Total field approach

simpegPF/notebooks/Boundary_conditions.ipynb

@@ -26,7 +26,7 @@
        ]
       }
      ],
-     "prompt_number": 13
+     "prompt_number": 2
     },
     {
      "cell_type": "markdown",
@@ -76,7 +76,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 14
+     "prompt_number": 3
     },
     {
      "cell_type": "code",
@@ -89,7 +89,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 15
+     "prompt_number": 4
     },
     {
      "cell_type": "code",
@@ -108,10 +108,10 @@
       {
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 16,
+       "prompt_number": 5,
        "text": [
-        "[<matplotlib.lines.Line2D at 0x355bd10>,\n",
-        " <matplotlib.lines.Line2D at 0x3586810>]"
+        "[<matplotlib.lines.Line2D at 0x4663e10>,\n",
+        " <matplotlib.lines.Line2D at 0x4667050>]"
        ]
       },
       {
@@ -119,11 +119,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3506210>"
+        "<matplotlib.figure.Figure at 0x40a7c90>"
        ]
       }
      ],
-     "prompt_number": 16
+     "prompt_number": 5
     },
     {
      "cell_type": "markdown",
@@ -144,16 +144,13 @@
       "qa = lambda x: -np.sin(x)\n",
       "M.setCellGradBC('neumann')\n",
       "G = M.cellGrad\n",
-      "e = np.ones(n)\n",
-      "e1 = np.zeros(n)\n",
-      "e1[1] = 1.\n",
       "D = M.faceDiv\n",
       "Mf = M.getFaceMass()"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 17
+     "prompt_number": 6
     },
     {
      "cell_type": "code",
@@ -164,12 +161,12 @@
       "bc[0] = -1.\n",
       "bc[n-1] = -1.\n",
       "bc = bc/h\n",
-      "A = D*G"
+      "A = -D*D.T"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 18
+     "prompt_number": 7
     },
     {
      "cell_type": "code",
@@ -182,7 +179,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 19
+     "prompt_number": 8
     },
     {
      "cell_type": "code",
@@ -202,22 +199,22 @@
       {
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 20,
+       "prompt_number": 9,
        "text": [
-        "[<matplotlib.lines.Line2D at 0x4197b50>,\n",
-        " <matplotlib.lines.Line2D at 0x35a7c90>]"
+        "[<matplotlib.lines.Line2D at 0x4a75b90>,\n",
+        " <matplotlib.lines.Line2D at 0x48d6c90>]"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAsUAAAE1CAYAAAAPsZ2cAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x18bd150>"
+        "<matplotlib.figure.Figure at 0x3389190>"
        ]
       }
      ],
-     "prompt_number": 20
+     "prompt_number": 9
     },
     {
      "cell_type": "markdown",
@@ -262,17 +259,9 @@
        "text": [
         "Populating the interactive namespace from numpy and matplotlib\n"
        ]
-      },
-      {
-       "output_type": "stream",
-       "stream": "stderr",
-       "text": [
-        "WARNING: pylab import has clobbered these variables: ['e']\n",
-        "`%pylab --no-import-all` prevents importing * from pylab and numpy\n"
-       ]
       }
      ],
-     "prompt_number": 21
+     "prompt_number": 10
     },
     {
      "cell_type": "markdown",
@@ -287,8 +276,10 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "n = 128\n",
-      "hx = np.ones(n)*np.pi/n\n",
+      "hxind = ((5,25,1.3),(20, 25),(5,25,1.3))\n",
+      "hx = Utils.meshTensors(hxind)\n",
+      "hx = hx/sum(hx)*np.pi\n",
+      "# hx = np.ones(n)*np.pi/n\n",
       "M3 = Mesh.TensorMesh([hx, hx, hx])\n",
       "M3.setCellGradBC('neumann')\n",
       "G = M3.cellGrad\n",
@@ -299,7 +290,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 22
+     "prompt_number": 11
     },
     {
      "cell_type": "code",
@@ -313,13 +304,13 @@
       {
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 23,
+       "prompt_number": 12,
        "text": [
-        "(6340608, 98304)"
+        "(83700, 5400)"
        ]
       }
      ],
-     "prompt_number": 23
+     "prompt_number": 12
     },
     {
      "cell_type": "markdown",
@@ -344,14 +335,14 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "(2113536, 3)\n",
-        "(2113536, 3)\n",
-        "(2113536, 3)\n",
-        "6340608\n"
+        "(27900, 3)\n",
+        "(27900, 3)\n",
+        "(27900, 3)\n",
+        "83700\n"
        ]
       }
      ],
-     "prompt_number": 24
+     "prompt_number": 13
     },
     {
      "cell_type": "code",
@@ -363,7 +354,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 25
+     "prompt_number": 14
     },
     {
      "cell_type": "code",
@@ -374,7 +365,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 26
+     "prompt_number": 15
     },
     {
      "cell_type": "code",
@@ -389,13 +380,42 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3587990>"
+        "<matplotlib.figure.Figure at 0x44a4910>"
        ]
       }
      ],
-     "prompt_number": 27
+     "prompt_number": 16
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "figsize(6,6)\n",
+      "M3.plotGrid()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stderr",
+       "text": [
+        "/usr/lib/pymodules/python2.7/matplotlib/lines.py:483: RuntimeWarning: invalid value encountered in greater_equal\n",
+        "  return np.alltrue(x[1:]-x[0:-1]>=0)\n"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAV0AAAFdCAYAAACgiL63AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x4a976d0>"
