{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Chapter 10 - Two-Dimensional Steady & Transient Heat Conduction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exa10.1  Page 195"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "the values of T from 1st element to last is:\n",
      "\n",
      "47.142854591 \t91.2499974482 \t182.857141581 \t57.3214260196 \t114.999997448 \t220.178570153 \t47.1428558669 \t91.2499987241 \t182.857142219 \t"
     ]
    }
   ],
   "source": [
    "from __future__ import division\n",
    "tnew=[]\n",
    "e=[]\n",
    "for i in range(1,10):\n",
    "    tnew.append(101)\n",
    "    e.append(1)                       #assumed values\n",
    "\n",
    "t=1e-6\n",
    "#since all the nodes are interior nodes so discretized eqn used is eqn 10.10\n",
    "while e[0]>t and e[1]>t and e[2]>t and e[3]>t and e[4]>t and e[5]>t and e[6]>t and e[7]>t and e[8]>t:\n",
    "    told=[]\n",
    "    for i in range(1,10):\n",
    "        told.append(tnew[i-1])\n",
    "    tnew[0]=(told[1]+40+told[3])/4                       #on solving eqns for various nodes we get,\n",
    "    tnew[1]=(tnew[0]+told[2]+told[4]+20)/4\n",
    "    tnew[2]=(tnew[1]+told[5]+420)/4\n",
    "    tnew[3]=(told[4]+tnew[0]+told[6]+20)/4\n",
    "    tnew[4]=(tnew[1]+told[7]+told[5]+tnew[3])/4\n",
    "    tnew[5]=(tnew[2]+tnew[4]+told[8]+400)/4\n",
    "    tnew[6]=(tnew[3]+told[7]+40)/4\n",
    "    tnew[7]=(tnew[4]+tnew[6]+told[8]+20)/4\n",
    "    tnew[8]=(tnew[5]+420+tnew[7])/4\n",
    "    e=[]\n",
    "    for i in range(1,10):\n",
    "        e.append(abs(tnew[i-1]-told[i-1]))\n",
    "    \n",
    "\n",
    "print \"the values of T from 1st element to last is:\\n\"\n",
    "for i in range(1,10):\n",
    "    print tnew[i-1],'\\t',"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exa10.2  Page 198"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "the values of T from 1st element to last is:\n",
      "\n",
      "145.833332213 \t141.666666106 \t120.833333053 \t141.666666527 \t145.833333263 \t"
     ]
    }
   ],
   "source": [
    "from __future__ import division\n",
    "tnew=[]\n",
    "e=[]\n",
    "for i in range(1,6):\n",
    "    tnew.append(101)\n",
    "    e.append(1)\n",
    "\n",
    "t=1e-6\n",
    "#since there is no source term so we get the following eqns.\n",
    "while e[0]>t and e[1]>t and e[2]>t and e[3]>t and e[4]>t:\n",
    "    told=[]\n",
    "    for i in range(1,6):\n",
    "        told.append(tnew[i-1])\n",
    "        \n",
    "    tnew[0]=(told[1]*2+300)/4\n",
    "    tnew[1]=(tnew[0]+told[2]+300)/4\n",
    "    tnew[2]=(tnew[1]+told[3]+200)/4\n",
    "    tnew[3]=(told[4]+tnew[2]+300)/4\n",
    "    tnew[4]=(2*tnew[3]+300)/4\n",
    "    \n",
    "    for i in range(1,6):\n",
    "        e[i-1]=(abs(tnew[0]-told[i-1]))\n",
    "    \n",
    "\n",
    "print \"the values of T from 1st element to last is:\\n\"\n",
    "for i in range(1,6):\n",
