{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Linear viscoelasticity\n", "\n", "In this numerical tour, we will explore the formulation of simple linear viscoelastic behaviours such as Maxwell, Kelvin-Voigt or Standard Linear Solid models. The formulation can also be quite easily extended to a generalized Maxwell model.\n", "\n", "## Constitutive evolution equations\n", "\n", "### 1D rheological formulation" ] }, { "cell_type": "raw", "metadata": { "raw_mimetype": "text/restructuredtext" }, "source": [ ".. image:: 1D_rheological_model.png\n", " :scale: 80%\n", " :align: center" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "We consider a 1D Linear Standard Solid model consisting of a spring of stiffness $E_0$ in parallel to a Maxwell arm (spring of stiffness $E_1$ in serie with a dashpot of viscosity $\\eta_1$). The uniaxial stress for this rheological model can be decomposed as the sum of a reversible and an irreversible stress:\n", "\n", "$$\\sigma = E_0\\varepsilon + E_1(\\varepsilon-\\varepsilon^v)$$\n", "\n", "whereas the evolution equation for the viscous internal strain is given by:\n", "\n", "$$\\dot{\\varepsilon}^v = \\dfrac{E_1}{\\eta_1}(\\varepsilon-\\varepsilon^v)$$\n", "\n", "The extension to a generalized Maxwell model with $N$ internal strains is given by:\n", "\n", "$$\\begin{align}\n", "\\sigma &= E_0\\varepsilon + \\sum_{i=1}^N E_i(\\varepsilon-\\varepsilon^{v,i}) \\\\\n", "\\dot{\\varepsilon}^{v,i} &= \\dfrac{E_i}{\\eta_i}(\\varepsilon-\\varepsilon^{v,i}) \\quad \\forall i=1,\\ldots, N\n", "\\end{align}$$\n", "\n", "\n", "### 3D generalization\n", "\n", "For the 3D case, isotropic viscoelasticity is characterized by two elastic moduli (resp. two viscosities, or equivalently two relaxation times) for each spring (resp. dashpot) element of the 1D model. Here, we will restrict to a simpler case in which one modulus is common to all elements (similar Poisson ratio for all elements), that is:\n", "\n", "$$\\begin{align}\n", "\\boldsymbol{\\sigma} = E_0\\mathbb{c}:\\boldsymbol{\\varepsilon} + E_1\\mathbb{c}:(\\boldsymbol{\\varepsilon}-\\boldsymbol{\\varepsilon}^v) \\\\\n", "\\dot{\\boldsymbol{\\varepsilon}}^v = \\dfrac{E_1}{\\eta_1}\\mathbb{c}:(\\boldsymbol{\\varepsilon}-\\boldsymbol{\\varepsilon}^v)\n", "\\end{align}$$\n", "\n", "where $\\mathbb{c} = \\dfrac{\\nu}{(1+\\nu)(1-2\\nu)}\\mathbf{1}\\otimes\\mathbf{1} + \\dfrac{1}{1+\\nu}\\mathbb{I}$ with $\\mathbf{1}$ and $\\mathbb{I}$ being respectively the 2nd and 4th order identity tensors and $\\nu$ being the Poisson ratio.\n", "\n", "\n", "## Problem position\n", "\n", "We consider here a 2D rectangular domain of dimensions $L\\times H$. The constitutive relations are written in plane stress conditions, the unitary stiffness tensor is therefore $\\mathbb{c} = \\dfrac{\\nu}{1-\\nu^2}\\mathbf{1}\\otimes\\mathbf{1} + \\dfrac{1}{1+\\nu}\\mathbb{I}$. The boundary conditions consist of symmetry planes on $x=0$ and $y=0$ and smooth contact with a plane with imposed vertical displacement on $y=H$ or imposed vertical uniform traction depending on the load case. The solution will therefore be homogeneous in the sample. " ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [], "source": [ "from dolfin import *\n", "from ufl import replace\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "%matplotlib notebook\n", "\n", "L, H = 0.1, 0.2\n", "mesh = RectangleMesh(Point(0., 0.), Point(L, H), 5, 10)\n", "\n", "E0 = Constant(70e3)\n", "E1 = Constant(20e3)\n", "eta1 = Constant(1e3)\n", "nu = Constant(0.)\n", "dt = Constant(0.) # time increment\n", "sigc = 100. # imposed creep stress\n", "epsr = 1e-3 # imposed relaxation strain\n", "\n", "def left(x, on_boundary):\n", " return near(x[0], 0.) and on_boundary\n", "def bottom(x, on_boundary):\n", " return near(x[1], 0.) and on_boundary\n", "class Top(SubDomain):\n", " def inside(self, x, on_boundary):\n", " return near(x[1], H) and on_boundary\n", "\n", "facets = MeshFunction(\"size_t\", mesh, 1)\n", "facets.set_all(0)\n", "Top().mark(facets, 1)\n", "ds = Measure(\"ds\", subdomain_data=facets) " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Time discretization and formulation of the variational problem\n", "\n", "Here we will discuss the time discretization of the viscoelastic constitutive equations and the formulation of the global problem using FEniCS.