+       ]
+      }
+     ],
+     "prompt_number": 17
     },
     {
      "cell_type": "markdown",
@@ -406,11 +426,11 @@
       "$$\\nabla\\cdot\\vec{B} = q$$\n",
       "$$\\nabla\\phi = \\frac{1}{\\mu}\\vec{B}$$\n",
       "\n",
-      "$$\\phi_{true} = sin(x)+sin(y)+sin(z)$$\n",
+      "$$\\phi_{true} = sin(x)-cos(y)-cos(z)$$\n",
       "\n",
-      "$$\\vec{B}_{true}  = cos(x)\\hat{i}+cos(y)\\hat{j}+cos(z)\\hat{k}$$\n",
+      "$$\\vec{B}_{true}  = cos(x)\\hat{i}+sin(y)\\hat{j}+sin(z)\\hat{k}$$\n",
       "\n",
-      "$$\\vec{q}_{true}  = -sin(x)-sin(y)-sin(z)$$\n",
+      "$$\\vec{q}_{true}  = -sin(x)+cos(y)+cos(z)$$\n",
       "\n",
       "In discrete form \n",
       "\n",
@@ -445,29 +465,66 @@
      "collapsed": false,
      "input": [
       "def phitrue(x,y,z):\n",
-      "    phi = np.sin(x)+np.sin(y)+np.sin(z)\n",
+      "    phi = np.sin(x)-np.cos(y)+np.cos(z)\n",
       "    return phi\n",
       "\n",
       "def Btrue(x,y,z):\n",
       "    Bx = np.cos(x)\n",
-      "    By = np.cos(y)\n",
-      "    Bz = np.cos(z)\n",
+      "    By = np.sin(y)\n",
+      "    Bz = np.sin(z)\n",
       "    B = np.r_[Bx, By, Bz]\n",
       "    return B\n",
       "def qtrue(x,y,z):\n",
-      "    q = -np.sin(x)-np.sin(y)-np.sin(z)\n",
+      "    q = -np.sin(x)+np.cos(y)+np.cos(z)\n",
       "    return q"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 28
+     "prompt_number": 18
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "# def phitrue(x,y,z):\n",
+      "#     phi = np.sin(x)+np.sin(y)+np.sin(z)\n",
+      "#     return phi\n",
+      "\n",
+      "# def Btrue(x,y,z):\n",
+      "#     Bx = np.cos(x)\n",
+      "#     By = np.cos(y)\n",
+      "#     Bz = np.cos(z)\n",
+      "#     B = np.r_[Bx, By, Bz]\n",
+      "#     return B\n",
+      "# def qtrue(x,y,z):\n",
+      "#     q = -np.sin(x)-np.sin(y)-np.sin(z)\n",
+      "#     return q"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 19
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Mc = Utils.sdiag(M3.vol) \n",
+      "Mfmu = Utils.sdiag(1/M3.getFaceMass().diagonal())\n",
+      "# A = Mc*D*Mfmu*G*Mc\n",
+      "\n",
+      "A = D*G\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 20
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "A = D*G\n",
       "qa = qtrue(M3.gridCC[:,0], M3.gridCC[:,1], M3.gridCC[:,2])\n",
       "Ba = Btrue(M3.gridFx[:,0], M3.gridFy[:,1], M3.gridFz[:,2])\n",
       "Phia = phitrue(M3.gridCC[:,0], M3.gridCC[:,1], M3.gridCC[:,2])"
@@ -475,7 +532,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 29
+     "prompt_number": 21
     },
     {
      "cell_type": "code",
@@ -487,7 +544,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 30
+     "prompt_number": 22
     },
     {
      "cell_type": "code",
@@ -500,7 +557,24 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 31
+     "prompt_number": 23
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "# Bbcx[indxd] = 1\n",
+      "# Bbcx[indxu] = -1\n",
+      "# Bbcy[indyd] = 1\n",
+      "# Bbcy[indyu] = -1\n",
+      "# Bbcz[indzd] = 1\n",
+      "# Bbcz[indzu] = -1\n",
+      "# Bbc = np.r_[Bbcx, Bbcy, Bbcz]"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 24
     },
     {
      "cell_type": "code",
@@ -508,16 +582,16 @@
      "input": [
       "Bbcx[indxd] = 1\n",
       "Bbcx[indxu] = -1\n",
-      "Bbcy[indyd] = 1\n",
-      "Bbcy[indyu] = -1\n",
-      "Bbcz[indzd] = 1\n",
-      "Bbcz[indzu] = -1\n",
+      "Bbcy[indyd] = 0\n",
+      "Bbcy[indyu] = 0\n",
+      "Bbcz[indzd] = 0\n",
+      "Bbcz[indzu] = 0\n",
       "Bbc = np.r_[Bbcx, Bbcy, Bbcz]"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 32
+     "prompt_number": 25
     },
     {
      "cell_type": "code",
@@ -532,24 +606,24 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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+       "png": "iVBORw0KGgoAAAANSUhEUgAAA5oAAAEHCAYAAADPkjk/AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x454bfd0>"
+        "<matplotlib.figure.Figure at 0x3f27a10>"
        ]
       }
      ],
-     "prompt_number": 33
+     "prompt_number": 26
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "m1 = sp.linalg.interface.aslinearoperator(Utils.sdiag(-1/A.diagonal()))"
+      "# m1 = sp.linalg.interface.aslinearoperator(Utils.sdiag(-1/A.diagonal()))"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 34
+     "prompt_number": 27
     },
     {
      "cell_type": "markdown",
@@ -562,11 +636,58 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "%%time\n",
+      "# %%time\n",
+      "# rhs = Mc*qa-Mc*Dbc*Bbc[ind]\n",
       "rhs = qa-Dbc*Bbc[ind]\n",
       "# Try to use Jacobi preconditioner, but does not work well...\n",
-      "# x, info = sp.linalg.minres(A, rhs, M = m1,tol=1e-6, maxiter = 1000, show=True)\n",
-      "x, info = sp.linalg.minres(A, rhs, tol=1e-6, maxiter = 1000, show=False)"
+      "# x, info = sp.linalg.minres(A, rhs, M = m1,tol=1e-6, maxiter = 2000, show=False)\n",
+      "# x, info = sp.linalg.minres(A, rhs, tol=1e-6, maxiter = 1000)\n",
+      "# x, info = sp.linalg.minres(A, rhs, tol=1e-6, maxiter = 2000, show=False)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 28
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "# rhs = Mc*qa-Mc*Dbc*Bbc[ind]\n",