    "    print tnew[i-1],'\\t',"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exa10.3  Page 201"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "the value of RED points respectively : 47.1429, 47.1429, 115.0000, 91.2500 & 47.1429\n",
      "the value of BLUE points respectively : 91.2500, 220.1786, 57.3214 & 91.2500\n"
     ]
    }
   ],
   "source": [
    "from __future__ import division\n",
    "from numpy import zeros\n",
    "tn=zeros([5,5])\n",
    "for j in range(1,6):\n",
    "    tn[j-1,0]=400                       #defining conditions\n",
    "\n",
    "for j in range(2,5):\n",
    "    tn[0,j-1]=20\n",
    "    tn[4,j-1]=20\n",
    "    tn[j-1,4]=20\n",
    "\n",
    "e=[]    \n",
    "for i in range(1,10):\n",
    "    e.append(1)\n",
    "\n",
    "for i in range(2,5):\n",
    "    tn[i-1,1:4]=60\n",
    "\n",
    "\n",
    "t1=1e-6\n",
    "t=zeros([4,4])\n",
    "while e[0]>t1 and e[1]>t1 and e[2]>t1 and e[3]>t1 and e[4]>t1 and e[5]>t1 and e[6]>t1 and e[7]>t1 and e[8]>t1:\n",
    "    for i in range(1,4):\n",
    "        for j in range(1,4):\n",
    "            t[i,j]=tn[i,j]\n",
    "        \n",
    "    #using red-black gauss-seidel method\n",
    "    for i in range(1,4):\n",
    "        for j in range(1,4):\n",
    "            tn[i,j]=(tn[i+1,j]+tn[i-1,j]+tn[i,j+1]+tn[i,j-1])/4\n",
    "    #now getting the absolute difference of the 9 variables\n",
    "    e[0]=abs(t[1,1]-tn[1,1])\n",
    "    e[1]=abs(t[1,2]-tn[1,2])\n",
    "    e[2]=abs(t[1,3]-tn[1,3])\n",
    "    e[3]=abs(t[2,1]-tn[2,1])\n",
    "    e[4]=abs(t[2,2]-tn[2,2])\n",
    "    e[5]=abs(t[2,3]-tn[2,3])\n",
    "    e[6]=abs(t[3,1]-tn[3,1])\n",
    "    e[7]=abs(t[3,2]-tn[3,2])\n",
    "    e[8]=abs(t[3,3]-tn[3,3])\n",
    "\n",
    "#now defining positions of the various nodes according to red black combination\n",
    "R1=t[1,3]\n",
    "R2=t[3,3]\n",
    "R3=t[2,2]\n",
    "R4=t[1,2]\n",
    "R5=t[1,3]\n",
    "B1=t[3,2]\n",
    "B2=t[2,1]\n",
    "B3=t[2,3]\n",
    "B4=t[1,2]\n",
    "print \"the value of RED points respectively : %.4f, %.4f, %.4f, %.4f & %.4f\"%(R1,R2,R3,R4,R5)\n",
    "\n",
    "print \"the value of BLUE points respectively : %.4f, %.4f, %.4f & %.4f\"%(B1, B2, B3, B4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exa10.8  Page 202"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "the values of T from 1st element to last is\n",
      "157.831321254\n",
      "192.469876146\n",
      "280.72288988\n",
      "184.939753978\n",
      "231.325296989\n",
      "330.421684639\n",
      "175.903610664\n",
      "217.469876356\n",
      "309.638552636\n"
     ]
    }
   ],
   "source": [
    "from __future__ import division\n",
    "from numpy import zeros\n",
    "tnew=[0];told=[0];e=[0];\n",
    "for i in range(1,10):\n",
    "    tnew.append(101)\n",
    "    told.append(0)\n",
    "    e.append(1)                    #assumed values\n",
    "\n",
    "t=1e-6\n",
    "while e[(1)]>t and e[(2)]>t and e[(3)]>t and e[(4)]>t and e[(5)]>t and e[(6)]>t and e[(7)]>t and e[(8)]>t and e[(9)]>t:\n",
    "    for i in range(1,10):\n",
    "        told[i]=tnew[(i)]\n",
