AFMEasyPlaneAnisotropySC

Calculates a residual contribution for the magnetic anisotropy energy.

Overview

Computes the residual and jacobian entries from the LLG equation due to a free energy density, $var element = document.getElementById("moose-equation-cbbfaa97-e1ff-45be-8ff7-1a76c69dcd79");katex.render("f_\\mathrm{easy-plane}", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ , corresponding to easy-plane magnetocrystalline anisotropy,

$var element = document.getElementById("moose-equation-1d3af183-f16a-4d7b-9abf-d1dbd9afb419");katex.render(" \\begin{aligned} f_\\mathrm{easy-plane} = \\sum\\limits_{\\eta = 1,2} K_1 \\left(\\textbf{m}_\\eta \\cdot \\hat{\\mathbf{P}}\\right)^2 \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$

where $var element = document.getElementById("moose-equation-7d7975cd-67f4-4f01-9bae-85f7685a1100");katex.render("\\hat{\\mathbf{P}}", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ is the director of the electric polarization field and sublattice indices $var element = document.getElementById("moose-equation-94b7b7e3-0e0a-4a26-ba4a-6e950685b1d2");katex.render("\\eta = 1,2", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ . The suffix SC corresponds to a strongly-coupled approach where $var element = document.getElementById("moose-equation-4ef8466e-b989-4685-b140-d9a08a165324");katex.render("\\mathbf{P}", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ is a MOOSE Variable. This requires us to compute off-diagonal jacobian entries for each component of $var element = document.getElementById("moose-equation-71823b4b-775c-40ad-b5db-71a1c1fd383a");katex.render("\\mathbf{P}", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ . The effective field due to this term can be calculated, in index notation,

$var element = document.getElementById("moose-equation-ad3cfb64-158f-4e50-a22d-2aa434a301ac");katex.render(" \\begin{aligned} H_{k\\eta}^\\mathrm{easy-plane} &= - \\frac{1}{\\mu_0 M_s} \\frac{\\delta f_\\mathrm{easy-plane}}{\\delta m_{k\\eta}}\\\\ &= - \\frac{2}{\\mu_0 M_s} \\hat{P}_k (m_{j\\eta} \\hat{P}_j)^2 \\\\ \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$

The effective field is evaluated at every time step of the two sublattice LLG-LLB equation,

$(1)var element = document.getElementById("moose-equation-75f43423-45cf-4f7d-bb97-4abc50ee7516");katex.render(" \\begin{aligned} \\frac{\\partial \\mathbf{m}_\\eta}{\\partial t} = -\\frac{\\gamma}{(1+\\alpha^2)} \\mathbf{m}_\\eta\\times \\mathbf{H}_\\eta - \\frac{\\gamma \\alpha}{1+\\alpha^2} \\mathbf{m}_\\eta \\times \\mathbf{m}_\\eta\\times \\mathbf{H}_\\eta, \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$

where $var element = document.getElementById("moose-equation-2a418b26-2a45-40f3-987c-760e2d5c4391");katex.render("\\gamma", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ is the electron gyromagnetic factor and $var element = document.getElementById("moose-equation-b00cbb50-89ef-4be5-90c2-7e24bcdf878f");katex.render("\\alpha", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ the phenomenological Gilbert damping parameter. Ignoring the time derivative, moving over the RHS to the LHS, and multiplying by a test function $var element = document.getElementById("moose-equation-5fe71f23-46d4-40d1-883b-c76f46439bca");katex.render("\\psi_h", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ , we have

$var element = document.getElementById("moose-equation-b85f337b-d2f4-420a-9591-5c3dee3057f7");katex.render(" \\begin{aligned} \\frac{\\gamma}{(1+\\alpha^2)}\\left(\\psi_h, \\mathbf{m}_\\eta \\times \\mathbf{H}_\\eta\\right) + \\frac{\\gamma \\alpha}{1+\\alpha^2} \\left(\\psi_h, \\mathbf{m}_\\eta \\times \\mathbf{m}_\\eta \\times \\mathbf{H}_\\eta \\right) = 0 \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$

In index notation, we have, with $var element = document.getElementById("moose-equation-42d059ee-dd2a-4279-b385-9c607c4bd011");katex.render("\\eta", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ a free index,

$var element = document.getElementById("moose-equation-c24bbb45-d5ed-47b5-8c4a-d3f1258d2fa7");katex.render(" \\begin{aligned} \\mathcal{R}_{m_{n\\eta}} = \\frac{\\gamma}{(1+\\alpha^2)}\\left(\\psi_h, \\epsilon_{ijn} m_{i\\eta} H_{j\\eta}\\right) + \\frac{\\gamma \\alpha}{1+\\alpha^2} \\left(\\psi_h, \\epsilon_{ijk} \\epsilon_{kln} m_{i\\eta} m_{j\\eta} H_{l\\eta} \\right) \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$

Substituting our definitions of $var element = document.getElementById("moose-equation-42c3258d-69bf-4d24-9a23-2f812b746fa0");katex.render("H_{n\\eta}", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ we have the residual contribution for the components of $var element = document.getElementById("moose-equation-18b87810-e1f8-4b3a-9f3f-f223f984323b");katex.render("\\mathbf{m}_\\eta", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ ,

$var element = document.getElementById("moose-equation-b89eba5b-b413-445e-9d17-6ae55428b1e8");katex.render(" \\begin{aligned} \\mathcal{R}_{m_{n\\eta}} = - \\frac{2 \\gamma}{\\mu_0 M_s (1+\\alpha^2)} \\left(\\psi_h, \\epsilon_{ijn} m_{i\\eta} \\hat{P}_j (m_{q\\eta} \\hat{P}_q)^2 \\right) - \\frac{2 \\gamma \\alpha}{\\mu_0 M_s \\left(1+\\alpha^2\\right)} \\left(\\psi_h, \\epsilon_{ijk} \\epsilon_{kln} m_{i\\eta} m_{j\\eta} \\hat{P}_l (m_{q\\eta} \\hat{P}_q)^2 \\right). \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$