+      "rhs = qa-Dbc*Bbc[ind]\n",
+      "\n",
+      "import petsc4py\n",
+      "import sys\n",
+      "petsc4py.init(sys.argv)\n",
+      "from petsc4py import PETSc\n",
+      "import PETScIO as IO\n",
+      "Apetsc = PETSc.Mat().createAIJ(size=A.shape,csr=(A.indptr, A.indices, A.data))\n",
+      "bpetsc = IO.arrayToVec(rhs)\n",
+      "xpetsc = IO.arrayToVec(0*rhs)\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 29
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "%%time\n",
+      "ksp = PETSc.KSP().create()\n",
+      "pc = PETSc.PC().create()\n",
+      "ksp.setOperators(Apetsc)\n",
+      "ksp.setType(ksp.Type.GMRES)\n",
+      "pc = ksp.getPC()\n",
+      "pc.setType(pc.Type.SOR)\n",
+      "OptDB = PETSc.Options()\n",
+      "OptDB[\"ksp_rtol\"] = 1e-8\n",
+      "OptDB[\"pc_factor_levels\"] = 1\n",
+      "ksp.setFromOptions()\n",
+      "ksp.view()\n",
+      "ksp.solve(bpetsc, xpetsc)\n",
+      "x = IO.vecToArray(xpetsc)"
      ],
      "language": "python",
      "metadata": {},
@@ -575,12 +696,12 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "CPU times: user 7.73 s, sys: 39.7 ms, total: 7.77 s\n",
-        "Wall time: 7.78 s\n"
+        "CPU times: user 11.1 s, sys: 409 \u00b5s, total: 11.1 s\n",
+        "Wall time: 11.1 s\n"
        ]
       }
      ],
-     "prompt_number": 36
+     "prompt_number": 30
     },
     {
      "cell_type": "markdown",
@@ -604,12 +725,12 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "4.0214495834e-05\n",
-        "0\n"
+        "0.000208241391358\n",
+        "<function info at 0x2c3fd70>\n"
        ]
       }
      ],
-     "prompt_number": 37
+     "prompt_number": 31
     },
     {
      "cell_type": "code",
@@ -624,11 +745,11 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "(2097152, 2097152) (2097152,) (2097152,) (2097152, 98304) (98304,)\n"
+        "(27000, 27000) (27000,) (27000,) (27000, 5400) (5400,)\n"
        ]
       }
      ],
-     "prompt_number": 38
+     "prompt_number": 32
     },
     {
      "cell_type": "code",
@@ -644,21 +765,21 @@
       {
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 39,
+       "prompt_number": 33,
        "text": [
-        "<matplotlib.collections.QuadMesh at 0x557a610>"
+        "<matplotlib.collections.QuadMesh at 0x3f27810>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x456e3d0>"
+        "<matplotlib.figure.Figure at 0x4675b50>"
        ]
       }
      ],
-     "prompt_number": 39
+     "prompt_number": 33
     },
     {
      "cell_type": "code",
@@ -672,19 +793,22 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x357b0d0>"
+        "<matplotlib.figure.Figure at 0x466be10>"
        ]
       }
      ],
-     "prompt_number": 40
+     "prompt_number": 34
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "M3.plotImage(G*x, imageType='F')"
+      "Bnum = G*x\n",
+      "# Bnum = G*x\n",
+      "\n",
+      "M3.plotImage(Bnum, imageType='F')"
      ],
      "language": "python",
      "metadata": {},
@@ -692,13 +816,33 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x4193710>"
+        "<matplotlib.figure.Figure at 0x85fcf90>"
        ]
       }
      ],
-     "prompt_number": 41
+     "prompt_number": 44
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "print Mc.diagonal()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "[ 0.01307576  0.01005828  0.00773714 ...,  0.00773714  0.01005828\n",
+        "  0.01307576]\n"
+       ]
+      }
+     ],
+     "prompt_number": 45
     },
     {
      "cell_type": "markdown",
@@ -711,7 +855,6 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "Bnum = G*x\n",
       "print np.linalg.norm(Ba[~ind]-Bnum[~ind])/np.linalg.norm(Ba[~ind])"
      ],
      "language": "python",
@@ -721,16 +864,17 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "6.64809418708e-06\n"
+        "0.00104365335183\n"
        ]
       }
      ],
-     "prompt_number": 42
+     "prompt_number": 46
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
+      "clf;\n",
       "plot(Ba[~ind]); hold\n",
       "plot(Bnum[~ind], 'r')"
      ],
@@ -740,21 +884,21 @@
       {
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 43,
+       "prompt_number": 47,
        "text": [
-        "[<matplotlib.lines.Line2D at 0x456da10>]"
+        "[<matplotlib.lines.Line2D at 0x6d16110>]"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2b211bd0>"
+        "<matplotlib.figure.Figure at 0x6d16190>"
        ]
       }
      ],
-     "prompt_number": 43
+     "prompt_number": 47
     },
     {
      "cell_type": "markdown",
@@ -776,11 +920,11 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "0.963186495974\n"
+        "1.31230938236\n"
        ]
       }
      ],
-     "prompt_number": 44
+     "prompt_number": 38
     },
     {
      "cell_type": "code",
@@ -795,21 +939,21 @@
       {
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 45,
+       "prompt_number": 39,
        "text": [
-        "[<matplotlib.lines.Line2D at 0x196102d0>]"
+        "[<matplotlib.lines.Line2D at 0x6d23f10>]"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3dfb3d0>"
+        "<matplotlib.figure.Figure at 0x4cc4650>"
        ]
       }
      ],
-     "prompt_number": 45
+     "prompt_number": 39
     },
     {
      "cell_type": "markdown",

