diff --git a/chapter3/multiple_dirichlet.ipynb b/chapter3/multiple_dirichlet.ipynb index 285622e1..c305aa65 100644 --- a/chapter3/multiple_dirichlet.ipynb +++ b/chapter3/multiple_dirichlet.ipynb @@ -6,7 +6,7 @@ "source": [ "# Setting multiple Dirichlet condition\n", "\n", - "In the previous section, we used a single function to $u_d$ to setting Dirichlet conditions on two parts of the boundary. However, it is often more practical to use multiple functins, one for each subdomain of the boundary. We consider a similar example to [the previous example](./neumann_dirichlet_code) and redefine it consist of two Dirichlet boundary conditions\n", + "In the previous section, we used a single function to $u_d$ to setting Dirichlet conditions on two parts of the boundary. However, it is often more practical to use multiple functions, one for each subdomain of the boundary. We consider a similar example to [the previous example](./neumann_dirichlet_code) and redefine it consist of two Dirichlet boundary conditions\n", "\n", "$$\n", "-\\nabla^2 u =f \\quad \\text{in } \\Omega,\n", @@ -49,7 +49,6 @@ "import pyvista\n", "\n", "\n", - "\n", "def u_exact(x):\n", " return 1 + x[0]**2 + 2 * x[1]**2\n", "\n", @@ -186,13 +185,6 @@ "else:\n", " figure = plotter.screenshot(\"multiple_dirichlet.png\")" ] - }, - { - "cell_type": "code", - "execution_count": null, - "metadata": {}, - "outputs": [], - "source": [] } ], "metadata": { diff --git a/chapter3/multiple_dirichlet.py b/chapter3/multiple_dirichlet.py index 5bd5be4f..2d0e243f 100644 --- a/chapter3/multiple_dirichlet.py +++ b/chapter3/multiple_dirichlet.py @@ -6,7 +6,7 @@ # extension: .py # format_name: light # format_version: '1.5' -# jupytext_version: 1.14.7 +# jupytext_version: 1.15.2 # kernelspec: # display_name: Python 3 (ipykernel) # language: python @@ -15,7 +15,7 @@ # # Setting multiple Dirichlet condition # -# In the previous section, we used a single function to $u_d$ to setting Dirichlet conditions on two parts of the boundary. However, it is often more practical to use multiple functins, one for each subdomain of the boundary. We consider a similar example to [the previous example](./neumann_dirichlet_code) and redefine it consist of two Dirichlet boundary conditions +# In the previous section, we used a single function to $u_d$ to setting Dirichlet conditions on two parts of the boundary. However, it is often more practical to use multiple functions, one for each subdomain of the boundary. We consider a similar example to [the previous example](./neumann_dirichlet_code) and redefine it consist of two Dirichlet boundary conditions # # $$ # -\nabla^2 u =f \quad \text{in } \Omega, @@ -52,7 +52,6 @@ import pyvista - def u_exact(x): return 1 + x[0]**2 + 2 * x[1]**2 @@ -123,6 +122,3 @@ def u_exact(x): plotter.show() else: figure = plotter.screenshot("multiple_dirichlet.png") -# - - - diff --git a/chapter3/neumann_dirichlet_code.ipynb b/chapter3/neumann_dirichlet_code.ipynb index 534ea39e..f043b527 100644 --- a/chapter3/neumann_dirichlet_code.ipynb +++ b/chapter3/neumann_dirichlet_code.ipynb @@ -7,7 +7,6 @@ }, "source": [ "# Combining Dirichlet and Neumann conditions\n", - "\n", "Author: Jørgen S. Dokken\n", "\n", "Let's return to the Poisson problem from the [Fundamentals chapter](./../chapter1/fundamentals.md) and see how to extend the mathematics and the implementation to handle Dirichlet condition in combination with a Neumann condition.\n", @@ -20,18 +19,15 @@ "is applied to the remaining sides $y=0$ and $y=1$.\n", "\n", "## The PDE problem\n", - "\n", "Let $\\Lambda_D$ and $\\Lambda_N$ denote parts of the boundary $\\partial \\Omega$ where the Dirichlet and Neumann conditions apply, respectively.