then((comm) => {\n",
" window.PyViz.comms[comm_id] = comm;\n",
" if (msg_handler) {\n",
" var messages = comm.messages[Symbol.asyncIterator]();\n",
" function processIteratorResult(result) {\n",
" var message = result.value;\n",
" var content = {data: message.data};\n",
" var metadata = message.metadata || {comm_id};\n",
" var msg = {content, metadata}\n",
" msg_handler(msg);\n",
" return messages.next().then(processIteratorResult);\n",
" }\n",
" return messages.next().then(processIteratorResult);\n",
" }\n",
" }) \n",
" var sendClosure = (data, metadata, buffers, disposeOnDone) => {\n",
" return comm_promise.then((comm) => {\n",
" comm.send(data, metadata, buffers, disposeOnDone);\n",
" });\n",
" };\n",
" var comm = {\n",
" send: sendClosure\n",
" };\n",
" }\n",
" window.PyViz.comms[comm_id] = comm;\n",
" return comm;\n",
" }\n",
" window.PyViz.comm_manager = new JupyterCommManager();\n",
" \n",
"\n",
"\n",
"var JS_MIME_TYPE = 'application/javascript';\n",
"var HTML_MIME_TYPE = 'text/html';\n",
"var EXEC_MIME_TYPE = 'application/vnd.holoviews_exec.v0+json';\n",
"var CLASS_NAME = 'output';\n",
"\n",
"/**\n",
" * Render data to the DOM node\n",
" */\n",
"function render(props, node) {\n",
" var div = document.createElement(\"div\");\n",
" var script = document.createElement(\"script\");\n",
" node.appendChild(div);\n",
" node.appendChild(script);\n",
"}\n",
"\n",
"/**\n",
" * Handle when a new output is added\n",
" */\n",
"function handle_add_output(event, handle) {\n",
" var output_area = handle.output_area;\n",
" var output = handle.output;\n",
" if ((output.data == undefined) || (!output.data.hasOwnProperty(EXEC_MIME_TYPE))) {\n",
" return\n",
" }\n",
" var id = output.metadata[EXEC_MIME_TYPE][\"id\"];\n",
" var toinsert = output_area.element.find(\".\" + CLASS_NAME.split(' ')[0]);\n",
" if (id !== undefined) {\n",
" var nchildren = toinsert.length;\n",
" var html_node = toinsert[nchildren-1].children[0];\n",
" html_node.innerHTML = output.data[HTML_MIME_TYPE];\n",
" var scripts = [];\n",
" var nodelist = html_node.querySelectorAll(\"script\");\n",
" for (var i in nodelist) {\n",
" if (nodelist.hasOwnProperty(i)) {\n",
" scripts.push(nodelist[i])\n",
" }\n",
" }\n",
"\n",
" scripts.forEach( function (oldScript) {\n",
" var newScript = document.createElement(\"script\");\n",
" var attrs = [];\n",
" var nodemap = oldScript.attributes;\n",
" for (var j in nodemap) {\n",
" if (nodemap.hasOwnProperty(j)) {\n",
" attrs.push(nodemap[j])\n",
" }\n",
" }\n",
" attrs.forEach(function(attr) { newScript.setAttribute(attr.name, attr.value) });\n",
" newScript.appendChild(document.createTextNode(oldScript.innerHTML));\n",
" oldScript.parentNode.replaceChild(newScript, oldScript);\n",
" });\n",
" if (JS_MIME_TYPE in output.data) {\n",
" toinsert[nchildren-1].children[1].textContent = output.data[JS_MIME_TYPE];\n",
" }\n",
" output_area._hv_plot_id = id;\n",
" if ((window.Bokeh !== undefined) && (id in Bokeh.index)) {\n",
" window.PyViz.plot_index[id] = Bokeh.index[id];\n",
" } else {\n",
" window.PyViz.plot_index[id] = null;\n",
" }\n",
" } else if (output.metadata[EXEC_MIME_TYPE][\"server_id\"] !== undefined) {\n",
" var bk_div = document.createElement(\"div\");\n",
" bk_div.innerHTML = output.data[HTML_MIME_TYPE];\n",
" var script_attrs = bk_div.children[0].attributes;\n",
" for (var i = 0; i < script_attrs.length; i++) {\n",
" toinsert[toinsert.length - 1].childNodes[1].setAttribute(script_attrs[i].name, script_attrs[i].value);\n",
" }\n",
" // store reference to server id on output_area\n",
" output_area._bokeh_server_id = output.metadata[EXEC_MIME_TYPE][\"server_id\"];\n",
" }\n",
"}\n",
"\n",
"/**\n",
" * Handle when an output is cleared or removed\n",
" */\n",
"function handle_clear_output(event, handle) {\n",
" var id = handle.cell.output_area._hv_plot_id;\n",
" var server_id = handle.cell.output_area._bokeh_server_id;\n",
" if (((id === undefined) || !(id in PyViz.plot_index)) && (server_id !== undefined)) { return; }\n",
