From 6469189565605264e53653db501097b846056539 Mon Sep 17 00:00:00 2001 From: lgmartinez1 Date: Fri, 8 May 2015 11:37:13 -0400 Subject: [PATCH] final project --- .../1D_Euler_FINAL-May7-checkpoint.ipynb | 1301 +++++++++++++++++ Martinez_1D_Euler/1D_Euler_FINAL-May7.ipynb | 1301 +++++++++++++++++ .../Legendre_Polynomial_Function_np3.py | 166 +++ .../Legendre_Polynomial_Function_np3.pyc | Bin 0 -> 4924 bytes Martinez_1D_Euler/Plot_Functions2.py | 232 +++ Martinez_1D_Euler/Plot_Functions2.pyc | Bin 0 -> 5986 bytes Martinez_1D_Euler/Residual_Function_np3.py | 349 +++++ Martinez_1D_Euler/Residual_Function_np3.pyc | Bin 0 -> 6661 bytes .../Runge_Kutta_Third_Order_np3.py | 53 + .../Runge_Kutta_Third_Order_np3.pyc | Bin 0 -> 1429 bytes Martinez_1D_Euler/Slide04.jpg | Bin 0 -> 31854 bytes Martinez_1D_Euler/Slide08.jpg | Bin 0 -> 12055 bytes Martinez_1D_Euler/Slide13.jpg | Bin 0 -> 29003 bytes Martinez_1D_Euler/Slide14.jpg | Bin 0 -> 29515 bytes Martinez_1D_Euler/Slide15.jpg | Bin 0 -> 25493 bytes Martinez_1D_Euler/Slide16.jpg | Bin 0 -> 30375 bytes Martinez_1D_Euler/U_initial_np3.py | 83 ++ Martinez_1D_Euler/U_initial_np3.pyc | Bin 0 -> 2216 bytes Martinez_1D_Euler/density.png | Bin 0 -> 14239 bytes Martinez_1D_Euler/energy.png | Bin 0 -> 14013 bytes Martinez_1D_Euler/meshing.py | 47 + Martinez_1D_Euler/meshing.pyc | Bin 0 -> 1212 bytes Martinez_1D_Euler/pressure.png | Bin 0 -> 14372 bytes Martinez_1D_Euler/velocity.png | Bin 0 -> 15207 bytes 24 files changed, 3532 insertions(+) create mode 100644 Martinez_1D_Euler/.ipynb_checkpoints/1D_Euler_FINAL-May7-checkpoint.ipynb create mode 100644 Martinez_1D_Euler/1D_Euler_FINAL-May7.ipynb create mode 100644 Martinez_1D_Euler/Legendre_Polynomial_Function_np3.py create mode 100644 Martinez_1D_Euler/Legendre_Polynomial_Function_np3.pyc create mode 100644 Martinez_1D_Euler/Plot_Functions2.py create mode 100644 Martinez_1D_Euler/Plot_Functions2.pyc create mode 100644 Martinez_1D_Euler/Residual_Function_np3.py create mode 100644 Martinez_1D_Euler/Residual_Function_np3.pyc create mode 100644 Martinez_1D_Euler/Runge_Kutta_Third_Order_np3.py create mode 100644 Martinez_1D_Euler/Runge_Kutta_Third_Order_np3.pyc create mode 100644 Martinez_1D_Euler/Slide04.jpg create mode 100644 Martinez_1D_Euler/Slide08.jpg create mode 100644 Martinez_1D_Euler/Slide13.jpg create mode 100644 Martinez_1D_Euler/Slide14.jpg create mode 100644 Martinez_1D_Euler/Slide15.jpg create mode 100644 Martinez_1D_Euler/Slide16.jpg create mode 100644 Martinez_1D_Euler/U_initial_np3.py create mode 100644 Martinez_1D_Euler/U_initial_np3.pyc create mode 100644 Martinez_1D_Euler/density.png create mode 100644 Martinez_1D_Euler/energy.png create mode 100644 Martinez_1D_Euler/meshing.py create mode 100644 Martinez_1D_Euler/meshing.pyc create mode 100644 Martinez_1D_Euler/pressure.png create mode 100644 Martinez_1D_Euler/velocity.png diff --git a/Martinez_1D_Euler/.ipynb_checkpoints/1D_Euler_FINAL-May7-checkpoint.ipynb b/Martinez_1D_Euler/.ipynb_checkpoints/1D_Euler_FINAL-May7-checkpoint.ipynb new file mode 100644 index 0000000..a025b8d --- /dev/null +++ b/Martinez_1D_Euler/.ipynb_checkpoints/1D_Euler_FINAL-May7-checkpoint.ipynb @@ -0,0 +1,1301 @@ +{ + "metadata": { + "name": "", + "signature": "sha256:8ef564186e5a734ce19b023798a6e7e0115273b8c7d2b7bb859c83f62cf371a0" + }, + "nbformat": 3, + "nbformat_minor": 0, + "worksheets": [ + { + "cells": [ + { + "cell_type": "heading", + "level": 1, + "metadata": {}, + "source": [ + "One-Dimensional Conservation Laws of Gas Dynamics " + ] + }, + { + "cell_type": "heading", + "level": 5, + "metadata": {}, + "source": [ + "By Luis Martinez" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "from IPython.display import Latex\n", + "from IPython.display import Image" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 1 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The gas dynamics equations, which constitue the conservation of mass, momentum, and energy, can be used to study shock waves. This notebook includes a numerical scheme to solve for the classical Sod's Shock Tube problem.\n", + "\n", + "The Euler equations may be used to model other interesting physical phenomena. For example The Euler equations with a gravitational source term are used to model atmospheric pphenomena essential to weather predicion, climate modeling, and astrophisical modeling such as simulating solar climate and supernova explosions [Kappeli, Mishra, 2013]" + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Principles of the Shock Tube Problem" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Initially, the tube is divided (by a diaphragm) into a high-pressure section (driver section) and a low pressure section (the driven section). There is no flow velocity anywhere inside the tube initially. \n", + "\n", + "As, the diaphragm is removed, the initial pressure discontinuity will propagate to the right as an unsteady normal shock wave traveling at velocity W. At the same time, an isentropic expansion wave will propagate to the left.\n", + "\n", + "We can see that shortly after the diaphragm is removed, there are 4 main regions in the tube.\n", + "\n", + "Region 1 is the undisturbed portion of the driven section (so we would expect initial conditions to still be true here)\n", + "\n", + "Region 2 : This section has been \u2018processed\u2019 by the shock wave moving through it, and the pressure here is equal to the pressure behind the shock.\n", + "\n", + "Region 3: Has been \u2018processed\u2019 by the expansion wave moving through the region. The pressure here is equal to P3. Regions 2 and 3 are said to be at the same pressure since the gas in these regions cannot support any pressure discontinuities\n", + "\n", + "Region 4: The undisturbed portion of the driver section (we should expect initial conditions).\n", + "\n", + "It is interesting that Regions 2 and 3 are at the same velocity and pressure. But there are differences. Since region 2 has been processed by a shock wave and region 3 by an expansion wave, the entropy, temperature, and density are different. This is why we see this contact surface to differentiate the regions.\n", + "\n", + "Both Pressure and velocity change discontinuously across the shock wave. When we look at these properties across the expansion wave, we see that the variations of pressure and velocity are finite and continuous.\n", + "\n", + "This information was taken from Anderson(1995).\n" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Image(filename='slide04.jpg')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "jpeg": 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8EeF/hf8A8FUvAuk/DfUvE8PhvW/hD4kvL62ufGOo6rb3E1ve6WkUgF1cShWQ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\n+tfSZFmay6hWrr42uWP6s8HNMC8ZVp038KfM/lsj0Xw/8If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+ "metadata": {}, + "output_type": "pyout", + "prompt_number": 2, + "text": [ + "" + ] + } + ], + "prompt_number": 2 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Governing Equations" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We look at one-dimensional conservation of mass, momentum, and energy:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\\frac{\\partial \\rho}{\\partial t} + \\frac{\\partial \\rho \\vec{v_x}}{\\partial x} = 0$$\n", + "\n", + "$$\\frac{\\partial \\rho \\vec{v_x}}{\\partial t} + \\frac{\\partial(\\rho \\vec{v_x}^2 +P)}{\\partial x} = 0$$\n", + "\n", + "$$\\frac{\\partial E}{\\partial t} +\\ \\frac{\\partial (E+P) \\vec{v_x}}{\\partial x} = 0$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can also relate an equation of state:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$ P = (\\gamma-1)(E - \\frac{\\rho v^2}{2})$$" + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "System of Equations" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "From the conservation Equations we can develop a system of equations:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\\frac{\\partial U}{\\partial t} + \\frac{\\partial f(U)}{\\partial x} = 0$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where U and f(U) are:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$ U = \\left(\n", + "\\begin{array}{c}\n", + "\\rho\\\\\n", + "\\rho \\vec{v_x}\\\\\n", + "E\n", + "\\end{array}\n", + "\\right)\n", + "\\,\\\n", + "f(U) = \\left(\n", + "\\begin{array}{c}\n", + "\\rho \\vec{v_x}\\\\\n", + "\\rho \\vec{v_x}^2 +\\ P\\\\\n", + "(E+P)\\vec{v_x}\n", + "\\end{array}\n", + "\\right)$$ " + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Discretization of System of Equations: The Discontinuous Galerkin Method" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The discontinuous Galerkin method represents unknowns like the finite element method by piecewise polynomial functions, but unlike finite element methods, the polynomials are discontinuous at cell interfaces. At these interfaces, a flux is defined in the same way as for finite volume methods [Sonnendrucker, 2013. Numerical Methods for Hyperbolic Systems]\n", + "\n", + "On each cell, the unknown U is represented by a linear combination of basis functions of the space of polynomials of degree 'np'. The chosen basis function for this solver is a Legendre polynomial, and the interpolation points are Gauss-Lobatto points." