{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "Source : \n", "\n", "gits https://gist.github.com/Juanlu001/10305c7730a5b17f6148\n", "original post https://jasmcole.com/2014/08/25/helmhurts/\n", "\n", "\n", "Other cool stuff:\n", "https://fenicsproject.org/qa/6386/helmholtz-equation-with-absorbent-subdomain" ] }, { "cell_type": "code", "execution_count": 91, "metadata": { "collapsed": true }, "outputs": [], "source": [ "%matplotlib notebook" ] }, { "cell_type": "code", "execution_count": 92, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from fenics import *" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First, we will solve this on the unit square. Let's build a mesh." ] }, { "cell_type": "code", "execution_count": 93, "metadata": { "collapsed": true }, "outputs": [], "source": [ "nx = ny = 30\n", "mesh = RectangleMesh(Point(-1, -1), Point(1, 1), nx, ny)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's define the function space upon which we will build." ] }, { "cell_type": "code", "execution_count": 94, "metadata": { "collapsed": true }, "outputs": [], "source": [ "V = FunctionSpace(mesh, 'P', 1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Our field will be $E$ and also, we'll define a test function.\n", "\n", "Woah I learned this the hard way:\n", "\n", "- the unknown is a trial function\n", "- the v field is a test function\n", "\n", "Later, I overwrite E with a function." ] }, { "cell_type": "code", "execution_count": 109, "metadata": { "collapsed": true }, "outputs": [], "source": [ "u = TrialFunction(V)\n", "v = TestFunction(V)" ] }, { "cell_type": "code", "execution_count": 110, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAABcAAAAcCAYAAACK7SRjAAAABHNCSVQICAgIfAhkiAAAAX9JREFU\nSInt1L9LVWEcBvCPGSEpLUE0hEncQUjwSpMIkoOIuyAEQv+AWKNDu0KDm2NuQVspgTQEgogtaYmL\nBhEFNmQmRoVKDu974HC8p+u59w4RPXB4z3l43uf7k8Nfin48wyf8xt2s4Fwd5m3YwAR+1OFTFQca\nnHlV/Bvmk8LUx3O0N/ALr9BU1Hw9nl052mlcwP2YRFWcT72/iefNCro+jOAJliPXhlIqyXaUsYsP\nlYJ9wdcM14RV/ERHir8tVJB95vIqeRkF11LcnchN5V06K2ai0VD8bsF7fMalombZVUz6ngz1Hq7j\nAfYz2mm8KBLslpD5I1zBN7xFcwXtIh4WMW/BkbDLszHQYI52B2NFzGFT2IwjLORorsbAo3iO73iH\ngWrmj+PFQ3TmaIajZkmorISneH3WCv6ESewJFSQYw8e0qNYfVxnzQt8TlLDdKPOVDNeDtXrNLwpZ\nZvt7yrwW9OIYrSnusjDgclpYS+bd2BLWL0GPsF2bNfj9RwNxAvh3UJ3D0p/VAAAAAElFTkSuQmCC\n", "text/latex": [ "${v_h^{1}}$" ], "text/plain": [ "Argument(FunctionSpace(Mesh(VectorElement(FiniteElement('Lagrange', triangle, 1), dim=2), 98), FiniteElement('Lagrange', triangle, 1)), 1, None)" ] }, "execution_count": 110, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u" ] }, { "cell_type": "code", "execution_count": 111, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAABcAAAAcCAYAAACK7SRjAAAABHNCSVQICAgIfAhkiAAAAbNJREFU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\n", "text/latex": [ "${v_h^{0}}$" ], "text/plain": [ "Argument(FunctionSpace(Mesh(VectorElement(FiniteElement('Lagrange', triangle, 1), dim=2), 98), FiniteElement('Lagrange', triangle, 1)), 0, None)" ] }, "execution_count": 111, "metadata": {}, "output_type": "execute_result" } ], "source": [ "v" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Our boundary conditions will be null values on the boundary." ] }, { "cell_type": "code", "execution_count": 112, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def boundary(x, on_boundary):\n", " \"Function to call for boundary conditions.\"\n", " return on_boundary" ] }, { "cell_type": "code", "execution_count": 124, "metadata": { "collapsed": false }, "outputs": [], "source": [ "u_D = Expression('0.', degree=2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can now define a Dirichlet boundary condition that gets applied to the function space V, and evaluates u_D when on the boundary." ] }, { "cell_type": "code", "execution_count": 125, "metadata": { "collapsed": true }, "outputs": [], "source": [ "bc = DirichletBC(V, u_D, boundary)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's now define the terms used in the equation: the speed of light." ] }, { "cell_type": "code", "execution_count": 166, "metadata": { "collapsed": false }, "outputs": [], "source": [ "one_over_c_squared = Constant(100.)