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Fix import bug of sys
This commit is contained in:
GudniRos
@@ -0,0 +1,95 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The Maxwell's equations are\n",
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"\n",
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"\\begin{align}\n",
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"\\frac{\\partial \\vec{B}}{\\partial t} &= - \\nabla \\times \\vec{E} \\hspace{2cm} \\text{Faraday equations}\\\\\n",
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"\\frac{\\partial \\vec{D}}{\\partial t} &= - \\nabla \\times \\vec{H} - \\vec{J} - \\vec{s} \\hspace{2cm} \\text{Ampere's law.}\n",
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"\\end{align}\n",
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"\n",
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"where $\\vec{E}$ is the electric field, $\\vec{H}$ is the magnetic field, $\\vec{J}$ is the electric flux, $\\vec{D}$ is the displacement flux, $\\vec{B}$ is the magnetic flux and $\\vec{s}$ is the source."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The constitutive realtations between the fields and flux are as follows.\n",
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"\n",
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"\\begin{align}\n",
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"\\vec{J} &= \\sigma \\vec{E}\\\\\n",
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"\\vec{B} &= \\mu \\vec{H}\\\\\n",
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"\\vec{D} &= \\epsilon \\vec{E}\n",
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"\\end{align}"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Using the contitutive relations, the Maxwell's equations can written in terms of the electric field and the magnetic flux \n",
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"\\begin{align}\n",
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"\\mu \\frac{\\partial \\vec{B}}{\\partial t} &= - \\nabla \\times \\vec{E} \\\\\n",
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"\\epsilon \\frac{\\partial \\vec{E}}{\\partial t} &= - \\nabla \\times \\frac{1}{\\mu} \\vec{B} - \\sigma \\vec{E} - \\vec{s} \n",
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"\\end{align}\n",
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"\n",
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"Futher by droping the $\\epsilon$ term and using a $e^{-i\\omega t}$ Fourier time relation, Maxwell's equations can be written in the frequency domain as\n",
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"\n",
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"\\begin{align}\n",
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"-i \\omega \\mu \\hat{\\vec{B}} &= - \\nabla \\times \\hat{\\vec{E}} \\\\\n",
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"- \\vec{s} &= - \\nabla \\times \\frac{1}{\\mu} \\hat{\\vec{B}} - \\sigma \\hat{\\vec{E}}\n",
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"\\end{align}\n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"\n",
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"\n",
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"The weak form is \n",
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"\\begin{align}\n",
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"-i \\omega \\mu (\\hat{\\vec{B}},\\hat{\\vec{F}}) + (\\nabla \\times \\hat{\\vec{E}},\\hat{\\vec{F}}) &= 0 \\\\\n",
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"(\\nabla \\times \\mu^{-1} \\hat{\\vec{B}},\\hat{\\vec{W}}) + (\\sigma \\hat{\\vec{E}},\\hat{\\vec{W}}) &= (\\hat{\\vec{s}},\\hat{\\vec{W}})\n",
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"\\end{align}\n",
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"\n",
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"\n",
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"$ (\\mu^{-1} B, \\nabla \\times W) + (\\sigma E, W) = (\\mu^{-1}B W )|_0^{end} $\n",
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"\n",
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"$ (i \\omega B,F) + (\\nabla \\times E , f) = 0 $"
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]
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@@ -0,0 +1,95 @@
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The Maxwell's equations are\n",
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"\n",
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"\\begin{align}\n",
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"\\frac{\\partial \\vec{B}}{\\partial t} &= - \\nabla \\times \\vec{E} \\hspace{2cm} \\text{Faraday equations}\\\\\n",
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"\\frac{\\partial \\vec{D}}{\\partial t} &= - \\nabla \\times \\vec{H} - \\vec{J} - \\vec{s} \\hspace{2cm} \\text{Ampere's law.}\n",
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"\\end{align}\n",
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"\n",
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"where $\\vec{E}$ is the electric field, $\\vec{H}$ is the magnetic field, $\\vec{J}$ is the electric flux, $\\vec{D}$ is the displacement flux, $\\vec{B}$ is the magnetic flux and $\\vec{s}$ is the source."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The constitutive realtations between the fields and flux are as follows.\n",
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"\n",
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"\\begin{align}\n",
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"\\vec{J} &= \\sigma \\vec{E}\\\\\n",
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"\\vec{B} &= \\mu \\vec{H}\\\\\n",
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"\\vec{D} &= \\epsilon \\vec{E}\n",
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"\\end{align}"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Using the contitutive relations, the Maxwell's equations can written in terms of the electric field and the magnetic flux \n",
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"\\begin{align}\n",
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"\\mu \\frac{\\partial \\vec{B}}{\\partial t} &= - \\nabla \\times \\vec{E} \\\\\n",