" var comm = window.PyViz.comm_manager.get_client_comm(\"hv-extension-comm\", \"hv-extension-comm\", function () {});\n",
" if (server_id !== null) {\n",
" comm.send({event_type: 'server_delete', 'id': server_id});\n",
" return;\n",
" } else if (comm !== null) {\n",
" comm.send({event_type: 'delete', 'id': id});\n",
" }\n",
" delete PyViz.plot_index[id];\n",
" if ((window.Bokeh !== undefined) & (id in window.Bokeh.index)) {\n",
" var doc = window.Bokeh.index[id].model.document\n",
" doc.clear();\n",
" const i = window.Bokeh.documents.indexOf(doc);\n",
" if (i > -1) {\n",
" window.Bokeh.documents.splice(i, 1);\n",
" }\n",
" }\n",
"}\n",
"\n",
"/**\n",
" * Handle kernel restart event\n",
" */\n",
"function handle_kernel_cleanup(event, handle) {\n",
" delete PyViz.comms[\"hv-extension-comm\"];\n",
" window.PyViz.plot_index = {}\n",
"}\n",
"\n",
"/**\n",
" * Handle update_display_data messages\n",
" */\n",
"function handle_update_output(event, handle) {\n",
" handle_clear_output(event, {cell: {output_area: handle.output_area}})\n",
" handle_add_output(event, handle)\n",
"}\n",
"\n",
"function register_renderer(events, OutputArea) {\n",
" function append_mime(data, metadata, element) {\n",
" // create a DOM node to render to\n",
" var toinsert = this.create_output_subarea(\n",
" metadata,\n",
" CLASS_NAME,\n",
" EXEC_MIME_TYPE\n",
" );\n",
" this.keyboard_manager.register_events(toinsert);\n",
" // Render to node\n",
" var props = {data: data, metadata: metadata[EXEC_MIME_TYPE]};\n",
" render(props, toinsert[0]);\n",
" element.append(toinsert);\n",
" return toinsert\n",
" }\n",
"\n",
" events.on('output_added.OutputArea', handle_add_output);\n",
" events.on('output_updated.OutputArea', handle_update_output);\n",
" events.on('clear_output.CodeCell', handle_clear_output);\n",
" events.on('delete.Cell', handle_clear_output);\n",
" events.on('kernel_ready.Kernel', handle_kernel_cleanup);\n",
"\n",
" OutputArea.prototype.register_mime_type(EXEC_MIME_TYPE, append_mime, {\n",
" safe: true,\n",
" index: 0\n",
" });\n",
"}\n",
"\n",
"if (window.Jupyter !== undefined) {\n",
" try {\n",
" var events = require('base/js/events');\n",
" var OutputArea = require('notebook/js/outputarea').OutputArea;\n",
" if (OutputArea.prototype.mime_types().indexOf(EXEC_MIME_TYPE) == -1) {\n",
" register_renderer(events, OutputArea);\n",
" }\n",
" } catch(err) {\n",
" }\n",
"}\n"
],
"application/vnd.holoviews_load.v0+json": ""
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"application/vnd.holoviews_exec.v0+json": "",
"text/html": [
"\n",
""
]
},
"metadata": {
"application/vnd.holoviews_exec.v0+json": {
"id": "323c7bee-1312-4818-8698-0250735bf20c"
}
},
"output_type": "display_data"
},
{
"data": {},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"application/vnd.holoviews_exec.v0+json": "",
"text/html": [
"\n",
""
],
"text/plain": [
"VTKRenderWindowSynchronized(vtkXOpenGLRenderWindow, color_mappers=[LinearColorMapper(id='00a...], sizing_mode='stretch_width')"
]
},
"metadata": {
"application/vnd.holoviews_exec.v0+json": {
"id": "34268dd5-ccf9-4606-bb1d-e74f4cf7ad35"
}
},
"output_type": "display_data"
}
],
"source": [
"import pyvista\n",
"pyvista.start_xvfb()\n",
"\n",
"grid = pyvista.UnstructuredGrid(*plot.vtk_mesh(V))\n",
"grid.point_data[\"u\"] = uh.x.array.real\n",
"grid.set_active_scalars(\"u\")\n",
"plotter = pyvista.Plotter()\n",
"plotter.add_mesh(grid, show_edges=True, show_scalar_bar=True)\n",
"plotter.view_xy()\n",
"if not pyvista.OFF_SCREEN:\n",
" plotter.show()\n",
"else:\n",
" figure = plotter.screenshot(\"nitsche.png\")"
]
},
{
"cell_type": "markdown",
"id": "choice-dealer",
"metadata": {},
"source": [
"```{bibliography}\n",
" :filter: cited and ({\"chapter1/nitsche\"} >= docnames)\n",
"```"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "9f22f7f7",
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"jupytext": {
"formats": "ipynb,py:light"
},
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.10.12"
}
},
"nbformat": 4,
"nbformat_minor": 5
}