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\\int_{I_i} \\frac{\\partial U(x,t)}{\\partial t} W(x)\\ dx\\ + \\int_{I_i} \\frac{\\partial f(U(x,t))}{\\partial x} W(x) dx = 0$$\n", + "\n", + "$$\\int_{I_i} \\frac{\\partial U(x,t)}{\\partial t} \\phi_l(x)\\ dx\\ + \\int_{I_i} \\frac{\\partial f(U(x,t))}{\\partial x} \\phi_l(x) dx = 0$$\n", + "\n", + "$$\\int_{I_i} \\frac{\\partial U(x,t)}{\\partial t} \\phi_l(x)\\ dx\\ + f(U)\\phi_l(x)\\ |_{I_i}\\ -\\ \\int_{I_i} f(U)\\frac{\\partial \\phi_l(x)}{\\partial x} dx = 0$$\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Note that the Integral I goes from [-0.5] t0 [0.5]. From the Galerkin Method we can estimate U(x,t):" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$U_{i}(x,t) = \\sum_{0}^{np} U_{l}^{i}(t)\\phi_l(x)$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The objective here is to obtain an expression for U(t). Using orthogonality, we can define the spectral coefficients, which in turn allow us to define U(t):" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\\alpha_l = \\int_{I_i} \\phi_l(x)\\ \\phi_l(x) dx\\\\$$\n", + "$$U_{l}^{i}(t) = \\frac{1}{\\alpha_l}\\ \\int_{I_i} U_{i}(x,t)\\ \\phi_l(x) dx\\\\$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Then, we can express the system of equations in a form that we can approximate:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\\frac{\\partial U_{l}^{i}(t)}{\\partial t} = \\frac{1}{\\alpha_l} \n", + "\\int_{I_i} f(U)\\frac{\\partial \\phi_l(x)}{\\partial x} dx \n", + "- f(U_{i+\\frac{1}{2}})\\phi_{i+\\frac{1}{2}}^l \n", + "+ f(U_{i-\\frac{1}{2}})\\phi_{i-\\frac{1}{2}}^l $$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The integral on the right-hand side can be approximated by Gauss-Lobatto quadratures and the fluxes are solved using the Rusanov flux. To march in time, a third-order runge-kutta method is used." + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "The Solver" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The solver is tested by applying it to the classic Sod's shock tube problem. It is executed using a third order polynomial approximation, with a domain of 10 cells." + ] + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Import Libraries" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import numpy\n", + "import sympy\n", + "from sympy.utilities.lambdify import lambdify\n", + "import math\n", + "from matplotlib import pyplot" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 3 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Import Scripts" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "from Legendre_Polynomial_Function_np3 import *\n", + "from meshing import *\n", + "from U_initial_np3 import *\n", + "from Runge_Kutta_Third_Order_np3 import *\n", + "from Residual_Function_np3 import *\n", + "from Plot_Functions2 import *" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 4 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Some Program Parameters" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "np = 3 # order of scheme\n", + "ncell = 10 #number of cells\n", + "nface = ncell+1 #cell boundaries\n", + "rk3 = 3 #This is because we are solving 3 equations(mass, momentum, energy)\n", + "dt = 0.01\n", + "\n", + "# for flux function:\n", + "normal = 1.0" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 5 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Legendre Polynomial and Derivative" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "fle_gauss = get_fle_gauss(np)\n", + "fle_grad_gauss = get_fle_derivative(np)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 6 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Meshing" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "x = get_x_cell(ncell)\n", + "dx = get_dx_cell(ncell)\n", + "f2c = get_face_to_cell(nface)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 7 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Initialize values of Mass, Momentum, and Energy" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Image(filename='slide08.jpg')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "jpeg": 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zSkEFgV9UYgfMW2gDknsO9fHuj/tgaNqt94R8Qv8ADPx3pvwW8VeJk8PeHPiP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\n9g/8FN/+h9/Y4/8ABJrP+NH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+ "metadata": {}, + "output_type": "pyout", + "prompt_number": 8, + "text": [ + "" + ] + } + ], + "prompt_number": 8 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The initial conditions are set as Sod's 1978 initial conditions. Using the mathematical relationships previously described, we can nitialize U, which contains mass, momentum, and energy:" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "aic = get_inverse_mass_matrix(np)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 9 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "U_initial = get_U_initial(np, rk3, ncell, fle_gauss, x ,dx, aic)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 10 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Third Order Runge Kutta Time Scheme" + ] + }, + { + "cell_type": "heading", + "level": 5, + "metadata": {}, + "source": [ + "Stages of RK Scheme:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "\n", + "$$U^1 = U^n\\ +\\ \\Delta R(U^n) \\\\$$\n", + "$$U^2 = \\frac{3}{4}U^n\\ +\\ \\frac{1}{4}\\ [U^1\\ +\\ \\Delta R(U^1)] \\\\$$\n", + "$$U^3 = \\frac{1}{3}U^n\\ +\\ \\frac{2}{3}\\ [U^2\\ +\\ \\Delta R(U^2)] \\\\$$\n" + ] + }, + { + "cell_type": "heading", + "level": 5, + "metadata": {}, + "source": [ + "Initialize Runge-Kutta Scheme:" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Q_rk = U_initial" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 11 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Qf = get_Qf(ncell, rk3, np, fle_gauss, Q_rk)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 12 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Boundary Conditions" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Qbleft, Qbright = get_Qbright_Qbleft(rk3,np, ncell, Qf)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 13 + }, + { + "cell_type": "heading", + "level": 5, + "metadata": {}, + "source": [ + "First Cell Left Edge" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Qfl, Qfr = get_Qfl_Qfr_left(rk3, Qf, np, Qbleft)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 14 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Fl = get_rusanov_flux(Qfl, Qfr, normal, rk3)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 15 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "bflux = get_bflux_left(Fl)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 16 + }, + { + "cell_type": "heading", + "level": 5, + "metadata": {}, + "source": [ + "Last Cell Right Edge" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Qfl = get_Qfl_right(Qf,np, ncell)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 17 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Fl = get_rusanov_flux(Qfl, Qbright, normal, rk3)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 18 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "bflux = get_bflux_right(rk3, ncell, Fl)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 19 + }, + { + "cell_type": "heading", + "level": 5, + "metadata": {}, + "source": [ + "Interior Edge Flux" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "bflux = get_bflux_interior(nface, f2c, Qf, np, get_rusanov_flux)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 20 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "All Volume Flux" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "volflux = get_volflux(rk3, np, ncell, Qf)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 21 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Initialize h matrix" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "h = get_initial_hmatrix(rk3, np, ncell, bflux, fle_gauss)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 22 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Volume Inegral by Gauss Lobatto" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "h = get_GL_hmatrix(rk3, np, ncell, h, volflux,volcoef, fle_grad_gauss)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 23 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Compute The Residual Function" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "residual = get_residual_matrix(rk3, np, ncell, aic, h, dx)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 24 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Runge-Kutta Stages 1 through 3 for Initial Time" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Qrk1 = get_Qrk_stage1(Q_rk, ncell, rk3, np, dt, residual)\n", + "\n", + "# Pre-Stage 2 RK, follow