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"\\epsilon \\frac{\\partial \\vec{E}}{\\partial t} &= - \\nabla \\times \\frac{1}{\\mu} \\vec{B} - \\sigma \\vec{E} - \\vec{s} \n",
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"\\end{align}\n",
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"\n",
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"Futher by droping the $\\epsilon$ term and using a $e^{-i\\omega t}$ Fourier time relation, Maxwell's equations can be written in the frequency domain as\n",
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"\n",
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"\\begin{align}\n",
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"-i \\omega \\mu \\hat{\\vec{B}} &= - \\nabla \\times \\hat{\\vec{E}} \\\\\n",
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"- \\vec{s} &= - \\nabla \\times \\frac{1}{\\mu} \\hat{\\vec{B}} - \\sigma \\hat{\\vec{E}}\n",
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"\\end{align}\n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"\n",
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"\n",
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"The weak form is \n",
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"\\begin{align}\n",
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"-i \\omega \\mu (\\hat{\\vec{B}},\\hat{\\vec{F}}) + (\\nabla \\times \\hat{\\vec{E}},\\hat{\\vec{F}}) &= 0 \\\\\n",
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"(\\nabla \\times \\mu^{-1} \\hat{\\vec{B}},\\hat{\\vec{W}}) + (\\sigma \\hat{\\vec{E}},\\hat{\\vec{W}}) &= (\\hat{\\vec{s}},\\hat{\\vec{W}})\n",
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"\\end{align}\n",
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"\n",
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"\n",
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"$ (\\mu^{-1} B, \\nabla \\times W) + (\\sigma E, W) = (\\mu^{-1}B W )|_0^{end} $\n",
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"\n",
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"$ (i \\omega B,F) + (\\nabla \\times E , f) = 0 $"
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]
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},
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{
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+458
-440
@@ -1,447 +1,465 @@
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"signature": "sha256:74ba4ac0804b189d65dce7f7af6f88a605b83052cf38c5acb492600297a13398"
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"cell_type": "code",
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"input": [
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"%pylab inline"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"output_type": "stream",
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"stream": "stdout",
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"text": [
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"Populating the interactive namespace from numpy and matplotlib\n"
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]
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}
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],
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"prompt_number": 1
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Simple notebook on performing a 1D MT problem.\n"
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Populating the interactive namespace from numpy and matplotlib\n"
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"#from IPython.display import Latex"
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],
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"language": "python",
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"metadata": {},
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"outputs": [],
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"prompt_number": 1
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Maxwell's equations in 1D are as follows\n",
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"\n",
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"\n",
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"$i \\omega b = - \\partial_z e $ \n",
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"\n",
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"$ s = \\partial_z(\\mu^{-1} b) - \\sigma(z) e$\n",
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"\n",
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"$b(0) = 1 \\hspace{1cm} ;\\hspace{1cm} b(-\\infty) = 0$\n",
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"\n",
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"\n",
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"\n",
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"where $e = \\widehat{\\overrightarrow{E}}_x $ and $b = \\widehat{\\overrightarrow{B}}_y$\n",
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"\n",
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"In weak form the equations become\n",
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"\n",
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"$i \\omega(b,f) = - (\\partial_ze,f)$ \n",
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"\n",
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"$(s,w) = -(\\mu^{-1} b, \\partial_z w) - (\\sigma(z) e, w)$\n",
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"\n",
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"\n",
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"where f and w are abritrary functions living in same discritizational space as b and e, respectivily.\n",
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"\n",
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"We consider e on nodes and b on cell centers. This way the derivative of any nodal function becomes\n",
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"\n",
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"$ (\\partial_z u)_k \\approx $\n",
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"\n",
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"$h_{k}^{-1} ( u_{k+\\frac{1}{2}} - u_{k-\\frac{1}{2}}) + O(h^2) $\n",
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"\n",
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"Matrix form\n",
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"\n",