The jacobian contributions are calculated by,

$var element = document.getElementById("moose-equation-e1d99991-65c2-4779-b5e3-d3e09b82580e");katex.render(" \\begin{aligned} \\mathcal{J}_{m_{n\\eta},m_{\\beta\\eta}} &= \\frac{\\partial \\mathcal{R}_{m_{n\\eta}}}{\\partial m_{\\beta\\eta}} \\\\ &= \\frac{\\partial}{\\partial m_{\\beta\\eta}} \\left\\{- \\frac{2 \\gamma}{\\mu_0 M_s (1+\\alpha^2)} \\left(\\psi_h, \\epsilon_{ijn} m_{i\\eta} \\hat{P}_j (m_{q\\eta} \\hat{P}_q)^2 \\right) - \\frac{2 \\gamma \\alpha}{\\mu_0 M_s \\left(1+\\alpha^2\\right)} \\left(\\psi_h, \\epsilon_{ijk} \\epsilon_{kln} m_{i\\eta} m_{j\\eta} \\hat{P}_l (m_{q\\eta} \\hat{P}_q)^2 \\right)\\right\\} \\\\ &= - \\frac{2 \\gamma}{\\mu_0 M_s (1+\\alpha^2)} \\left( \\psi_h, \\epsilon_{ijn} \\hat{P}_j \\hat{P}_q m_{q\\eta} \\phi \\left\\{\\delta_{i\\beta} \\left(m_{r\\eta} \\hat{P}_r\\right) + 2 m_{i\\eta} \\hat{P}_\\beta \\right\\} \\right) - \\frac{2 \\gamma \\alpha}{\\mu_0 M_s (1+\\alpha^2)} \\left( \\psi_h, \\epsilon_{ijk} \\epsilon_{kln} \\hat{P}_l m_{q\\eta} \\hat{P}_q \\phi \\left\\{ m_{r\\eta} \\hat{P}_r \\left(\\delta_{i\\beta} + \\delta_{j\\beta}\\right) + 2 m_{i\\eta} m_{j\\eta} \\right\\} \\right). \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$

where we use $var element = document.getElementById("moose-equation-6bbce708-6098-40de-bf70-e5767f3dc2da");katex.render("\\partial m_{j\\eta} / \\partial m_{k\\eta} = \\delta_{jk} \\phi", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ with $var element = document.getElementById("moose-equation-0924b70a-9630-42b8-a09f-2927fc7d21b1");katex.render("\\phi", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ a shape function of the finite element method. Since we are computing the SC version of this Kernel, then $var element = document.getElementById("moose-equation-b62702ad-7bce-4185-9639-81b529ff774d");katex.render("\\mathbf{P}", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ is a MOOSE Variable. Therefore we need,

$var element = document.getElementById("moose-equation-8a9dc7ec-9ece-4321-a433-e19cf8f26fe3");katex.render(" \\begin{aligned} \\mathcal{J}_{m_{n\\eta},P_{\\beta}} &= \\frac{\\partial \\mathcal{R}_{m_{n\\eta}}}{\\partial P_{\\beta}} \\\\ \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$

Note that $var element = document.getElementById("moose-equation-4faf9d07-0130-4917-9154-338928ddde46");katex.render("\\hat{P}_q", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ is the component of the normalized director $var element = document.getElementById("moose-equation-4f959cd5-2192-4616-b2ba-e7f12e5b3519");katex.render("\\hat{\\mathbf{P}} = \\mathbf{P} / \\sqrt{P_k P_k}", element, {displayMode:false,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ so we will have many additional terms due to the chain rule.

Example Input File Syntax

Input Parameters

componentAn integer corresponding to the direction in order parameter space this kernel acts in (e.g. for unrotated functionals 0 for q_x, 1 for q_y, 2 for q_z).
C++ Type:unsigned int
Controllable:No
Description:An integer corresponding to the direction in order parameter space this kernel acts in (e.g. for unrotated functionals 0 for q_x, 1 for q_y, 2 for q_z).
mag1_xThe x component of the constrained 1st sublattice magnetization vector
C++ Type:std::vector<VariableName>
Controllable:No
Description:The x component of the constrained 1st sublattice magnetization vector
mag1_yThe y component of the constrained 1st sublattice magnetization vector
C++ Type:std::vector<VariableName>
Controllable:No
Description:The y component of the constrained 1st sublattice magnetization vector
mag1_zThe z component of the constrained 1st sublattice magnetization vector
C++ Type:std::vector<VariableName>
Controllable:No
Description:The z component of the constrained 1st sublattice magnetization vector
mag_subAn integer corresponding to the sublattice this Kernel acts on
C++ Type:unsigned int
Controllable:No
Description:An integer corresponding to the sublattice this Kernel acts on
polar_xThe x component of the polarization
C++ Type:std::vector<VariableName>
Controllable:No
Description:The x component of the polarization
polar_yThe y component of the polarization
C++ Type:std::vector<VariableName>
Controllable:No
Description:The y component of the polarization
polar_zThe z component of the polarization
C++ Type:std::vector<VariableName>
Controllable:No
Description:The z component of the polarization
variableThe name of the variable that this residual object operates on
C++ Type:NonlinearVariableName
Controllable:No
Description:The name of the variable that this residual object operates on

Required Parameters

blockThe list of blocks (ids or names) that this object will be applied
C++ Type:std::vector<SubdomainName>
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
displacementsThe displacements
C++ Type:std::vector<VariableName>
Controllable:No
Description:The displacements
mag2_xThe x component of the constrained 2nd sublattice magnetization vector
C++ Type:std::vector<VariableName>
Controllable:No
Description:The x component of the constrained 2nd sublattice magnetization vector
mag2_yThe y component of the constrained 2nd sublattice magnetization vector
C++ Type:std::vector<VariableName>
Controllable:No
Description:The y component of the constrained 2nd sublattice magnetization vector
mag2_zThe z component of the constrained 2nd sublattice magnetization vector
C++ Type:std::vector<VariableName>
Controllable:No
Description:The z component of the constrained 2nd sublattice magnetization vector
prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.

Optional Parameters

absolute_value_vector_tagsThe tags for the vectors this residual object should fill with the absolute value of the residual contribution
C++ Type:std::vector<TagName>
Controllable:No
Description:The tags for the vectors this residual object should fill with the absolute value of the residual contribution
extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the matrices this Kernel should fill
extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the vectors this Kernel should fill
matrix_tagssystemThe tag for the matrices this Kernel should fill
Default:system
C++ Type:MultiMooseEnum
Options:nontime, system
Controllable:No
Description:The tag for the matrices this Kernel should fill
vector_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime
C++ Type:MultiMooseEnum
Options:nontime, time
Controllable:No
Description:The tag for the vectors this Kernel should fill

Tagging Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
diag_save_inThe name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Controllable:No
Description:The name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
save_inThe name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Controllable:No
Description:The name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Controllable:No
Description:The seed for the master random number generator
use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Controllable:No
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

Input Files

(tutorial/BFO_P111_TO_P111b_switch_m1_a1.i)

(tutorial/BFO_P111_TO_P111b_switch_m1_a1.i)



alphadef = 0.003

endtdef = 0.00223

efreq = 600

Eadef = -1.8e3

[Mesh]
  [fileload]
    type = FileMeshGenerator
    file = out_BFOMDL_P111A111_m1.e
    use_for_exodus_restart = true
  []
[]