the steps as above to obtain new residual " + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 25 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Qf = get_Qf(ncell, rk3, np, fle_gauss, Qrk1)\n", + "\n", + "#BC\n", + "\n", + "Qbleft, Qbright = get_Qbright_Qbleft(rk3,np, ncell, Qf)\n", + "\n", + "#1st cell left edge\n", + "\n", + "Qfl, Qfr = get_Qfl_Qfr_left(rk3, Qf, np, Qbleft)\n", + "Fl = get_rusanov_flux(Qfl, Qfr, normal, rk3)\n", + "bflux = get_bflux_left(Fl)\n", + "\n", + "#last cell right edge\n", + "\n", + "Qfl = get_Qfl_right(Qf,np, ncell)\n", + "Fl = get_rusanov_flux(Qfl, Qbright, normal, rk3)\n", + "bflux = get_bflux_right(rk3, ncell, Fl)\n", + "\n", + "#Interior Edge Flux\n", + "\n", + "bflux = get_bflux_interior(nface, f2c, Qf, np, get_rusanov_flux)\n", + "volflux = get_volflux(rk3, np, ncell, Qf)\n", + "\n", + "#Initialize h matrix\n", + "\n", + "h = get_initial_hmatrix(rk3, np, ncell, bflux, fle_gauss)\n", + "\n", + "# Volume Flux GL\n", + "\n", + "h = get_GL_hmatrix(rk3, np, ncell, h, volflux,volcoef, fle_grad_gauss)\n", + "\n", + "#Compute residual (mass matrix)\n", + "\n", + "residual = get_residual_matrix(rk3, np, ncell, aic, h, dx)\n", + "\n", + "#Stage 2 Rk--------------------------------------------------------------------\n", + "\n", + "Qrk2 = get_Qrk_stage2(Q_rk, Qrk1, ncell, rk3, np, dt, residual)\n", + "\n", + "#Pre Stage 3 Rk-----------------------------------------------------------------\n", + "\n", + "Qf = get_Qf(ncell, rk3, np, fle_gauss, Qrk2)\n", + "\n", + "#BC\n", + "\n", + "Qbleft, Qbright = get_Qbright_Qbleft(rk3,np, ncell, Qf)\n", + "\n", + "#1st cell left edge\n", + "\n", + "Qfl, Qfr = get_Qfl_Qfr_left(rk3, Qf, np, Qbleft)\n", + "Fl = get_rusanov_flux(Qfl, Qfr, normal, rk3)\n", + "bflux = get_bflux_left(Fl)\n", + "\n", + "#last cell right edge\n", + "\n", + "Qfl = get_Qfl_right(Qf,np, ncell)\n", + "Fl = get_rusanov_flux(Qfl, Qbright, normal, rk3)\n", + "bflux = get_bflux_right(rk3, ncell, Fl)\n", + "\n", + "#Interior Edge Flux\n", + "\n", + "bflux = get_bflux_interior(nface, f2c, Qf, np, get_rusanov_flux)\n", + "volflux = get_volflux(rk3, np, ncell, Qf)\n", + "\n", + "#Initialize h matrix\n", + "\n", + "h = get_initial_hmatrix(rk3, np, ncell, bflux, fle_gauss)\n", + "\n", + "# Volume Flux GL\n", + "\n", + "h = get_GL_hmatrix(rk3, np, ncell, h, volflux,volcoef, fle_grad_gauss)\n", + "\n", + "#Compute residual (mass matrix)\n", + "\n", + "residual = get_residual_matrix(rk3, np, ncell, aic, h, dx)\n", + "\n", + "#Stage 3 Rk-------------------------------------------------------------\n", + "\n", + "Qrk3= get_Qrk_stage3(Q_rk, Qrk2, ncell, rk3, np, dt, residual)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 26 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Post-Processing" + ] + }, + { + "cell_type": "heading", + "level": 5, + "metadata": {}, + "source": [ + "Plots for Density, Velocity, Pressure, Energy" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "x_density2, density2 = plot_density2(ncell,np, Qrk3, x, dx)\n", + "x_vel2, vel2 = plot_velocity2(ncell, np, Qrk3, x, dx)\n", + "x_pressure2, pressure2 = plot_pressure2(ncell,np, Qrk3, x, dx)\n", + "x_energy2, energy2 = plot_energy2(ncell,np, Qrk3, x, dx)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 27 + }, + { + "cell_type": "heading", + "level": 5, + "metadata": {}, + "source": [ + "Density" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "%matplotlib inline\n", + "\n", + "pyplot.figure(figsize=(5,5))\n", + "pyplot.title('Density Profile at t = 0.01 s', fontsize=20)\n", + "pyplot.xlabel('x', fontsize=20)\n", + "pyplot.ylabel('Density', fontsize=20)\n", + "pyplot.scatter(x_density2[20:30],density2[20:30])\n", + "pyplot.plot(x_density2[20:30],density2[20:30], \n", + " linewidth=1, color = 'r', linestyle='--')\n", + "\n", + "pyplot.savefig(\"density.png\") ;" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "display_data", + "png": 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+ "text": [ + "" + ] + } + ], + "prompt_number": 29 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "%matplotlib inline\n", + "\n", + "pyplot.figure(figsize=(5,5))\n", + "pyplot.title('Pressure Profile at t = 0.01 s', fontsize=20)\n", + "pyplot.xlabel('x', fontsize=20)\n", + "pyplot.ylabel('Pressure', fontsize=20)\n", + "pyplot.scatter(x_pressure2[20:30],pressure2[20:30])\n", + "pyplot.plot(x_pressure2[20:30],pressure2[20:30], \n", + " linewidth=1, color = 'r', linestyle='--')\n", + "\n", + "pyplot.savefig(\"pressure.png\");" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "display_data", + "png": 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+ "text": [ + "" + ] + } + ], + "prompt_number": 31 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Comparison with Gudunov's Method:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Sod (1978) examined several finite difference methods by appliying them to the shock tube problem. I have compared my results with the Gudunov method:" + ] + }, + { + "cell_type": "heading", + "level": 5, + "metadata": {}, + "source": [ + "Density" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Image(filename='Slide13.jpg')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "jpeg": 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+ "metadata": {}, + "output_type": "pyout", + "prompt_number": 34, + "text": [ + "" + ] + } + ], + "prompt_number": 34 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Conclusions and On-going Work" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Several things can be done to further test the capabilities of the solver presented in this notebook. A more refined mesh would add a better understanding of localized behavior of the properties examined. \n", + "\n", + "Additionally, I would like to experiment with different time steps and see the effect on the stability of the system. Several numerical conditions should be improved, such as the handling of negative densities.\n", + "\n", + "As on-going work I will attempt to model reflective waves and move on to problems with source terms, eventually moving on to 2 and 3 dimensional problems" + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "References" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "[Anderson, 1995]. Computational Fluid Dynamics, The Basics With Applications. John D. Anderson, Jr.1995.\n", + "\n", + "[Boyd,2001]. Chebyshev and Fourier Specral Methods. John P. Boyd. Second Edition, 2001.\n", + "\n", + "[Kappeli, Mishra, 2013]. Well Balanced Schemes for the Euler Equations with Gravitation. Journal of Computational Physics. R Kappeli, S. Mishra. 2013.\n", + "\n", + "[Liang]. Numerical Solution techniques in Mechanical and Aerospace Engineering. Discontinuous Galerkin Method for Hyperbolic PDEs. no date.\n", + "\n", + "[Sod, 1978].A Survey of Several Finite Difference Methods for Systems of Nonlinear Hyperbolic Conservation Laws. Gary A. Sod. 1978\n", + "\n", + "[Sonnendrucker, 2013]. Numerical Methods for Hyperbolic Systems. Max-Planck-Institut fur Plasmaphysik und Zentrum Mathematik, TU Munchen. July 2013." + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "from IPython.core.display import HTML\n", + "def css_styling():\n", + " styles = open('../styles/custom.css', 'r').read()\n", + " return HTML(styles)\n", + "css_styling()" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "html": [ + "\n", + "\n", + "\n", + "\n", + "\n" + ], + "metadata": {}, + "output_type": "pyout", + "prompt_number": 35, + "text": [ + "" + ] + } + ], + "prompt_number": 35 + } + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/Martinez_1D_Euler/1D_Euler_FINAL-May7.ipynb b/Martinez_1D_Euler/1D_Euler_FINAL-May7.ipynb new file mode 100644 index 0000000..a025b8d --- /dev/null +++ b/Martinez_1D_Euler/1D_Euler_FINAL-May7.ipynb @@ -0,0 +1,1301 @@ +{ + "metadata": { + "name": "", + "signature": "sha256:8ef564186e5a734ce19b023798a6e7e0115273b8c7d2b7bb859c83f62cf371a0" + }, + "nbformat": 3, + "nbformat_minor": 0, + "worksheets": [ + { + "cells": [ + { + "cell_type": "heading", + "level": 1, + "metadata": {}, + "source": [ + "One-Dimensional Conservation Laws of Gas Dynamics " + ] + }, + { + "cell_type": "heading", + "level": 5, + "metadata": {}, + "source": [ + "By Luis Martinez" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "from IPython.display import Latex\n", + "from IPython.display import Image" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 1 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The gas dynamics equations, which constitue the conservation of mass, momentum, and energy, can be used to study shock waves. This notebook includes a numerical scheme to solve for the classical Sod's Shock Tube problem.