simpegPF/notebooks/MagForward_total.ipynb

@@ -0,0 +1,662 @@
+{
+ "metadata": {
+  "name": ""
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+  {
+   "cells": [
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from SimPEG import *\n",
+      "from test_boundary import spheremodel, MagSphereAnalFun\n",
+      "%pylab inline"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "Populating the interactive namespace from numpy and matplotlib\n"
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stderr",
+       "text": [
+        "WARNING: pylab import has clobbered these variables: ['axes', 'info', 'flag']\n",
+        "`%pylab --no-import-all` prevents importing * from pylab and numpy\n"
+       ]
+      }
+     ],
+     "prompt_number": 29
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "### Step:1 Generating mesh and operators"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "hxind = ((10,25,1.3),(41, 25),(10,25,1.3))\n",
+      "hyind = ((10,25,1.3),(41, 25),(10,25,1.3))\n",
+      "hzind = ((10,25,1.3),(40, 25),(10,25,1.3))\n",
+      "hx, hy, hz = Utils.meshTensors(hxind, hyind, hzind)\n",
+      "M3 = Mesh.TensorMesh([hx, hy, hz], [-sum(hx)/2,-sum(hy)/2,-sum(hz)/2])"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 30
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "### Step2: Compute Boundary indicies and set $\\mathbf{B}_{bc}$"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "BC = [['neumann', 'neumann'], ['neumann', 'neumann'], ['neumann', 'neumann']]\n",
+      "indxd, indxu, indyd, indyu, indzd, indzu = M3.faceBoundaryInd\n",
+      "ind = np.r_[(indxd | indxu), (indyd | indyu), (indzd | indzu)]"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 31
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Box = 1 # Primary field in x-direction (background)\n",
+      "Boy = 0 # Primary field in y-direction (background)\n",
+      "Boz = 0 # Primary field in z-direction (background)\n",
+      "\n",
+      "Bbcx = np.zeros(np.prod(M3.nFx))\n",
+      "Bbcy = np.zeros(np.prod(M3.nFy))\n",
+      "Bbcz = np.zeros(np.prod(M3.nFz))\n",
+      "\n",
+      "Bbcx[indxd] = Box\n",
+      "Bbcx[indxu] = Box\n",
+      "Bbcy[indyd] = Boy\n",
+      "Bbcy[indyu] = Boy\n",
+      "Bbcz[indzd] = Boz\n",
+      "Bbcz[indzu] = Boz\n",
+      "\n",
+      "Bbc = np.r_[Bbcx, Bbcy, Bbcz]"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 32
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "### Step3: Generating model $\\mu$\n",
+      "\n",
+      "$\\mu = \\mu_0(1+\\chi)$"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "mu0 = 4*np.pi*1e-7\n",
+      "chibkg = 0.\n",
+      "chiblk = 0.01\n",
+      "chi = np.ones(M3.nC)*chibkg"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 33
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "sph_ind = spheremodel(M3, 0, 0, 0, 100)\n",
+      "chi[sph_ind] = chiblk\n",
+      "mu = (1.+chi)*mu0"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 34
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "figsize(10,10)\n",
+      "M3.plotGrid()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0xa514ed0>"
+       ]
+      }
+     ],
+     "prompt_number": 35
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "M3.plotImage(np.log10(mu), imageType='CC')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 36,
+       "text": [
+        "<matplotlib.collections.QuadMesh at 0xc576050>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0xc9469d0>"
+       ]
+      }
+     ],
+     "prompt_number": 36
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "### Step4: get system matrix $\\mathbf{A}$ and right handside\n",
+      "#### 4.1 Backgrounds: Weak formulation for Poisson equation with cell-centered system\n",
+      "\n",
+      "$$\\nabla \\phi = \\frac{1}{\\mu}\\mathbf{B}$$\n",
+      "\n",
+      "$$\\nabla \\cdot \\mathbf{B} = q $$\n",
+      "\n",
+      "\n",
+      "$$ (\\phi)_{\\partial\\Omega} = \\phi_{BC}$$\n",
+      "or\n",
+      "$$ (\\mathbf{B}\\cdot{\\bar{\\mathbf{n}}})_{\\partial\\Omega} = B_{BC}$$\n",
+      "\n",
+      "In discretized form we have \n",
+      "\n",
+      "$$\\mathbf{diag(v)}\\mathbf{D}\\mathbf{P}_{in}^T \\mathbf{P}_{in}\\mathbf{B} +  \\mathbf{diag(v)}\\mathbf{D} \\mathbf{P}_{out}^T B_{BC}= \\mathbf{diag(v)}q $$\n",
+      "\n",
+      "$$\\mathbf{M}^f_{\\frac{1}{\\mu}}\\mathbf{B} = -\\mathbf{P}_{in}\\mathbf{P}_{in}^T\\mathbf{D}^T\\mathbf{diag(v)}\\phi + \\mathbf{P}_{BC}\\phi_{BC}$$\n",
+      "\n",
+      "$$\\mathbf{B} = (\\mathbf{M}^f_{\\frac{1}{\\mu}})^{-1}(-\\mathbf{P}_{in}\\mathbf{P}_{in}^T\\mathbf{D}\\mathbf{diag(v)}\\phi +\\mathbf{P}_{BC}\\phi_{BC})$$\n",
+      "\n",
+      "$$\\mathbf{diag(v)}\\mathbf{D}\\mathbf{P}_{in}^T \\mathbf{P}_{in} (\\mathbf{M}^f_{\\frac{1}{\\mu}})^{-1}(-\\mathbf{P}_{in}\\mathbf{P}_{in}^T\\mathbf{D}\\mathbf{diag(v)}\\phi +\\mathbf{P}_{BC}\\phi_{BC}) +  \\mathbf{diag(v)}\\mathbf{D} \\mathbf{P}_{out}^T B_{BC}= \\mathbf{diag(v)}q $$\n",
+      "\n",
+      "$$-\\mathbf{diag(v)}\\mathbf{D}\\mathbf{P}_{in}^T \\mathbf{P}_{in} (\\mathbf{M}^f_{\\frac{1}{\\mu}})^{-1}\\mathbf{P}_{in}\\mathbf{P}_{in}^T\\mathbf{D}\\mathbf{diag(v)}\\phi  = \\mathbf{diag(v)}q -\\mathbf{diag(v)}\\mathbf{D}\\mathbf{P}_{in}^T \\mathbf{P}_{in} (\\mathbf{M}^f_{\\frac{1}{\\mu}})^{-1}\\mathbf{P}_{BC}\\phi_{BC} - \\mathbf{diag(v)}\\mathbf{D} \\mathbf{P}_{out}^T B_{BC}$$\n",
+      "\n",
+      "$$\\mathbf{A} = -\\mathbf{diag(v)}\\mathbf{D}\\mathbf{P}_{in}^T \\mathbf{P}_{in} (\\mathbf{M}^f_{\\frac{1}{\\mu}})^{-1}\\mathbf{P}_{in}\\mathbf{P}_{in}^T\\mathbf{D}\\mathbf{diag(v)}$$\n",
+      "\n",