    "    #using eqn 10.10 for the interior nodes and convective boundary conditions for corner nodes\n",
    "    tnew[(1)]=(told[(2)]+50+.5*told[(4)]+100/3)*3/7\n",
    "    tnew[(2)]=(tnew[(1)]+told[(3)]+told[(5)]+100)/4\n",
    "    tnew[(3)]=(tnew[(2)]+told[(6)]+600)/4\n",
    "    tnew[(4)]=(told[(5)]+.5*tnew[(1)]+.5*told[(7)]+100/3)*3/7\n",
    "    tnew[(5)]=(tnew[(2)]+told[(8)]+told[(6)]+tnew[(4)])/4\n",
    "    tnew[(6)]=(tnew[(3)]+tnew[(5)]+told[(9)]+500)/4\n",
    "    tnew[(7)]=(.5*tnew[(4)]+.5*told[(8)]+100/3)*3/4\n",
    "    tnew[(8)]=(tnew[(5)]+.5*tnew[(7)]+.5*told[(9)]+100/3)*3/7\n",
    "    tnew[(9)]=(tnew[(6)]+100/3+.5*tnew[(8)]+250)*3/7\n",
    "    for i in range(1,10):\n",
    "        e[i]=abs(tnew[i]-told[i])\n",
    "    \n",
    "\n",
    "print \"the values of T from 1st element to last is\"\n",
    "for i in range(1,10):\n",
    "    print tnew[i]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exa10.9  Page 211"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "the values of T from 1st element to last is\n",
      "83.6250164077\n",
      "83.3426002934\n",
      "81.8505046198\n",
      "86.8385228347\n",
      "86.864496399\n",
      "86.0434332286\n",
      "86.7854379748\n",
      "87.2768602399\n",
      "88.5496879219\n"
     ]
    }
   ],
   "source": [
    "from __future__ import division\n",
    "from numpy import zeros\n",
    "tnew=[0];told=[0];e=[0];\n",
    "for i in range(1,10):\n",
    "    tnew.append(101)\n",
    "    told.append(0)\n",
    "    e.append(1)                    #assumed values\n",
    "\n",
    "t=1e-6\n",
    "while e[(1)]>t and e[(2)]>t and e[(3)]>t and e[(4)]>t and e[(5)]>t and e[(6)]>t and e[(7)]>t and e[(8)]>t and e[(9)]>t:\n",
    "    for i in range(1,10):\n",
    "        told[(i)]=tnew[(i)]\n",
    "    #using the basic discretization eqn. for all the nodes since the boundary conditions vary for each point\n",
    "    tnew[(1)]=(told[(2)]+1.25+told[(4)])/2.05\n",
    "    tnew[(2)]=(.5*tnew[(1)]+.5*told[(3)]+told[(5)]+1.25)/2.05\n",
    "    tnew[(3)]=(.5*tnew[(2)]+.5*told[(6)]+1.25)/1.05\n",
    "    tnew[(4)]=(told[(5)]+.5*tnew[(1)]+45)/2\n",
    "    tnew[(5)]=(tnew[(2)]+told[(8)]+90+tnew[(4)])/4\n",
    "    tnew[(6)]=(.5*tnew[(3)]+tnew[(5)]+.5*told[(7)]+91.25)/3.05\n",
    "    tnew[(7)]=(.5*tnew[(6)]+.5*told[(8)]+91.25)/2.05\n",
    "    tnew[(8)]=(91.25+.5*tnew[(7)]+.5*told[(9)])/2.05\n",
    "    tnew[(9)]=(47.125+.5*tnew[(8)])/1.025\n",
    "    for i in range(1,10):\n",
    "        e[i]=abs(tnew[i]-told[i])\n",
    "    \n",
    "\n",
    "print \"the values of T from 1st element to last is\"\n",
    "for i in range(1,10):\n",
    "    print told[i]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exa10.10  Page 213"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "the soln of eg 10.10-->Alternating Direction Implicit Method\n",
      "the soln by ADI method is\n",
      "20.0 20.0 20.0\n",
      "--------------\n",
      "20.0 20.0 20.0\n",
      "--------------\n",
      "48.1904761905 43.6666666667 105.0\n"
