{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import bempp.api\n", "import numpy as np\n", "import fractalpp" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Set problem parameters:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "prefractal_level = 3\n", "#inc wave params:\n", "kwave = 10\n", "#mesh parameter\n", "h = 2*np.pi/(10*kwave) # ten wavelengths long\n", "inc_dir = np.sqrt(1/3)*np.array([1,1,-1])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now create the mesh using ```fractalpp.mesh.koch```\n", "We are including the optional parameter ```h_max```, which will keep subdividing until all elements are less than this diameter. Other optional paramteres, and their defaults, are:\n", "```h_max=math.inf, scale = 1, apex_angle = math.pi/3, shift = np.array([0,0]), mid_refine=True```\n", "The final one of these avoivds refinement in the centre of the fractal, and only refines close to the edges." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "koch_mesh = fractalpp.mesh.koch(prefractal_level,h_max = h)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now plot the mesh. When ```prefractal_level``` is high, I would advise not running the below cell of code, as the plot function will takes ages and it's not necessary." ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/html": [ " \n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "application/vnd.plotly.v1+json": { "config": { "linkText": "Export to plot.ly", "plotlyServerURL": "https://plot.ly", "showLink": false }, "data": [ { "facecolor": [ "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 173)", "rgb(255, 222, 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "koch_mesh.plot()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now make a function for plotting slices in this notebook, which we can use for each BC. We can also obtain our approximation to $u^s(x)\\approx\\mathcal{D}\\phi(x) - \\mathcal{S}\\psi(x)$. The next few cells of code plot the solution in a slice $[-1,1]\\times[-1,1]\\times\\{z\\}\\subset\\mathbb{R}^3$ for some $z$." ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "def slice(u_s,z_val = -0.3):\n", " # z_val is the z value of square slice\n", " n_grid_points = 150\n", " plot_grid = np.mgrid[-1:1:n_grid_points*1j, -1:1:n_grid_points*1j]\n", " points = np.vstack((plot_grid[0].ravel(),\n", " plot_grid[1].ravel(),\n", " z_val*np.ones(plot_grid[0].size)))\n", " scattered_field_at_points = (u_s(points)).reshape((n_grid_points,n_grid_points))\n", " %matplotlib inline\n", " from matplotlib import pylab as plt\n", "\n", " plt.imshow(np.abs(scattered_field_at_points), extent=(-1,1,-1,1))\n", " plt.title('Slice of scattered field')\n", " plt.colorbar()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Sound-soft problem" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "%%capture\n", "# create incident wave\n", "uinc = fractalpp.solve.u_i(kwave,inc_dir)\n", "# solve impedance problem via BIE formulation\n", "sol = fractalpp.solve.dirichlet(koch_mesh,uinc)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "sol.neumann_jump.plot()" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "slice(sol.u_s)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Impedance problem\n", "Now solve the impedance problem on the mesh, using our new impedance BIE." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "#impedance parameters:\n", "lambda_plus = kwave*(1.5+1.5j)\n", "lambda_minus = kwave*(1+1j)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "%%capture\n", "# create incident wave\n", "uinc = fractalpp.solve.u_i(kwave,inc_dir)\n", "# solve impedance problem via BIE formulation\n", "imp_sol = fractalpp.solve.impedance(koch_mesh,uinc,lambda_plus,lambda_minus)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The above ```fractalpp.solve.impedance``` has the optional parameter ```Hassen=False```, which if set to ```True``` will implement the BIE from the Hassen et al paper, which is not well-posed for $\\lambda_-+\\lambda_+$=0.\n", "\n", "From the ```imp_sol``` object we have created, we can access the components of the solution $\\left(\\phi,\\psi\\right) \\approx \\left([u],[\\partial_nu]\\right)$, via ```imp_sol.Dirichlet_jump```\n", "and ```imp_sol.Neumann_jump```. For example, we can run the following to get a surface plot:" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "imp_sol.Dirichlet_jump.plot()" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "imp_sol.Neumann_jump.plot()" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "slice(imp_sol.u_s,z_val=0.1)" ] } ], "metadata": { "kernelspec": { "display_name": "bempp3", "language": "python", "name": "bempp3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.9.6" } }, "nbformat": 4, "nbformat_minor": 4 }