+      "$$\\mathbf{rhs} = \\mathbf{diag(v)}q -\\mathbf{diag(v)}\\mathbf{D}\\mathbf{P}_{in}^T \\mathbf{P}_{in} (\\mathbf{M}^f_{\\frac{1}{\\mu}})^{-1}\\mathbf{P}_{BC}\\phi_{BC} - \\mathbf{diag(v)}\\mathbf{D} \\mathbf{P}_{out}^T B_{BC} $$\n",
+      "\n",
+      "In magnetostatic case we do not have $\\mathbf{q}$ and $\\phi_{BC}$ and with\n",
+      "\n",
+      "$$ \\mathbf{Div} = \\mathbf{diag(v)}\\mathbf{D}\\mathbf{P}_{in}^T \\mathbf{P}_{in}$$\n",
+      "\n",
+      "$$\\mathbf{A} = - D(\\mathbf{M}^f_{\\frac{1}{\\mu}})^{-1}D^{T}$$\n",
+      "\n",
+      "$$\\mathbf{rhs} = - \\mathbf{diag(v)}\\mathbf{D} \\mathbf{P}_{out}^T B_{BC} $$\n",
+      "\n",
+      "$$\\mathbf{B} = (\\mathbf{M}^f_{\\frac{1}{\\mu}})^{-1}(-\\mathbf{Div}^{T}\\phi)$$\n",
+      "\n",
+      "### Things to remember\n",
+      "\n",
+      "- $\\mathbf{P}_{BC}$ contains $\\mathbf{diag}(\\mathbf{area}_{BC})$ and a weired projection matrix, which has some +1 and -1 ....\n",
+      "- Forward differential problem in weak form does not work well.., although inverse works really well\n",
+      "- Mesh.getFaceInnerProduct or Mesh.getFaceMass should be used since we integrate $(\\frac{1}{\\mu}\\mathbf{B}, \\mathbf{F})$ in the cell, which means we need averaging from faces to cell centers\n",
+      "- Weak formulation is a fancy representation of Finite Volume Approach. In terms of final system matrix \\mathbf{A} is going to be exactly same, I guess.."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Dface = M3.faceDiv\n",
+      "Pbc,Pin, Pout = M3.getBCProjWF(BC, discretization='CC')\n",
+      "Mc = Utils.sdiag(M3.vol)\n",
+      "D = Mc*Dface*Pin.T*Pin\n",
+      "MfmuI = Utils.sdiag(1/M3.getFaceInnerProduct(mu = 1/mu).diagonal())\n",
+      "A = -D*MfmuI*D.T\n",
+      "rhs = -Mc*Dface*Pout.T*Bbc[ind]"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 37
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "import petsc4py\n",
+      "import sys\n",
+      "from sys import getrefcount\n",
+      "petsc4py.init(sys.argv)\n",
+      "from petsc4py import PETSc\n",
+      "import PETScIO as IO\n",
+      "Apetsc = PETSc.Mat().createAIJ(size=A.shape,csr=(A.indptr, A.indices, A.data))\n",
+      "bpetsc = IO.arrayToVec(rhs)\n",
+      "xpetsc = IO.arrayToVec(0*rhs)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 38
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "# u1 = PETSc.Vec().createSeq(M3.nC)\n",
+      "# u1.set(1)\n",
+      "# u1.normalize()\n",
+      "# basis = [u1]\n",
+      "# nullsp = PETSc.NullSpace().create(False, basis, comm=PETSc.COMM_SELF)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 39
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "%%time\n",
+      "ksp = PETSc.KSP().create()\n",
+      "pc = PETSc.PC().create()\n",
+      "ksp.setOperators(Apetsc)\n",
+      "ksp.setType(ksp.Type.BCGS)\n",
+      "pc = ksp.getPC()\n",
+      "pc.setType(pc.Type.SOR)\n",
+      "OptDB = PETSc.Options()\n",
+      "OptDB[\"ksp_rtol\"] = 1e-8\n",
+      "OptDB[\"pc_factor_levels\"] = 1\n",
+      "ksp.setFromOptions()\n",
+      "ksp.view()\n",
+      "ksp.solve(bpetsc, xpetsc)\n",
+      "print ksp.its# print ksp.its\n",
+      "phi = IO.vecToArray(xpetsc)\n",
+      "print np.linalg.norm(A*phi-rhs)/norm(rhs)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "116\n",
+        "4.24430002044e-09\n",
+        "CPU times: user 1.65 s, sys: 0 ns, total: 1.65 s\n",
+        "Wall time: 1.66 s\n"
+       ]
+      }
+     ],
+     "prompt_number": 40
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "%%time\n",
+      "m1 = sp.linalg.interface.aslinearoperator(Utils.sdiag(-1/A.diagonal()))\n",
+      "phi, info = sp.linalg.bicgstab(A, rhs, tol=1e-6, maxiter = 1000, M =m1)\n",
+      "# m1 = sp.linalg.interface.aslinearoperator(Utils.sdiag(np.sqrt(1/(Utils.mkvc(-A.diagonal())))))\n",
+      "# m2 = m1\n",
+      "# phi, info = sp.linalg.qmr(A, rhs, tol = 1e-5, M1 = m1, M2=m2)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "CPU times: user 2.13 s, sys: 0 ns, total: 2.13 s\n",
+        "Wall time: 2.13 s\n"
+       ]
+      }
+     ],
+     "prompt_number": 41
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "print info\n",
+      "print M3.nC"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "0\n",
+        "223260\n"
+       ]
+      }
+     ],
+     "prompt_number": 42
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "B = MfmuI*(-D.T*phi)\n",
+      "rhsa = A*phi\n",
+      "print info\n",
+      "print np.linalg.norm(rhs-rhsa)/np.linalg.norm(rhs)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "0\n",
+        "9.47100435444e-07\n"
+       ]
+      }
+     ],
+     "prompt_number": 43
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "#### BICG, qmr or BICGstab with Jacobi preconditioner works well (scipy), minres does not work ... Need to play with some other iterative solvers. Remeber that our system has null-space, which is constant"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "figsize(16,5)\n",
+      "M3.plotImage(B, imageType='F')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0xc4624d0>"
+       ]
+      }
+     ],
+     "prompt_number": 44
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "### Step4: Compute analytic function (sphere in whole space)\n",
+      "Outside of the sphere $(r>R)$\n",
+      "\n",
+      "$$\\mathbf{H}_1 = H_0 \\hat{x} + H_0\\frac{R}{r^5}\\frac{\\mu_2-\\mu_1}{\\mu_2+2\\mu_1}[(2x^2-y^2-z^2)\\hat{x}+(3xy)\\hat{y}+(3xz)\\hat{z}]$$\n",
+      "\n",
+      "$$H_{x1} = H_0 + H_0\\frac{R}{r^5}\\frac{\\mu_2-\\mu_1}{\\mu_2+2\\mu_1}(2x^2-y^2-z^2)$$\n",
+      "\n",
+      "$$H_{y1} =  H_0\\frac{R}{r^5}\\frac{\\mu_2-\\mu_1}{\\mu_2+2\\mu_1}(3xy)$$\n",