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"$ e_z \\approx \\textbf{L}^{-1} \\textbf{G} e = \\begin{bmatrix} h_1^{-1} & & & \\\\\\\\ & h_2^{-1} & & \\\\\\\\ & & \\ddots & \\\\\\\\ & & & h_n^{-1} \\end{bmatrix}^{(n,n)}\n",
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"\\begin{bmatrix} -1 & 1 & & & & \\\\\\\\ & -1 & 1 & & & \\\\\\\\ & & \\ddots & \\ddots & \\\\\\\\ & & & -1 & 1 \\end{bmatrix}^{(n,n+1)} \n",
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"\\begin{bmatrix} e_1 \\\\\\\\ \\\\\\\\ \\vdots \\\\\\\\ \\\\\\\\ e_{n+1} \\end{bmatrix}^{(n+1,1)} $\n",
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"\n",
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"where $ \\textbf{L} = diag(h) $ is the cell size and $ \\textbf{G}$ is the gradient operator with -1,1 representing the topology of the mesh, taking the difference between adjoint cells.\n",
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"\n",
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"We need to compute 2 inner products, on cell centers and from nodes to cell centers.\n",
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"\n",
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"Cell centers inner product is\n",
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"\n",
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"$ (b,f) \\approx \\sum\\limits_k h_k \\textbf{b}_k \\textbf{f}_k + O(h^2) $\n",
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"\n",
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"and in matrix from\n",
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"\n",
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"$ (b,) \\approx \\textbf{b}^T \\textbf{M}^f \\textbf{f}$ and $ (\\mu^{-1} b,f) \\approx \\textbf{b}^T \\textbf{M}_{\\mu}^f \\textbf{f}$ \n",
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"\n",
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"where $ \\textbf{M}_{\\mu}^f = diag(\\textbf{h} \\odot \\mu^{-1}) $ and $ \\textbf{M}^f = diag(\\textbf{h}) $ are the matrices.\n",
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"Nodes to cell centers inner product is\n",
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"\n",
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"$ (\\sigma e, w) \\approx \\sum\\limits_k \\frac{h_k \\sigma_k}{4} ( e_{k+\\frac{1}{2}} w_{k+\\frac{1}{2}} + e_{k+\\frac{1}{2}} w_{k+\\frac{1}{2}} )$\n",
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"\n",
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"and in matrix from\n",
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"\n",
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"$ (\\sigma e, w ) \\approx (\\textbf{h} \\odot \\sigma )^T ( \\textbf{A}_v (\\textbf{e} \\odot \\textbf{w} = \\textbf{w}^T diag(\\textbf{A}_v^T (\\textbf{h} \\odot \\sigma)) \\textbf{e} $\n",
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"\n",
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"Here $\\odot$ is a point wise Hadamard product and $\\textbf{A}_v$ is the averaging operator/matrix from nodes to cell centers\n",
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"\n",
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"$ \\textbf{A}_v = \\begin{bmatrix} \\frac{1}{2} & \\frac{1}{2} & & \\\\\\\\ & \\ddots & \\ddots & \\\\\\\\ & & \\frac{1}{2} & \\frac{1}{2} \\end{bmatrix} ^{(n+1 , n)} $\n",
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"\n",
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"The sigma mass matrix is defined as \n",
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"\n",
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"$ \\textbf{M}_{\\sigma}^{e} = diag(\\textbf{A}_v^T (\\textbf{h} \\odot \\sigma)) $\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"In the MT problem there is no source in the domain, so $ s = 0 $. How ever the boundary conditions provide the right hand side where \n",
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"\n",
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"$ (\\partial_z \\mu^{-1} b, w ) = - (\\mu^{-1} b, \\partial_z w ) + (\\mu^{-1} b w )|_0^{end} $\n",
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"\n",
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"where \n",
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"\n",
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"$ (\\mu^{-1} b w )|_0^{end} = \\textbf{bc}^T (\\textbf{BC w}) $\n",
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"\n",
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"here $\\textbf{BC}$ is an matrix operator that extracts the boundary elements from $ \\textbf{w}$ and $\\textbf{bc} $ are the known boundary condintions. For the 1D case with homogenous boundary conditions we have \n",
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"\n",
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"$ \\textbf{B} = \\begin{bmatrix} -1 & 0 \\\\\\\\ \\vdots & \\vdots \\\\\\\\ 0 & 1 \\end{bmatrix} $ \n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The weak form is \n",
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"\n",
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"$ (\\mu^{-1} b, \\partial_z w) + (\\sigma e, w) = (\\mu^{-1} b w )|_0^{end} $\n",
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"\n",
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"$ (i \\omega b,f) + (\\partial_z e , f) = 0 $\n",
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"\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Using the above matrix represntation we get the Maxwells equations in following form\n",
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"\n",
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"$ i \\omega \\textbf{f}^T \\textbf{M}^f \\textbf{b} + \\textbf{f}^T \\textbf{M}^f \\textbf{L}^{-1} \\textbf{G} \\textbf{e} = 0 $ \n",
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"\n",
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"$ \\textbf{w}^T \\textbf{G}^T \\textbf{L}^{-1} \\textbf{M}^f_{\\mu} \\textbf{b} + \\textbf{w}^T \\textbf{M}_{\\sigma}^e \\textbf{e} = \\textbf{w}^T \\textbf{bc}^T \\textbf{BC} $ \n",
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"\n",
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"Here we use that \n",
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"\n",