\n", "\n", "### A first approach\n", "\n", "A first approach for formulating the time-discretized equations consists in approximating the viscous strain evolution equation using an implicit backward-Euler scheme at a given time $t_{n+1}$:\n", "\n", "$$\\begin{equation}\n", "\\dfrac{\\varepsilon^{v,n+1}-\\varepsilon^{v,n}}{\\Delta t} \\approx \\dot{\\varepsilon}^{v,n+1}= \\dfrac{E_1}{\\eta_1}(\\varepsilon^{n+1}-\\varepsilon^{v, n+1})\n", "\\end{equation}$$\n", "\n", "which can be rewritten as:\n", "$$\\begin{equation}\n", "\\varepsilon^{v,n+1}= \\left(1+\\dfrac{\\Delta tE_1}{\\eta_1}\\right)^{-1}\\left(\\varepsilon^{v,n} +\\dfrac{\\Delta t E_1}{\\eta_1}\\varepsilon^{n+1}\\right)\n", "\\end{equation}$$\n", "\n", "**Note:** In the 3D case, the previous expression is more involved since one must invert the following 4-th order tensor: $\\mathbb{I} + \\dfrac{\\Delta tE_1}{\\eta_1}\\mathbb{c}$\n", "\n", "Introducing $\\tau = \\eta_1/E_1$ and plugging the previous relation into the stress-strain relation, one obtains:\n", "\n", "$$\\begin{align}\n", "\\sigma_{n+1} &= (E_0+E_1)\\varepsilon^{n+1} - E_1\\dfrac{\\Delta t/\\tau}{1+\\Delta t/\\tau}\\varepsilon^{n+1} - E_1\\varepsilon^{v,n} \\\\\n", "&= \\left(E_0 + \\dfrac{E_1}{1+\\Delta t/\\tau}\\right)\\varepsilon^{n+1} - \\dfrac{E_1}{1+\\Delta t/\\tau}\\varepsilon^{v,n}\n", "\\end{align}$$\n", "\n", "A possible solution for solving these simple linear viscoelastic behaviours would then be to formulate exactly the problem associated with the previous stress-strain relation, taking into account the modified elasticity tensor depending on the time step $\\Delta t$ and the value of the previous viscous strain. As a result, one would have a pure displacement problem. After solving for $u$ at a given time step, the new viscous strain would have to be updated using the previous relations.\n", "\n", "### A mixed approach\n", "\n", "One problematic aspect of the previous approach is that it requires inverting the modified elasticity tensor which poses no problem in the present case but may not be possible for more general non-linear behaviours for instance. We hence propose a more general approach which amounts to solving both the displacement-problem and the viscous strain update in a monolothic way. The formulation can be derived from an incremental variational principle by defining the following incremental potential:\n", "\n", "$$\\begin{equation}\n", "\\mathcal{E} = \\int_{\\Omega} w(\\boldsymbol{\\varepsilon}, \\dot{\\boldsymbol{\\varepsilon}}^v)d\\Omega + \\Delta t \\int_{\\Omega} \\phi(\\dot{\\boldsymbol{\\varepsilon}}^v)d\\Omega - W_{ext}(\\boldsymbol{u})\n", "\\end{equation}$$\n", "\n", "where $W_{ext}(\\boldsymbol{u})$ is the work of external forces and the strain energy density $w(\\boldsymbol{\\varepsilon}, \\boldsymbol{\\varepsilon}^v)$ and the dissipation potential $\\phi(\\dot{\\boldsymbol{\\varepsilon}}^v)$ of the Linear Standard Solid model are respectively given by:\n", "\n", "$$\\begin{align*}\n", "w(\\boldsymbol{\\varepsilon}, \\boldsymbol{\\varepsilon}^v) &= \\dfrac{1}{2}\\boldsymbol{\\varepsilon}:(E_0\\mathbb{c}):\\boldsymbol{\\varepsilon} + \\dfrac{1}{2}(\\boldsymbol{\\varepsilon}-\\boldsymbol{\\varepsilon}^v):(E_1\\mathbb{c}):(\\boldsymbol{\\varepsilon}-\\boldsymbol{\\varepsilon}^v) \\\\\n", "\\phi(\\dot{\\boldsymbol{\\varepsilon}}^v) &= \\dfrac{1}{2}\\eta_1 \\dot{\\boldsymbol{\\varepsilon}}^v:\\dot{\\boldsymbol{\\varepsilon}}^v\n", "\\end{align*}$$\n", "\n", "Introducing the backward-Euler approximation \n", "\n", "$$\\begin{equation}\n", "\\dot{\\boldsymbol{\\varepsilon}}^v \\approx \\dfrac{\\boldsymbol{\\varepsilon}^{v,n+1}-\\boldsymbol{\\varepsilon}^{v,n}}{\\Delta t}\n", "\\end{equation}$$\n", "\n", "into the dissipation potential, the new total and viscous strain variables are obtained as the solution to the following minimization problem:\n", "\n", "$$\\begin{equation}\n", "\\min_{\\boldsymbol{\\varepsilon}^{n+1},\\: \\boldsymbol{\\varepsilon}^{v,n+1}} \\mathcal{E}_{n+1} = \\int_{\\Omega} w(\\boldsymbol{\\varepsilon}^{n+1}, \\boldsymbol{\\varepsilon}^{v,n+1})d\\Omega + \\Delta t \\int_{\\Omega} \\phi\\left(\\dfrac{\\boldsymbol{\\varepsilon}^{v,n+1}-\\boldsymbol{\\varepsilon}^{v,n}}{\\Delta t}\\right)d\\Omega\n", "\\end{equation}$$\n", "\n", "which depends on the knowledge of the previous viscous strain $\\boldsymbol{\\varepsilon}^{v,n}$.\n", "\n", "It can be shown that the optimality conditions of the previous minimization problem correspond exactly to the relations of the first approach.\n", "\n", "In the FEniCS implementation, we use a CG1 interpolation for the displacement and a DG0 interpolation for the viscoelastic strain. We then define appropriate functions and form the previous incremental potential. We use the `derivative` function for automatic differentiation. Since the considered problem is linear, we also extract the corresponding bilinear and linear parts of the potential derivative." ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [], "source": [ "Ve = VectorElement(\"CG\", mesh.ufl_cell(), 1)\n", "Qe = TensorElement(\"DG\", mesh.ufl_cell(), 0)\n", "W = FunctionSpace(mesh, MixedElement([Ve, Qe]))\n", "w = Function(W, name=\"Variables at current step\")\n", "(u, epsv) = split(w)\n", "w_old = Function(W, name=\"Variables at previous step\")\n", "(u_old, epsv_old) = split(w_old)\n", "w_ = TestFunction(W)\n", "(u_, epsv_) = split(w_)\n", "dw = TrialFunction(W)\n", "\n", "def eps(u):\n", " return sym(grad(u))\n", "def dotC(e):\n", " return nu/(1+nu)/(1-nu)*tr(e)*Identity(2) + 1/(1+nu)*e\n", "def sigma(u, epsv): \n", " return E0*dotC(eps(u)) + E1*dotC(eps(u) - epsv)\n", "def strain_energy(u, epsv):\n", " e = eps(u)\n", " return 0.5*(E0*inner(e,dotC(e)) + E1*inner(e-epsv, dotC(e-epsv)))\n", "def dissipation_potential(depsv):\n", " return 0.5*eta1*inner(depsv, depsv)\n", "\n", "Traction = Constant(0.)\n", "incremental_potential = strain_energy(u, epsv)*dx \\\n", " + dt*dissipation_potential((epsv-epsv_old)/dt)*dx \\\n", " - Traction*u[1]*ds(1)\n", "F = derivative(incremental_potential, w, w_)\n", "form = replace(F, {w: dw})" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Solutions of elementary tests\n", "\n", "In the following, we will consider three elementary tests of viscoelastic behaviour, namely a relaxation, creep and recovery test. We implement a function for setting up the boundary conditions and loading parameters for these three different tests and perform the time integration. We keep track of the average vertical stress and strain states (the problem is such that all fields are uniform in the sample) and compare the evolutions with analytical solutions given below. Parameters have been taken as $E_0= 70\\text{ GPa}$, $E_1= 20\\text{ GPa}$, $\\eta_1 = 1 \\text{ GPa.s}$." ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [], "source": [ "dimp = Constant(H*epsr) # imposed vertical displacement instead\n", "bcs = [DirichletBC(W.sub(0).sub(0), Constant(0), left),\n", " DirichletBC(W.sub(0).sub(1), Constant(0), bottom),\n", " DirichletBC(W.sub(0).sub(1), dimp, facets, 1)]\n", "\n", "def viscoelastic_test(case, Nsteps=50, Tend=1.):\n", " # Solution fields are initialized to zero\n", " w.interpolate(Constant((0.,)*6))\n", " \n", " # Define proper loading and BCs\n", " if case in [\"creep\", \"recovery\"]: # imposed traction on top\n", " Traction.assign(Constant(sigc))\n", " bc = bcs[:2] # remove the last boundary conditions from bcs\n", " t0 = Tend/2. # traction goes to zero at t0 for recovery test\n", " elif case == \"relaxation\":\n", " Traction.assign(Constant(0.)) # no traction on top\n", " bc = bcs\n", "\n", " # Time-stepping loop\n", " time = np.linspace(0, Tend, Nsteps+1)\n", " Sigyy = np.zeros((Nsteps+1, ))\n", " Epsyy = np.zeros((Nsteps+1, 2))\n", " for (i, dti) in enumerate(np.diff(time)):\n", " if i>0 and i % (Nsteps/5) == 0:\n", " print(\"Increment {}/{}\".format(i, Nsteps))\n", " dt.assign(dti)\n", " if case == \"recovery\" and time[i+1] > t0:\n", " Traction.assign(Constant(0.))