[GlobalParams]
  len_scale = 1.0

  mag1_x = mag1_x
  mag1_y = mag1_y
  mag1_z = mag1_z

  mag2_x = mag2_x
  mag2_y = mag2_y
  mag2_z = mag2_z

  polar_x = polar_x
  polar_y = polar_y
  polar_z = polar_z

  antiphase_A_x = antiphase_A_x
  antiphase_A_y = antiphase_A_y
  antiphase_A_z = antiphase_A_z

  displacements = 'u_x u_y u_z'

  E_x = E_x
  E_y = E_y
  E_z = E_z

[]

[Functions]
  [./bc_func_1]
    type = ParsedFunction
    value = 'st'
    vars = 'st '
    vals = '5e2'
  [../]
[]

[Materials]

  [./constants] # Constants used in other material properties
    type = GenericConstantMaterial
    prop_names = '  alpha      De       D0          g0mu0Ms        g0           K1        K1c      Kt     '
    prop_values = '0.003     3.7551    0.003       48291.9      48291.9      -5.0068  -0.00550748 -0.000365997 '
  [../]

  [./a_long]
    type = GenericFunctionMaterial
    prop_names = 'alpha_long'
    prop_values = 'bc_func_1'
  [../]

  [./Landau_P]
    type = GenericConstantMaterial
    prop_names = 'alpha1 alpha11 alpha12 alpha111 alpha112 alpha123 alpha1111 alpha1112 alpha1122 alpha1123'
    prop_values = '-2.81296e3 1.72351e3 2.24147e3 0.0 0.0 0.0 0.0 0.0 0.0 0.0'
  [../]

  [./Landau_A]
    type = GenericConstantMaterial
    prop_names = 'beta1 beta11 beta12 beta111 beta112 beta123 beta1111 beta1112 beta1122 beta1123'
    prop_values = '-0.0137763e3 0.0000349266e3 0.0000498846e3 0.0 0.0 0.0 0.0 0.0 0.0 0.0'
  [../]

  [./P_A_couple]
    type = GenericConstantMaterial
    prop_names = 't1111 t1122 t1212 t42111111 t24111111 t42111122 t24112222 t42112233 t24112233 t42112211 t24111122 t42111212   t42123312 t24121112 t24121233 t6211111111 t2611111111 t6211111122 t2611222222 t4411111111 t4411112222'
    prop_values = '0.012516e3 0.0180504e3 -0.036155e3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0'
  [../]

  [./mat_C]
    type = GenericConstantMaterial
    prop_names = 'C11 C12 C44'
    prop_values = '295.179e3 117.567e3 74.0701e3'
  [../]

  [./mat_Q]
    type = GenericConstantMaterial
    prop_names = 'Q11 Q12 Q44'   
    prop_values = '-0.0603833 0.0111245 -0.0175686'
  [../]

  [./mat_R]
    type = GenericConstantMaterial
    prop_names = 'R11 R12 R44'
    prop_values = '-0.0000878064 0.0000295306 0.0000627962'
  [../]

  [./mat_q]
    type = GenericConstantMaterial
    prop_names = 'q11 q12 q44'
    prop_values = '-30.4162e3 -5.01496e3 -10.4105e3'   

#the point is the following: use a slightly different definition of Q_ij than Hlinka

  [../]
  [./mat_r]
    type = GenericConstantMaterial
    prop_names = 'r11 r12 r44'
    prop_values = '-0.0379499e3 0.00373096e3 0.0372105e3' 
  [../]
  [./elasticity_tensor_1]
    type = ComputeElasticityTensor
    fill_method = symmetric9
    C_ijkl = '295.179e3 117.567e3 117.567e3 295.179e3 117.567e3 295.179e3 74.0701e3 74.0701e3 74.0701e3'
  [../]


  [./strain]
    type = ComputeSmallStrain
    global_strain = global_strain
  [../]

  [./global_strain]
    type = ComputeGlobalStrain
    scalar_global_strain = global_strain
    global_strain_uo = global_strain_uo
  [../]

  [./stress]
    type = ComputeLinearElasticStress
  [../]

  [./permitivitty_1]

    ###############################################
    ##
    ##  so-called background dielectric constant
    ##  (it encapsulates the motion of core electrons
    ##  at high frequency) = e_b*e_0 (here we use 
    ##  e_b = 10), see PRB. 74, 104014, (2006)
    ##
    ###############################################

    type = GenericConstantMaterial
    prop_names = 'permittivity'
    prop_values = '0.00008854187'
  [../]

[]



[Variables]

  [./mag1_x]
    order = FIRST
    family = LAGRANGE
    initial_from_file_var = mag1_x
    initial_from_file_timestep = 'LATEST'
  [../]
  [./mag1_y]
    order = FIRST
    family = LAGRANGE
    initial_from_file_var = mag1_y
    initial_from_file_timestep = 'LATEST'
  [../]
  [./mag1_z]
    order = FIRST
    family = LAGRANGE
    initial_from_file_var = mag1_z
    initial_from_file_timestep = 'LATEST'
  [../]

  [./mag2_x]
    order = FIRST
    family = LAGRANGE
    initial_from_file_var = mag2_x
    initial_from_file_timestep = 'LATEST'
  [../]
  [./mag2_y]
    order = FIRST
    family = LAGRANGE
    initial_from_file_var = mag2_y
    initial_from_file_timestep = 'LATEST'
  [../]
  [./mag2_z]
    order = FIRST
    family = LAGRANGE
    initial_from_file_var = mag2_z
    initial_from_file_timestep = 'LATEST'
  [../]


  [./u_x]
  [../]
  [./u_y]
  [../]
  [./u_z]
  [../]
  [./global_strain]
    order = SIXTH
    family = SCALAR
  [../]
  [./polar_x]
    order = FIRST
    family = LAGRANGE
    initial_from_file_var = polar_x
    initial_from_file_timestep = 'LATEST'
  [../]
  [./polar_y]
    order = FIRST
    family = LAGRANGE
    initial_from_file_var = polar_y
    initial_from_file_timestep = 'LATEST'
  [../]
  [./polar_z]
    order = FIRST
    family = LAGRANGE
    initial_from_file_var = polar_z
    initial_from_file_timestep = 'LATEST'
  [../]
  [./antiphase_A_x]
    order = FIRST
    family = LAGRANGE
    initial_from_file_var = antiphase_A_x
    initial_from_file_timestep = 'LATEST'
  [../]
  [./antiphase_A_y]
    order = FIRST
    family = LAGRANGE
    initial_from_file_var = antiphase_A_y
    initial_from_file_timestep = 'LATEST'
  [../]
  [./antiphase_A_z]
    order = FIRST
    family = LAGRANGE
    initial_from_file_var = antiphase_A_z
    initial_from_file_timestep = 'LATEST'
  [../]