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, let's write the problem:" ] }, { "cell_type": "code", "execution_count": 167, "metadata": { "collapsed": false }, "outputs": [], "source": [ "a = Constant(-1) * dot(grad(u), grad(v))*dx + one_over_c_squared*u*v*dx\n", "L = v*dx" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Applying a point source:\n", "https://fenicsproject.org/qa/2893/point-source-in-rhs-of-heat-eq" ] }, { "cell_type": "code", "execution_count": 168, "metadata": { "collapsed": false }, "outputs": [], "source": [ "A, b = assemble_system(a, L, bc)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "And the point source." ] }, { "cell_type": "code", "execution_count": 169, "metadata": { "collapsed": false }, "outputs": [], "source": [ "source = Point(0.0, 0.0)" ] }, { "cell_type": "code", "execution_count": 170, "metadata": { "collapsed": true }, "outputs": [], "source": [ "delta = PointSource(V, source, 1)\n", "delta.apply(b)" ] }, { "cell_type": "code", "execution_count": 171, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "1" ] }, "execution_count": 171, "metadata": {}, "output_type": "execute_result" } ], "source": [ "E = Function(V)\n", "solve(A, E.vector(), b)" ] }, { "cell_type": "code", "execution_count": 172, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import matplotlib.pyplot as plt" ] }, { "cell_type": "code", "execution_count": 173, "metadata": { "collapsed": false }, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $('
');\n", " this._root_extra_style(this.root)\n", " this.root.attr('style', 'display: inline-block');\n", "\n", " $(parent_element).append(this.root);\n", "\n", " this._init_header(this);\n", " this._init_canvas(this);\n", " this._init_toolbar(this);\n", "\n", " var fig = this;\n", "\n", " this.waiting = false;\n", "\n", " this.ws.onopen = function () {\n", " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", " fig.send_message(\"send_image_mode\", {});\n", " if (mpl.ratio != 1) {\n", " fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n", " }\n", " fig.send_message(\"refresh\", {});\n", " }\n", "\n", " this.imageObj.onload = function() {\n", " if (fig.image_mode == 'full') {\n", " // Full images could contain transparency (where diff images\n", " // almost always do), so we need to clear the canvas so that\n", " // there is no ghosting.\n", " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", " this.ws.close();\n", " }\n", "\n", " this.ws.onmessage = this._make_on_message_function(this);\n", "\n", " this.ondownload = ondownload;\n", "}\n", "\n", "mpl.figure.prototype._init_header = function() {\n", " var titlebar = $(\n", " '');\n", " var titletext = $(\n", " '');\n", " titlebar.append(titletext)\n", " this.root.append(titlebar);\n", " this.header = titletext[0];\n", "}\n", "\n", "\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "\n", "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "mpl.figure.prototype._init_canvas = function() {\n", " var fig = this;\n", "\n", " var canvas_div = $('');\n", "\n", " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", "\n", " function canvas_keyboard_event(event) {\n", " return fig.key_event(event, event['data']);\n", " }\n", "\n", " canvas_div.keydown('key_press', canvas_keyboard_event);\n", " canvas_div.keyup('key_release', canvas_keyboard_event);\n", " this.canvas_div = canvas_div\n", " this._canvas_extra_style(canvas_div)\n", " this.root.append(canvas_div);\n", "\n", " var canvas = $('');\n", " canvas.addClass('mpl-canvas');\n", " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", "\n", " this.canvas = canvas[0];\n", " this.context = canvas[0].getContext(\"2d\");\n", "\n", " var backingStore = this.context.backingStorePixelRatio ||\n", "\tthis.context.webkitBackingStorePixelRatio ||\n", "\tthis.context.mozBackingStorePixelRatio ||\n", "\tthis.context.msBackingStorePixelRatio ||\n", "\tthis.context.oBackingStorePixelRatio ||\n", "\tthis.context.backingStorePixelRatio || 