+      "\n",
+      "$$H_{z1} =  H_0\\frac{R}{r^5}\\frac{\\mu_2-\\mu_1}{\\mu_2+2\\mu_1}(3xz)$$\n",
+      "\n",
+      "Inside of the sphere $(r\\le R)$\n",
+      "\n",
+      "$$\\mathbf{H}_2 = H_0\\frac{3\\mu_1}{\\mu_2+2\\mu_1}\\hat{x}$$\n",
+      "\n",
+      "$$H_{x2} = H_0\\frac{3\\mu_1}{\\mu_2+2\\mu_1}$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "### Step5: Projection to receiver plane"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "xr = np.linspace(-400, 400, 41)\n",
+      "yr = np.linspace(-400, 400, 41)\n",
+      "X, Y = np.meshgrid(xr, yr)\n",
+      "Z = np.ones((size(xr), size(yr)))*50"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 45
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "rxLoc = np.c_[Utils.mkvc(X), Utils.mkvc(Y), Utils.mkvc(Z)]\n",
+      "Qfx = M3.getInterpolationMat(rxLoc,'Fx')\n",
+      "Qfy = M3.getInterpolationMat(rxLoc,'Fy')\n",
+      "Qfz = M3.getInterpolationMat(rxLoc,'Fz')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 46
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Bxr = np.reshape(Qfx*B, (size(xr), size(yr)), order='F')\n",
+      "Byr = np.reshape(Qfy*B, (size(xr), size(yr)), order='F')\n",
+      "Bzr = np.reshape(Qfz*B, (size(xr), size(yr)), order='F')\n",
+      "H0 = Box/mu0\n",
+      "flag = 'secondary'\n",
+      "if flag=='secondary':\n",
+      "    Bxr = Bxr-Box\n",
+      "\n",
+      "Bxra, Byra, Bzra = MagSphereAnalFun(X, Y, Z, 100, 0., 0., 0., mu0, mu0*(1+chiblk), H0, flag)\n",
+      "\n",
+      "Bxra = np.reshape(Bxra, (size(xr), size(yr)), order='F')\n",
+      "Byra = np.reshape(Byra, (size(xr), size(yr)), order='F')\n",
+      "Bzra = np.reshape(Bzra, (size(xr), size(yr)), order='F')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 47
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "figsize(10, 4)\n",
+      "plot(Utils.mkvc(Bxra))\n",
+      "plot(Utils.mkvc(Bxr), 'k:')\n",
+      "plot(Utils.mkvc(Byra))\n",
+      "plot(Utils.mkvc(Byr), 'k:')\n",
+      "plot(Utils.mkvc(Bzra))\n",
+      "plot(Utils.mkvc(Bzr), 'k:')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 48,
+       "text": [
+        "[<matplotlib.lines.Line2D at 0xc44e810>]"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAm8AAAD9CAYAAADqB5DpAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0xa52a7d0>"
+       ]
+      }
+     ],
+     "prompt_number": 48
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "fig, ax = subplots(3,2, figsize = (10,15))\n",
+      "dat1 = ax[0,0].imshow(Bxr); fig.colorbar(dat1, ax=ax[0,0])\n",
+      "dat2 = ax[0,1].imshow(Bxra); fig.colorbar(dat2, ax=ax[0,1])\n",
+      "dat3 = ax[1,0].imshow(Byr); fig.colorbar(dat3, ax=ax[1,0])\n",
+      "dat4 = ax[1,1].imshow(Byra); fig.colorbar(dat4, ax=ax[1,1])\n",
+      "dat5 = ax[2,0].imshow(Bzr); fig.colorbar(dat5, ax=ax[2,0])\n",
+      "dat6 = ax[2,1].imshow(Bzra); fig.colorbar(dat6, ax=ax[2,1])"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 49,
+       "text": [
+        "<matplotlib.colorbar.Colorbar instance at 0x47e1518>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAl8AAANWCAYAAADAx+lhAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0xa52a950>"
+       ]
+      }
+     ],
+     "prompt_number": 49
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "fig, axes = subplots(1,3, figsize=(14,5))\n",
+      "epsx = np.linalg.norm(Utils.mkvc(Bxr))*1e-6\n",
+      "epsy = np.linalg.norm(Utils.mkvc(Byr))*1e-6\n",
+      "epsz = np.linalg.norm(Utils.mkvc(Bzr))*1e-6\n",
+      "axes[0].plot(Y[:,1], Bxra[:,1], 'b', Y[:,1], Bxr[:,1], 'r.')\n",
+      "axes[1].plot(Y[:,1], Byra[:,1], 'b', Y[:,1], Byr[:,1], 'r.')\n",
+      "axes[2].plot(Y[:,1], Bzra[:,1], 'b', Y[:,1], Bzr[:,1], 'r.')\n",
+      "print X[:,1]"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "[-380. -380. -380. -380. -380. -380. -380. -380. -380. -380. -380. -380.\n",
+        " -380. -380. -380. -380. -380. -380. -380. -380. -380. -380. -380. -380.\n",
+        " -380. -380. -380. -380. -380. -380. -380. -380. -380. -380. -380. -380.\n",
+        " -380. -380. -380. -380. -380.]\n"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAA1gAAAE1CAYAAAD6akEFAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x152faf50>"
+       ]
+      }
+     ],
+     "prompt_number": 50
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "fig, axes = subplots(1,3, figsize=(14,5))\n",
+      "epsx = np.linalg.norm(Utils.mkvc(Bxr))*1e-6\n",
+      "epsy = np.linalg.norm(Utils.mkvc(Byr))*1e-6\n",
+      "epsz = np.linalg.norm(Utils.mkvc(Bzr))*1e-6\n",
+      "axes[0].plot(Y[:,1], abs((Bxr[:,1]-Bxra[:,1])/(Bxra[:,1]+epsx)), 'r.')\n",
+      "axes[1].plot(Y[:,1], abs((Byr[:,1]-Byra[:,1])/(Byra[:,1]+epsy)), 'r.')\n",
+      "axes[2].plot(Y[:,1], abs((Bzr[:,1]-Bzra[:,1])/(Bzra[:,1]+epsz)), 'r.')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 51,
+       "text": [
+        "[<matplotlib.lines.Line2D at 0xf0bfc50>]"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x15309ad0>"
+       ]
+      }
+     ],
+     "prompt_number": 51
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "### Thoughts\n",
+      "\n",
+      "- It works well with non-uniform mesh!!\n",
+      "- Actual accuray is ~10% relative error in secondary fields. As we pad more we can get better accuracy, since we did not consider secondary field at the boudnary ($\\partial\\Omega$). \n",
+      "- Here, we can try primary secondary field approach to get better accuracy. \n",
+      "- In addition, we can use the congruous sphere method to handle this secondary fields at boundaries"
+     ]
+    }
+   ],
+   "metadata": {}
+  }
+ ]
+}