     ]
    }
   ],
   "source": [
    "from __future__ import division\n",
    "from numpy import zeros\n",
    "print \"the soln of eg 10.10-->Alternating Direction Implicit Method\"\n",
    "nmax=25\n",
    "a=[0,1,1]                                #defining variables\n",
    "b=[-4,-4,-4]\n",
    "c=[1,1]\n",
    "t=zeros([6,6])\n",
    "tstar=zeros([6,6])\n",
    "t[1,2]=20 ; t[1,3]=20 ; t[1,4]=20; t[2,1]=20; t[3,1]=20 ; t[4,1]=20 ; t[5,2]=20 ; t[5,3]=20 ; t[5,4]=20 ; t[2,5]=400; t[3,5]=400; t[4,5]=400\n",
    "tstar[1,2]=20 ; tstar[1,3]=20; tstar[1,4]=20 ; tstar[2,1]=20 ; tstar[3,1]=20 ; tstar[4,1]=20 ; tstar[5,2]=20 ; tstar[5,3]=20 ; tstar[5,4]=20 ; tstar[2,5]=400 ; tstar[3,5]=400 ; tstar[4,5]=400\n",
    "for i in range(2,5):\n",
    "    for j in range(2,5):\n",
    "        t[i,j]=20\n",
    "    \n",
    "\n",
    "#solving by TDMA Method\n",
    "d=zeros(4)\n",
    "for nn in range(1,nmax+1):\n",
    "    for jj in range(2,5):\n",
    "        d[(1)]=-t[1,jj]-t[2,jj+1]-tstar[2,jj-1]\n",
    "        d[(2)]=-t[3,jj+1]-tstar[3,jj-1]\n",
    "        d[(3)]=-t[5,jj]-t[4,jj+1]-tstar[4,jj-1]\n",
    "        i=1 ; n=3\n",
    "    Beta=[0,b[i]]\n",
    "    Gamma=[0,d[i]/Beta[i]]\n",
    "    i1=i+1\n",
    "    for j in range(i1,n+1):\n",
    "        Beta.append(b[(j-1)]-a[(j-1)]*c[(j-2)]/Beta[(j-1)])\n",
    "        Gamma.append((d[(j)]-a[(j-1)]*Gamma[(j-1)])/Beta[(j)])\n",
    "    #end \n",
    "    x=zeros(n)\n",
    "    x[-1]=Gamma[n-1]\n",
    "    n1=n-i\n",
    "    for k in range(1,n1+1):\n",
    "        j=n-k\n",
    "        x[j-1]=Gamma[(j)]-c[(j-1)]*x[j-1]/Beta[j]\n",
    "    #end\n",
    "    tstar[2,jj]=x[0]\n",
    "    tstar[3,jj]=x[1]\n",
    "    tstar[4,jj]=x[2]\n",
    "#end\n",
    "    for ii in range(2,5):\n",
    "        d[(1)]=-t[ii,1]-tstar[ii+1,2]-tstar[ii-1,2]\n",
    "        d[(2)]=-tstar[ii+1,3]-tstar[ii-1,3]\n",
    "        d[(3)]=-t[ii,5]-tstar[ii+1,4]-tstar[ii-1,4]\n",
    "    i=1 ; n=3\n",
    "    Beta[(i)]=b[i]\n",
    "    Gamma[i]=d[(i)]/Beta[(i)]\n",
    "    i1=i+1\n",
    "    for j in range(i1,n+1):\n",
    "        Beta[(j)]=b[(j-1)]-a[(j-1)]*c[j-2]/Beta[(j-1)]\n",
    "        Gamma[(j)]=(d[(j)]-a[(j-1)]*Gamma[(j-1)])/Beta[(j)]\n",
    "    #end \n",
    "    \n",
    "    x[-1]=Gamma[(n)]\n",
    "    n1=n-i\n",
    "    for k in range(1,n1+1):\n",
    "        j=n-k\n",
    "        x[j]=Gamma[(j)]-c[(j-1)]*x[(j)]/Beta[(j)]\n",
    "    #end\n",
    "    t[ii,2]=x[0]\n",
    "    t[ii,3]=x[1]\n",
    "    t[ii,4]=x[2]\n",
    "\n",
    "\n",
    "print \"the soln by ADI method is\"\n",
    "print t[2,4],t[2,3],t[2,2]\n",
    "print \"--------------\"\n",
    "print t[3,4],t[3,3],t[3,2]\n",
    "print \"--------------\"\n",
    "print t[4,4],t[4,3],t[4,2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exa 10.11  Page 215"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "the soln by ADI is\n",
      "1.58563151796\n",
      "1.90275782155\n",
      "2.98098725376\n",
      "5.25161158747\n",
      "9.62288055617\n",