\n", "The complete boundary-value problem can be written as\n", "\n", "$$\n", "-\\nabla^2 u =f \\qquad \\text{in } \\Omega,\n", "$$\n", - "\n", "$$\n", "u=u_D \\qquad\\text{on } \\Lambda_D,\n", "$$\n", - "\n", "$$\n", "-\\frac{\\partial u}{\\partial n}=g \\qquad \\text{on }\\Lambda_N\n", "$$\n", @@ -41,14 +37,12 @@ "$$\n", "f(x,y)=-6,\n", "$$\n", - "\n", "$$\n", "g(x,y)=\\begin{cases}\n", "0, & y=0,\\\\\n", "-4, & y=1,\n", "\\end{cases}\n", "$$\n", - "\n", "$$\n", "u_D(x,y)=1+x^2+2y^2.\n", "$$\n", @@ -60,7 +54,6 @@ "$$\n", "\n", "## The variational formulation\n", - "\n", "The first task is to derive the variational formulatin. This time we cannot omit the boundary term arising from integration by parts, because $v$ is only zero on $\\Lambda_D$. We have\n", "\n", "$$\n", @@ -79,20 +72,17 @@ "$$\n", " \\int_\\Omega \\nabla u \\cdot \\nabla v~\\mathrm{d} x = \\int_\\Omega fv~\\mathrm{d} x - \\int_{\\Lambda_N}gv~\\mathrm{d}s.\n", "$$\n", - "\n", "Expressing this equation in the standard notation $a(u,v)=L(v)$ is straight-forward with\n", "\n", "$$\n", " a(u,v) = \\int_{\\Omega} \\nabla u \\cdot \\nabla v ~\\mathrm{d} x,\\\\\n", "$$\n", - "\n", "$$\n", "L(v) = \\int_{\\Omega} fv ~\\mathrm{d} x - \\int_{\\Lambda_N} gv~\\mathrm{d} s.\n", "$$\n", "\n", "## Implementation\n", - "\n", - "As in the previous example, we define our mesh,function space and bilinear form $a(u,v)$.\n" + "As in the previous example, we define our mesh,function space and bilinear form $a(u,v)$." ] }, { @@ -125,7 +115,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Now we get to the Neumann and Dirichlet boundary condition. As previously, we use a Python-function to define the boundary where we should have a Dirichlet condition. Then, with this function, we locate degrees of freedom that fullfils this condition.\n" + "Now we get to the Neumann and Dirichlet boundary condition. As previously, we use a Python-function to define the boundary where we should have a Dirichlet condition. Then, with this function, we locate degrees of freedom that fullfils this condition." ] }, { @@ -152,7 +142,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "The next step is to define the Neumann condition. We first define $g$ uses `UFL`s `SpatialCoordinate`-function, and then in turn create a boundary integration measure `ds`. As the test function $v$ is zero on the boundary integrals over the Dirichlet boundary dissapears, and wee can integrate `g*v*ds` over the entire boundary.\n" + "The next step is to define the Neumann condition. We first define $g$ uses `UFL`s `SpatialCoordinate`-function, and then in turn create a boundary integration measure `ds`. As the test function $v$ is zero on the boundary integrals over the Dirichlet boundary disappears, and we can integrate `g*v*ds` over the entire boundary." ] }, { @@ -171,7 +161,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "We can now assemble and solve the linear system of equations\n" + "We can now assemble and solve the linear system of equations" ] }, { @@ -213,8 +203,7 @@ "metadata": {}, "source": [ "## Visualization\n", - "\n", - "To look at the actual solution, run the script as a python script with `off_screen=True` or as a Jupyter notebook with `off_screen=False`\n" + "To look at the actual solution, run the script as a python script with `off_screen=True` or as a Jupyter notebook with `off_screen=False`" ] }, { @@ -224,7 +213,291 @@ "outputs": [ { "data": { - "application/javascript": "(function(root) {\n function now() {\n return new Date();\n }\n\n var force = true;\n var py_version = '3.2.1'.replace('rc', '-rc.').replace('.dev', '-dev.');\n var is_dev = 