{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "# Boundary Conditions: Deduction for the Lattice D2Q5 - Poisson Equation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [],
   "source": [
    "from __future__ import division\n",
    "from sympy import *\n",
    "import numpy as np\n",
    "from sympy import S, collect, expand, factor, Wild\n",
    "from sympy import fraction, Rational, Symbol\n",
    "from sympy import solve\n",
    "#-------------------------------------------------Symbols----------------------------------------------\n",
    "p, dpx, dpy = symbols('\\widehat{p} \\\\partial_{x}\\widehat{p} \\\\partial_{y}\\widehat{p}')\n",
    "ux, uy, tau, cs = symbols('u_{x} u_{y} \\\\tau c_{s}')\n",
    "wi, cx, cy = symbols('w_{i} c_{x} c_{y}')\n",
    "fi, f0, f1, f2, f3, f4 = symbols('f_{i} f_{0} f_{1} f_{2} f_{3} f_{4}')\n",
    "#-------------------------------------------------Functions----------------------------------------------\n",
    "feq = Function('feq')(wi, cx, cy)\n",
    "f = Function('f')(fi)\n",
    "#-------------------------------------------------Arrays----------------------------------------------\n",
    "fi=np.array([f0,f1,f2,f3,f4])\n",
    "wi=np.array([Rational(2,6),Rational(1,6),Rational(1,6),Rational(1,6),Rational(1,6)])\n",
    "cx=np.array([0,1,0,-1,0])\n",
    "cy=np.array([0,0,1,0,-1])\n",
    "#-------------------------------------------------Calc.Func------------------------------------------------\n",
    "f= fi\n",
    "feq=p*wi\n",
    "feq[0]=p*(wi[0]-1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "## Boundary Layers"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "### Pressure Moment"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\widehat{p} + \\frac{3 f_{1}}{2} + \\frac{3 f_{2}}{2} + \\frac{3 f_{3}}{2} + \\frac{3 f_{4}}{2}$"
      ],
      "text/plain": [
       "-\\widehat{p} + 3*f_{1}/2 + 3*f_{2}/2 + 3*f_{3}/2 + 3*f_{4}/2"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "An0=simplify(sum(f[1:])/(1-wi[0])-p)\n",
    "An0\n",
    "#print(latex(An2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "### Camada Norte "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "editable": true,
    "scrolled": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f_{4} = \\frac{2 \\widehat{p}}{3} - f_{1} - f_{2} - f_{3}$"
      ],
      "text/plain": [
       "Eq(f_{4}, 2*\\widehat{p}/3 - f_{1} - f_{2} - f_{3})"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sols = solve([An0], [f4], dict=True)\n",
    "# sols\n",
    "b1=Array(sols[0][f4])\n",
    "b0 = Array(f4)\n",
    "Eq(b0,b1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "### Camada Sul"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f_{2} = \\frac{2 \\widehat{p}}{3} - f_{1} - f_{3} - f_{4}$"
      ],
      "text/plain": [
       "Eq(f_{2}, 2*\\widehat{p}/3 - f_{1} - f_{3} - f_{4})"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sols = solve([An0], [f2], dict=True)\n",
    "# sols\n",
    "b1=Array(sols[0][f2])\n",
    "b0 = Array(f2)\n",
    "Eq(b0,b1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "### Camada Oeste"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "editable": true,
    "scrolled": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f_{1} = \\frac{2 \\widehat{p}}{3} - f_{2} - f_{3} - f_{4}$"
      ],
      "text/plain": [
       "Eq(f_{1}, 2*\\widehat{p}/3 - f_{2} - f_{3} - f_{4})"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sols = solve([An0], [f1], dict=True)\n",
    "# sols\n",
    "b1=Array(sols[0][f1])\n",
    "b0 = Array(f1)\n",