      "17.8433017473\n",
      "33.2010436374\n",
      "61.8392029825\n",
      "945.175268307\n",
      "-467.149084346\n"
     ]
    }
   ],
   "source": [
    "from __future__ import division\n",
    "from numpy import arange,zeros\n",
    "a=[0,0];b=[0,0];c=[0,0]\n",
    "for k in range(2,11):\n",
    "    a.append(-1)\n",
    "    b.append(2.4)\n",
    "    c.append(-1)\n",
    "\n",
    "alpha=1 ; delta_t=.05 ; delta_x=.1\n",
    "m=alpha*delta_t/delta_x**2\n",
    "b[1]=2.4\n",
    "c[1]=-2\n",
    "t=zeros([12,12])\n",
    "tstar=zeros([12,12])\n",
    "\n",
    "for k in range(1,12):\n",
    "    t[11,k]=400 ; tstar[11,k]=400 ;  t[k,11]=400 ; tstar[k,11]=400\n",
    "\n",
    "\n",
    "for i in range(1,11):\n",
    "    for j in range(1,11):\n",
    "        t[i,j]=0\n",
    "        tstar[i,j]=0\n",
    "    \n",
    "d=zeros(11)\n",
    "for tm in arange(.05,5+.05,.05):\n",
    "    for jj in range(1,11):\n",
    "        if jj==1:\n",
    "            for ii in range(1,11):\n",
    "                d[(ii)]=2*t[ii,2]-1.6*t[ii,1]\n",
    "            d[(10)]=d[(10)]+400\n",
    "        else:\n",
    "            for ii in range(1,11):\n",
    "                d[(ii)]=t[ii,jj+1]+t[ii,jj-1]-1.6*t[ii,jj]\n",
    "            d[(10)]=d[(10)]+400\n",
    "        i=1 ; n=10\n",
    "        Beta=[0,b[i]]\n",
    "        Gamma=[0,d[i]/Beta[i]]\n",
    "        i1=i+1\n",
    "        for j in range(i1,n+1):\n",
    "            Beta.append(b[(j)]-a[(j)]*c[(j-1)]/Beta[(j-1)])\n",
    "            Gamma.append((d[(j)]-a[(j)]*Gamma[(j-1)])/Beta[(j)])\n",
    "    \n",
    "        x=zeros(n+1)\n",
    "        x[(n)]=Gamma[(n)] \n",
    "        x[(j)]=Gamma[(j)]-c[(j)]*x[(j)]/Beta[(j)]\n",
    "        for count in range(1,11):\n",
    "            tstar[count,jj]=x[(count)]\n",
    "    \n",
    "    for ii in range(1,11):\n",
    "        if ii==1:\n",
    "            for jj in range(1,11):\n",
    "                d[(jj)]=2*tstar[ii+1,1]-1.6*tstar[ii,1]\n",
    "            d[(10)]=d[(10)]+400\n",
    "        else:\n",
    "            for jj in range(1,11):\n",
    "                d[(jj)]=tstar[ii-1,jj]+tstar[ii+1,jj]-1.6*tstar[ii,jj]\n",
    "                d[(10)]=d[(10)]+400\n",
    "                \n",
    "        i=1 ;  n=10\n",
    "        Beta[(i)]=b[i]\n",
    "        Gamma[(i)]=d[(i)]/Beta[(i)]\n",
    "        i1=i+1\n",
    "        for j in range(i1,n+1):\n",
    "            Beta[(j)]=b[(j)]-a[(j)]*c[(j-1)]/Beta[(j-1)]\n",
    "            Gamma[(j)]=(d[(j)]-a[(j)]*Gamma[(j-1)])/Beta[(j)]\n",
    "        \n",
    "        x[(n)]=Gamma[(n)]\n",
    "        n1=n-i\n",
    "        for k in range(1,n1+1):\n",
    "            j =n-k\n",
    "            x[(j)]=Gamma[(j)]-c[(j)]*x[(j+1)]/Beta[(j)]\n",
    "    \n",
    "        for count in range(1,11):\n",
    "            t[ii,count]=x[count]\n",
    "  \n",
    "\n",
    "print \"the soln by ADI is\"\n",
    "for i in range(1,11):\n",
    "    print t[i,i]\n"
   ]
  }
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