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"\ton_load()\n", + " })\n", + " require([\"jspanel-tooltip\"], function() {\n", + "\ton_load()\n", + " })\n", + " require([\"jspanel-hint\"], function() {\n", + "\ton_load()\n", + " })\n", + " require([\"jspanel-layout\"], function() {\n", + "\ton_load()\n", + " })\n", + " require([\"jspanel-contextmenu\"], function() {\n", + "\ton_load()\n", + " })\n", + " require([\"jspanel-dock\"], function() {\n", + "\ton_load()\n", + " })\n", + " require([\"gridstack\"], function(GridStack) {\n", + "\twindow.GridStack = GridStack\n", + "\ton_load()\n", + " })\n", + " require([\"notyf\"], function() {\n", + "\ton_load()\n", + " })\n", + " root._bokeh_is_loading = css_urls.length + 10;\n", + " } else {\n", + " root._bokeh_is_loading = css_urls.length + js_urls.length + js_modules.length + Object.keys(js_exports).length;\n", + " }\n", + "\n", + " var existing_stylesheets = []\n", + " var links = document.getElementsByTagName('link')\n", + " for (var i = 0; i < links.length; i++) {\n", + " var link = links[i]\n", + " if (link.href != null) {\n", + "\texisting_stylesheets.push(link.href)\n", + " }\n", + " }\n", + " for (var i = 0; i < css_urls.length; i++) {\n", + " var url = css_urls[i];\n", + " if (existing_stylesheets.indexOf(url) !== -1) {\n", + "\ton_load()\n", + "\tcontinue;\n", + " }\n", + " const element = document.createElement(\"link\");\n", + " element.onload = on_load;\n", + " element.onerror = on_error;\n", + " element.rel = \"stylesheet\";\n", + " element.type = \"text/css\";\n", + " element.href = url;\n", + " console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n", + " document.body.appendChild(element);\n", + " } if (((window['vtk'] !== undefined) && (!(window['vtk'] instanceof HTMLElement))) || window.requirejs) {\n", + " var urls = ['https://cdn.holoviz.org/panel/1.2.1/dist/bundled/abstractvtkplot/vtk.js@20.0.1/vtk.js'];\n", + " for (var i = 0; i < urls.length; i++) {\n", + " skip.push(urls[i])\n", + " }\n", + " } if (((window['jsPanel'] !== undefined) && (!(window['jsPanel'] instanceof HTMLElement))) || window.requirejs) {\n", + " var urls = ['https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/jspanel.js', 'https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/modal/jspanel.modal.js', 'https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/tooltip/jspanel.tooltip.js', 'https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/hint/jspanel.hint.js', 'https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/layout/jspanel.layout.js', 'https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/contextmenu/jspanel.contextmenu.js', 'https://cdn.holoviz.org/panel/1.2.1/dist/bundled/floatpanel/jspanel4@4.12.0/dist/extensions/dock/jspanel.dock.js'];\n", + " for (var i = 0; i < urls.length; i++) {\n", + " skip.push(urls[i])\n", + " }\n", + " } if (((window['GridStack'] !== undefined) && (!(window['GridStack'] instanceof HTMLElement))) || window.requirejs) {\n", + " var urls = ['https://cdn.holoviz.org/panel/1.2.1/dist/bundled/gridstack/gridstack@7.2.3/dist/gridstack-all.js'];\n", + " for (var i = 0; i < urls.length; i++) {\n", + " skip.push(urls[i])\n", + " }\n", + " } if (((window['Notyf'] !== undefined) && (!(window['Notyf'] instanceof HTMLElement))) || window.requirejs) {\n", + " var urls = ['https://cdn.holoviz.org/panel/1.2.1/dist/bundled/notificationarea/notyf@3/notyf.min.js'];\n", + " for (var i = 0; i < urls.length; i++) {\n", + " skip.push(urls[i])\n", + " }\n", + " } var existing_scripts = []\n", + " var scripts = document.getElementsByTagName('script')\n", + " for (var i = 0; i < scripts.length; i++) {\n", + " var script = scripts[i]\n", + " if (script.src != null) {\n", + "\texisting_scripts.push(script.src)\n", + " }\n", + " }\n", + " for (var i = 0; i < js_urls.length; i++) {\n", + " var url = js_urls[i];\n", + " if (skip.indexOf(url) !