    "Eq(b0,b1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "### Camada Leste "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "editable": true,
    "scrolled": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f_{3} = \\frac{2 \\widehat{p}}{3} - f_{1} - f_{2} - f_{4}$"
      ],
      "text/plain": [
       "Eq(f_{3}, 2*\\widehat{p}/3 - f_{1} - f_{2} - f_{4})"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sols = solve([An0], [f3], dict=True)\n",
    "# sols\n",
    "b1=Array(sols[0][f3])\n",
    "b0 = Array(f3)\n",
    "Eq(b0,b1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "## Corner Boundaries"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "### Camada Noroeste"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\tau c_{s}^{2} \\left(\\partial_{x}\\widehat{p} + \\partial_{y}\\widehat{p}\\right) + f_{1} + f_{2} - f_{3} - f_{4}$"
      ],
      "text/plain": [
       "\\tau*c_{s}**2*(\\partial_{x}\\widehat{p} + \\partial_{y}\\widehat{p}) + f_{1} + f_{2} - f_{3} - f_{4}"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Bn0=simplify(sum(f*(cx+cy))+(dpx+dpy)*cs**2*tau)\n",
    "Bn0\n",
    "#print(latex(An2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "editable": true,
    "scrolled": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}f_{1}\\\\f_{4}\\end{matrix}\\right] = \\left[\\begin{matrix}- \\frac{\\partial_{x}\\widehat{p} \\tau c_{s}^{2}}{2} - \\frac{\\partial_{y}\\widehat{p} \\tau c_{s}^{2}}{2} + \\frac{\\widehat{p}}{3} - f_{2}\\\\\\frac{\\partial_{x}\\widehat{p} \\tau c_{s}^{2}}{2} + \\frac{\\partial_{y}\\widehat{p} \\tau c_{s}^{2}}{2} + \\frac{\\widehat{p}}{3} - f_{3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Eq([[f_{1}], [f_{4}]], Matrix([\n",
       "[-\\partial_{x}\\widehat{p}*\\tau*c_{s}**2/2 - \\partial_{y}\\widehat{p}*\\tau*c_{s}**2/2 + \\widehat{p}/3 - f_{2}],\n",
       "[ \\partial_{x}\\widehat{p}*\\tau*c_{s}**2/2 + \\partial_{y}\\widehat{p}*\\tau*c_{s}**2/2 + \\widehat{p}/3 - f_{3}]]))"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sols = solve([An0, Bn0], [f1,f4], dict=True)\n",
    "# sols\n",
    "b1=Matrix([[sols[0][f1]],[sols[0][f4]]])\n",
    "b0 = Array([[f1],[f4]])\n",
    "Eq(b0,b1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "## Camada Nordeste"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\tau c_{s}^{2} \\left(\\partial_{x}\\widehat{p} - \\partial_{y}\\widehat{p}\\right) + f_{1} - f_{2} - f_{3} + f_{4}$"
      ],
      "text/plain": [
       "\\tau*c_{s}**2*(\\partial_{x}\\widehat{p} - \\partial_{y}\\widehat{p}) + f_{1} - f_{2} - f_{3} + f_{4}"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Bn0=simplify(sum(f*(cx-cy))+(dpx-dpy)*cs**2*tau)\n",
    "Bn0\n",
    "#print(latex(An2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "editable": true,
    "scrolled": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}f_{3}\\\\f_{4}\\end{matrix}\\right] = \\left[\\begin{matrix}\\frac{\\partial_{x}\\widehat{p} \\tau c_{s}^{2}}{2} - \\frac{\\partial_{y}\\widehat{p} \\tau c_{s}^{2}}{2} + \\frac{\\widehat{p}}{3} - f_{2}\\\\- \\frac{\\partial_{x}\\widehat{p} \\tau c_{s}^{2}}{2} + \\frac{\\partial_{y}\\widehat{p} \\tau c_{s}^{2}}{2} + \\frac{\\widehat{p}}{3} - f_{1}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Eq([[f_{3}], [f_{4}]], Matrix([\n",
       "[ \\partial_{x}\\widehat{p}*\\tau*c_{s}**2/2 - \\partial_{y}\\widehat{p}*\\tau*c_{s}**2/2 + \\widehat{p}/3 - f_{2}],\n",
       "[-\\partial_{x}\\widehat{p}*\\tau*c_{s}**2/2 + \\partial_{y}\\widehat{p}*\\tau*c_{s}**2/2 + \\widehat{p}/3 - f_{1}]]))"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sols = solve([An0, Bn0], [f3,f4], dict=True)\n",