simpegPF/notebooks/test_boundary.py

@@ -1,78 +1,142 @@
 
 from SimPEG.Utils.sputils import kron3, speye, sdiag
+from SimPEG import *
 import numpy as np
 import scipy.sparse as sp
 def ddxFaceDivBC(n, bc):
 
-  ij   = (np.array([0, n-1]),np.array([0, 1]))
-  vals = np.zeros(2)
+    ij   = (np.array([0, n-1]),np.array([0, 1]))
+    vals = np.zeros(2)
 
-  # Set the first side
-  if(bc[0] == 'dirichlet'):
-    vals[0] = 0
-  elif(bc[0] == 'neumann'):
-    vals[0] = -1
-  # Set the second side
-  if(bc[1] == 'dirichlet'):
-    vals[1] = 0
-  elif(bc[1] == 'neumann'):
-    vals[1] = 1
-  D = sp.csr_matrix((vals, ij), shape=(n,2))
-  return D
+    # Set the first side
+    if(bc[0] == 'dirichlet'):
+        vals[0] = 0
+    elif(bc[0] == 'neumann'):
+        vals[0] = -1
+    # Set the second side
+    if(bc[1] == 'dirichlet'):
+        vals[1] = 0
+    elif(bc[1] == 'neumann'):
+        vals[1] = 1
+    D = sp.csr_matrix((vals, ij), shape=(n,2))
+    return D
 
 
 def faceDivBC(mesh, BC, ind):
-  """
-  The facd divergence boundary condtion matrix
+    """
+    The facd divergence boundary condtion matrix
 
-  """
-  # The number of cell centers in each direction
-  n = mesh.nCv
-  # Compute faceDivergence operator on faces
-  if(mesh.dim == 1):
-    D = ddxFaceDivBC(n[0], BC[0])
-  elif(mesh.dim == 2):
-    D1 = sp.kron(speye(n[1]), ddxFaceDivBC(n[0]), BC[0])
-    D2 = sp.kron(ddxFaceDivBC(n[1], BC[1]), speye(n[0]))
-    D = sp.hstack((D1, D2), format="csr")
-  elif(mesh.dim == 3):
-    D1 = kron3(speye(n[2]), speye(n[1]), ddxFaceDivBC(n[0], BC[0]))
-    D2 = kron3(speye(n[2]), ddxFaceDivBC(n[1], BC[1]), speye(n[0]))
-    D3 = kron3(ddxFaceDivBC(n[2], BC[2]), speye(n[1]), speye(n[0]))
-  D = sp.hstack((D1, D2, D3), format="csr")
-  # Compute areas of cell faces & volumes
-  S = mesh.area[ind]
-  V = mesh.vol
-  mesh._faceDiv = sdiag(1/V)*D*sdiag(S)
+    .. math::
 
-  return mesh._faceDiv
+
+
+    """
+    # The number of cell centers in each direction
+    n = mesh.nCv
+    # Compute faceDivergence operator on faces
+    if(mesh.dim == 1):
+        D = ddxFaceDivBC(n[0], BC[0])
+    elif(mesh.dim == 2):
+        D1 = sp.kron(speye(n[1]), ddxFaceDivBC(n[0]), BC[0])
+        D2 = sp.kron(ddxFaceDivBC(n[1], BC[1]), speye(n[0]))
+        D = sp.hstack((D1, D2), format="csr")
+    elif(mesh.dim == 3):
+        D1 = kron3(speye(n[2]), speye(n[1]), ddxFaceDivBC(n[0], BC[0]))
+        D2 = kron3(speye(n[2]), ddxFaceDivBC(n[1], BC[1]), speye(n[0]))
+        D3 = kron3(ddxFaceDivBC(n[2], BC[2]), speye(n[1]), speye(n[0]))
+    D = sp.hstack((D1, D2, D3), format="csr")
+    # Compute areas of cell faces & volumes
+    S = mesh.area[ind]
+    V = mesh.vol
+    mesh._faceDiv = sdiag(1/V)*D*sdiag(S)
+
+    return mesh._faceDiv
 
 
 def faceBCind(mesh):
-  """
-  Find indices of boundary faces in each direction
+    """
+    Find indices of boundary faces in each direction
 