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"$ (\\textbf{b},\\textbf{f}) \\approx \\textbf{b}^T \\textbf{M}^f \\textbf{f} = \\textbf{f}^T \\textbf{M}^f \\textbf{b} $ \n",
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"\n",
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"since $\\textbf{M}^f$ is a symmetric diaoganal matrix of size n by n and $\\textbf{b}$ and $\\textbf{f}$ are vectors of length n."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"We eliminate testing vectors and get system of equations to solve\n",
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"\n",
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"$ i \\omega \\textbf{b} + \\textbf{L}^{-1} \\textbf{G} \\textbf{e} = 0 $ \n",
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"\n",
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"$ \\textbf{G}^T \\textbf{L}^{-1} \\textbf{M}^f_{\\mu} \\textbf{b} + \\textbf{M}_{\\sigma}^e \\textbf{e} = \\textbf{bc}^T \\textbf{BC} $ \n",
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"\n",
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"and as $ \\textbf{A} \\textbf{x} = \\textbf{bc} $ system that we will solve, where \n",
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"\n",
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"$ \\textbf{A} = \\begin{bmatrix} \\textbf{G}^T \\textbf{L}^{-1} \\textbf{M}^f_{\\mu} & \\textbf{M}_{\\sigma}^e \\\\\\\\ i \\omega & \\textbf{L}^{-1} \\textbf{G} \\end{bmatrix} $\n",
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"\n",
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"$ \\textbf{x} = \\begin{bmatrix} \\textbf{b} \\\\\\\\ \\textbf{e} \\end{bmatrix} $\n",
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"\n",
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"$ \\textbf{bc} = \\begin{bmatrix} \\textbf{bc}^T \\textbf{BC} \\\\\\\\ \\textbf{0} \\end{bmatrix} $ "
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]
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||||
},
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||||
{
|
||||
"cell_type": "code",
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||||
"collapsed": false,
|
||||
"input": [
|
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"import sys\n",
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"sys.path.append('C:/GudniWork/Codes/python/simpeg')\n",
|
||||
"import SimPEG as simpeg, numpy as np, scipy, scipy.sparse as sp"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"stream": "stdout",
|
||||
"text": [
|
||||
"Efficiency Warning: Interpolation will be slow, use setup.py!\n",
|
||||
"\n",
|
||||
" python setup.py build_ext --inplace\n",
|
||||
" \n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 2
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We have\n",
|
||||
"\n",
|
||||
"$ i \\omega b = - \\partial_z e \\hspace{1cm} ; \\hspace{1cm} \\partial_z(\\mu^{-1} b) - \\sigma(z) e \\hspace{1cm} ;\\hspace{1cm} b(0) = 1 \\hspace{1cm} ;\\hspace{1cm} b(-\\infty) = 0 $\n",
|
||||
"\n",
|
||||
" \n",
|
||||
"To deal with boundary: we assume that below depth L both $ \\sigma $ and $ \\mu $ are constants ($ z < - L $). At the boundary we have that \n",
|
||||
"\n",
|
||||
"$ e = c \\exp(ikz) \\hspace{0.2cm} where \\hspace{0.2cm} k = \\sqrt{i\\omega\\mu\\sigma} $. \n",
|
||||
"\n",
|
||||
"Therefore for $ z < - L $ we have that\n",
|
||||
"\n",
|
||||
"$\\omega b - k e = 0 $.\n",
|
||||
"\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We discretize the e field on the nodes and b field at the cell centers. The system we want to solve is \n",
|
||||
"\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"$\\begin{bmatrix} i \\omega & \\frac{\\partial}{\\partial z} \\\\\\\\ \\frac{1}{\\mu} \\frac{\\partial}{\\partial z} & -\\sigma \\end{bmatrix} \n",
|
||||
"\\begin{bmatrix} b \\\\\\\\ e \\end{bmatrix} = \\begin{bmatrix} s1 \\\\\\\\ s2 \\end{bmatrix}$\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"prompt_number": 2
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
" # Set up the problem\n",
|
||||
"mu = 4*np.pi*1e-7\n",
|
||||
"eps0 = 8.85e-12\n",
|
||||
"# Frequency\n",
|
||||
"fr = np.array([1e1]) #np.logspace(0,5,200) #np.array([2000]) #np.logspace(-4,5,82)\n",
|
||||
"omega = 2*np.pi*fr\n",
|
||||
"# Mesh\n",
|
||||
"sig0 = 1e-2\n",
|
||||
"#L = 3*np.sqrt(2/(mu*omega[0]*sig0))\n",
|
||||
"#nn=np.ceil(np.log(0.3*L + 1)/np.log(1.3))\n",
|
||||
"#h = 5*(1.3**(np.arange(nn+1)))\n",
|
||||
"\n",
|
||||
"h = np.ones(18)\n",
|
||||
"x0 = np.array([0])\n",
|
||||
"#sig = sig0*np.ones((len(h),1)) \n",
|
||||
"#sig[0:50] = 0.1\n",
|
||||
"#sig[50:100] = 1\n",
|
||||
"# Make the mesh\n",
|
||||
"mesh = simpeg.Mesh.TensorMesh([h],x0)\n",
|
||||
"sig = np.zeros(mesh.nC) + 1e-8\n",
|
||||
"sig[mesh.vectorCCx<=0] = 1e-2"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"prompt_number": 3
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"fr,omega"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"metadata": {},
|
||||
"output_type": "pyout",
|
||||
"prompt_number": 4,
|
||||
"text": [
|
||||
"(array([ 10.]), array([ 62.83185307]))"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 4
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"# Make the operators\n",
|
||||
"G = mesh.nodalGrad\n",
|
||||
"Av = mesh.aveN2CC\n",
|
||||
"Li = scipy.sparse.spdiags(1/mesh.hx,0,mesh.nNx,mesh.nNx)\n",
|
||||
"Mmu = scipy.sparse.spdiags(mesh.hx/mu,0,mesh.nCx,mesh.nCx)\n",
|
||||
"Msig = scipy.sparse.spdiags(Av.T.dot(mesh.hx*sig.ravel()),0,mesh.nNx,mesh.nNx)\n",
|
||||
"# The boundaries\n",
|
||||
"bc_b = np.zeros((mesh.nCx,1))\n",
|
||||
"bc_b[0] = -1 # Set the top b field to 1\n",
|
||||
"bc_e = np.zeros((mesh.nNx,1))\n",
|
||||
"# Make the sparse matrix\n",
|
||||
"bc = sp.vstack((bc_b,bc_e))\n"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"prompt_number": 5
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"b = np.empty((mesh.nCx,len(omega)),dtype=np.complex64)\n",
|
||||
"e = np.empty((mesh.nNx,len(omega)),dtype=np.complex64)\n",
|
||||
"# Loop all the frequencies\n",
|
||||
"for nrOm, om in enumerate(omega):\n",
|
||||
" # Left hand side\n",
|
||||
" A = sp.vstack((sp.hstack(( -G.conj().T.dot(Mmu), - Msig)), sp.hstack((1j*om*scipy.sparse.identity(mesh.nCx) , G))))\n",
|
||||
" #A = A.tocsr\n",
|
||||