    "# sols\n",
    "b1=Matrix([[sols[0][f3]],[sols[0][f4]]])\n",
    "b0 = Array([[f3],[f4]])\n",
    "Eq(b0,b1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "## Camada Sudoeste"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\tau c_{s}^{2} \\left(\\partial_{x}\\widehat{p} - \\partial_{y}\\widehat{p}\\right) + f_{1} - f_{2} - f_{3} + f_{4}$"
      ],
      "text/plain": [
       "\\tau*c_{s}**2*(\\partial_{x}\\widehat{p} - \\partial_{y}\\widehat{p}) + f_{1} - f_{2} - f_{3} + f_{4}"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Bn0=simplify(sum(f*(cx-cy))+(dpx-dpy)*cs**2*tau)\n",
    "Bn0\n",
    "#print(latex(An2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "editable": true,
    "scrolled": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}f_{1}\\\\f_{2}\\end{matrix}\\right] = \\left[\\begin{matrix}- \\frac{\\partial_{x}\\widehat{p} \\tau c_{s}^{2}}{2} + \\frac{\\partial_{y}\\widehat{p} \\tau c_{s}^{2}}{2} + \\frac{\\widehat{p}}{3} - f_{4}\\\\\\frac{\\partial_{x}\\widehat{p} \\tau c_{s}^{2}}{2} - \\frac{\\partial_{y}\\widehat{p} \\tau c_{s}^{2}}{2} + \\frac{\\widehat{p}}{3} - f_{3}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Eq([[f_{1}], [f_{2}]], Matrix([\n",
       "[-\\partial_{x}\\widehat{p}*\\tau*c_{s}**2/2 + \\partial_{y}\\widehat{p}*\\tau*c_{s}**2/2 + \\widehat{p}/3 - f_{4}],\n",
       "[ \\partial_{x}\\widehat{p}*\\tau*c_{s}**2/2 - \\partial_{y}\\widehat{p}*\\tau*c_{s}**2/2 + \\widehat{p}/3 - f_{3}]]))"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sols = solve([An0, Bn0], [f1,f2], dict=True)\n",
    "# sols\n",
    "b1=Matrix([[sols[0][f1]],[sols[0][f2]]])\n",
    "b0 = Array([[f1],[f2]])\n",
    "Eq(b0,b1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": []
   },
   "source": [
    "## Camada Sudeste "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "editable": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\tau c_{s}^{2} \\left(\\partial_{x}\\widehat{p} + \\partial_{y}\\widehat{p}\\right) + f_{1} + f_{2} - f_{3} - f_{4}$"
      ],
      "text/plain": [
       "\\tau*c_{s}**2*(\\partial_{x}\\widehat{p} + \\partial_{y}\\widehat{p}) + f_{1} + f_{2} - f_{3} - f_{4}"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Bn0=simplify(sum(f*(cx+cy))+(dpx+dpy)*cs**2*tau)\n",
    "Bn0\n",
    "#print(latex(An2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "editable": true,
    "scrolled": true,
    "slideshow": {
     "slide_type": ""
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}f_{2}\\\\f_{3}\\end{matrix}\\right] = \\left[\\begin{matrix}- \\frac{\\partial_{x}\\widehat{p} \\tau c_{s}^{2}}{2} - \\frac{\\partial_{y}\\widehat{p} \\tau c_{s}^{2}}{2} + \\frac{\\widehat{p}}{3} - f_{1}\\\\\\frac{\\partial_{x}\\widehat{p} \\tau c_{s}^{2}}{2} + \\frac{\\partial_{y}\\widehat{p} \\tau c_{s}^{2}}{2} + \\frac{\\widehat{p}}{3} - f_{4}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Eq([[f_{2}], [f_{3}]], Matrix([\n",
       "[-\\partial_{x}\\widehat{p}*\\tau*c_{s}**2/2 - \\partial_{y}\\widehat{p}*\\tau*c_{s}**2/2 + \\widehat{p}/3 - f_{1}],\n",
       "[ \\partial_{x}\\widehat{p}*\\tau*c_{s}**2/2 + \\partial_{y}\\widehat{p}*\\tau*c_{s}**2/2 + \\widehat{p}/3 - f_{4}]]))"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sols = solve([An0, Bn0], [f2,f3], dict=True)\n",
    "# sols\n",
    "b1=Matrix([[sols[0][f2]],[sols[0][f3]]])\n",
    "b0 = Array([[f2],[f3]])\n",
    "Eq(b0,b1)"
   ]
  }
 ],
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