\n", " w_old.assign(w)\n", " solve(lhs(form) == rhs(form), w, bc)\n", " # get average stress/strain\n", " Sigyy[i+1] = assemble(sigma(u, epsv)[1, 1]*dx)/L/H\n", " Epsyy[i+1, 0] = assemble(eps(u)[1, 1]*dx)/L/H\n", " Epsyy[i+1, 1] = assemble(epsv[1, 1]*dx)/L/H\n", " \n", " # Define analytical solutions\n", " if case == \"creep\":\n", " if float(E0) == 0.:\n", " eps_an = sigc*(1./float(E1)+time/float(eta1))\n", " else:\n", " Estar = float(E0*E1/(E0+E1))\n", " tau = float(eta1)/Estar\n", " eps_an = sigc/float(E0)*(1-float(Estar/E0)*np.exp(-time/tau))\n", " sig_an = sigc + 0*time\n", " elif case == \"relaxation\":\n", " if float(E1) == 0.:\n", " sig_an = epsr*float(E0) + 0*time\n", " else:\n", " tau = float(eta1/E1)\n", " sig_an = epsr*(float(E0) + float(E1)*np.exp(-time/tau))\n", " eps_an = epsr + 0*time\n", " \n", " elif case == \"recovery\":\n", " Estar = float(E0*E1/(E0+E1))\n", " tau = float(eta1)/Estar\n", " eps_an = sigc/float(E0)*(1-float(E1/(E0+E1))*np.exp(-time/tau))\n", " sig_an = sigc + 0*time\n", " time2 = time[time > t0]\n", " sig_an[time > t0] = 0.\n", " eps_an[time > t0] = sigc*float(E1/E0/(E0+E1))*(np.exp(-(time2-t0)/tau)\n", " - np.exp(-time2/tau))\n", " \n", " # Plot strain evolution\n", " plt.figure()\n", " plt.plot(time, 100*eps_an, label=\"analytical solution\")\n", " plt.plot(time, 100*Epsyy[:, 0], '.', label=\"FE solution\")\n", " plt.plot(time, 100*Epsyy[:, 1], '--', linewidth=1, label=\"viscous strain\")\n", " plt.ylim(0, 1.2*max(Epsyy[:, 0])*100)\n", " plt.xlabel(\"Time\")\n", " plt.ylabel(\"Vertical strain [\\%]\")\n", " plt.title(case.capitalize() + \" test\")\n", " plt.legend()\n", " plt.show()\n", " \n", " # Plot stress evolution\n", " plt.figure()\n", " plt.plot(time, sig_an, label=\"analytical solution\")\n", " plt.plot(time, Sigyy, '.', label=\"FE solution\")\n", " plt.ylim(0, 1.2*max(Sigyy))\n", " plt.ylabel(\"Vertical stress\")\n", " plt.xlabel(\"Time\")\n", " plt.title(case.capitalize() + \" test\")\n", " plt.legend()\n", " plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Relaxation test\n", "\n", "First we consider a relaxation test in which the top surface is instantaneously displaced and maintained fixed, generating a uniform and constant uniaxial strain state $\\varepsilon_{yy} = \\varepsilon_r$. Resolving the 1D evolution equation for this test gives:\n", "\n", "$$\\begin{equation}\n", "\\sigma_{yy}(t) = E_0\\varepsilon_r + E_1\\varepsilon_r \\exp(-t/\\tau) \\quad \\text{with } \\tau = \\dfrac{\\eta_1}{E_1}\n", "\\end{equation}$$\n", "\n", "The analytical solution is well reproduced by the FE solution as seen below. The instantaneous stress being $\\sigma_{yy}(t=0^+) = (E_0+E_1)\\varepsilon_r$ whereas the long term stress is being given by $\\sigma_{yy}(t=\\infty) = E_0\\varepsilon_r$." ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Increment 10/50\n", "Increment 20/50\n", "Increment 30/50\n", "Increment 40/50\n" ] }, { "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. 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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 = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "}\n", "\n", "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n", " // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n", " // which will in turn request a refresh of the image.