[]


[AuxVariables]
  [./mag1_s]
    order = FIRST
    family = LAGRANGE
  [../]
  [./mag2_s]
    order = FIRST
    family = LAGRANGE
  [../]

  [./Neel_L_x]
    order = FIRST
    family = LAGRANGE
  [../]
  [./Neel_L_y]
    order = FIRST
    family = LAGRANGE
  [../]
  [./Neel_L_z]
    order = FIRST
    family = LAGRANGE
  [../]


  [./SSMag_x]
    order = FIRST
    family = LAGRANGE
  [../]
  [./SSMag_y]
    order = FIRST
    family = LAGRANGE
  [../]
  [./SSMag_z]
    order = FIRST
    family = LAGRANGE
  [../]

  [./ph]
    order = FIRST
    family = LAGRANGE
  [../]

  [./th1]
    order = FIRST
    family = LAGRANGE
  [../]
  [./th2]
    order = FIRST
    family = LAGRANGE
  [../]

  [./disp_x]
  [../]
  [./disp_y]
  [../]
  [./disp_z]
  [../]
  [./s00]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./s01]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./s10]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./s11]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./e00]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./e01]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./e10]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./e11]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./e22]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./e12]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./e21]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./e02]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./e20]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./E_x]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./E_y]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./E_z]
    order = CONSTANT
    family = MONOMIAL
  [../]

  [./sublat1_phi]
    order = FIRST
    family = LAGRANGE
  [../]
  [./sublat1_th]
    order = FIRST
    family = LAGRANGE
  [../]
  [./sublat2_phi]
    order = FIRST
    family = LAGRANGE
  [../]
  [./sublat2_th]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Kernels]
  [./TensorMechanics]
  [../]

  [./rotostr_ux]
    type = RotostrictiveCouplingDispDerivative
    variable = u_x
    component = 0
  [../]
  [./rotostr_uy]
    type = RotostrictiveCouplingDispDerivative
    variable = u_y
    component = 1
  [../]
  [./rotostr_uz]
    type = RotostrictiveCouplingDispDerivative
    variable = u_z
    component = 2
  [../]

  [./electrostr_ux]
    type = ElectrostrictiveCouplingDispDerivative
    variable = u_x
    component = 0
  [../]
  [./electrostr_uy]
    type = ElectrostrictiveCouplingDispDerivative
    variable = u_y
    component = 1
  [../]
  [./electrostr_uz]
    type = ElectrostrictiveCouplingDispDerivative
    variable = u_z
    component = 2
  [../]

  ### Operators for the polar field: ###
  [./bed_x]
    type = BulkEnergyDerivativeEighth
    variable = polar_x
    component = 0
  [../]
  [./bed_y]
    type = BulkEnergyDerivativeEighth
    variable = polar_y
    component = 1
  [../]
  [./bed_z]
    type = BulkEnergyDerivativeEighth
    variable = polar_z
    component = 2
  [../]

  [./roto_polar_coupled_x]
    type = RotoPolarCoupledEnergyPolarDerivativeAlt
    variable = polar_x
    component = 0
  [../]
  [./roto_polar_coupled_y]
    type = RotoPolarCoupledEnergyPolarDerivativeAlt
    variable = polar_y
    component = 1
  [../]
  [./roto_polar_coupled_z]
    type = RotoPolarCoupledEnergyPolarDerivativeAlt
    variable = polar_z
    component = 2
  [../]
  [./roto_dis_coupled_x]
    type = RotoPolarCoupledEnergyDistortDerivativeAlt
    variable = antiphase_A_x
    component = 0
  [../]
  [./roto_dis_coupled_y]
    type = RotoPolarCoupledEnergyDistortDerivativeAlt
    variable = antiphase_A_y
    component = 1
  [../]
  [./roto_dis_coupled_z]
    type = RotoPolarCoupledEnergyDistortDerivativeAlt
    variable = antiphase_A_z
    component = 2
  [../]

  [./electrostr_polar_coupled_x]
    type = ElectrostrictiveCouplingPolarDerivative
    variable = polar_x
    component = 0
  u_x = disp_x
  u_y = disp_y
  u_z = disp_z

  [../]
  [./electrostr_polar_coupled_y]
    type = ElectrostrictiveCouplingPolarDerivative
    variable = polar_y
    component = 1
  u_x = disp_x
  u_y = disp_y
  u_z = disp_z

  [../]
  [./electrostr_polar_coupled_z]
    type = ElectrostrictiveCouplingPolarDerivative
    variable = polar_z
    component = 2
  u_x = disp_x
  u_y = disp_y
  u_z = disp_z

  [../]


  #Operators for the AFD field

  [./rbed_x]
    type = RotoBulkEnergyDerivativeEighthAlt
    variable = antiphase_A_x
    component = 0
  [../]
  [./rbed_y]
    type = RotoBulkEnergyDerivativeEighthAlt
    variable = antiphase_A_y
    component = 1
  [../]
  [./rbed_z]
    type = RotoBulkEnergyDerivativeEighthAlt
    variable = antiphase_A_z
    component = 2
  [../]

  [./rotostr_dis_coupled_x]
    type = RotostrictiveCouplingDistortDerivative
    variable = antiphase_A_x
    component = 0
  u_x = disp_x
  u_y = disp_y
  u_z = disp_z
  [../]
  [./rotostr_dis_coupled_y]
    type = RotostrictiveCouplingDistortDerivative
    variable = antiphase_A_y
    component = 1
  u_x = disp_x
  u_y = disp_y
  u_z = disp_z
  [../]
  [./rotostr_dis_coupled_z]
    type = RotostrictiveCouplingDistortDerivative
    variable = antiphase_A_z
    component = 2
  u_x = disp_x
  u_y = disp_y
  u_z = disp_z
  [../]

  [./polar_electric_px]
     type = PolarElectricPStrongEConst
     variable = polar_x
     component = 0
  [../]
  [./polar_electric_py]
     type = PolarElectricPStrongEConst
     variable = polar_y
     component = 1
  [../]
  [./polar_electric_pz]
     type = PolarElectricPStrongEConst
     variable = polar_z
     component = 2
  [../]


  #---------------------------------------#
  #                                       #
  #          Time dependence              #
  #                                       #
  #---------------------------------------#

  [./mag1_x_time]
    type = TimeDerivative
    variable = mag1_x
  [../]
  [./mag1_y_time]
    type = TimeDerivative
    variable = mag1_y
  [../]
  [./mag1_z_time]
    type = TimeDerivative
    variable = mag1_z
  [../]