== -1 || existing_scripts.indexOf(url) !== -1) {\n", + "\tif (!window.requirejs) {\n", + "\t on_load();\n", + "\t}\n", + "\tcontinue;\n", + " }\n", + " var element = document.createElement('script');\n", + " element.onload = on_load;\n", + " element.onerror = on_error;\n", + " element.async = false;\n", + " element.src = url;\n", + " console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n", + " document.head.appendChild(element);\n", + " }\n", + " for (var i = 0; i < js_modules.length; i++) {\n", + " var url = js_modules[i];\n", + " if (skip.indexOf(url) !== -1 || existing_scripts.indexOf(url) !== -1) {\n", + "\tif (!window.requirejs) {\n", + "\t on_load();\n", + "\t}\n", + "\tcontinue;\n", + " }\n", + " var element = document.createElement('script');\n", + " element.onload = on_load;\n", + " element.onerror = on_error;\n", + " element.async = false;\n", + " element.src = url;\n", + " element.type = \"module\";\n", + " console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n", + " document.head.appendChild(element);\n", + " }\n", + " for (const name in js_exports) {\n", + " var url = js_exports[name];\n", + " if (skip.indexOf(url) >= 0 || root[name] != null) {\n", + "\tif (!window.requirejs) {\n", + "\t on_load();\n", + "\t}\n", + "\tcontinue;\n", + " }\n", + " var element = document.createElement('script');\n", + " element.onerror = on_error;\n", + " element.async = false;\n", + " element.type = \"module\";\n", + " console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n", + " element.textContent = `\n", + " import ${name} from \"${url}\"\n", + " window.${name} = ${name}\n", + " window._bokeh_on_load()\n", + " `\n", + " document.head.appendChild(element);\n", + " }\n", + " if (!js_urls.length && !js_modules.length) {\n", + " on_load()\n", + " }\n", + " };\n", + "\n", + " function inject_raw_css(css) {\n", + " const element = document.createElement(\"style\");\n", + " element.appendChild(document.createTextNode(css));\n", + " document.body.appendChild(element);\n", + " }\n", + "\n", + " var js_urls = [\"https://cdn.holoviz.org/panel/1.2.1/dist/bundled/abstractvtkplot/vtk.js@20.0.1/vtk.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.2.1.min.js\", \"https://cdn.holoviz.org/panel/1.2.1/dist/panel.min.js\"];\n", + " var js_modules = [];\n", + " var js_exports = {};\n", + " var css_urls = [];\n", + " var inline_js = [ function(Bokeh) {\n", + " Bokeh.set_log_level(\"info\");\n", + " },\n", + "function(Bokeh) {} // ensure no trailing comma for IE\n", + " ];\n", + "\n", + " function run_inline_js() {\n", + " if ((root.Bokeh !== undefined) || (force === true)) {\n", + " for (var i = 0; i < inline_js.length; i++) {\n", + " inline_js[i].call(root, root.Bokeh);\n", + " }\n", + " // Cache old bokeh versions\n", + " if (Bokeh != undefined && !reloading) {\n", + "\tvar NewBokeh = root.Bokeh;\n", + "\tif (Bokeh.versions === undefined) {\n", + "\t Bokeh.versions = new Map();\n", + "\t}\n", + "\tif (NewBokeh.version !== Bokeh.version) {\n", + "\t Bokeh.versions.set(NewBokeh.version, NewBokeh)\n", + "\t}\n", + "\troot.Bokeh = Bokeh;\n", + " }} else if (Date.now() < root._bokeh_timeout) {\n", + " setTimeout(run_inline_js, 100);\n", + " } else if (!root._bokeh_failed_load) {\n", + " console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n", + " root._bokeh_failed_load = true;\n", + " }\n", + " root._bokeh_is_initializing = false\n", + " }\n", + "\n", + " function load_or_wait() {\n", + " // Implement a backoff loop that tries to ensure we do not load multiple\n", + " // versions of Bokeh and its dependencies at the same time.