diff --git a/applications/CHAC_anisotropy/CHAC_anisotropy.md b/applications/CHAC_anisotropy/CHAC_anisotropy.md new file mode 100644 index 000000000..932b5d5c3 --- /dev/null +++ b/applications/CHAC_anisotropy/CHAC_anisotropy.md @@ -0,0 +1,123 @@ +## PRISMS-PF: CHAC Anisotropy (with Coupled CH-AC Dynamics) +Consider a free energy expression of the form: + +$$ +\begin{equation} + \Pi(c, \eta, \nabla \eta) = \int_{\Omega} \left( f_{\alpha}(1-H) + f_{\beta}H \right) + \frac{1}{2} | \gamma( \mathbf{n} ) \nabla \eta |^2 ~dV +\end{equation} +$$ + +where $f_{\alpha}$ and $f_{\beta}$ are the free energy densities corresponding to $\alpha$ and $\beta$ phases, respectively, and are functions of composition $c$. $H$ is a function of the structural order parameter $\eta$. The interface normal vector $\mathbf{n}$ is given by + +$$ +\begin{equation} +\mathbf{n} = \frac{\nabla \eta}{|\nabla \eta|} +\end{equation} +$$ + +for $\nabla \eta \ne \mathbf{0}$, and $\mathbf{n} = \mathbf{0}$ when $\nabla \eta = \mathbf{0}$. + +## Variational Treatment + +Following standard variational arguments (see Cahn-Hilliard formulation), we obtain the chemical potentials: + +$$ +\begin{align} + \mu_{c} &= (f_{\alpha,c}(1-H)+f_{\beta,c}H) \\ + \mu_{\eta} &= (f_{\beta,c}-f_{\alpha,c})H_{,\eta} - \nabla \cdot \mathbf{m} +\end{align} +$$ + +The components of the anisotropic gradient $\mathbf{m}$ are given by + +$$ +\begin{equation} +m_i = \gamma(\mathbf{n}) \left( \nabla \eta + |\nabla \eta| (\delta_{ij}-n_i n_j) \frac{\partial \gamma (\mathbf{n})}{n_j} \right), +\end{equation} +$$ + +where $\delta_{ij}$ is the Kronecker delta. + +## Kinetics + +Now the PDE for Cahn-Hilliard dynamics is given by: + +$$ +\begin{align} + \frac{\partial c}{\partial t} &= -~\nabla \cdot (-M_c\nabla \mu_c)\\ + &=M_c~\nabla \cdot (\nabla (f_{\alpha,c}(1-H)+f_{\beta,c}H)) + \end{align} +$$ + +and the PDE for Allen-Cahn dynamics is given by: + +$$ +\begin{align} + \frac{\partial \eta}{\partial t} &= -M_\eta \mu_\eta \\ + &=-M_\eta ~ ((f_{\beta,c}-f_{\alpha,c})H_{,\eta} - \nabla \cdot \mathbf{m}) +\end{align} +$$ + +where $M_c$ and $M_\eta$ are the constant mobilities. + +## Time discretization + +Considering forward Euler explicit time stepping, we have the time discretized kinetics equation: + +$$ +\begin{align} + \eta^{n+1} &= \eta^{n} - \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n - \nabla \cdot \mathbf{m}^n) \\ + c^{n+1} &= c^{n} + \Delta t M_{\eta}~\nabla \cdot (\nabla (f_{\alpha,c}^n(1-H^{n})+f_{\beta,c}^n H^{n})) +\end{align} +$$ + +## Weak formulation +In the weak formulation, considering an arbitrary variation $w$, the above equations can be expressed as residual equations: + +$$ +\begin{align} +\int_{\Omega} w \eta^{n+1} ~dV &= \int_{\Omega} w \eta^{n} - w \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n - \kappa \Delta \eta^n) ~dV +\end{align} +$$ + +$$ +\begin{align} +&= \int_{\Omega} w \left( \eta^{n} - \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n) \right)+ \nabla w \cdot (- \Delta t M_{\eta}) \mathbf{m}^n ~dV +\end{align} +$$ + +$$ +\begin{align} +r_{\eta} &= \eta^{n} - \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n) +\end{align} +$$ + +$$ +\begin{align} +r_{\eta x} &= (- \Delta t M_{\eta}) \mathbf{m}^n +\end{align} +$$ + +and + +$$ +\begin{align} +\int_{\Omega} w c^{n+1} ~dV &= \int_{\Omega} w c^{n} + w \Delta t M_{c}~ \nabla \cdot (\nabla (f_{\alpha,c}^n(1-H^{n})+f_{\beta,c}^n H^{n})) ~dV\\ +&= \int_{\Omega} w c^{n} + \nabla w (-\Delta t M_{c})~ [~(f_{\alpha,cc}^n(1-H^{n})+f_{\beta,cc}^n H^{n}) \nabla c + ~((f_{\beta,c}^n-f_{\alpha,c}^n)H^{n}_{,\eta} \nabla \eta) ] ~dV +\end{align} +$$ + +$$ +\begin{align} +r_c &= c^{n} +\end{align} +$$ + +$$ +\begin{align} +r_{cx} &= (-\Delta t M_{c})~ [~(f_{\alpha,cc}^n(1-H^{n})+f_{\beta,cc}^n H^{n}) \nabla c + ~((f_{\beta,c}^n-f_{\alpha,c}^n)H^{n}_{,\eta} \nabla \eta) ] +\end{align} +$$ + +The above values of $r_{\eta}$, $r_{\eta x}$, $r_{c}$ and $r_{cx}$ are used to define the residuals in the following equations file: +`applications/CHAC\_anisotropy/equations.h` diff --git a/applications/CHAC_anisotropy/CHAC_anisotropy.pdf b/applications/CHAC_anisotropy/CHAC_anisotropy.pdf deleted file mode 100644 index 7358fbc6a..000000000 Binary files a/applications/CHAC_anisotropy/CHAC_anisotropy.pdf and /dev/null differ diff --git a/applications/CHAC_anisotropy/tex_files/CHAC_anisotropy.tex b/applications/CHAC_anisotropy/tex_files/CHAC_anisotropy.tex deleted file mode 100644 index f9404c970..000000000 --- a/applications/CHAC_anisotropy/tex_files/CHAC_anisotropy.tex +++ /dev/null @@ -1,242 +0,0 @@ -\documentclass[10pt]{article} -\usepackage{amsmath} -\usepackage{bm} -\usepackage{bbm} -\usepackage{mathrsfs} -\usepackage{graphicx} -\usepackage{wrapfig} -\usepackage{subcaption} -\usepackage{epsfig} -\usepackage{amsfonts} -\usepackage{amssymb} -\usepackage{amsmath} -\usepackage{wrapfig} -\usepackage{graphicx} -\usepackage{psfrag} -\newcommand{\sun}{\ensuremath{\odot}} % sun symbol is \sun -\let\vaccent=\v % rename builtin command \v{} to \vaccent{} -\renewcommand{\v}[1]{\ensuremath{\mathbf{#1}}} % for vectors -\newcommand{\gv}[1]{\ensuremath{\mbox{\boldmath$ #1 $}}} -\newcommand{\grad}[1]{\gv{\nabla} #1} -\renewcommand{\baselinestretch}{1.2} -\jot 5mm -\graphicspath{{./figures/}} -%text dimensions -\textwidth 6.5 in -\oddsidemargin .2 in -\topmargin -0.2 in -\textheight 8.5 in -\headheight 0.2in -\overfullrule = 0pt -\pagestyle{plain} -\def\newpar{\par\vskip 0.5cm} -\begin{document} -% -%---------------------------------------------------------------------- -% Define symbols -%---------------------------------------------------------------------- -% -\def\iso{\mathbbm{1}} -\def\half{{\textstyle{1 \over 2}}} -\def\third{{\textstyle{1 \over 3}}} -\def\fourth{{\textstyle{{1 \over 4}}}} -\def\twothird{{\textstyle {{2 \over 3}}}} -\def\ndim{{n_{\rm dim}}} -\def\nint{n_{\rm int}} -\def\lint{l_{\rm int}} -\def\nel{n_{\rm el}} -\def\nf{n_{\rm f}} -\def\DIV {\hbox{\af div}} -\def\GRAD{\hbox{\af Grad}} -\def\sym{\mathop{\rm sym}\nolimits} -\def\tr{\mathop{\rm tr}\nolimits} -\def\dev{\mathop{\rm dev}\nolimits} -\def\Dev{\mathop{\rm Dev}\nolimits} -\def\DEV{\mathop {\rm DEV}\nolimits} -\def\bfb {{\bi b}} -\def\Bnabla{\nabla} -\def\bG{{\bi G}} -\def\jmpdelu{{\lbrack\!\lbrack \Delta u\rbrack\!\rbrack}} -\def\jmpudot{{\lbrack\!\lbrack\dot u\rbrack\!\rbrack}} -\def\jmpu{{\lbrack\!\lbrack u\rbrack\!\rbrack}} -\def\jmphi{{\lbrack\!\lbrack\varphi\rbrack\!\rbrack}} -\def\ljmp{{\lbrack\!\lbrack}} -\def\rjmp{{\rbrack\!