  [./mag2_x_time]
    type = TimeDerivative
    variable = mag2_x
  [../]
  [./mag2_y_time]
    type = TimeDerivative
    variable = mag2_y
  [../]
  [./mag2_z_time]
    type = TimeDerivative
    variable = mag2_z
  [../]


  #---------------------------------------#
  #                                       #
  #     AFM sublattice exchange           #
  #                                       #
  #---------------------------------------#

  [./afmex1_x]
    type = AFMSublatticeSuperexchange
    variable = mag1_x
    mag_sub = 0
    component = 0
  [../]
  [./afmex1_y]
    type = AFMSublatticeSuperexchange
    variable = mag1_y
    mag_sub = 0
    component = 1
  [../]
  [./afmex1_z]
    type = AFMSublatticeSuperexchange
    variable = mag1_z
    mag_sub = 0
    component = 2
  [../]

  [./afmex2_x]
    type = AFMSublatticeSuperexchange
    variable = mag2_x
    mag_sub = 1
    component = 0
  [../]
  [./afmex2_y]
    type = AFMSublatticeSuperexchange
    variable = mag2_y
    mag_sub = 1
    component = 1
  [../]
  [./afmex2_z]
    type = AFMSublatticeSuperexchange
    variable = mag2_z
    mag_sub = 1
    component = 2
  [../]

  #---------------------------------------#
  #                                       #
  #     AFM sublattice DMI                #
  #        !isStronglyCoupled=true        #
  #---------------------------------------#

  [./afmdmi1_x]
    type = AFMSublatticeDMInteractionSC
    variable = mag1_x
    mag_sub = 0
    component = 0
  [../]
  [./afmdmi1_y]
    type = AFMSublatticeDMInteractionSC
    variable = mag1_y
    mag_sub = 0
    component = 1
  [../]
  [./afmdmi1_z]
    type = AFMSublatticeDMInteractionSC
    variable = mag1_z
    mag_sub = 0
    component = 2
  [../]

  [./afmdmi2_x]
    type = AFMSublatticeDMInteractionSC
    variable = mag2_x
    mag_sub = 1
    component = 0
  [../]
  [./afmdmi2_y]
    type = AFMSublatticeDMInteractionSC
    variable = mag2_y
    mag_sub = 1
    component = 1
  [../]
  [./afmdmi2_z]
    type = AFMSublatticeDMInteractionSC
    variable = mag2_z
    mag_sub = 1
    component = 2
  [../]

  #---------------------------------------#
  #                                       #
  #   Magnetocrystalline anisotropy for   #
  #   the AFM sublattice in easy-plane    #
  #        !isStronglyCoupled=true        #
  #---------------------------------------#

  [./afma1_x]
    type = AFMEasyPlaneAnisotropySC
    variable = mag1_x
    mag_sub = 0
    component = 0
  [../]
  [./afma1_y]
    type = AFMEasyPlaneAnisotropySC
    variable = mag1_y
    mag_sub = 0
    component = 1
  [../]
  [./afma1_z]
    type = AFMEasyPlaneAnisotropySC
    variable = mag1_z
    mag_sub = 0
    component = 2
  [../]


  [./afma2_x]
    type = AFMEasyPlaneAnisotropySC
    variable = mag2_x
    mag_sub = 1
    component = 0
  [../]
  [./afma2_y]
    type = AFMEasyPlaneAnisotropySC
    variable = mag2_y
    mag_sub = 1
    component = 1
  [../]
  [./afma2_z]
    type = AFMEasyPlaneAnisotropySC
    variable = mag2_z
    mag_sub = 1
    component = 2
  [../]


  #---------------------------------------#
  #                                       #
  #   Single-ion anisotropy environment   #
  #   for the AFM sublattice in the       #
  #   degenerate easy-plane               #
  #          !isStronglyCoupled=true      #
  #---------------------------------------#

  [./afmsia1_x]
    type = AFMSingleIonCubicSixthAnisotropySC
    variable = mag1_x
    mag_sub = 0
    component = 0
  [../]
  [./afmsia1_y]
    type = AFMSingleIonCubicSixthAnisotropySC
    variable = mag1_y
    mag_sub = 0
    component = 1
  [../]
  [./afmsia1_z]
    type = AFMSingleIonCubicSixthAnisotropySC
    variable = mag1_z
    mag_sub = 0
    component = 2
  [../]

  [./afmsia2_x]
    type = AFMSingleIonCubicSixthAnisotropySC
    variable = mag2_x
    mag_sub = 1
    component = 0
  [../]
  [./afmsia2_y]
    type = AFMSingleIonCubicSixthAnisotropySC
    variable = mag2_y
    mag_sub = 1
    component = 1
  [../]
  [./afmsia2_z]
    type = AFMSingleIonCubicSixthAnisotropySC
    variable = mag2_z
    mag_sub = 1
    component = 2
  [../]

  #---------------------------------------#
  #                                       #
  #          LLB constraint terms         #
  #                                       #
  #---------------------------------------#

  [./llb1_x]
    type = LongitudinalLLB
    variable = mag1_x
    mag_x = mag1_x
    mag_y = mag1_y
    mag_z = mag1_z
    component = 0
  [../]
  [./llb1_y]
    type = LongitudinalLLB
    variable = mag1_y
    mag_x = mag1_x
    mag_y = mag1_y
    mag_z = mag1_z
    component = 1
  [../]

  [./llb1_z]
    type = LongitudinalLLB
    variable = mag1_z
    mag_x = mag1_x
    mag_y = mag1_y
    mag_z = mag1_z
    component = 2
  [../]

  [./llb2_x]
    type = LongitudinalLLB
    variable = mag2_x
    mag_x = mag2_x
    mag_y = mag2_y
    mag_z = mag2_z
    component = 0
  [../]
  [./llb2_y]
    type = LongitudinalLLB
    variable = mag2_y
    mag_x = mag2_x
    mag_y = mag2_y
    mag_z = mag2_z
    component = 1
  [../]

  [./llb2_z]
    type = LongitudinalLLB
    variable = mag2_z
    mag_x = mag2_x
    mag_y = mag2_y
    mag_z = mag2_z
    component = 2
  [../]

  #---------------------------------------#
  #                                       #
  #     Time dependence                   #
  #                                       #
  #---------------------------------------#

  [./polar_x_time]
    type = TimeDerivativeScaled
    variable=polar_x
    time_scale = 0.005
    block = '0'
  [../]
  [./polar_y_time]
    type = TimeDerivativeScaled
    variable=polar_y
    time_scale = 0.005
    block = '0'
  [../]
  [./polar_z_time]
    type = TimeDerivativeScaled
    variable = polar_z
    time_scale = 0.005
    block = '0'
  [../]