" # Solve the system\n",
|
||||
" bef = scipy.sparse.linalg.spsolve(A,bc)\n",
|
||||
" # Sort the output\n",
|
||||
" b[:,nrOm] = bef[0:mesh.nCx]\n",
|
||||
" e[:,nrOm] = bef[mesh.nCx::]\n"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"stream": "stderr",
|
||||
"text": [
|
||||
"/home/Gudni/anaconda/lib/python2.7/site-packages/scipy/sparse/linalg/dsolve/linsolve.py:90: SparseEfficiencyWarning: spsolve requires A be CSC or CSR matrix format\n",
|
||||
" SparseEfficiencyWarning)\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 6
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"import matplotlib.pyplot as plt\n",
|
||||
"# Plot the solution\n",
|
||||
"z=e[0,:]/(b[0,:]/mu)\n",
|
||||
"app_res = ((1./(8e-7*np.pi**2))/fr)*np.abs(z)**2\n",
|
||||
"app_phs = np.arctan(z.imag/z.real)*(180/np.pi)\n",
|
||||
"ax_res = plt.subplot(2,1,1)\n",
|
||||
"ax_res.loglog(fr,app_res)\n",
|
||||
"ax_phs = plt.subplot(2,1,2)\n",
|
||||
"ax_phs.semilogx(fr,app_phs)"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"metadata": {},
|
||||
"output_type": "pyout",
|
||||
"prompt_number": 7,
|
||||
"text": [
|
||||
"[<matplotlib.lines.Line2D at 0x7fa8671e1fd0>]"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 7
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"plt.show()"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"prompt_number": 8
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"# Calculate the impedance\n",
|
||||
"z = e[0,:]/(b[0,:]/mu)\n",
|
||||
"z"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"metadata": {},
|
||||
"output_type": "pyout",
|
||||
"prompt_number": 9,
|
||||
"text": [
|
||||
"array([-5714285.5+0.00050081j], dtype=complex64)"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 9
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"app_res"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"metadata": {},
|
||||
"output_type": "pyout",
|
||||
"prompt_number": 10,
|
||||
"text": [
|
||||
"array([ 4.13555827e+17])"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 10
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"prompt_number": 10
|
||||
}
|
||||
],
|
||||
"metadata": {}
|
||||
"source": [
|
||||
"%pylab inline"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Simple notebook on performing a 1D MT problem.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 1,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"#from IPython.display import Latex"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Maxwell's equations in 1D are as follows\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"$i \\omega b = - \\partial_z e $ \n",
|
||||
"\n",
|
||||
"$ s = \\partial_z(\\mu^{-1} b) - \\sigma(z) e$\n",
|
||||
"\n",
|
||||
"$b(0) = 1 \\hspace{1cm} ;\\hspace{1cm} b(-\\infty) = 0$\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"where $e = \\widehat{\\overrightarrow{E}}_x $ and $b = \\widehat{\\overrightarrow{B}}_y$\n",
|
||||
"\n",
|
||||
"In weak form the equations become\n",
|
||||
"\n",
|
||||
"$i \\omega(b,f) = - (\\partial_ze,f)$ \n",
|
||||
"\n",
|
||||
"$(s,w) = -(\\mu^{-1} b, \\partial_z w) - (\\sigma(z) e, w)$\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"where f and w are abritrary functions living in same discritizational space as b and e, respectivily.\n",
|
||||
"\n",
|
||||
"We consider e on nodes and b on cell centers. This way the derivative of any nodal function becomes\n",
|
||||
"\n",
|
||||
"$ (\\partial_z u)_k \\approx $\n",
|
||||
"\n",
|
||||
"$h_{k}^{-1} ( u_{k+\\frac{1}{2}} - u_{k-\\frac{1}{2}}) + O(h^2) $\n",
|
||||
"\n",
|
||||
"Matrix form\n",
|
||||
"\n",
|
||||
"$ e_z \\approx \\textbf{L}^{-1} \\textbf{G} e = \\begin{bmatrix} h_1^{-1} & & & \\\\\\\\ & h_2^{-1} & & \\\\\\\\ & & \\ddots & \\\\\\\\ & & & h_n^{-1} \\end{bmatrix}^{(n,n)}\n",
|
||||
"\\begin{bmatrix} -1 & 1 & & & & \\\\\\\\ & -1 & 1 & & & \\\\\\\\ & & \\ddots & \\ddots & \\\\\\\\ & & & -1 & 1 \\end{bmatrix}^{(n,n+1)} \n",
|
||||
"\\begin{bmatrix} e_1 \\\\\\\\ \\\\\\\\ \\vdots \\\\\\\\ \\\\\\\\ e_{n+1} \\end{bmatrix}^{(n+1,1)} $\n",
|
||||
"\n",
|
||||
"where $ \\textbf{L} = diag(h) $ is the cell size and $ \\textbf{G}$ is the gradient operator with -1,1 representing the topology of the mesh, taking the difference between adjoint cells.\n",
|
||||
"\n",
|
||||
"We need to compute 2 inner products, on cell centers and from nodes to cell centers.\n",
|
||||
"\n",
|
||||
"Cell centers inner product is\n",
|
||||
"\n",
|
||||
"$ (b,f) \\approx \\sum\\limits_k h_k \\textbf{b}_k \\textbf{f}_k + O(h^2) $\n",
|
||||
"\n",
|
||||
"and in matrix from\n",
|
||||
"\n",
|
||||
"$ (b,) \\approx \\textbf{b}^T \\textbf{M}^f \\textbf{f}$ and $ (\\mu^{-1} b,f) \\approx \\textbf{b}^T \\textbf{M}_{\\mu}^f \\textbf{f}$ \n",
|
||||
"\n",
|
||||
"where $ \\textbf{M}_{\\mu}^f = diag(\\textbf{h} \\odot \\mu^{-1}) $ and $ \\textbf{M}^f = diag(\\textbf{h}) $ are the matrices.\n",
|
||||
"Nodes to cell centers inner product is\n",
|
||||
"\n",
|
||||
"$ (\\sigma e, w) \\approx \\sum\\limits_k \\frac{h_k \\sigma_k}{4} ( e_{k+\\frac{1}{2}} w_{k+\\frac{1}{2}} + e_{k+\\frac{1}{2}} w_{k+\\frac{1}{2}} )$\n",
|
||||
"\n",
|
||||
"and in matrix from\n",
|
||||
"\n",
|
||||
"$ (\\sigma e, w ) \\approx (\\textbf{h} \\odot \\sigma )^T ( \\textbf{A}_v (\\textbf{e} \\odot \\textbf{w} = \\textbf{w}^T diag(\\textbf{A}_v^T (\\textbf{h} \\odot \\sigma)) \\textbf{e} $\n",
|
||||
"\n",
|
||||
"Here $\\odot$ is a point wise Hadamard product and $\\textbf{A}_v$ is the averaging operator/matrix from nodes to cell centers\n",
|
||||
"\n",
|
||||
"$ \\textbf{A}_v = \\begin{bmatrix} \\frac{1}{2} & \\frac{1}{2} & & \\\\\\\\ & \\ddots & \\ddots & \\\\\\\\ & & \\frac{1}{2} & \\frac{1}{2} \\end{bmatrix} ^{(n+1 , n)} $\n",
|
||||
"\n",
|
||||
"The sigma mass matrix is defined as \n",
|
||||
"\n",
|
||||
"$ \\textbf{M}_{\\sigma}^{e} = diag(\\textbf{A}_v^T (\\textbf{h} \\odot \\sigma)) $\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"In the MT problem there is no source in the domain, so $ s = 0 $. How ever the boundary conditions provide the right hand side where \n",
|
||||
"\n",
|
||||
"$ (\\partial_z \\mu^{-1} b, w ) = - (\\mu^{-1} b, \\partial_z w ) + (\\mu^{-1} b w )|_0^{end} $\n",
|
||||
"\n",
|
||||
"where \n",
|
||||
"\n",
|
||||
"$ (\\mu^{-1} b w )|_0^{end} = \\textbf{bc}^T (\\textbf{BC w}) $\n",
|
||||
"\n",
|
||||