\n", + "\n", + "The Euler equations may be used to model other interesting physical phenomena. For example The Euler equations with a gravitational source term are used to model atmospheric pphenomena essential to weather predicion, climate modeling, and astrophisical modeling such as simulating solar climate and supernova explosions [Kappeli, Mishra, 2013]" + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Principles of the Shock Tube Problem" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Initially, the tube is divided (by a diaphragm) into a high-pressure section (driver section) and a low pressure section (the driven section). There is no flow velocity anywhere inside the tube initially. \n", + "\n", + "As, the diaphragm is removed, the initial pressure discontinuity will propagate to the right as an unsteady normal shock wave traveling at velocity W. At the same time, an isentropic expansion wave will propagate to the left.\n", + "\n", + "We can see that shortly after the diaphragm is removed, there are 4 main regions in the tube.\n", + "\n", + "Region 1 is the undisturbed portion of the driven section (so we would expect initial conditions to still be true here)\n", + "\n", + "Region 2 : This section has been \u2018processed\u2019 by the shock wave moving through it, and the pressure here is equal to the pressure behind the shock.\n", + "\n", + "Region 3: Has been \u2018processed\u2019 by the expansion wave moving through the region. The pressure here is equal to P3. Regions 2 and 3 are said to be at the same pressure since the gas in these regions cannot support any pressure discontinuities\n", + "\n", + "Region 4: The undisturbed portion of the driver section (we should expect initial conditions).\n", + "\n", + "It is interesting that Regions 2 and 3 are at the same velocity and pressure. But there are differences. Since region 2 has been processed by a shock wave and region 3 by an expansion wave, the entropy, temperature, and density are different. This is why we see this contact surface to differentiate the regions.\n", + "\n", + "Both Pressure and velocity change discontinuously across the shock wave. When we look at these properties across the expansion wave, we see that the variations of pressure and velocity are finite and continuous.\n", + "\n", + "This information was taken from Anderson(1995).\n" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Image(filename='slide04.jpg')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "jpeg": 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8EeF/hf8A8FUvAuk/DfUvE8PhvW/hD4kvL62ufGOo6rb3E1ve6WkUgF1cShWQ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\n+tfSZFmay6hWrr42uWP6s8HNMC8ZVp038KfM/lsj0Xw/8If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+ "metadata": {}, + "output_type": "pyout", + "prompt_number": 2, + "text": [ + "" + ] + } + ], + "prompt_number": 2 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Governing Equations" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We look at one-dimensional conservation of mass, momentum, and energy:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\\frac{\\partial \\rho}{\\partial t} + \\frac{\\partial \\rho \\vec{v_x}}{\\partial x} = 0$$\n", + "\n", + "$$\\frac{\\partial \\rho \\vec{v_x}}{\\partial t} + \\frac{\\partial(\\rho \\vec{v_x}^2 +P)}{\\partial x} = 0$$\n", + "\n", + "$$\\frac{\\partial E}{\\partial t} +\\ \\frac{\\partial (E+P) \\vec{v_x}}{\\partial x} = 0$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can also relate an equation of state:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$ P = (\\gamma-1)(E - \\frac{\\rho v^2}{2})$$" + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "System of Equations" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "From the conservation Equations we can develop a system of equations:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\\frac{\\partial U}{\\partial t} + \\frac{\\partial f(U)}{\\partial x} = 0$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "where U and f(U) are:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$ U = \\left(\n", + "\\begin{array}{c}\n", + "\\rho\\\\\n", + "\\rho \\vec{v_x}\\\\\n", + "E\n", + "\\end{array}\n", + "\\right)\n", + "\\,\\\n", + "f(U) = \\left(\n", + "\\begin{array}{c}\n", + "\\rho \\vec{v_x}\\\\\n", + "\\rho \\vec{v_x}^2 +\\ P\\\\\n", + "(E+P)\\vec{v_x}\n", + "\\end{array}\n", + "\\right)$$ " + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Discretization of System of Equations: The Discontinuous Galerkin Method" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The discontinuous Galerkin method represents unknowns like the finite element method by piecewise polynomial functions, but unlike finite element methods, the polynomials are discontinuous at cell interfaces. At these interfaces, a flux is defined in the same way as for finite volume methods [Sonnendrucker, 2013. Numerical Methods for Hyperbolic Systems]\n", + "\n", + "On each cell, the unknown U is represented by a linear combination of basis functions of the space of polynomials of degree 'np'. The chosen basis function for this solver is a Legendre polynomial, and the interpolation points are Gauss-Lobatto points." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\\int_{I_i} \\frac{\\partial U(x,t)}{\\partial t} W(x)\\ dx\\ + \\int_{I_i} \\frac{\\partial f(U(x,t))}{\\partial x} W(x) dx = 0$$\n", + "\n", + "$$\\int_{I_i} \\frac{\\partial U(x,t)}{\\partial t} \\phi_l(x)\\ dx\\ + \\int_{I_i} \\frac{\\partial f(U(x,t))}{\\partial x} \\phi_l(x) dx = 0$$\n", + "\n", + "$$\\int_{I_i} \\frac{\\partial U(x,t)}{\\partial t} \\phi_l(x)\\ dx\\ + f(U)\\phi_l(x)\\ |_{I_i}\\ -\\ \\int_{I_i} f(U)\\frac{\\partial \\phi_l(x)}{\\partial x} dx = 0$$\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Note that the Integral I goes from [-0.5] t0 [0.5]. From the Galerkin Method we can estimate U(x,t):" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$U_{i}(x,t) = \\sum_{0}^{np} U_{l}^{i}(t)\\phi_l(x)$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The objective here is to obtain an expression for U(t). Using orthogonality, we can define the spectral coefficients, which in turn allow us to define U(t):" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\\alpha_l = \\int_{I_i} \\phi_l(x)\\ \\phi_l(x) dx\\\\$$\n", + "$$U_{l}^{i}(t) = \\frac{1}{\\alpha_l}\\ \\int_{I_i} U_{i}(x,t)\\ \\phi_l(x) dx\\\\$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Then, we can express the system of equations in a form that we can approximate:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "$$\\frac{\\partial U_{l}^{i}(t)}{\\partial t} = \\frac{1}{\\alpha_l} \n", + "\\int_{I_i} f(U)\\frac{\\partial \\phi_l(x)}{\\partial x} dx \n", + "- f(U_{i+\\frac{1}{2}})\\phi_{i+\\frac{1}{2}}^l \n", + "+ f(U_{i-\\frac{1}{2}})\\phi_{i-\\frac{1}{2}}^l $$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The integral on the right-hand side can be approximated by Gauss-Lobatto quadratures and the fluxes are solved using the Rusanov flux. To march in time, a third-order runge-kutta method is used." + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "The Solver" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The solver is tested by applying it to the classic Sod's shock tube problem. It is executed using a third order polynomial approximation, with a domain of 10 cells." + ] + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Import Libraries" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "import numpy\n", + "import sympy\n", + "from sympy.utilities.lambdify import lambdify\n", + "import math\n", + "from matplotlib import pyplot" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 3 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Import Scripts" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "from Legendre_Polynomial_Function_np3 import *\n", + "from meshing import *\n", + "from U_initial_np3 import *\n", + "from Runge_Kutta_Third_Order_np3 import *\n", + "from Residual_Function_np3 import *\n", + "from Plot_Functions2 import *" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 4 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Some Program Parameters" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "np = 3 # order of scheme\n", + "ncell = 10 #number of cells\n", + "nface = ncell+1 #cell boundaries\n", + "rk3 = 3 #This is because we are solving 3 equations(mass, momentum, energy)\n", + "dt = 0.01\n", + "\n", + "# for flux function:\n", + "normal = 1.0" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 5 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Legendre Polynomial and Derivative" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "fle_gauss = get_fle_gauss(np)\n", + "fle_grad_gauss = get_fle_derivative(np)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 6 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Meshing" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "x = get_x_cell(ncell)\n", + "dx = get_dx_cell(ncell)\n", + "f2c = get_face_to_cell(nface)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 7 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Initialize values of Mass, Momentum, and Energy" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Image(filename='slide08.jpg')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "jpeg": 