  [./a_x_time]
    type = TimeDerivativeScaled
    variable = antiphase_A_x
    time_scale = 0.00005
    block = '0'
  [../]
  [./a_y_time]
    type = TimeDerivativeScaled
    variable = antiphase_A_y
    time_scale = 0.00005
    block = '0'
  [../]
  [./a_z_time]
    type = TimeDerivativeScaled
    variable = antiphase_A_z
    time_scale = 0.00005
    block = '0'
  [../]
[]


[AuxKernels]

  [./mag1_mag]
    type = VectorMag
    variable = mag1_s
    vector_x = mag1_x
    vector_y = mag1_y
    vector_z = mag1_z
    execute_on = 'initial timestep_end final'
  [../]

  [./mag2_mag]
    type = VectorMag
    variable = mag2_s
    vector_x = mag2_x
    vector_y = mag2_y
    vector_z = mag2_z
    execute_on = 'initial timestep_end final'
  [../]


  [./Neel_Lx]
    type = VectorDiffOrSum
    variable = Neel_L_x
    var1 = mag1_x
    var2 = mag2_x
    diffOrSum = 0
    execute_on = 'initial timestep_end final'
  [../]
  [./Neel_Ly]
    type = VectorDiffOrSum
    variable = Neel_L_y
    var1 = mag1_y
    var2 = mag2_y
    diffOrSum = 0
    execute_on = 'initial timestep_end final'
  [../]
  [./Neel_Lz]
    type = VectorDiffOrSum
    variable = Neel_L_z
    var1 = mag1_z
    var2 = mag2_z
    diffOrSum = 0
    execute_on = 'initial timestep_end final'
  [../]

  [./smallSignalMag_x]
    type = VectorDiffOrSum
    variable = SSMag_x
    var1 = mag1_x
    var2 = mag2_x
    diffOrSum = 1
    execute_on = 'initial timestep_end final'
  [../]
  [./smallSignalMag_y]
    type = VectorDiffOrSum
    variable = SSMag_y
    var1 = mag1_y
    var2 = mag2_y
    diffOrSum = 1
    execute_on = 'initial timestep_end final'
  [../]
  [./smallSignalMag_z]
    type = VectorDiffOrSum
    variable = SSMag_z
    var1 = mag1_z
    var2 = mag2_z
    diffOrSum = 1
    execute_on = 'initial timestep_end final'
  [../]


  [./phc]
    type = AngleBetweenTwoVectors
    variable = ph
    var1x = mag1_x
    var1y = mag1_y
    var1z = mag1_z
    var2x = mag2_x
    var2y = mag2_y
    var2z = mag2_z

    execute_on = 'initial timestep_end final'
  [../]

  [./th1c]
    type = AngleBetweenTwoVectors
    variable = th1
    var1x = mag1_x
    var1y = mag1_y
    var1z = mag1_z
    var2x = polar_x
    var2y = polar_y
    var2z = polar_z

    execute_on = 'initial timestep_end final'
  [../]

  [./th2c]
    type = AngleBetweenTwoVectors
    variable = th2
    var1x = mag2_x
    var1y = mag2_y
    var1z = mag2_z
    var2x = polar_x
    var2y = polar_y
    var2z = polar_z

    execute_on = 'initial timestep_end final'
  [../]


  [./disp_x]
    type = GlobalDisplacementAux
    variable = disp_x
    scalar_global_strain = global_strain
    global_strain_uo = global_strain_uo
    component = 0
  [../]
  [./disp_y]
    type = GlobalDisplacementAux
    variable = disp_y
    scalar_global_strain = global_strain
    global_strain_uo = global_strain_uo
    component = 1
  [../]
  [./disp_z]
    type = GlobalDisplacementAux
    variable = disp_z
    scalar_global_strain = global_strain
    global_strain_uo = global_strain_uo
    component = 2
  [../]
  [./s00]
    type = RankTwoAux
    variable = s00
    rank_two_tensor = stress
    index_i = 0
    index_j = 0
  [../]
  [./s01]
    type = RankTwoAux
    variable = s01
    rank_two_tensor = stress
    index_i = 0
    index_j = 1
  [../]
  [./s10]
    type = RankTwoAux
    variable = s10
    rank_two_tensor = stress
    index_i = 1
    index_j = 0
  [../]
  [./s11]
    type = RankTwoAux
    variable = s11
    rank_two_tensor = stress
    index_i = 1
    index_j = 1
  [../]
  [./e00]
    type = RankTwoAux
    variable = e00
    rank_two_tensor = total_strain
    index_i = 0
    index_j = 0
  [../]
  [./e01]
    type = RankTwoAux
    variable = e01
    rank_two_tensor = total_strain
    index_i = 0
    index_j = 1
  [../]
  [./e10]
    type = RankTwoAux
    variable = e10
    rank_two_tensor = total_strain
    index_i = 1
    index_j = 0
  [../]
  [./e11]
    type = RankTwoAux
    variable = e11
    rank_two_tensor = total_strain
    index_i = 1
    index_j = 1
  [../]
  [./e12]
    type = RankTwoAux
    variable = e12
    rank_two_tensor = total_strain
    index_i = 1
    index_j = 2
  [../]
  [./e21]
    type = RankTwoAux
    variable = e21
    rank_two_tensor = total_strain
    index_i = 2
    index_j = 1
  [../]
  [./e20]
    type = RankTwoAux
    variable = e20
    rank_two_tensor = total_strain
    index_i = 2
    index_j = 0
  [../]
  [./e02]
    type = RankTwoAux
    variable = e02
    rank_two_tensor = total_strain
    index_i = 0
    index_j = 2
  [../]
  [./e22]
    type = RankTwoAux
    variable = e22
    rank_two_tensor = total_strain
    index_i = 2
    index_j = 2
  [../]

  [./ez]
    type = HarmonicFieldAux
    variable = E_z
    amplitude = ${Eadef}
    correction = 1.0
    frequency = ${efreq}
    tshift = 0.0
    ton = 0.0
    toff = 0.000944
    execute_on = 'initial timestep_end final'
  [../]


  [./mcsublat1_phi]
    type = SphericalCoordinateVector
    variable = sublat1_phi
    component = 0
    var1x = mag1_x
    var1y = mag1_y
    var1z = mag1_z
    execute_on = 'initial timestep_end final'
  [../]
  [./mcsublat1_th]
    type = SphericalCoordinateVector
    variable = sublat1_th
    component = 1
    var1x = mag1_x
    var1y = mag1_y
    var1z = mag1_z
    execute_on = 'initial timestep_end final'
  [../]
  [./mcsublat2_phi]
    type = SphericalCoordinateVector
    variable = sublat2_phi
    component = 0
    var1x = mag2_x
    var1y = mag2_y
    var1z = mag2_z
    execute_on = 'initial timestep_end final'
  [../]
  [./mcsublat2_th]
    type = SphericalCoordinateVector
    variable = sublat2_th
    component = 1
    var1x = mag2_x
    var1y = mag2_y
    var1z = mag2_z
    execute_on = 'initial timestep_end final'
  [../]
[]