"here $\\textbf{BC}$ is an matrix operator that extracts the boundary elements from $ \\textbf{w}$ and $\\textbf{bc} $ are the known boundary condintions. For the 1D case with homogenous boundary conditions we have \n",
|
||||
"\n",
|
||||
"$ \\textbf{B} = \\begin{bmatrix} -1 & 0 \\\\\\\\ \\vdots & \\vdots \\\\\\\\ 0 & 1 \\end{bmatrix} $ \n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"The weak form is \n",
|
||||
"\n",
|
||||
"$ (\\mu^{-1} b, \\partial_z w) + (\\sigma e, w) = (\\mu^{-1} b w )|_0^{end} $\n",
|
||||
"\n",
|
||||
"$ (i \\omega b,f) + (\\partial_z e , f) = 0 $\n",
|
||||
"\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Using the above matrix represntation we get the Maxwells equations in following form\n",
|
||||
"\n",
|
||||
"$ i \\omega \\textbf{f}^T \\textbf{M}^f \\textbf{b} + \\textbf{f}^T \\textbf{M}^f \\textbf{L}^{-1} \\textbf{G} \\textbf{e} = 0 $ \n",
|
||||
"\n",
|
||||
"$ \\textbf{w}^T \\textbf{G}^T \\textbf{L}^{-1} \\textbf{M}^f_{\\mu} \\textbf{b} + \\textbf{w}^T \\textbf{M}_{\\sigma}^e \\textbf{e} = \\textbf{w}^T \\textbf{bc}^T \\textbf{BC} $ \n",
|
||||
"\n",
|
||||
"Here we use that \n",
|
||||
"\n",
|
||||
"$ (\\textbf{b},\\textbf{f}) \\approx \\textbf{b}^T \\textbf{M}^f \\textbf{f} = \\textbf{f}^T \\textbf{M}^f \\textbf{b} $ \n",
|
||||
"\n",
|
||||
"since $\\textbf{M}^f$ is a symmetric diaoganal matrix of size n by n and $\\textbf{b}$ and $\\textbf{f}$ are vectors of length n."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We eliminate testing vectors and get system of equations to solve\n",
|
||||
"\n",
|
||||
"$ i \\omega \\textbf{b} + \\textbf{L}^{-1} \\textbf{G} \\textbf{e} = 0 $ \n",
|
||||
"\n",
|
||||
"$ \\textbf{G}^T \\textbf{L}^{-1} \\textbf{M}^f_{\\mu} \\textbf{b} + \\textbf{M}_{\\sigma}^e \\textbf{e} = \\textbf{bc}^T \\textbf{BC} $ \n",
|
||||
"\n",
|
||||
"and as $ \\textbf{A} \\textbf{x} = \\textbf{bc} $ system that we will solve, where \n",
|
||||
"\n",
|
||||
"$ \\textbf{A} = \\begin{bmatrix} \\textbf{G}^T \\textbf{L}^{-1} \\textbf{M}^f_{\\mu} & \\textbf{M}_{\\sigma}^e \\\\\\\\ i \\omega & \\textbf{L}^{-1} \\textbf{G} \\end{bmatrix} $\n",
|
||||
"\n",
|
||||
"$ \\textbf{x} = \\begin{bmatrix} \\textbf{b} \\\\\\\\ \\textbf{e} \\end{bmatrix} $\n",
|
||||
"\n",
|
||||
"$ \\textbf{bc} = \\begin{bmatrix} \\textbf{bc}^T \\textbf{BC} \\\\\\\\ \\textbf{0} \\end{bmatrix} $ "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 2,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Efficiency Warning: Interpolation will be slow, use setup.py!\n",
|
||||
"\n",
|
||||
" python setup.py build_ext --inplace\n",
|
||||
" \n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"import sys\n",
|
||||
"sys.path.append('C:/GudniWork/Codes/python/simpeg')\n",
|
||||
"import SimPEG as simpeg, numpy as np, scipy, scipy.sparse as sp"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We have\n",
|
||||
"\n",
|
||||
"$ i \\omega b = - \\partial_z e \\hspace{1cm} ; \\hspace{1cm} \\partial_z(\\mu^{-1} b) - \\sigma(z) e \\hspace{1cm} ;\\hspace{1cm} b(0) = 1 \\hspace{1cm} ;\\hspace{1cm} b(-\\infty) = 0 $\n",
|
||||
"\n",
|
||||
" \n",
|
||||
"To deal with boundary: we assume that below depth L both $ \\sigma $ and $ \\mu $ are constants ($ z < - L $). At the boundary we have that \n",
|
||||
"\n",
|
||||
"$ e = c \\exp(ikz) \\hspace{0.2cm} where \\hspace{0.2cm} k = \\sqrt{i\\omega\\mu\\sigma} $. \n",
|
||||
"\n",
|
||||
"Therefore for $ z < - L $ we have that\n",
|
||||
"\n",
|
||||
"$\\omega b - k e = 0 $.\n",
|
||||
"\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We discretize the e field on the nodes and b field at the cell centers. The system we want to solve is \n",
|
||||
"\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"$\\begin{bmatrix} i \\omega & \\frac{\\partial}{\\partial z} \\\\\\\\ \\frac{1}{\\mu} \\frac{\\partial}{\\partial z} & -\\sigma \\end{bmatrix} \n",
|
||||
"\\begin{bmatrix} b \\\\\\\\ e \\end{bmatrix} = \\begin{bmatrix} s1 \\\\\\\\ s2 \\end{bmatrix}$\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 2,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 3,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
" # Set up the problem\n",
|
||||
"mu = 4*np.pi*1e-7\n",
|
||||
"eps0 = 8.85e-12\n",
|
||||
"# Frequency\n",
|
||||
"fr = np.array([1e1]) #np.logspace(0,5,200) #np.array([2000]) #np.logspace(-4,5,82)\n",
|
||||
"omega = 2*np.pi*fr\n",
|
||||
"# Mesh\n",
|
||||
"sig0 = 1e-2\n",
|
||||
"#L = 3*np.sqrt(2/(mu*omega[0]*sig0))\n",
|
||||
"#nn=np.ceil(np.log(0.3*L + 1)/np.log(1.3))\n",
|
||||
"#h = 5*(1.3**(np.arange(nn+1)))\n",
|
||||
"\n",
|
||||
"h = np.ones(18)\n",
|
||||
"x0 = np.array([0])\n",
|
||||
"#sig = sig0*np.ones((len(h),1)) \n",
|
||||
"#sig[0:50] = 0.1\n",
|
||||
"#sig[50:100] = 1\n",
|
||||
"# Make the mesh\n",
|
||||
"mesh = simpeg.Mesh.TensorMesh([h],x0)\n",
|
||||
"sig = np.zeros(mesh.nC) + 1e-8\n",
|
||||
"sig[mesh.vectorCCx<=0] = 1e-2"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"(array([ 10.]), array([ 62.83185307]))"
|
||||
]
|
||||
},
|
||||
"execution_count": 4,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"fr,omega"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 5,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# Make the operators\n",
|
||||
"G = mesh.nodalGrad\n",
|
||||
"Av = mesh.aveN2CC\n",
|
||||
"Li = scipy.sparse.spdiags(1/mesh.hx,0,mesh.nNx,mesh.nNx)\n",
|
||||
"Mmu = scipy.sparse.spdiags(mesh.hx/mu,0,mesh.nCx,mesh.nCx)\n",
|
||||
"Msig = scipy.sparse.spdiags(Av.T.dot(mesh.hx*sig.ravel()),0,mesh.nNx,mesh.nNx)\n",
|
||||
"# The boundaries\n",
|
||||
"bc_b = np.zeros((mesh.nCx,1))\n",
|
||||
"bc_b[0] = -1 # Set the top b field to 1\n",
|
||||
"bc_e = np.zeros((mesh.nNx,1))\n",
|
||||
"# Make the sparse matrix\n",
|
||||
"bc = sp.vstack((bc_b,bc_e))\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 6,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stderr",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"/home/Gudni/anaconda/lib/python2.7/site-packages/scipy/sparse/linalg/dsolve/linsolve.py:90: SparseEfficiencyWarning: spsolve requires A be CSC or CSR matrix format\n",
|
||||
" SparseEfficiencyWarning)\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"b = np.empty((mesh.nCx,len(omega)),dtype=np.complex64)\n",
|
||||
"e = np.empty((mesh.nNx,len(omega)),dtype=np.complex64)\n",
|
||||
"# Loop all the frequencies\n",