1;\n", "\n", " mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n", "\n", " var rubberband = $('');\n", " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", "\n", " var pass_mouse_events = true;\n", "\n", " canvas_div.resizable({\n", " start: function(event, ui) {\n", " pass_mouse_events = false;\n", " },\n", " resize: function(event, ui) {\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " stop: function(event, ui) {\n", " pass_mouse_events = true;\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " });\n", "\n", " function mouse_event_fn(event) {\n", " if (pass_mouse_events)\n", " return fig.mouse_event(event, event['data']);\n", " }\n", "\n", " rubberband.mousedown('button_press', mouse_event_fn);\n", " rubberband.mouseup('button_release', mouse_event_fn);\n", " // Throttle sequential mouse events to 1 every 20ms.\n", " rubberband.mousemove('motion_notify', mouse_event_fn);\n", "\n", " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", "\n", " canvas_div.on(\"wheel\", function (event) {\n", " event = event.originalEvent;\n", " event['data'] = 'scroll'\n", " if (event.deltaY < 0) {\n", " event.step = 1;\n", " } else {\n", " event.step = -1;\n", " }\n", " mouse_event_fn(event);\n", " });\n", "\n", " canvas_div.append(canvas);\n", " canvas_div.append(rubberband);\n", "\n", " this.rubberband = rubberband;\n", " this.rubberband_canvas = rubberband[0];\n", " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", " this.rubberband_context.strokeStyle = \"#000000\";\n", "\n", " this._resize_canvas = function(width, height) {\n", " // Keep the size of the canvas, canvas container, and rubber band\n", " // canvas in synch.\n", " canvas_div.css('width', width)\n", " canvas_div.css('height', height)\n", "\n", " canvas.attr('width', width * mpl.ratio);\n", " canvas.attr('height', height * mpl.ratio);\n", " canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n", "\n", " rubberband.attr('width', width);\n", " rubberband.attr('height', height);\n", " }\n", "\n", " // Set the figure to an initial 600x600px, this will subsequently be updated\n", " // upon first draw.\n", " this._resize_canvas(600, 600);\n", "\n", " // Disable right mouse context menu.\n", " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", " return false;\n", " });\n", "\n", " function set_focus () {\n", " canvas.focus();\n", " canvas_div.focus();\n", " }\n", "\n", " window.setTimeout(set_focus, 100);\n", "}\n", "\n", "mpl.figure.prototype._init_toolbar = function() {\n", " var fig = this;\n", "\n", " var nav_element = $('')\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\n", " for(var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " // put a spacer in here.\n", " continue;\n", " }\n", " var button = $('');\n", " button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n", " 'ui-button-icon-only');\n", " button.attr('role', 'button');\n", " button.attr('aria-disabled', 'false');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", "\n", " var icon_img = $('');\n", " icon_img.addClass('ui-button-icon-primary ui-icon');\n", " icon_img.addClass(image);\n", " icon_img.addClass('ui-corner-all');\n", "\n", " var tooltip_span = $('');\n", " tooltip_span.addClass('ui-button-text');\n", " tooltip_span.html(tooltip);\n", "\n", " button.append(icon_img);\n", " button.append(tooltip_span);\n", "\n", " nav_element.append(button);\n", " }\n", "\n", " var fmt_picker_span = $('');\n", "\n", " var fmt_picker = $('');\n", " fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n", " fmt_picker_span.append(fmt_picker);\n", " nav_element.append(fmt_picker_span);\n", " this.format_dropdown = fmt_picker[0];\n", "\n", " for (var ind in mpl.extensions) {\n", " var fmt = mpl.extensions[ind];\n", " var option = $(\n", " '', {selected: fmt === mpl.default_extension}).html(fmt);\n", " fmt_picker.append(option)\n", " }\n", "\n", " // Add hover states to the ui-buttons\n", " $( \".ui-button\" ).hover(\n", " function() { $(this).addClass(\"ui-state-hover\");},\n", " function() { $(this).removeClass(\"ui-state-hover\");}\n", " );\n", "\n", " var status_bar = $('