[ScalarKernels]
  [./global_strain]
    type = GlobalStrain
    variable = global_strain
    global_strain_uo = global_strain_uo
  [../]
[]

[BCs]
  [./Periodic]
    [./xyz]
      auto_direction = 'x y z'
      variable = 'u_x u_y u_z polar_x polar_y polar_z antiphase_A_x antiphase_A_y antiphase_A_z mag1_x mag1_y mag1_z mag2_x mag2_y mag2_z'
    [../]
  [../]

  # fix center point location
  [./centerfix_x]
    type = DirichletBC
    boundary = 100
    variable = u_x
    value = 0
  [../]
  [./centerfix_y]
    type = DirichletBC
    boundary = 100
    variable = u_y
    value = 0
  [../]
  [./centerfix_z]
    type = DirichletBC
    boundary = 100
    variable = u_z
    value = 0
  [../]
[]


[Postprocessors]

 #---------------------------------------#
  #                                       #
  #     Average Mk = |m_k| and along      # 
  #     other directions                  #
  #                                       #
  #---------------------------------------#

  [./M1]
    type = ElementAverageValue
    variable = mag1_s
    execute_on = 'initial timestep_end final'
  [../]
  [./M2]
    type = ElementAverageValue
    variable = mag2_s
    execute_on = 'initial timestep_end final'
  [../]

  [./<m1x>]
    type = ElementAverageValue
    variable = mag1_x
    execute_on = 'initial timestep_end final'
  [../]
  [./<m1y>]
    type = ElementAverageValue
    variable = mag1_y
    execute_on = 'initial timestep_end final'
  [../]
  [./<m1z>]
    type = ElementAverageValue
    variable = mag1_z
    execute_on = 'initial timestep_end final'
  [../]

  [./<m2x>]
    type = ElementAverageValue
    variable = mag2_x
    execute_on = 'initial timestep_end final'
  [../]
  [./<m2y>]
    type = ElementAverageValue
    variable = mag2_y
    execute_on = 'initial timestep_end final'
  [../]
  [./<m2z>]
    type = ElementAverageValue
    variable = mag2_z
    execute_on = 'initial timestep_end final'
  [../]

  [./<Lx>]
    type = ElementAverageValue
    variable = Neel_L_x
    execute_on = 'initial timestep_end final'
  [../]
  [./<Ly>]
    type = ElementAverageValue
    variable = Neel_L_y
    execute_on = 'initial timestep_end final'
  [../]
  [./<Lz>]
    type = ElementAverageValue
    variable = Neel_L_z
    execute_on = 'initial timestep_end final'
  [../]

  [./<SSmx>]
    type = ElementAverageValue
    variable = SSMag_x
    execute_on = 'initial timestep_end final'
  [../]
  [./<SSmy>]
    type = ElementAverageValue
    variable = SSMag_y
    execute_on = 'initial timestep_end final'
  [../]
  [./<SSmz>]
    type = ElementAverageValue
    variable = SSMag_z
    execute_on = 'initial timestep_end final'
  [../]

  [./<ph>]
    type = ElementAverageValue
    variable = ph
    execute_on = 'initial timestep_end final'
  [../]
  [./<th1>]
    type = ElementAverageValue
    variable = th1
    execute_on = 'initial timestep_end final'
  [../]
  [./<th2>]
    type = ElementAverageValue
    variable = th2
    execute_on = 'initial timestep_end final'
  [../]


  [./<sl1phi>]
    type = ElementAverageValue
    variable = sublat1_phi
    execute_on = 'initial timestep_end final'
  [../]
  [./<sl1th>]
    type = ElementAverageValue
    variable = sublat1_th
    execute_on = 'initial timestep_end final'
  [../]
  [./<sl2phi>]
    type = ElementAverageValue
    variable = sublat2_phi
    execute_on = 'initial timestep_end final'
  [../]
  [./<sl2th>]
    type = ElementAverageValue
    variable = sublat2_th
    execute_on = 'initial timestep_end final'
  [../]


  #---------------------------------------#
  #                                       #
  #   Calculate exchange energy of        #
  #   the magnetic body                   #
  #                                       #
  #---------------------------------------#

  [./FafmSLexch]
    type = AFMSublatticeSuperexchangeEnergy
    execute_on = 'initial timestep_end final'
    mag1_x = mag1_x
    mag1_y = mag1_y
    mag1_z = mag1_z
    mag2_x = mag2_x
    mag2_y = mag2_y
    mag2_z = mag2_z
    energy_scale = 6241.51

  [../]
  [./FafmSLdmi]
    type = AFMSublatticeDMInteractionEnergy
    execute_on = 'initial timestep_end final'
    energy_scale = 6241.51
  [../]

  #---------------------------------------#
  #                                       #
  #   Calculate excess energy from missed #
  #   LLB targets                         #
  #                                       #
  #---------------------------------------#

  [./Fllb1]
    type = MagneticExcessLLBEnergy
    mag_x = mag1_x
    mag_y = mag1_y
    mag_z = mag1_z
    execute_on = 'initial timestep_end final'
  [../]
  [./Fllb2]
    type = MagneticExcessLLBEnergy
    mag_x = mag2_x
    mag_y = mag2_y
    mag_z = mag2_z
    execute_on = 'initial timestep_end final'
  [../]

  #---------------------------------------#
  #                                       #
  #   Calculate the anisotropy energy     #
  #                                       #
  #---------------------------------------#

  [./Fa1]
    type = AFMEasyPlaneAnisotropyEnergy
    execute_on = 'initial timestep_end final'
    mag_x = mag1_x
    mag_y = mag1_y
    mag_z = mag1_z
    energy_scale = 6241.51
  [../]
  [./Fa2]
    type = AFMEasyPlaneAnisotropyEnergy
    execute_on = 'initial timestep_end final'
    mag_x = mag2_x
    mag_y = mag2_y
    mag_z = mag2_z
    energy_scale = 6241.51
  [../]

  [./Fsia1]
    type = AFMSingleIonCubicSixthAnisotropyEnergy
    execute_on = 'initial timestep_end final'
    mag_x = mag1_x
    mag_y = mag1_y
    mag_z = mag1_z
    energy_scale = 6241.51
  [../]
  [./Fsia2]
    type = AFMSingleIonCubicSixthAnisotropyEnergy
    execute_on = 'initial timestep_end final'
    mag_x = mag2_x
    mag_y = mag2_y
    mag_z = mag2_z
    energy_scale = 6241.51
  [../]

