From a84bc5cbc19122f11c514280ec7ad19d48337213 Mon Sep 17 00:00:00 2001 From: Lindsey Heagy Date: Mon, 6 Jul 2015 19:04:24 -0500 Subject: [PATCH] documentation updates --- docs/api_FDEM.rst | 67 ++++++++++++-- docs/images/ebjhdiscretizations.png | Bin 0 -> 37799 bytes docs/images/finitevolrealestate.png | Bin 0 -> 29996 bytes simpegEM/Base.py | 45 ++++++++- simpegEM/FDEM/FDEM.py | 138 ++++++++++++++++++---------- 5 files changed, 188 insertions(+), 62 deletions(-) create mode 100644 docs/images/ebjhdiscretizations.png create mode 100644 docs/images/finitevolrealestate.png diff --git a/docs/api_FDEM.rst b/docs/api_FDEM.rst index 2c063216..1848c569 100644 --- a/docs/api_FDEM.rst +++ b/docs/api_FDEM.rst @@ -36,7 +36,7 @@ In the frequency domain, Maxwell's equations are given by .. math :: \curl \vec{E} = - i \omega \vec{B} \\ - \curl \vec{H} = \vec{J} + i \omega \vec{D} + \vec{J}_s \\ + \curl \vec{H} = \vec{J} + i \omega \vec{D} + \vec{S} \\ \div \vec{B} = 0 \\ @@ -50,7 +50,7 @@ where: - \\(\\vec{D}\\) : electric displacement / electric flux density (\\(C/m^2\\)) - \\(\\vec{J}\\) : electric current density (\\(A/m^2\\)) - \\(\\rho_f\\) : free charge density -The source term is \\(\\vec{J}_s\\) +The source term is \\(\\vec{S}\\) Constitutive Relations @@ -71,7 +71,6 @@ where: - \\(\\varepsilon\\) : dielectric permittivity \\(F/m\\) \\(\\sigma\\), \\(\\mu\\), \\(\\varepsilon\\) are physical properties which depend on the material. \\(\\sigma\\) describes how easily electric current passes through a material, \\(\\mu\\) describes how easily a material is magnetized, and \\(\\varepsilon\\) describes how easily a material is electrically polarized. In most geophysical applications of EM, \\(\\sigma\\) is the the primary physical property of interest, and \\(\\mu\\), \\(\\varepsilon\\) are assumed to have their free-space values \\(\\mu_0 = 4\\pi \\times 10^{-7} H/m \\), \\(\\varepsilon_0 = 8.85 \\times 10^{-12} F/m\\) -For a more complete discussion of physical properties see `GPG `_ Quasi-static Approximation @@ -80,17 +79,65 @@ For the frequency range typical of most geophysical surveys, the contribution of .. math :: \nabla \times \vec{E} = -i \omega \vec{B} \\ - \nabla \times \vec{H} = \vec{J} + \vec{J}_s + \nabla \times \vec{H} = \vec{J} + \vec{S} -Fields from a Dipole --------------------- +Implementation in simpegEM +========================== +We consider two formulations in simpegEM, both first-order and both in terms of one field and one flux. We allow for the definition of magnetic and electric sources (see for example: Ward and Hohmann, starting on page 144). The E-B formulation is in terms of the electric field and the magnetic flux: -Forward Problem -=============== +.. math :: + \nabla \times \vec{E} + i \omega \vec{B} = \vec{S}_m \\ + \nabla \times \mu^{-1} \vec{B} - \sigma \vec{E} = \vec{S}_e -Inverse Problem -=============== +The H-J formulation is in terms of the current density and the magnetic field: + +.. math :: + \nabla \times \sigma^{-1} \vec{J} + i \omega \mu \vec{H} = \vec{S}_m \\ + \nabla \times \vec{H} - \vec{J} = \vec{S}_e + + +Discretizing +------------ +For both formulations, we use a finite volume discretization +and discretize fields on cell edges, fluxes on cell faces and +physical properties in cell centers. This is particularly +important when using symmetry to reduce the dimensionality of a problem +(for instance on a 2D CylMesh, there are \\(r\\), \\(z\\) faces and \\(\\theta\\) edges) + +.. figure:: ./images/finitevolrealestate.png + :align: center + :scale: 60 % + +For the two formulations, the discretization of the physical properties, fields and fluxes are summarized below. + +.. figure:: ./images/ebjhdiscretizations.png + :align: center + :scale: 60 % + +Note that resistivity is the inverse of conductivity, \\(\\rho = \\sigma^{-1}\\). + + +E-B Formulation: +**************** + +.. math :: + \mathbf{C} \mathbf{e} + i \omega \mathbf{b} = \mathbf{s_m} \\ + \mathbf{C^T} \mathbf{M^f_{\mu^{-1}}} \mathbf{b} - \mathbf{M^e_\sigma} \mathbf{e} = \mathbf{s_e} + +H-J Formulation: +**************** + +.. math :: + \mathbf{C^T} \mathbf{M^f_\rho} \mathbf{j} + i \omega \mathbf{M^e_\mu} \mathbf{h} = \mathbf{s_m} \\ + \mathbf{C} \mathbf{h} - \mathbf{j} = \mathbf{s_e} + + +.. Forward Problem +.. =============== + +.. Inverse Problem +.. =============== API === diff --git a/docs/images/ebjhdiscretizations.png b/docs/images/ebjhdiscretizations.png new file mode 100644 index 0000000000000000000000000000000000000000..85bfbd361b723745160fadc193c247cd9de478fe GIT binary patch literal 37799 zcmeFZRa9Ns7A=Ys90I|e;2tc%#zSy-x8Uw>AxLn71`iq>g1bY4ySqCZcesnHQ*}7^ z`~AGuF0Jtad#x#B4x4@SvA@d8ilHFlBSJtxph$>+P=tVh8UcO@U%>(YS@!;}1OWjJ zHWv|*mk<#lm3Oc;F}E^?fOsDfp|WT@uZ$(|)mA$PTOhL{67@}3nTnEDJpq1651d47 z?yfL?{oo8;&L3wRe5Gm5nsIBDj7un6G;U6;QOA*&>mgx*ZP&Ea#J z6Eusx{9I^PTBg*f(^D5y=F7e9%6~k{A=s;yyq+Y1r@wTn!;ujam64Rdt4fA2EJS@S zn4jk<+n~(^#nclhHr1&|*1p6AKezwn=&V|h5B)Y>*h-DhoA|`sFRp;M6peK0;|-X0 zS!m$|v_`~46{2q7maxvy$wo+m_Fid^#xNl)XS$khjk4>3jrGO~`Hb*$HH94G9)81s zhl0ytvcZ^ogPi=+Fo!Lt=84D2?&%g))LLWIq^@i41iEjjWxYffGeyhRH6E4ko0H}q zZORBYX2H|%$(}hKH-EUW_mz%Gm%i3nTn84&x0nsaQPo*HxDW9c2BlgyaE0;(;9x8l zjtj?ZQwYYJI2P-LcGF-mV~`e5&8Qyz)gEsyu~hj^m^U}4=6imHZFv`yYEGZw2H)0+T9pKS za$CmqP)>hZA$TN*;u1>uJpBqg1ex5`kemPDc{>>)RW|M=fR^q&i#I^FZ 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