\n", + " // In recent versions we use the root._bokeh_is_initializing flag\n", + " // to determine whether there is an ongoing attempt to initialize\n", + " // bokeh, however for backward compatibility we also try to ensure\n", + " // that we do not start loading a newer (Panel>=1.0 and Bokeh>3) version\n", + " // before older versions are fully initialized.\n", + " if (root._bokeh_is_initializing && Date.now() > root._bokeh_timeout) {\n", + " root._bokeh_is_initializing = false;\n", + " root._bokeh_onload_callbacks = undefined;\n", + " console.log(\"Bokeh: BokehJS was loaded multiple times but one version failed to initialize.\");\n", + " load_or_wait();\n", + " } else if (root._bokeh_is_initializing || (typeof root._bokeh_is_initializing === \"undefined\" && root._bokeh_onload_callbacks !== undefined)) {\n", + " setTimeout(load_or_wait, 100);\n", + " } else {\n", + " Bokeh = root.Bokeh;\n", + " bokeh_loaded = Bokeh != null && (Bokeh.version === py_version || (Bokeh.versions !== undefined && Bokeh.versions.has(py_version)));\n", + " root._bokeh_is_initializing = true\n", + " root._bokeh_onload_callbacks = []\n", + " if (!reloading && (!bokeh_loaded || is_dev)) {\n", + "\troot.Bokeh = undefined;\n", + " }\n", + " load_libs(css_urls, js_urls, js_modules, js_exports, function() {\n", + "\tconsole.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n", + "\trun_inline_js();\n", + " });\n", + " }\n", + " }\n", + " // Give older versions of the autoload script a head-start to ensure\n", + " // they initialize before we start loading newer version.\n", + " setTimeout(load_or_wait, 100)\n", + "}(window));" + ], "application/vnd.holoviews_load.v0+json": "" }, "metadata": {}, @@ -232,7 +505,247 @@ }, { "data": { - "application/javascript": "\nif ((window.PyViz === undefined) || (window.PyViz instanceof HTMLElement)) {\n window.PyViz = {comms: {}, comm_status:{}, kernels:{}, receivers: {}, plot_index: []}\n}\n\n\n function JupyterCommManager() {\n }\n\n 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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 */\nfunction 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 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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": {}, diff --git a/chapter3/neumann_dirichlet_code.py b/chapter3/neumann_dirichlet_code.py index 66fbaef2..f4adc919 100644 --- a/chapter3/neumann_dirichlet_code.py +++ b/chapter3/neumann_dirichlet_code.py @@ -6,7 +6,7 @@ # extension: .py # format_name: light # format_version: '1.5' -# jupytext_version: 1.14.7 +# jupytext_version: 1.15.2 # kernelspec: # display_name: Python 3 (ipykernel) # language: python @@ -23,7 +23,7 @@ # -\frac{\partial u}{\partial n}=g # $$ # -# is applied to the remaining sides $y=0$ and $y=1$. +# is applied to the remaining sides $y=0$ and $y=1$. # # ## The PDE problem # Let $\Lambda_D$ and $\Lambda_N$ denote parts of the boundary $\partial \Omega$ where the Dirichlet and Neumann conditions apply, respectively. @@ -131,7 +131,7 @@ def boundary_D(x): bc = dirichletbc(u_bc, dofs_D) # - -# The next step is to define the Neumann condition. We first define $g$ uses `UFL`s `SpatialCoordinate`-function, and then in turn create a boundary integration measure `ds`. As the test function $v$ is zero on the boundary integrals over the Dirichlet boundary dissapears, and wee can integrate `g*v*ds` over the entire boundary. +# The next step is to define the Neumann condition. We first define $g$ uses `UFL`s `SpatialCoordinate`-function, and then in turn create a boundary integration measure `ds`. As the test function $v$ is zero on the boundary integrals over the Dirichlet boundary disappears, and we can integrate `g*v*ds` over the entire boundary. x = SpatialCoordinate(mesh) g = -4 * x[1]