  #---------------------------------------#
  #                                       #
  #   add all the energy contributions    #
  #   and calculate their percent change  #
  #                                       #
  #---------------------------------------#

  [./FtotMAG]
    type = LinearCombinationPostprocessor 
    pp_names = 'FafmSLexch FafmSLdmi Fa1 Fa2 Fsia1 Fsia2'
    pp_coefs = ' 1.0 1.0 1.0 1.0 1.0 1.0'
    execute_on = 'initial timestep_end final'
  [../]

  [./FtotLLB]
    type = LinearCombinationPostprocessor 
    pp_names = 'Fllb1 Fllb2'
    pp_coefs = ' 1.0 1.0'
    execute_on = 'initial timestep_end final'
  [../]

  [./Px]
    type = ElementAverageValue
    variable = polar_x
    execute_on = 'initial timestep_end final'
  [../]
  [./Py]
    type = ElementAverageValue
    variable = polar_y
    execute_on = 'initial timestep_end final'
  [../]
  [./Pz]
    type = ElementAverageValue
    variable = polar_z
    execute_on = 'initial timestep_end final'
  [../]

  [./Ax]
    type = ElementAverageValue
    variable = antiphase_A_x
    execute_on = 'initial timestep_end final'
  [../]
  [./Ay]
    type = ElementAverageValue
    variable = antiphase_A_y
    execute_on = 'initial timestep_end final'
  [../]
  [./Az]
    type = ElementAverageValue
    variable = antiphase_A_z
    execute_on = 'initial timestep_end final'
  [../]

  [./e00]
    type = ElementAverageValue
    variable = e00
    execute_on = 'initial timestep_end final'
  [../]
  [./e11]
    type = ElementAverageValue
    variable = e11
    execute_on = 'initial timestep_end final'
  [../]
  [./e22]
    type = ElementAverageValue
    variable = e22
    execute_on = 'initial timestep_end final'
  [../]
  [./e01]
    type = ElementAverageValue
    variable = e01
    execute_on = 'initial timestep_end final'
  [../]
  [./e12]
    type = ElementAverageValue
    variable = e12
    execute_on = 'initial timestep_end final'
  [../]
  [./e02]
    type = ElementAverageValue
    variable = e02
    execute_on = 'initial timestep_end final'
  [../]

  [./Ez]
    type = ElementAverageValue
    variable = E_z
    execute_on = 'initial timestep_end final'
  [../]

  [./dt]
     type = TimestepSize
  [../]

  [./FbP]
    type = BulkEnergyEighth
    execute_on = 'initial timestep_end final'
    energy_scale = 6.24151
  [../]
  [./FbA]
    type = RotoBulkEnergyEighth
    execute_on = 'initial timestep_end final'
    energy_scale = 6.24151
  [../]
  [./FcPA]
    type = RotoPolarCoupledEnergyEighth
    execute_on = 'initial timestep_end final'
    energy_scale = 6.24151
  [../]
  [./FcPu]
    type = ElectrostrictiveCouplingEnergy
    execute_on = 'initial timestep_end final'
    energy_scale = 6.24151
    u_x = disp_x
    u_y = disp_y
    u_z = disp_z
  [../]
  [./FcAu]
    type = RotostrictiveCouplingEnergy
    execute_on = 'initial timestep_end final'
    energy_scale = 6.24151
    u_x = disp_x
    u_y = disp_y
    u_z = disp_z
  [../]

  [./Felu]
    type = ElasticEnergy
    execute_on = 'initial timestep_end final'
    energy_scale = 6.24151
  [../]

  [./scSSMag_x]
    type = LinearCombinationPostprocessor
    pp_names = '<SSmx>'
    pp_coefs = ' 100 '
    execute_on = 'initial timestep_end final'
  [../]

  [./scSSMag_y]
    type = LinearCombinationPostprocessor
    pp_names = '<SSmy>'
    pp_coefs = ' 100 '
    execute_on = 'initial timestep_end final'
  [../]

  [./scSSMag_z]
    type = LinearCombinationPostprocessor
    pp_names = '<SSmz>'
    pp_coefs = ' 100 '
    execute_on = 'initial timestep_end final'
  [../]


  [./rA_x]
    type = LinearCombinationPostprocessor
    pp_names = 'Ax'
    pp_coefs = '0.017453277'
    execute_on = 'initial timestep_end final'
  [../]
  [./rA_y]
    type = LinearCombinationPostprocessor
    pp_names = 'Ay'
    pp_coefs = '0.017453277'
    execute_on = 'initial timestep_end final'
  [../]
  [./rA_z]
    type = LinearCombinationPostprocessor
    pp_names = 'Az'
    pp_coefs = '0.017453277'
    execute_on = 'initial timestep_end final'
  [../]


  [./FtotFER]
    type = LinearCombinationPostprocessor
    pp_names = 'FbP FbA FcPA FcPu FcAu Felu'
    pp_coefs = ' 1 1 1 1 1 1 '
    execute_on = 'initial timestep_end final'
  
    ##########################################
    #
    # NOTE: Ferret output is in attojoules
    #
    ##########################################
  [../]
  [./perc_change]
    type = EnergyRatePostprocessor
    postprocessor = FtotFER
    execute_on = 'initial timestep_end final'
    dt = dt
  [../]
  [./elapsed]
    type = PerfGraphData
    section_name = "Root"  # for profiling the problem
    data_type = total
  [../]
[]

[UserObjects]
  [./global_strain_uo]
    type = GlobalBFOMaterialRVEUserObject
    execute_on = 'Initial Linear Nonlinear'
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
    petsc_options_iname = '-ksp_gmres_restart -snes_atol  -snes_rtol  -ksp_rtol -pc_type '
    petsc_options_value = '    121            1e-8          1e-7       1e-5    bjacobi'
  [../]
[]

[Executioner]
  type = Transient            
  solve_type = 'PJFNK'


  [./TimeIntegrator]
    type = ImplicitEuler
  [../]
  dtmin = 1e-14
  dtmax = 1.0e-6

  [./TimeStepper]
    type = IterationAdaptiveDT
    optimal_iterations = 18     #usually 8-16
    linear_iteration_ratio = 100
    dt = 1.0e-8
  [../]

  num_steps = 150000

  end_time = ${endtdef}

[]

[Outputs]
  print_linear_residuals = false
  perf_graph_live = false
  [./out]
    type = Exodus
    file_base = out_P111-P111b-BFOMDL_m1_a1
    elemental_as_nodal = true
  [../]
  [./outCSV]
    type = CSV
    file_base = out_P111-P111b-BFOMDL_m1_a1
  [../]
#
[]