\n", " this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n", "}\n", "\n", "mpl.figure.prototype.send_message = function(type, properties) {\n", " properties['type'] = type;\n", " properties['figure_id'] = this.id;\n", " this.ws.send(JSON.stringify(properties));\n", "}\n", "\n", "mpl.figure.prototype.send_draw_message = function() {\n", " if (!this.waiting) {\n", " this.waiting = true;\n", " this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n", " }\n", "}\n", "\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " var format_dropdown = fig.format_dropdown;\n", " var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n", " fig.ondownload(fig, format);\n", "}\n", "\n", "\n", "mpl.figure.prototype.handle_resize = function(fig, msg) {\n", " var size = msg['size'];\n", " if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n", " fig._resize_canvas(size[0], size[1]);\n", " fig.send_message(\"refresh\", {});\n", " };\n", "}\n", "\n", "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n", " var x0 = msg['x0'] / mpl.ratio;\n", " var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n", " var x1 = msg['x1'] / mpl.ratio;\n", " var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n", " x0 = Math.floor(x0) + 0.5;\n", " y0 = Math.floor(y0) + 0.5;\n", " x1 = Math.floor(x1) + 0.5;\n", " y1 = Math.floor(y1) + 0.5;\n", " var min_x = Math.min(x0, x1);\n", " var min_y = Math.min(y0, y1);\n", " var width = Math.abs(x1 - x0);\n", " var height = Math.abs(y1 - y0);\n", "\n", " fig.rubberband_context.clearRect(\n", " 0, 0, fig.canvas.width, fig.canvas.height);\n", "\n", " fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n", "}\n", "\n", "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n", " // Updates the figure title.\n", " fig.header.textContent = msg['label'];\n", "}\n", "\n", "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n", " var cursor = msg['cursor'];\n", " switch(cursor)\n", " {\n", " case 0:\n", " cursor = 'pointer';\n", " break;\n", " case 1:\n", " cursor = 'default';\n", " break;\n", " case 2:\n", " cursor = 'crosshair';\n", " break;\n", " case 3:\n", " cursor = 'move';\n", " break;\n", " }\n", " fig.rubberband_canvas.style.cursor = cursor;\n", "}\n", "\n", "mpl.figure.prototype.handle_message = function(fig, msg) {\n", " fig.message.textContent = msg['message'];\n", "}\n", "\n", "mpl.figure.prototype.handle_draw = function(fig, msg) {\n", " // Request the server to send over a new figure.\n", " fig.send_draw_message();\n", "}\n", "\n", "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n", " fig.image_mode = msg['mode'];\n", "}\n", "\n", "mpl.figure.prototype.updated_canvas_event = function() {\n", " // Called whenever the canvas gets updated.\n", " this.send_message(\"ack\", {});\n", "}\n", "\n", "// A function to construct a web socket function for onmessage handling.\n", "// Called in the figure constructor.\n", "mpl.figure.prototype._make_on_message_function = function(fig) {\n", " return function socket_on_message(evt) {\n", " if (evt.data instanceof Blob) {\n", " /* FIXME: We get \"Resource interpreted as Image but\n", " * transferred with MIME type text/plain:\" errors on\n", " * Chrome. But how to set the MIME type? It doesn't seem\n", " * to be part of the websocket stream */\n", " evt.data.type = \"image/png\";\n", "\n", " /* Free the memory for the previous frames */\n", " if (fig.imageObj.src) {\n", " (window.URL || window.webkitURL).revokeObjectURL(\n", " fig.imageObj.src);\n", " }\n", "\n", " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", " fig.waiting = false;\n", " return;\n", " }\n", "\n", " var msg = JSON.parse(evt.data);\n", " var msg_type = msg['type'];\n", "\n", " // Call the \"handle_{type}\" callback, which takes\n", " // the figure and JSON message as its only arguments.