|
||||
"for nrOm, om in enumerate(omega):\n",
|
||||
" # Left hand side\n",
|
||||
" A = sp.vstack((sp.hstack(( -G.conj().T.dot(Mmu), - Msig)), sp.hstack((1j*om*scipy.sparse.identity(mesh.nCx) , G))))\n",
|
||||
" #A = A.tocsr\n",
|
||||
" # Solve the system\n",
|
||||
" bef = scipy.sparse.linalg.spsolve(A,bc)\n",
|
||||
" # Sort the output\n",
|
||||
" b[:,nrOm] = bef[0:mesh.nCx]\n",
|
||||
" e[:,nrOm] = bef[mesh.nCx::]\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 7,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"[<matplotlib.lines.Line2D at 0x7fa8671e1fd0>]"
|
||||
]
|
||||
},
|
||||
"execution_count": 7,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"import matplotlib.pyplot as plt\n",
|
||||
"# Plot the solution\n",
|
||||
"z=e[0,:]/(b[0,:]/mu)\n",
|
||||
"app_res = ((1./(8e-7*np.pi**2))/fr)*np.abs(z)**2\n",
|
||||
"app_phs = np.arctan(z.imag/z.real)*(180/np.pi)\n",
|
||||
"ax_res = plt.subplot(2,1,1)\n",
|
||||
"ax_res.loglog(fr,app_res)\n",
|
||||
"ax_phs = plt.subplot(2,1,2)\n",
|
||||
"ax_phs.semilogx(fr,app_phs)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 8,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 9,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"array([-5714285.5+0.00050081j], dtype=complex64)"
|
||||
]
|
||||
},
|
||||
"execution_count": 9,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# Calculate the impedance\n",
|
||||
"z = e[0,:]/(b[0,:]/mu)\n",
|
||||
"z"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 10,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"array([ 4.13555827e+17])"
|
||||
]
|
||||
},
|
||||
"execution_count": 10,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"app_res"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 10,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
}
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 2",
|
||||
"language": "python",
|
||||
"name": "python2"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 2
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython2",
|
||||
"version": "2.7.9"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 0
|
||||
}
|
||||
|
||||
@@ -0,0 +1,925 @@
|
||||
{
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 1,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Populating the interactive namespace from numpy and matplotlib\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"%pylab inline"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 2,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"import SimPEG as simpeg\n",
|
||||
"import simpegMT as simpegmt\n",
|
||||
"from scipy.constants import mu_0\n",
|
||||
"\n",
|
||||
"def omega(freq):\n",
|
||||
" \"\"\"Change frequency to angular frequency, omega\"\"\"\n",
|
||||
" return 2.*np.pi*freq"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 115,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"#Define the mesh\n",
|
||||
"z = 100.\n",
|
||||
"hz = [(z,5,-1.5),(z,10),(z,5,1.5)]\n",
|
||||
"M = simpeg.Mesh.TensorMesh([hz],'C')\n",
|
||||
"# sig = np.zeros(M.nC) + 1e-8\n",
|
||||
"conds = [1,1e-2]\n",
|
||||
"elev = 300\n",
|
||||
"sig = np.zeros(M.nC) + conds[1]\n",
|
||||
"sig[np.logical_and(M.gridCC>-200,M.gridCC<0)] = conds[0]\n",
|
||||
"sig[M.gridCC>elev] = 1e-8\n",
|
||||
"sig[M.gridCC<-500] = 1e-1\n",
|
||||
"sig[M.gridCC<-900] = conds[1]"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 116,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"array([-2478.125, -1718.75 , -1212.5 , -875. , -650. , -500. ,\n",
|
||||
" -400. , -300. , -200. , -100. , 0. , 100. ,\n",
|
||||
" 200. , 300. , 400. , 500. , 650. , 875. ,\n",
|
||||
" 1212.5 , 1718.75 , 2478.125])"
|
||||
]
|
||||
},
|
||||
"execution_count": 116,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"M.vectorNx"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 141,
|
||||
"metadata": {
|
||||
"collapsed": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"\n",
|
||||
"# Calculate the analytic fields\n",
|
||||
"freqs = np.logspace(4,-4,33)\n",
|
||||
"Zana = []\n",
|
||||
"for freq in freqs:\n",
|
||||
" Ed, Eu, Hd, Hu = simpegmt.Utils.getEHfields(M,sig,freq,np.array([elev]))\n",
|
||||
" Zana.append((Ed + Eu)/(Hd + Hu))\n",
|
||||
"ZanaArr = np.concatenate(Zana)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 169,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# Calculate the synthetic solution\n",
|
||||
"Zsyn = [] \n",
|
||||
"Qex = M.getInterpolationMat(np.array([elev]),'Ex')\n",
|
||||
"Qfx = M.getInterpolationMat(np.array([elev]),'Fx')\n",
|
||||
"for freq in freqs:\n",
|
||||
" e = simpegmt.Utils.get1DEfields(M,sig,freq,sourceAmp=None)\n",
|
||||
" h = -(M.nodalGrad*e)/(1j*omega(freq)*mu_0)\n",
|
||||
" Zsyn.append((Qfx*e).conj()/(Qex*h).conj())\n",
|
||||
"ZsynArr = np.concatenate(Zsyn)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 170,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"(array([12]),)"
|
||||
]
|
||||
},
|
||||
"execution_count": 170,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"np.where(freqs==10)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 183,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"(0.000197894493906+0.000181285727378j)\n",
|
||||
"[ 1.00551946+0.00563967j] [ 2776.87342066-2515.31936068j]\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"print ZsynArr[-1]\n",
|
||||
"print (Qfx*e).conj(), (Qex*h).conj()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 171,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def appResPhs(freq,z):\n",
|
||||
" app_res = ((1./(8e-7*np.pi**2))/freq)*np.abs(z)**2\n",
|
||||
" app_phs = np.arctan2(z.imag,z.real)*(180/np.pi)\n",
|
||||
" return app_res, app_phs\n",
|
||||
"appAna_r, appAna_p = appResPhs(freqs,ZanaArr)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 172,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"[ 99.9982606 100.03796243 99.63440411 99.33759941 106.29388449\n",
|
||||
" 120.23667968 122.60558712 101.23963102 70.5016044 45.28589223\n",
|
||||
" 28.43953028 17.6198611 10.84432222 6.83742416 4.62137881\n",
|
||||
" 3.61487549 3.53761463 4.30735297 5.99370277 8.77556001\n",
|
||||
" 12.87536967 18.46051811 25.52731592 33.81862185 42.83565417\n",
|
||||
" 51.95659295 60.60094607 68.35309357 75.00153541 80.50894752\n",
|
||||
" 84.9530758 88.47009307 91.21391269]\n",
|
||||