-  """
-  if(mesh.dim==1):
-    indxd = (mesh.gridFx[:,0]==min(mesh.gridFx[:,0]))
-    indxu = (mesh.gridFx[:,0]==max(mesh.gridFx[:,0]))
-    return indxd, indxu
-  elif(mesh.dim==1):
-    indxd = (mesh.gridFx[:,0]==min(mesh.gridFx[:,0]))
-    indxu = (mesh.gridFx[:,0]==max(mesh.gridFx[:,0]))
-    indyd = (mesh.gridFy[:,1]==min(mesh.gridFy[:,1]))
-    indyu = (mesh.gridFy[:,1]==max(mesh.gridFy[:,1]))
-    return indxd, indxu, indyd, indyu
-  elif(mesh.dim==3):
-    indxd = (mesh.gridFx[:,0]==min(mesh.gridFx[:,0]))
-    indxu = (mesh.gridFx[:,0]==max(mesh.gridFx[:,0]))
-    indyd = (mesh.gridFy[:,1]==min(mesh.gridFy[:,1]))
-    indyu = (mesh.gridFy[:,1]==max(mesh.gridFy[:,1]))
-    indzd = (mesh.gridFz[:,2]==min(mesh.gridFz[:,2]))
-    indzu = (mesh.gridFz[:,2]==max(mesh.gridFz[:,2]))
-    return indxd, indxu, indyd, indyu, indzd, indzu
+    """
+    if(mesh.dim==1):
+        indxd = (mesh.gridFx[:,0]==min(mesh.gridFx[:,0]))
+        indxu = (mesh.gridFx[:,0]==max(mesh.gridFx[:,0]))
+        return indxd, indxu
+    elif(mesh.dim==1):
+        indxd = (mesh.gridFx[:,0]==min(mesh.gridFx[:,0]))
+        indxu = (mesh.gridFx[:,0]==max(mesh.gridFx[:,0]))
+        indyd = (mesh.gridFy[:,1]==min(mesh.gridFy[:,1]))
+        indyu = (mesh.gridFy[:,1]==max(mesh.gridFy[:,1]))
+        return indxd, indxu, indyd, indyu
+    elif(mesh.dim==3):
+        indxd = (mesh.gridFx[:,0]==min(mesh.gridFx[:,0]))
+        indxu = (mesh.gridFx[:,0]==max(mesh.gridFx[:,0]))
+        indyd = (mesh.gridFy[:,1]==min(mesh.gridFy[:,1]))
+        indyu = (mesh.gridFy[:,1]==max(mesh.gridFy[:,1]))
+        indzd = (mesh.gridFz[:,2]==min(mesh.gridFz[:,2]))
+        indzu = (mesh.gridFz[:,2]==max(mesh.gridFz[:,2]))
+        return indxd, indxu, indyd, indyu, indzd, indzu
 
+
+def spheremodel(mesh, x0, y0, z0, r):
+    """
+        Generate model indicies for sphere
+        - (x0, y0, z0 ): is the center location of sphere
+        - r: is the radius of the sphere
+        - it returns logical indicies of cell-center model
+    """
+    ind = np.sqrt((mesh.gridCC[:,0]-x0)**2+(mesh.gridCC[:,1]-y0)**2+(mesh.gridCC[:,2]-z0)**2 ) < r
+    return ind
   
 
 
-    
+def MagSphereAnalFun(x, y, z, R, x0, y0, z0, mu1, mu2, H0, flag):
+    """ 
+        Analytic function for Magnetics problem. The set up here is 
+        magnetic sphere in whole-space.
+        - (x0,y0,z0)
+        - (x0, y0, z0 ): is the center location of sphere
+        - r: is the radius of the sphere
+
+    .. math::
+    
+        \mathbf{H}^p = H_0\hat{x}
+
+                
+    """
+    if (~np.size(x)==np.size(y)==np.size(z)):
+        print "Specify same size of x, y, z"
+        return
+    dim = x.shape
+    x = Utils.mkvc(x)
+    y = Utils.mkvc(y)
+    z = Utils.mkvc(z)
+    
+    ind = np.sqrt((x-x0)**2+(y-y0)**2+(z-z0)**2 ) < R
+    r = Utils.mkvc(np.sqrt((x-x0)**2+(y-y0)**2+(z-z0)**2 ))
+    Bx = np.zeros(x.size)
+    By = np.zeros(x.size)
+    Bz = np.zeros(x.size)
+
+    # Inside of the sphere
+    rf2 = 3*mu1/(mu2+2*mu1)
+    if (flag == 'total'):
+        Bx[ind] = mu2*H0*(rf2)
+    elif (flag == 'secondary'):
+        Bx[ind] = mu2*H0*(rf2)-mu1*H0
+        
+    By[ind] = 0.
+    Bz[ind] = 0.
+    # Outside of the sphere
+    rf1 = (mu2-mu1)/(mu2+2*mu1)
+    if (flag == 'total'):
+        Bx[~ind] = mu1*(H0+H0/r[~ind]**5*(R**3)*rf1*(2*x[~ind]**2-y[~ind]**2-z[~ind]**2))
+    elif (flag == 'secondary'):
+        Bx[~ind] = mu1*(H0/r[~ind]**5*(R**3)*rf1*(2*x[~ind]**2-y[~ind]**2-z[~ind]**2))
+    
+    By[~ind] = mu1*(H0/r[~ind]**5*(R**3)*rf1*(3*x[~ind]*y[~ind]))
+    Bz[~ind] = mu1*(H0/r[~ind]**5*(R**3)*rf1*(3*x[~ind]*z[~ind]))
+    
+    return np.reshape(Bx, x.shape, order='F'), np.reshape(By, x.shape, order='F'), np.reshape(Bz, x.shape, order='F')
+    
+