\n", " try {\n", " var callback = fig[\"handle_\" + msg_type];\n", " } catch (e) {\n", " console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n", " return;\n", " }\n", "\n", " if (callback) {\n", " try {\n", " // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n", " callback(fig, msg);\n", " } catch (e) {\n", " console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n", " }\n", " }\n", " };\n", "}\n", "\n", "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n", "mpl.findpos = function(e) {\n", " //this section is from http://www.quirksmode.org/js/events_properties.html\n", " var targ;\n", " if (!e)\n", " e = window.event;\n", " if (e.target)\n", " targ = e.target;\n", " else if (e.srcElement)\n", " targ = e.srcElement;\n", " if (targ.nodeType == 3) // defeat Safari bug\n", " targ = targ.parentNode;\n", "\n", " // jQuery normalizes the pageX and pageY\n", " // pageX,Y are the mouse positions relative to the document\n", " // offset() returns the position of the element relative to the document\n", " var x = e.pageX - $(targ).offset().left;\n", " var y = e.pageY - $(targ).offset().top;\n", "\n", " return {\"x\": x, \"y\": y};\n", "};\n", "\n", "/*\n", " * return a copy of an object with only non-object keys\n", " * we need this to avoid circular references\n", " * http://stackoverflow.com/a/24161582/3208463\n", " */\n", "function simpleKeys (original) {\n", " return Object.keys(original).reduce(function (obj, key) {\n", " if (typeof original[key] !== 'object')\n", " obj[key] = original[key]\n", " return obj;\n", " }, {});\n", "}\n", "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", " if (name === 'button_press')\n", " {\n", " this.canvas.focus();\n", " this.canvas_div.focus();\n", " }\n", "\n", " var x = canvas_pos.x * mpl.ratio;\n", " var y = canvas_pos.y * mpl.ratio;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", " step: event.step,\n", " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", " * to control all of the cursor setting manually through the\n", " * 'cursor' event from matplotlib */\n", " event.preventDefault();\n", " return false;\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " // Handle any extra behaviour associated with a key event\n", "}\n", "\n", "mpl.figure.prototype.key_event = function(event, name) {\n", "\n", " // Prevent repeat events\n", " if (name == 'key_press')\n", " {\n", " if (event.which === this._key)\n", " return;\n", " else\n", " this._key = event.which;\n", " }\n", " if (name == 'key_release')\n", " this._key = null;\n", "\n", " var value = '';\n", " if (event.ctrlKey && event.which != 17)\n", " value += \"ctrl+\";\n", " if (event.altKey && event.which != 18)\n", " value += \"alt+\";\n", " if (event.shiftKey && event.which != 16)\n", " value += \"shift+\";\n", "\n", " value += 'k';\n", " value += event.which.toString();\n", "\n", " this._key_event_extra(event, name);\n", "\n", " this.send_message(name, {key: value,\n", " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n", " if (name == 'download') {\n", " this.handle_save(this, null);\n", " } else {\n", " this.send_message(\"toolbar_button\", {name: name});\n", " }\n", "};\n", "\n", "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n", " this.message.textContent = tooltip;\n", "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", " // object with the appropriate methods. Currently this is a non binary\n", " // socket, so there is still some room for performance tuning.\n", " var ws = {};\n", "\n", " ws.close = function() {\n", " comm.close()\n", " };\n", " ws.send = function(m) {\n", " //console.log('sending', m);\n", " comm.send(m);\n", " };\n", " // Register the callback with on_msg.\n", " comm.on_msg(function(msg) {\n", " //console.log('receiving', msg['content']['data'], msg);\n", " // Pass the mpl event to the overridden (by mpl) onmessage function.\n", " ws.onmessage(msg['content']['data'])\n", " });\n", " return ws;\n", "}\n", "\n", "mpl.mpl_figure_comm = function(comm, msg) {\n", " // This is the function which gets called when the mpl process\n", " // starts-up an IPython Comm through the \"matplotlib\" channel.\n", "\n", " var id = msg.content.data.id;\n", " // Get hold of the div created by the display call when the Comm\n", " // socket was opened in Python.