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zSkEFgV9UYgfMW2gDknsO9fHuj/tgaNqt94R8Qv8ADPx3pvwW8VeJk8PeHPiP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\n9g/8FN/+h9/Y4/8ABJrP+NH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+ "metadata": {}, + "output_type": "pyout", + "prompt_number": 8, + "text": [ + "" + ] + } + ], + "prompt_number": 8 + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The initial conditions are set as Sod's 1978 initial conditions. Using the mathematical relationships previously described, we can nitialize U, which contains mass, momentum, and energy:" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "aic = get_inverse_mass_matrix(np)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 9 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "U_initial = get_U_initial(np, rk3, ncell, fle_gauss, x ,dx, aic)\n" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 10 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Third Order Runge Kutta Time Scheme" + ] + }, + { + "cell_type": "heading", + "level": 5, + "metadata": {}, + "source": [ + "Stages of RK Scheme:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n", + "\n", + "$$U^1 = U^n\\ +\\ \\Delta R(U^n) \\\\$$\n", + "$$U^2 = \\frac{3}{4}U^n\\ +\\ \\frac{1}{4}\\ [U^1\\ +\\ \\Delta R(U^1)] \\\\$$\n", + "$$U^3 = \\frac{1}{3}U^n\\ +\\ \\frac{2}{3}\\ [U^2\\ +\\ \\Delta R(U^2)] \\\\$$\n" + ] + }, + { + "cell_type": "heading", + "level": 5, + "metadata": {}, + "source": [ + "Initialize Runge-Kutta Scheme:" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Q_rk = U_initial" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 11 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Qf = get_Qf(ncell, rk3, np, fle_gauss, Q_rk)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 12 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Boundary Conditions" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Qbleft, Qbright = get_Qbright_Qbleft(rk3,np, ncell, Qf)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 13 + }, + { + "cell_type": "heading", + "level": 5, + "metadata": {}, + "source": [ + "First Cell Left Edge" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Qfl, Qfr = get_Qfl_Qfr_left(rk3, Qf, np, Qbleft)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 14 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Fl = get_rusanov_flux(Qfl, Qfr, normal, rk3)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 15 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "bflux = get_bflux_left(Fl)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 16 + }, + { + "cell_type": "heading", + "level": 5, + "metadata": {}, + "source": [ + "Last Cell Right Edge" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Qfl = get_Qfl_right(Qf,np, ncell)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 17 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Fl = get_rusanov_flux(Qfl, Qbright, normal, rk3)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 18 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "bflux = get_bflux_right(rk3, ncell, Fl)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 19 + }, + { + "cell_type": "heading", + "level": 5, + "metadata": {}, + "source": [ + "Interior Edge Flux" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "bflux = get_bflux_interior(nface, f2c, Qf, np, get_rusanov_flux)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 20 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "All Volume Flux" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "volflux = get_volflux(rk3, np, ncell, Qf)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 21 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Initialize h matrix" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "h = get_initial_hmatrix(rk3, np, ncell, bflux, fle_gauss)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 22 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Volume Inegral by Gauss Lobatto" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "h = get_GL_hmatrix(rk3, np, ncell, h, volflux,volcoef, fle_grad_gauss)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 23 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Compute The Residual Function" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "residual = get_residual_matrix(rk3, np, ncell, aic, h, dx)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 24 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Runge-Kutta Stages 1 through 3 for Initial Time" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Qrk1 = get_Qrk_stage1(Q_rk, ncell, rk3, np, dt, residual)\n", + "\n", + "# Pre-Stage 2 RK, follow the steps as above to obtain new residual " + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 25 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Qf = get_Qf(ncell, rk3, np, fle_gauss, Qrk1)\n", + "\n", + "#BC\n", + "\n", + "Qbleft, Qbright = get_Qbright_Qbleft(rk3,np, ncell, Qf)\n", + "\n", + "#1st cell left edge\n", + "\n", + "Qfl, Qfr = get_Qfl_Qfr_left(rk3, Qf, np, Qbleft)\n", + "Fl = get_rusanov_flux(Qfl, Qfr, normal, rk3)\n", + "bflux = get_bflux_left(Fl)\n", + "\n", + "#last cell right edge\n", + "\n", + "Qfl = get_Qfl_right(Qf,np, ncell)\n", + "Fl = get_rusanov_flux(Qfl, Qbright, normal, rk3)\n", + "bflux = get_bflux_right(rk3, ncell, Fl)\n", + "\n", + "#Interior Edge Flux\n", + "\n", + "bflux = get_bflux_interior(nface, f2c, Qf, np, get_rusanov_flux)\n", + "volflux = get_volflux(rk3, np, ncell, Qf)\n", + "\n", + "#Initialize h matrix\n", + "\n", + "h = get_initial_hmatrix(rk3, np, ncell, bflux, fle_gauss)\n", + "\n", + "# Volume Flux GL\n", + "\n", + "h = get_GL_hmatrix(rk3, np, ncell, h, volflux,volcoef, fle_grad_gauss)\n", + "\n", + "#Compute residual (mass matrix)\n", + "\n", + "residual = get_residual_matrix(rk3, np, ncell, aic, h, dx)\n", + "\n", + "#Stage 2 Rk--------------------------------------------------------------------\n", + "\n", + "Qrk2 = get_Qrk_stage2(Q_rk, Qrk1, ncell, rk3, np, dt, residual)\n", + "\n", + "#Pre Stage 3 Rk-----------------------------------------------------------------\n", + "\n", + "Qf = get_Qf(ncell, rk3, np, fle_gauss, Qrk2)\n", + "\n", + "#BC\n", + "\n", + "Qbleft, Qbright = get_Qbright_Qbleft(rk3,np, ncell, Qf)\n", + "\n", + "#1st cell left edge\n", + "\n", + "Qfl, Qfr = get_Qfl_Qfr_left(rk3, Qf, np, Qbleft)\n", + "Fl = get_rusanov_flux(Qfl, Qfr, normal, rk3)\n", + "bflux = get_bflux_left(Fl)\n", + "\n", + "#last cell right edge\n", + "\n", + "Qfl = get_Qfl_right(Qf,np, ncell)\n", + "Fl = get_rusanov_flux(Qfl, Qbright, normal, rk3)\n", + "bflux = get_bflux_right(rk3, ncell, Fl)\n", + "\n", + "#Interior Edge Flux\n", + "\n", + "bflux = get_bflux_interior(nface, f2c, Qf, np, get_rusanov_flux)\n", + "volflux = get_volflux(rk3, np, ncell, Qf)\n", + "\n", + "#Initialize h matrix\n", + "\n", + "h = get_initial_hmatrix(rk3, np, ncell, bflux, fle_gauss)\n", + "\n", + "# Volume Flux GL\n", + "\n", + "h = get_GL_hmatrix(rk3, np, ncell, h, volflux,volcoef, fle_grad_gauss)\n", + "\n", + "#Compute residual (mass matrix)\n", + "\n", + "residual = get_residual_matrix(rk3, np, ncell, aic, h, dx)\n", + "\n", + "#Stage 3 Rk-------------------------------------------------------------\n", + "\n", + "Qrk3= get_Qrk_stage3(Q_rk, Qrk2, ncell, rk3, np, dt, residual)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 26 + }, + { + "cell_type": "heading", + "level": 3, + "metadata": {}, + "source": [ + "Post-Processing" + ] + }, + { + "cell_type": "heading", + "level": 5, + "metadata": {}, + "source": [ + "Plots for Density, Velocity, Pressure, Energy" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "x_density2, density2 = plot_density2(ncell,np, Qrk3, x, dx)\n", + "x_vel2, vel2 = plot_velocity2(ncell, np, Qrk3, x, dx)\n", + "x_pressure2, pressure2 = plot_pressure2(ncell,np, Qrk3, x, dx)\n", + "x_energy2, energy2 = plot_energy2(ncell,np, Qrk3, x, dx)" + ], + "language": "python", + "metadata": {}, + "outputs": [], + "prompt_number": 27 + }, + { + "cell_type": "heading", + "level": 5, + "metadata": {}, + "source": [ + "Density" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "%matplotlib inline\n", + "\n", + "pyplot.figure(figsize=(5,5))\n", + "pyplot.title('Density Profile at t = 0.01 s', fontsize=20)\n", + "pyplot.xlabel('x', fontsize=20)\n", + "pyplot.ylabel('Density', fontsize=20)\n", + "pyplot.scatter(x_density2[20:30],density2[20:30])\n", + "pyplot.plot(x_density2[20:30],density2[20:30], \n", + " linewidth=1, color = 'r', linestyle='--')\n", + "\n", + "pyplot.savefig(\"density.png\") ;" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "display_data", + "png": 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+ "text": [ + "" + ] + } + ], + "prompt_number": 29 + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "%matplotlib inline\n", + "\n", + "pyplot.figure(figsize=(5,5))\n", + "pyplot.title('Pressure Profile at t = 0.01 s', fontsize=20)\n", + "pyplot.xlabel('x', fontsize=20)\n", + "pyplot.ylabel('Pressure', fontsize=20)\n", + "pyplot.scatter(x_pressure2[20:30],pressure2[20:30])\n", + "pyplot.plot(x_pressure2[20:30],pressure2[20:30], \n", + " linewidth=1, color = 'r', linestyle='--')\n", + "\n", + "pyplot.savefig(\"pressure.png\");" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "metadata": {}, + "output_type": "display_data", + "png": 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+ "text": [ + "" + ] + } + ], + "prompt_number": 31 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Comparison with Gudunov's Method:" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Sod (1978) examined several finite difference methods by appliying them to the shock tube problem. I have compared my results with the Gudunov method:" + ] + }, + { + "cell_type": "heading", + "level": 5, + "metadata": {}, + "source": [ + "Density" + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "Image(filename='Slide13.jpg')" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "jpeg": 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+ "metadata": {}, + "output_type": "pyout", + "prompt_number": 34, + "text": [ + "" + ] + } + ], + "prompt_number": 34 + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "Conclusions and On-going Work" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Several things can be done to further test the capabilities of the solver presented in this notebook. A more refined mesh would add a better understanding of localized behavior of the properties examined. \n", + "\n", + "Additionally, I would like to experiment with different time steps and see the effect on the stability of the system. Several numerical conditions should be improved, such as the handling of negative densities.\n", + "\n", + "As on-going work I will attempt to model reflective waves and move on to problems with source terms, eventually moving on to 2 and 3 dimensional problems" + ] + }, + { + "cell_type": "heading", + "level": 2, + "metadata": {}, + "source": [ + "References" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "[Anderson, 1995]. Computational Fluid Dynamics, The Basics With Applications. John D. Anderson, Jr.1995.\n", + "\n", + "[Boyd,2001]. Chebyshev and Fourier Specral Methods. John P. Boyd. Second Edition, 2001.\n", + "\n", + "[Kappeli, Mishra, 2013]. Well Balanced Schemes for the Euler Equations with Gravitation. Journal of Computational Physics. R Kappeli, S. Mishra. 2013.\n", + "\n", + "[Liang]. Numerical Solution techniques in Mechanical and Aerospace Engineering. Discontinuous Galerkin Method for Hyperbolic PDEs. no date.\n", + "\n", + "[Sod, 1978].A Survey of Several Finite Difference Methods for Systems of Nonlinear Hyperbolic Conservation Laws. Gary A. Sod. 1978\n", + "\n", + "[Sonnendrucker, 2013]. Numerical Methods for Hyperbolic Systems. Max-Planck-Institut fur Plasmaphysik und Zentrum Mathematik, TU Munchen. July 2013." + ] + }, + { + "cell_type": "code", + "collapsed": false, + "input": [ + "from IPython.core.display import HTML\n", + "def css_styling():\n", + " styles = open('../styles/custom.css', 'r').read()\n", + " return HTML(styles)\n", + "css_styling()" + ], + "language": "python", + "metadata": {}, + "outputs": [ + { + "html": [ + "\n", + "\n", + "\n", + "\n", + "\n" + ], + "metadata": {}, + "output_type": "pyout", + "prompt_number": 35, + "text": [ + "" + ] + } + ], + "prompt_number": 35 + } + ], + "metadata": {} + } + ] +} \ No newline at end of file diff --git a/Martinez_1D_Euler/Legendre_Polynomial_Function_np3.py b/Martinez_1D_Euler/Legendre_Polynomial_Function_np3.py new file mode 100644 index 0000000..fc333e3 --- /dev/null +++ b/Martinez_1D_Euler/Legendre_Polynomial_Function_np3.py @@ -0,0 +1,166 @@ +# Legendre Polynomial Function + +import sympy +import numpy +from sympy.utilities.lambdify import lambdify + +#------------------------------------------------------- +np = 3 +xp = sympy.symbols('xp') +#------------------------------------------------------- + +# Store the Quadrature Points and Weights for 4 point quadrature(np=3), fifth order polynomial: + +#Quadrature Points + +gauss = numpy.zeros((np+1, np+1), dtype=float) +gauss[0,0]=-0.5 +gauss[1,0]=0.5 +gauss[2,0]= -1 * numpy.sqrt(5)/10 +gauss[3,0]= numpy.sqrt(5)/10.0 + +# Corresponding Quadrature weights + +gauss[0,1]=1.0/12.0 +gauss[1,1]=gauss[0,1] +gauss[2,1]=5.0/12.0 +gauss[3,1]=gauss[2,1] + +#---------------------------------------------------------- +#Legendre Polynomial Terms +fle = [1.0]*(np+1) + +fle[0] = 1.0 +fle[1] = xp +fle[2] = (xp**2)-(1.0/12.0) +fle[3] = (xp**3)-(0.15*xp) + +#--------------------------------------------------------- +fle_eval = [1.0]*(np+1) + +for i in range(np+1): + fle_eval[i] = lambdify((xp),fle[i]) +#---------------------------------------------------------- +fle_0_eval = fle_eval[0] +fle_1_eval = fle_eval[1] +fle_2_eval = fle_eval[2] +fle_3_eval = fle_eval[3] + +#---------------------------------------------------------- +def fle0(xp): + f0 = fle_0_eval(xp) + return f0 + +def fle1(xp): + f1 = fle_1_eval(xp) + return f1 + +def fle2(xp): + f2 = fle_2_eval(xp) + return f2 + +def fle3(xp): + f3 = fle_3_eval(xp) + return f3 + +#---------------------------------------------------------------------- +#Legendre Polynomial Derivative + +fle_grad = [0.0]*(np+1) +fle_grad[0] = 0.0 +fle_grad[1] = 1 +fle_grad[2] = 2*xp +fle_grad[3] = (3*(xp**2))-(0.15) + +#-------------------------------------------------------------------- +fle_grad_eval = [1.0]*(np+1) + +for i in range(np+1): + fle_grad_eval[i] = lambdify((xp),fle_grad[i]) +#-------------------------------------------------------------------- +fle_grad_0_eval = fle_grad_eval[0] +fle_grad_1_eval = fle_grad_eval[1] +fle_grad_2_eval = fle_grad_eval[2] +fle_grad_3_eval = fle_grad_eval[3] + + +#-------------------------------------------------------------------- +def fle0_grad(xp): + fg0 = fle_grad_0_eval(xp) + return fg0 + +def fle1_grad(xp): + fg1 = fle_grad_1_eval(xp) + return fg1 + +def fle2_grad(xp): + fg2 = fle_grad_2_eval(xp) + return fg2 + +def fle3_grad(xp): + fg3 = fle_grad_3_eval(xp) + return fg3 + +#---------------------------------------------------------------------- +#------------------------------------------------------------- +def get_fle_gauss(np): + """ + fle_gauss => the Legendre Polynomial function evaluated at gauss points + + """ + fle_ncb = [1.0]*(np+1) + fle_ncb = [fle0(gauss[0,0]),fle1(gauss[0,0]),fle2(gauss[0,0]),fle3(gauss[0,0])] + + fle_pcb = [1.0]*(np+1) + fle_pcb = [fle0(gauss[1,0]),fle1(gauss[1,0]),fle2(gauss[1,0]),fle3(gauss[1,0])] + + fle_nip = [1.0]*(np+1) + fle_nip = [fle0(gauss[2,0]),fle1(gauss[2,0]),fle2(gauss[2,0]),fle3(gauss[2,0])] + + fle_pip = [1.0]*(np+1) + fle_pip = [fle0(gauss[3,0]),fle1(gauss[3,0]),fle2(gauss[3,0]),fle3(gauss[3,0])] + + +#------------------------------------------------------------------- + fle_gauss = numpy.zeros((np+1,np+1), dtype=float) + + for j in range(np+1): + fle_gauss[j,0] = fle_ncb[j] + fle_gauss[j,1] = fle_pcb[j] + fle_gauss[j,2] = fle_nip[j] + fle_gauss[j,3] = fle_pip[j] + + + return fle_gauss +#------------------------------------------------------------------- +#------------------------------------------------------------------- +def get_fle_derivative(np): + """ fle_grad_gauss => the Legendre Polynomial derivative of function evaluated + at gauss points + """ + + fle_grad_ncb = [1.0]*(np+1) + fle_grad_ncb = [fle0_grad(gauss[0,0]),fle1_grad(gauss[0,0]),fle2_grad(gauss[0,0]),fle3_grad(gauss[0,0])] + + fle_grad_pcb = [1.0]*(np+1) + fle_grad_pcb = [fle0_grad(gauss[1,0]),fle1_grad(gauss[1,0]),fle2_grad(gauss[1,0]),fle3_grad(gauss[1,0])] + + fle_grad_nip = [1.0]*(np+1) + fle_grad_nip = [fle0_grad(gauss[2,0]),fle1_grad(gauss[2,0]),fle2_grad(gauss[2,0]),fle3_grad(gauss[2,0])] + + fle_grad_pip = [1.0]*(np+1) + fle_grad_pip = [fle0_grad(gauss[3,0]),fle1_grad(gauss[3,0]),fle2_grad(gauss[3,0]),fle3_grad(gauss[3,0])] + +#---------------------------------------------------------------------- + fle_grad_gauss = numpy.zeros((np+1,np+1), dtype=float) + + for j in range(np+1): + fle_grad_gauss[j,0] = fle_grad_ncb[j] + fle_grad_gauss[j,1] = fle_grad_pcb[j] + fle_grad_gauss[j,2] = fle_grad_nip[j] + fle_grad_gauss[j,3] = fle_grad_pip[j] + +#----------------------------------------------------------------------- + return fle_grad_gauss + + \ No newline at end of file diff --git a/Martinez_1D_Euler/Legendre_Polynomial_Function_np3.pyc b/Martinez_1D_Euler/Legendre_Polynomial_Function_np3.pyc new file mode 100644 index 0000000000000000000000000000000000000000..ec05475e6b2a68fadd9f1b7b30c527a4318c78c3 GIT binary patch literal 4924 zcmeHK-E!Mh5MKQ_i5%z0i4)t+pW2kxTr}iohKo*UI@1o^Fbq|JFdjcpEL6t{jx^{PCL&018R1lpW|K8atj!s%18tr#vmjdM zg2NoW{D|~KcLgN#@W~QYv53QFaHS`y{Msg+V{ZHTBWo8d)-PA5Sc&z^)oI~!;IJ5U 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/dev/null +++ b/Martinez_1D_Euler/Plot_Functions2.py @@ -0,0 +1,232 @@ + +# Legendre Polynomial Function for Plots + +import sympy +import numpy +from sympy.utilities.lambdify import lambdify + +#------------------------------------------------------- +np = 3 +xp = sympy.symbols('xp') +#------------------------------------------------------- + + +#Quadrature Points + +gauss_plot = numpy.zeros((np), dtype=float) +gauss_plot[0]=-1./3. +gauss_plot[1]=0. +gauss_plot[2]=1./3. + +#---------------------------------------------------------- +#Legendre Polynomial Terms +fle = [1.0]*(np+1) + +fle[0] = 1.0 +fle[1] = xp +fle[2] = (xp**2)-(1.0/12.0) +fle[3] = (xp**3)-(0.15*xp) + +#--------------------------------------------------------- +fle_eval = [1.0]*(np+1) + +for i in range(np+1): + fle_eval[i] = lambdify((xp),fle[i]) +#---------------------------------------------------------- +fle_0_eval = fle_eval[0] +fle_1_eval = fle_eval[1] +fle_2_eval = fle_eval[2] +fle_3_eval = fle_eval[3] + +#---------------------------------------------------------- +def fle0(xp): + f0 = fle_0_eval(xp) + return f0 + +def fle1(xp): + f1 = fle_1_eval(xp) + return f1 + +def fle2(xp): + f2 = fle_2_eval(xp) + return f2 + +def fle3(xp): + f3 = fle_3_eval(xp) + return f3 + +#---------------------------------------------------------------------- + + +fle_np = [1.0]*(np+1) +fle_np = [fle0(gauss_plot[0]),fle1(gauss_plot[0]),fle2(gauss_plot[0]),fle3(gauss_plot[0])] + +fle_zero = [1.0]*(np+1) +fle_zero = [fle0(gauss_plot[1]),fle1(gauss_plot[1]),fle2(gauss_plot[1]),fle3(gauss_plot[1])] + +fle_pp = [1.0]*(np+1) +fle_pp = [fle0(gauss_plot[2]),fle1(gauss_plot[2]),fle2(gauss_plot[2]),fle3(gauss_plot[2])] + +#------------------------------------------------------------------- +fle_gauss_plot = numpy.zeros((np+1,np), dtype=float) + +for j in range(np+1): + fle_gauss_plot[j,0] = fle_np[j] + fle_gauss_plot[j,1] = fle_zero[j] + fle_gauss_plot[j,2] = fle_pp[j] + +#----------------------------------------------------------------------------- +def plot_density2(ncell,np, Q, x, dx): + + + rho1 = numpy.zeros((ncell), dtype=float) + rho2 = numpy.zeros((ncell), dtype=float) + rho3 = numpy.zeros((ncell), dtype=float) + + xx1 = numpy.zeros((ncell), dtype=float) + xx2 = numpy.zeros((ncell), dtype=float) + xx3 = numpy.zeros((ncell), dtype=float) + + for i in range(ncell): + for j in range(np): + + rho1[i] += fle_gauss_plot[j,0]*Q[0,j,i] + rho2[i] += fle_gauss_plot[j,1]*Q[0,j,i] + rho3[i] += fle_gauss_plot[j,2]*Q[0,j,i] + + xx1[i] = x[i] + (-1./3.)*dx[i] + xx2[i] = x[i] + (0.)*dx[i] + xx3[i] = x[i] + (1./3.)*dx[i] + + + rho_t2 = numpy.concatenate((rho1,rho2, rho3)) + xx_t2 = numpy.concatenate((xx1, xx2, xx3)) + + return xx_t2, rho_t2 + +#----------------------------------------------------------------------------- +def plot_velocity2(ncell,np, Q, x, dx): + + rho1 = numpy.zeros((ncell), dtype=float) + rho2 = numpy.zeros((ncell), dtype=float) + rho3 = numpy.zeros((ncell), dtype=float) + + mom1 = numpy.zeros((ncell), dtype=float) + mom2 = numpy.zeros((ncell), dtype=float) + mom3 = numpy.zeros((ncell), dtype=float) + + vel1 = numpy.zeros((ncell), dtype=float) + vel2 = numpy.zeros((ncell), dtype=float) + vel3 = numpy.zeros((ncell), dtype=float) + + xx1 = numpy.zeros((ncell), dtype=float) + xx2 = numpy.zeros((ncell), dtype=float) + xx3 = numpy.zeros((ncell), dtype=float) + + for i in range(ncell): + for j in range(np): + + rho1[i] += fle_gauss_plot[j,0]*Q[0,j,i] + rho2[i] += fle_gauss_plot[j,1]*Q[0,j,i] + rho3[i] += fle_gauss_plot[j,2]*Q[0,j,i] + + mom1[i] += fle_gauss_plot[j,0]*Q[1,j,i] + mom2[i] += fle_gauss_plot[j,1]*Q[1,j,i] + mom3[i] += fle_gauss_plot[j,2]*Q[1,j,i] + + vel1[i] = mom1[i]/rho1[i] + vel2[i] = mom2[i]/rho2[i] + vel3[i] = mom3[i]/rho3[i] + + xx1[i] = x[i] + (-1./3.)*dx[i] + xx2[i] = x[i] + (0.)*dx[i] + xx3[i] = x[i] + (1./3.)*dx[i] + + vel_t2 = numpy.concatenate((vel1,vel2, vel3)) + xx_t2 = numpy.concatenate((xx1, xx2, xx3)) + + return xx_t2, vel_t2 + +#------------------PLOT Pressure!----------------------------------------------- + +def plot_pressure2(ncell,np, Q, x, dx): + + gamma = 1.4 + + rho1 = numpy.zeros((ncell), dtype=float) + rho2 = numpy.zeros((ncell), dtype=float) + rho3 = numpy.zeros((ncell), dtype=float) + + mom1 = numpy.zeros((ncell), dtype=float) + mom2 = numpy.zeros((ncell), dtype=float) + mom3 = numpy.zeros((ncell), dtype=float) + + en1 = numpy.zeros((ncell), dtype=float) + en2 = numpy.zeros((ncell), dtype=float) + en3 = numpy.zeros((ncell), dtype=float) + + press1 = numpy.zeros((ncell), dtype=float) + press2 = numpy.zeros((ncell), dtype=float) + press3 = numpy.zeros((ncell), dtype=float) + + xx1 = numpy.zeros((ncell), dtype=float) + xx2 = numpy.zeros((ncell), dtype=float) + xx3 = numpy.zeros((ncell), dtype=float) + + for i in range(ncell): + for j in range(np): + + rho1[i] += fle_gauss_plot[j,0]*Q[0,j,i] + rho2[i] += fle_gauss_plot[j,1]*Q[0,j,i] + rho3[i] += fle_gauss_plot[j,2]*Q[0,j,i] + + mom1[i] += fle_gauss_plot[j,0]*Q[1,j,i] + mom2[i] += fle_gauss_plot[j,1]*Q[1,j,i] + mom3[i] += fle_gauss_plot[j,2]*Q[1,j,i] + + en1[i] += fle_gauss_plot[j,0]*Q[2,j,i] + en2[i] += fle_gauss_plot[j,1]*Q[2,j,i] + en3[i] += fle_gauss_plot[j,2]*Q[2,j,i] + + press1[i] = (gamma-1.)*(en1[i]- 0.5*(mom1[i]**2)/rho1[i]) + press2[i] = (gamma-1.)*(en2[i]- 0.5*(mom2[i]**2)/rho2[i]) + press3[i] = (gamma-1.)*(en3[i]- 0.5*(mom3[i]**2)/rho3[i]) + + xx1[i] = x[i] + (-1./3.)*dx[i] + xx2[i] = x[i] + (0.)*dx[i] + xx3[i] = x[i] + (1./3.)*dx[i] + + press_t2 = numpy.concatenate((press1,press2, press3)) + xx_t2 = numpy.concatenate((xx1, xx2, xx3)) + + return xx_t2, press_t2 + +#------------------PLOT Energy!----------------------------------------------- + +def plot_energy2(ncell,np, Q, x, dx): + + en1 = numpy.zeros((ncell), dtype=float) + en2 = numpy.zeros((ncell), dtype=float) + en3 = numpy.zeros((ncell), dtype=float) + + xx1 = numpy.zeros((ncell), dtype=float) + xx2 = 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dtype=float) + +volflux = numpy.zeros((rk3,np+1,ncell), dtype=float) +h = numpy.zeros((rk3,np,ncell), dtype=float) + +#f2c = numpy.empty((nface, 2), dtype=float) + +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- + +#--------------------------------------------------------------- + +def get_Qf(ncell, rk3, np, fle_gauss, Q): + + #Here we approximate U(x,t) = sum (basis_function(x)* U(t)) + + for i in range(ncell): + for iv in range(rk3): + for k in range(np): + Qf[iv,0,i] += fle_gauss[k,0]*Q[iv,k,i] + Qf[iv,1,i] += fle_gauss[k,2]*Q[iv,k,i] + Qf[iv,2,i] += fle_gauss[k,3]*Q[iv,k,i] + Qf[iv,3,i] += fle_gauss[k,1]*Q[iv,k,i] + + return Qf +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- + +#-----------BOUNDARY CONDITIONS--------------------------------- + +def get_Qbright_Qbleft(rk3,np, ncell, Qf): + + #Apply Boundary Conditions + + # Try to change the boundary conditions to see the impact to the solver + #currently in comment block are the reflecting boundary conditions + + + Qbleft = [0.0]*(rk3) + Qbright = [0.0]*(rk3) +# +# Qbleft[0] = Qf[0,0,0] +# Qbleft[1] = -Qf[1,0,0] +# Qbleft[2] = Qf[2,0,0] +# +# Qbright[0] = Qf[0,np, ncell-1] +# Qbright[1] = -Qf[1, np, ncell-1] +# Qbright[2] = Qf[2, np, ncell-1] + + Qbleft[0] = 1.25 + Qbleft[1] = 0.0 + Qbleft[2] = 2.5 + + Qbright[0] = 0.125 + Qbright[1] = 0.0 + Qbright[2] = 0.25 + + return Qbleft, Qbright +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- + +#------------------BC FOR 1st CELL LEFT EDGE--------------------- + +def get_Qfl_Qfr_left(rk3,Qf,np, Qbleft): + + + for iv in range(rk3): + Qfl[iv] = Qbleft[iv] + Qfr[iv] = Qf[iv,0,0] + + return Qfl, Qfr +#--------------------------------------------------------------- + +#Fl = get_rusanov_flux(Qfl, Qfr, normal, rk3) + +def get_bflux_left(Fl): + bflux[:,0,0] = Fl[:] + return bflux +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- + +#------------------BC FOR LAST CELL RIGHT EDGE------------------- + +def get_Qfl_right(Qf,np, ncell): + + # Qfl = [1.0]*(rk3) + + Qfl[:] = Qf[:,np, ncell-1] + return Qfl + +#--------------------------------------------------------------- +#Fl = get_rusanov_flux(Qfl, Qbright, normal, rk3) + + +def get_bflux_right(rk3, ncell, Fl): + for i in range(rk3): + bflux[i,ncell-1,1] = Fl[i] + + return bflux + +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- + +#------------------Interior Edge Flux---------------------------- + +def get_bflux_interior(nface, f2c, Qf, np, get_rusanov_flux): + + normal = 1.0 + + ic_left = numpy.empty((nface-1), dtype=float) + ic_right = numpy.empty((nface-1), dtype=float) + Fl = numpy.empty((rk3), dtype = float) + + for i in range(1,nface-1): + for j in range(rk3): + + ic_left[i] = f2c[i,0] + ic_right[i] = f2c[i,1] + + ic_l = ic_left[i] + ic_r = ic_right[i] + + Qfl[j] = Qf[j, np, ic_l] + Qfr[j] = Qf[j,0, ic_r] + + #----------------------------------# + #Incorporate Rusanov flux function: + + "Returns the fluxes" + gamma = 1.4 + + rhol = numpy.abs(Qfl[0]) + ul = Qfl[1]/rhol + pl = (gamma-1.0)*(Qfl[2]-(0.5*rhol*(ul**2))) + + rhor = numpy.abs(Qfr[0]) + ur = Qfr[1]/rhor + pr = (gamma-1.0)*(Qfr[2]-(0.5*rhor*(ur**2))) + + + c_a = numpy.sqrt(((gamma*numpy.abs(pl+pr))/(rhol+rhor))) + eigv = 0.5*(numpy.absolute(ul+ur)) + c_a + + + "Calculate the flux components" + + Fl[0] = Qfl[0]*ul + Qfr[0]*ur + Fl[1] = Qfl[1]*ul + Qfr[1]*ur + (pl+pr) + Fl[2] = (Qfl[2]+pl)*ul + (Qfr[2]+pr)*ur + + for iv in range(rk3): + Fl[iv] = (Fl[iv]*normal - (Qfr[iv]-Qfl[iv]))*eigv*0.5 + + #---------------------------------------# + for j in range(rk3): + bflux[j,ic_l,1] = Fl[j] + bflux[j,ic_r,0] = Fl[j] + + return bflux + +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- + +#----------------Compute All Volume Fluxes---------------------- + + + + +def get_volflux(rk3, np, ncell, Qf): + + volflux = numpy.zeros((rk3,np+1,ncell), dtype=float) + Fl = numpy.empty((rk3), dtype = float) + + for i in range(ncell): + for j in range(np+1): + for iv in range(rk3): + Qfl[iv] = Qf[iv,j,i] + + gamma = 1.4 + rhol = numpy.abs(Qfl[0]) + ul = Qfl[1]/rhol + pl = (gamma-1.0)*(Qfl[2]-(0.5*Qfl[0]*(ul**2))) + + Fl[0] = Qfl[0]*ul + Fl[1] = Qfl[1]*ul + pl + Fl[2] = (Qfl[2]+pl)*ul + + + for iv in range(rk3): + volflux[iv,j,i] = Fl[iv] + + return volflux + + +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- + +#-----------Get Initial h Matrix (RHS)-------------------------- + +def get_initial_hmatrix(rk3, np, ncell, bflux, fle_gauss): + + bl = [0.0]*(np) + br = [0.0]*(np) + + for k in range(np): + bl[k] = fle_gauss[k,0] + br[k] = fle_gauss[k,1] + + h = numpy.zeros((rk3,np,ncell), dtype=float) + + for i in range(ncell): + for iv in range(rk3): + for k in range(np): + h[iv,k,i] = (bflux[iv,i,0]*bl[k]) - (bflux[iv,i,1]*br[k]) + return h + +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- + +#--------------Volume Integral by Gauss Lobatto---------------- + +def get_GL_hmatrix(rk3, np, ncell, h, volflux, volcoef, fle_grad_gauss): + for i in range(ncell): + for iv in range(rk3): + for j in range(1, np): + h[iv,j,i] = (h[iv,j,i] + + volflux[iv,0,i]*volcoef[0]*fle_grad_gauss[j,0] + + volflux[iv,1,i]*volcoef[1]*fle_grad_gauss[j,2] + + volflux[iv,2,i]*volcoef[2]*fle_grad_gauss[j,3] + + volflux[iv,3,i]*volcoef[3]*fle_grad_gauss[j,1] ) + + return h + +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- + +#-------------Compute the Mass Matrix -------------------------- + +def get_residual_matrix(rk3, np, ncell, aic, h, dx): + + residual = numpy.zeros((rk3,np,ncell)) + + for i in range(ncell): + for iv in range(rk3): + for k in range(np): + residual[iv,k,i] = aic[k,k]*h[iv,k,i]/dx[i] + + return residual + +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- +#--------------------------------------------------------------- + +def get_rusanov_flux(Ql, Qr, unitnorm, rk3): + + "Returns the fluxes" + gamma = 1.4 + + rhol = numpy.abs(Ql[0]) + ul = Ql[1]/rhol + pl = (gamma-1.0)*(Ql[2]-(0.5*rhol*(ul**2))) + + rhor = numpy.abs(Qr[0]) + ur = Qr[1]/rhor + pr = (gamma-1.0)*(Qr[2]-(0.5*rhor*(ur**2))) + + c_a = numpy.sqrt(((gamma*numpy.abs(pl+pr))/(rhol+rhor))) + eigv = 0.5*(numpy.absolute(ul+ur)) + c_a + + + "Calculate the flux components" + + Fl = numpy.empty((rk3), dtype = float) + + Fl[0] = Ql[0]*ul + Qr[0]*ur + Fl[1] = Ql[1]*ul + Qr[1]*ur + (pl+pr) + Fl[2] = (Ql[2]+pl)*ul + (Qr[2]+pr)*ur + + for iv in range(rk3): + Fl[iv] = (Fl[iv]*unitnorm - (Qr[iv]-Ql[iv]))*eigv*0.5 + + return Fl +#---------------------------------------------------------------- +#---------------------------------------------------------------- +#---------------------------------------------------------------- +#---------------------------------------------------------------- + +#All volume flux: + +def get_volume_flux(Ql): + + "Returns the volume fluxes" + + gamma = 1.4 + rhol = numpy.abs(Ql[0]) + ul = Ql[1]/rhol + pl = (gamma-1.0)*(Ql[2]-(0.5*Ql[0]*(ul**2))) + + "Calculate the flux components" + Fl[0] = numpy.abs(Ql[0]*ul) + Fl[1] = Ql[1]*ul + pl + Fl[2] = (Ql[2]+pl)*ul + + return Fl +#-------------------------------------------------------------------- + diff --git a/Martinez_1D_Euler/Residual_Function_np3.pyc b/Martinez_1D_Euler/Residual_Function_np3.pyc new file mode 100644 index 0000000000000000000000000000000000000000..76d4372f29169b9de3343eb24e63bfbc6f5f34f8 GIT binary patch literal 6661 zcmbtY&yQPI6+Z9Tn?R0_P=Z zrsc8Y>|}IBmMdmN%;K>V#N3mJ`MC|SbLKen!rbhWw?A`i#Yy3oCCJH#P~3_HdF`x9 zP|(hr1ZTA4NwBP)q6912S(ji{J0%I$v{RPA(@sT#qIS+ou&$kR5|p&_f&^vloTs(! zRVy%G`!oFAZ70y($5;Q_uxK>;WYdzXH`<)EBkxAr#i*S{6E2_$!Viv`4ffLbF5!C% z-(dx;#;Z;!G7%Uy6P+qN!fKhnYbRnSG7VHrKYIcx``dS~~}sM`qJz3y)J=-Snz55g2CvGeX$>oR)7 zWi*mpbNG2(KR4V4V*uAt5HaH*uqCt%|RWGK4vzG!*?HJ8s 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get_Qrk_stage2(Q_rk, Qrk1, ncell, rk3, np, dt, residual): + + Qrk2 = numpy.zeros((rk3,np, ncell), dtype=float) + + for i in range(ncell): + for iv in range(rk3): + for k in range(np): + Qrk2[iv,k,i] = (0.75*Q_rk[iv,k,i])+0.25*(Qrk1[iv,k,i]+(dt*residual[iv,k,i])) + + return Qrk2 + +#------------------------------------------------------ +#------------------------------------------------------ +# -----------------Stage 3 Runge Kutta----------------- + +def get_Qrk_stage3(Q_rk, Qrk2, ncell, rk3, np, dt, residual): + + Qrk3 = numpy.zeros((rk3,np, ncell), dtype=float) + + for i in range(ncell): + for iv in range(rk3): + for k in range(np): + Qrk3[iv,k,i] = ((1.0/3.0)*(Q_rk[iv,k,i])) + ((2.0/3.0)*(Qrk2[iv,k,i]+(dt*residual[iv,k,i]))) + + return Qrk3 +#------------------------------------------------------ +#ctime = ctime + 0.5*dt +#------------------------------------------------------ +'''residnorm = 0.0 + +for i in range(ncell): + for k in range(np): + residnorm = residnorm + 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G6aNE-P-C_L literal 0 HcmV?d00001 diff --git a/Martinez_1D_Euler/U_initial_np3.py b/Martinez_1D_Euler/U_initial_np3.py new file mode 100644 index 0000000..a56b328 --- /dev/null +++ b/Martinez_1D_Euler/U_initial_np3.py @@ -0,0 +1,83 @@ +import numpy + +np = 3 + +aic = numpy.zeros((np, np), dtype=float) + + +# Store the spectral coefficients for np = 3 and a fifth order polynomial: +volcoef = [0.0]*(np+1) + +volcoef[0] = 1.0/12.0 +volcoef[1] = 5.0/12.0 +volcoef[2] = 5.0/12.0 +volcoef[3] = 1.0/12.0 + +# Store the Quadrature Points and Weights for 4 point quadrature(np=3), fifth order polynomial: + +#Quadrature Points + +gauss = numpy.zeros((np+1, np+1), dtype=float) +gauss[0,0]=-0.5 +gauss[1,0]=0.5 +gauss[2,0]= -1 * numpy.sqrt(5)/10 +gauss[3,0]= numpy.sqrt(5)/10.0 + +# Corresponding Quadrature weights + +gauss[0,1]=1.0/12.0 +gauss[1,1]=gauss[0,1] +gauss[2,1]=5.0/12.0 +gauss[3,1]=gauss[2,1] + +#Inverse Mass Matrix +aic_3 = numpy.zeros((np,np), dtype=float) +aic_3[0,0]=1.0 +aic_3[1,1]=12.0 +aic_3[2,2]=180.0 + +#--------------------------------------- +"""Defines inverse mass matrix and GL x points and weights +""" +def get_inverse_mass_matrix(np): + + #Third Order:Store the Inverse of the Mass Matrix + + for i in range(np): + for j in range(np): + aic[i,j] = aic_3[i,j] + + return aic +#------------------------------------------------------------------ + +def get_U_initial(np, rk3, ncell, fle_gauss, x , dx, aic): + block = 0.50 + + qinit = numpy.zeros((np+1,rk3,ncell), dtype=float) + U_initial = numpy.empty((rk3,np,ncell), dtype=float) + atemp = numpy.empty((np,rk3), dtype=float) + + for i in range(ncell): + for j in range(np): + for k in range(3): + + xeval = x[i] + gauss[j,0] * dx[i] + if (xeval < block): + qinit[k,0,i] = 1.25 + qinit[k,1,i] = 0.0 + qinit[k,2,i] = 2.5 + + if (xeval >= block): + qinit[k,0,i] = 0.125 + qinit[k,1,i] = 0.0 + qinit[k,2,i] = 0.25 + + atemp[k,j] = ((qinit[0,k,i]*fle_gauss[j,0])*gauss[0,1] + (qinit[1,k,i]*fle_gauss[j,1])*gauss[1,1] + + (qinit[2,k,i]*fle_gauss[j,2])*gauss[2,1] + (qinit[3,k,i]*fle_gauss[j,3])*gauss[3,1]) + + U_initial[0,j,i]=aic[j,j]*atemp[0,j] + U_initial[1,j,i]=aic[j,j]*atemp[1,j] + U_initial[2,j,i]=aic[j,j]*atemp[2,j] + + return U_initial +#----------------------------------------------------------------------- diff --git a/Martinez_1D_Euler/U_initial_np3.pyc b/Martinez_1D_Euler/U_initial_np3.pyc new file mode 100644 index 0000000000000000000000000000000000000000..be7562c33f47d7744deb5e2460efa28731c85b7f GIT binary patch literal 2216 zcmah~-EJFI5T0|q{fbPZ~Q@yhrtMBF8 z0@MWvulv;FR}Z!%Tu2|pwtj5;un^#O5o(;X0sROLs=_@0cT2dB!QG}GP)xYKBTR$K z`*N_4db@w&bL5u524xT@xL%r5F!FJ67?eoA9VX0R6L&g32-pBF254-%~1@H3eF8VFf?YEM1X(<=B~Jgk&t-(o|Yz^2N{ zogo>y=G`Q%5?^oRfa|@f38ExPqBKalXsxGKn9{yT7B(pC>%ihAQq8vBG%2)1l@$0- zwP{i4DWK3pg8Ei%+7w|`6QwOv)Z!-KI!6$o>vT6e6yaB;bE36YxOY*FqBi3IKy-Wo z6BW)TQ2`R8>ZxjukYhh(TE9?EBSrkDpJs}>;RhEVs}Q*C){9WCEbtD}K(F>X2mqtx zJyd*+1l8vJ7zXFt&gxPHf$gL)lxZ7g<*;3WAuV89tRAj@q~i$nY9)$Zgg!*zVnu*x zPmz+My$40~1*k~>21N!1s7MKP^*wrIjzH?@5x&|-dBgJkxABmSgJP5ak@kMp_STKI z|0T8mdN|-_lC?5CVo^GaNh}H^7r7T>FpmRi()r~Dx{N2tcV@ltg`0VcNg$lLSO(WN zI}q`7a_Rd6?l7DGMx?jb=Lce&PmSjVb*N{a84FT6JVc+M>q0x55U;((^jy}gxY=b* z*7u^H1lZW-plk`01Bh3I=4#^o-^^$O&NCx(`Z*Lnj|);2{ZKtnea#bNPw%T1ztugR zyh}gnN9vI}(1q&qx%OXQRa)~$ou67zG*wv>Y}M3o=1K$r}g zAU4lJIW3#d#^%3?_$YW-UM;-o!p!UzcuXf$X|)b)UTm}3z7LXDTeyE7)4a>@V&PnR l<{5|LKL%;(itbyt^^THnS?>Xl`=id9A=UyL0_S$=|HP@VTjPZ{5eOJg`RYg)F1|k#+MXIcX)LPm#VP9V#e&pk&p**_@&t22GuD5ON?UudU$x@yS zkx~a7UlYXq=!`um;bX49vocVxq}g1guhL{!-zU|~Pz_K>`#e%&{=}(%dudoMt zkd*XQ?#jmICM6{$7CSq8DQ1vJKaZA~nc3;UGjq9kC|A#~u<)Aevu9~eOGCXDLUAHC zKWk5v!UI2!V*L(F

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