"[ 44.9980325 45.00483104 45.04645859 44.4151918 43.72907936\n",
|
||||
" 46.72033941 54.60169587 63.94626768 71.19189243 75.42083811\n",
|
||||
" 77.37907592 77.73640994 76.058035 71.73541117 64.19535734\n",
|
||||
" 53.36978096 41.16658711 30.81763511 24.08982259 20.77904552\n",
|
||||
" 19.99288987 20.90312313 22.8822938 25.46014498 28.27778379\n",
|
||||
" 31.06840674 33.65137133 35.92447275 37.84909319 39.43106457\n",
|
||||
" 40.70229309 41.70642543 42.48936455]\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"print appAna_r\n",
|
||||
"print appAna_p"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 173,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"appSyn_r, appSyn_p = appResPhs(freqs,ZsynArr)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 174,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"[ 148.47846913 175.57907495 170.44187937 154.82139762 151.64437874\n",
|
||||
" 157.04097616 142.11024881 103.42128503 66.07607566 41.13098488\n",
|
||||
" 26.10536117 16.64437152 10.50362449 6.72485409 4.58294174\n",
|
||||
" 3.60236081 3.53787296 4.31623649 6.00810219 8.793518\n",
|
||||
" 12.8970497 18.48663881 25.55782824 33.85235462 42.87059902\n",
|
||||
" 51.99052864 60.6320452 68.38024378 75.02434148 80.52754625\n",
|
||||
" 84.96791113 88.48173456 91.22293906]\n",
|
||||
"[ 24.90447028 35.50740917 43.45028334 46.88513715 48.17221695\n",
|
||||
" 52.65990261 61.40691455 69.96355574 75.14671736 77.09349558\n",
|
||||
" 77.37322373 76.95784679 75.19022669 71.03384758 63.68398281\n",
|
||||
" 53.00055411 40.91266512 30.67116253 24.02349762 20.75523371\n",
|
||||
" 19.98819197 20.90730695 22.89080611 25.4705669 28.28863141\n",
|
||||
" 31.07873966 33.66064252 35.93243359 37.85570553 39.43641995\n",
|
||||
" 40.70654829 41.709758 42.49194635]\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"print appSyn_r\n",
|
||||
"print appSyn_p"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 175,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": [
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|
||||
],
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x7fc5b9632550>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"loglog(freqs,np.abs(ZanaArr),freqs,np.abs(ZsynArr))\n",
|
||||
"gca().invert_xaxis()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 176,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
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||||
"image/png": [
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||||
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|
||||
],
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x7fc5b8713f90>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"loglog(freqs,appAna_r,'bo--',freqs,appSyn_r,'gx:')\n",
|
||||
"gca().invert_xaxis()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 177,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": [
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||||
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],
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"text/plain": [
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||||
"<matplotlib.figure.Figure at 0x7fc5b8c06050>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"semilogx(freqs,appAna_p,'bo--',freqs,appSyn_p,'gx:')\n",
|
||||
"gca().invert_xaxis()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {
|
||||
"collapsed": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {
|
||||
"collapsed": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {
|
||||
"collapsed": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"kernelspec": {
|
||||
"display_name": "Python 2",
|
||||
"language": "python",
|
||||
"name": "python2"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 2
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython2",
|
||||
"version": "2.7.9"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 0
|
||||
}
|
||||
@@ -1,7 +1,7 @@
|
||||
from SimPEG import Survey, Problem, Utils, Models, np, sp, SolverLU as SimpegSolver
|
||||
from scipy.constants import mu_0
|
||||
from SurveyMT import SurveyMT, FieldsMT
|
||||
import multiprocessing
|
||||
import multiprocessing, sys
|
||||
|
||||
def omega(freq):
|
||||
"""Change frequency to angular frequency, omega"""
|
||||
|
||||
@@ -0,0 +1,48 @@
|
||||
import unittest
|
||||
from SimPEG import *
|
||||
import simpegMT as MT
|
||||
|
||||
TOL = 1e-6
|
||||
|
||||
def appResPhs(freq,z):
|
||||
app_res = ((1./(8e-7*np.pi**2))/freq)*np.abs(z)**2
|
||||
app_phs = np.arctan2(-z.imag,z.real)*(180/np.pi)
|
||||
return app_res, app_phs
|
||||
|
||||
def appResNorm(sigmaHalf):
|
||||
nFreq = 26
|
||||
|
||||
m1d = Mesh.TensorMesh([[(100,5,1.5),(100.,10),(100,5,1.5)]], x0=['C'])
|
||||
sigma = np.zeros(m1d.nC) + sigmaHalf
|
||||
sigma[m1d.gridCC[:]>200] = 1e-8
|
||||
|
||||
# Calculate the analytic fields
|
||||
freqs = np.logspace(4,-4,nFreq)
|
||||
Z = []
|
||||
for freq in freqs:
|
||||
Ed, Eu, Hd, Hu = MT.Utils.getEHfields(m1d,sigma,freq,np.array([200]))
|
||||
Z.append((Ed + Eu)/(Hd + Hu))
|
||||
|
||||
Zarr = np.concatenate(Z)
|
||||
|
||||
app_r, app_p = appResPhs(freqs,Zarr)
|
||||
|
||||
return np.linalg.norm(np.abs(app_r - np.ones(nFreq)/sigmaHalf)) / np.log10(sigmaHalf)
|
||||
|
||||
|
||||
class TestAnalytics(unittest.TestCase):
|
||||
|
||||
def setUp(self):
|
||||
pass
|
||||
def test_appRes2en1(self):self.assertLess(appResNorm(2e-1), TOL)
|
||||
def test_appRes2en2(self):self.assertLess(appResNorm(2e-2), TOL)
|
||||
def test_appRes2en3(self):self.assertLess(appResNorm(2e-3), TOL)
|
||||
def test_appRes2en4(self):self.assertLess(appResNorm(2e-4), TOL)
|
||||
def test_appRes2en5(self):self.assertLess(appResNorm(2e-5), TOL)
|
||||
def test_appRes2en6(self):self.assertLess(appResNorm(2e-6), TOL)
|
||||
|
||||
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
unittest.main()
|
||||
@@ -21,7 +21,7 @@ def get1DEfields(m1d,sigma,freq,sourceAmp=1.0):
|
||||
|
||||
# Set the boundary conditions
|
||||
Ed, Eu, Hd, Hu = getEHfields(m1d,sigma,freq,m1d.vectorNx)
|
||||
Etot = Ed + Eu
|
||||
Etot = (Ed + Eu).conj()
|
||||
if sourceAmp is not None:
|
||||
Etot = ((Etot/Etot[-1])*sourceAmp) # Scale the fields to be equal to sourceAmp at the top
|
||||
## Note: need to use conjugate of the analytic solution. It is derived with e^iwt
|
||||
|
||||
Reference in New Issue
Block a user