\n", " var element = $(\"#\" + id);\n", " var ws_proxy = comm_websocket_adapter(comm)\n", "\n", " function ondownload(figure, format) {\n", " window.open(figure.imageObj.src);\n", " }\n", "\n", " var fig = new mpl.figure(id, ws_proxy,\n", " ondownload,\n", " element.get(0));\n", "\n", " // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n", " // web socket which is closed, not our websocket->open comm proxy.\n", " ws_proxy.onopen();\n", "\n", " fig.parent_element = element.get(0);\n", " fig.cell_info = mpl.find_output_cell(\"\");\n", " if (!fig.cell_info) {\n", " console.error(\"Failed to find cell for figure\", id, fig);\n", " return;\n", " }\n", "\n", " var output_index = fig.cell_info[2]\n", " var cell = fig.cell_info[0];\n", "\n", "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", " var width = fig.canvas.width/mpl.ratio\n", " fig.root.unbind('remove')\n", "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", " // Re-enable the keyboard manager in IPython - without this line, in FF,\n", " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('');\n", " fig.close_ws(fig, msg);\n", "}\n", "\n", "mpl.figure.prototype.close_ws = function(fig, msg){\n", " fig.send_message('closing', msg);\n", " // fig.ws.close()\n", "}\n", "\n", "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n", " // Turn the data on the canvas into data in the output cell.\n", " var width = this.canvas.width/mpl.ratio\n", " var dataURL = this.canvas.toDataURL();\n", " this.cell_info[1]['text/html'] = '';\n", "}\n", "\n", "mpl.figure.prototype.updated_canvas_event = function() {\n", " // Tell IPython that the notebook contents must change.\n", " IPython.notebook.set_dirty(true);\n", " this.send_message(\"ack\", {});\n", " var fig = this;\n", " // Wait a second, then push the new image to the DOM so\n", " // that it is saved nicely (might be nice to debounce this).\n", " setTimeout(function () { fig.push_to_output() }, 1000);\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) { continue; };\n", "\n", " var button = $('');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", " nav_element.append(button);\n", " }\n", "\n", " // Add the status bar.\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "\n", " // Add the close button to the window.\n", " var buttongrp = $('');\n", " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", "mpl.figure.prototype._root_extra_style = function(el){\n", " var fig = this\n", " el.on(\"remove\", function(){\n", "\tfig.close_ws(fig, {});\n", " });\n", "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", " el.attr('tabindex', 0)\n", " // reach out to IPython and tell the keyboard manager to turn it's self\n", " // off when our div gets focus\n", "\n", " // location in version 3\n", " if (IPython.notebook.keyboard_manager) {\n", " IPython.notebook.keyboard_manager.register_events(el);\n", " }\n", " else {\n", " // location in version 2\n", " IPython.keyboard_manager.register_events(el);\n", " }\n", "\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " var manager = IPython.notebook.keyboard_manager;\n", " if (!manager)\n", " manager = IPython.keyboard_manager;\n", "\n", " // Check for shift+enter\n", " if (event.shiftKey && event.which == 13) {\n", " this.canvas_div.blur();\n", " event.shiftKey = false;\n", " // Send a \"J\" for go to next cell\n", " event.which = 74;\n", " event.keyCode = 74;\n", " manager.command_mode();\n", " manager.handle_keydown(event);\n", " }\n", "}\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " fig.ondownload(fig, null);\n", "}\n", "\n", "\n", "mpl.find_output_cell = function(html_output) {\n", " // Return the cell and output element which can be found *uniquely* in the notebook.\n", " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", " // IPython event is triggered only after the cells have been serialised, which for\n", " // our purposes (turning an active figure into a static one), is too late.\n", " var cells = IPython.notebook.get_cells();\n", " var ncells = cells.length;\n", " for (var i=0; i