\rbrack}} -\def\sign{{\rm sign}} -\def\nn{{n+1}} -\def\na{{n+\vartheta}} -\def\nna{{n+(1-\vartheta)}} -\def\nt{{n+{1\over 2}}} -\def\nb{{n+\beta}} -\def\nbb{{n+(1-\beta)}} -%--------------------------------------------------------- -% Bold Face Math Characters: -% All In Format: \B***** . -%--------------------------------------------------------- -\def\bOne{\mbox{\boldmath$1$}} -\def\BGamma{\mbox{\boldmath$\Gamma$}} -\def\BDelta{\mbox{\boldmath$\Delta$}} -\def\BTheta{\mbox{\boldmath$\Theta$}} -\def\BLambda{\mbox{\boldmath$\Lambda$}} -\def\BXi{\mbox{\boldmath$\Xi$}} -\def\BPi{\mbox{\boldmath$\Pi$}} -\def\BSigma{\mbox{\boldmath$\Sigma$}} -\def\BUpsilon{\mbox{\boldmath$\Upsilon$}} -\def\BPhi{\mbox{\boldmath$\Phi$}} -\def\BPsi{\mbox{\boldmath$\Psi$}} -\def\BOmega{\mbox{\boldmath$\Omega$}} -\def\Balpha{\mbox{\boldmath$\alpha$}} -\def\Bbeta{\mbox{\boldmath$\beta$}} -\def\Bgamma{\mbox{\boldmath$\gamma$}} -\def\Bdelta{\mbox{\boldmath$\delta$}} -\def\Bepsilon{\mbox{\boldmath$\epsilon$}} -\def\Bzeta{\mbox{\boldmath$\zeta$}} -\def\Beta{\mbox{\boldmath$\eta$}} -\def\Btheta{\mbox{\boldmath$\theta$}} -\def\Biota{\mbox{\boldmath$\iota$}} -\def\Bkappa{\mbox{\boldmath$\kappa$}} -\def\Blambda{\mbox{\boldmath$\lambda$}} -\def\Bmu{\mbox{\boldmath$\mu$}} -\def\Bnu{\mbox{\boldmath$\nu$}} -\def\Bxi{\mbox{\boldmath$\xi$}} -\def\Bpi{\mbox{\boldmath$\pi$}} -\def\Brho{\mbox{\boldmath$\rho$}} -\def\Bsigma{\mbox{\boldmath$\sigma$}} -\def\Btau{\mbox{\boldmath$\tau$}} -\def\Bupsilon{\mbox{\boldmath$\upsilon$}} -\def\Bphi{\mbox{\boldmath$\phi$}} -\def\Bchi{\mbox{\boldmath$\chi$}} -\def\Bpsi{\mbox{\boldmath$\psi$}} -\def\Bomega{\mbox{\boldmath$\omega$}} -\def\Bvarepsilon{\mbox{\boldmath$\varepsilon$}} -\def\Bvartheta{\mbox{\boldmath$\vartheta$}} -\def\Bvarpi{\mbox{\boldmath$\varpi$}} -\def\Bvarrho{\mbox{\boldmath$\varrho$}} -\def\Bvarsigma{\mbox{\boldmath$\varsigma$}} -\def\Bvarphi{\mbox{\boldmath$\varphi$}} -\def\bone{\mathbf{1}} -\def\bzero{\mathbf{0}} -%--------------------------------------------------------- -% Bold Face Math Italic: -% All In Format: \b* . -%--------------------------------------------------------- -\def\bA{\mbox{\boldmath$ A$}} -\def\bB{\mbox{\boldmath$ B$}} -\def\bC{\mbox{\boldmath$ C$}} -\def\bD{\mbox{\boldmath$ D$}} -\def\bE{\mbox{\boldmath$ E$}} -\def\bF{\mbox{\boldmath$ F$}} -\def\bG{\mbox{\boldmath$ G$}} -\def\bH{\mbox{\boldmath$ H$}} -\def\bI{\mbox{\boldmath$ I$}} -\def\bJ{\mbox{\boldmath$ J$}} -\def\bK{\mbox{\boldmath$ K$}} -\def\bL{\mbox{\boldmath$ L$}} -\def\bM{\mbox{\boldmath$ M$}} -\def\bN{\mbox{\boldmath$ N$}} -\def\bO{\mbox{\boldmath$ O$}} -\def\bP{\mbox{\boldmath$ P$}} -\def\bQ{\mbox{\boldmath$ Q$}} -\def\bR{\mbox{\boldmath$ R$}} -\def\bS{\mbox{\boldmath$ S$}} -\def\bT{\mbox{\boldmath$ T$}} -\def\bU{\mbox{\boldmath$ U$}} -\def\bV{\mbox{\boldmath$ V$}} -\def\bW{\mbox{\boldmath$ W$}} -\def\bX{\mbox{\boldmath$ X$}} -\def\bY{\mbox{\boldmath$ Y$}} -\def\bZ{\mbox{\boldmath$ Z$}} -\def\ba{\mbox{\boldmath$ a$}} -\def\bb{\mbox{\boldmath$ b$}} -\def\bc{\mbox{\boldmath$ c$}} -\def\bd{\mbox{\boldmath$ d$}} -\def\be{\mbox{\boldmath$ e$}} -\def\bff{\mbox{\boldmath$ f$}} -\def\bg{\mbox{\boldmath$ g$}} -\def\bh{\mbox{\boldmath$ h$}} -\def\bi{\mbox{\boldmath$ i$}} -\def\bj{\mbox{\boldmath$ j$}} -\def\bk{\mbox{\boldmath$ k$}} -\def\bl{\mbox{\boldmath$ l$}} -\def\bm{\mbox{\boldmath$ m$}} -\def\bn{\mbox{\boldmath$ n$}} -\def\bo{\mbox{\boldmath$ o$}} -\def\bp{\mbox{\boldmath$ p$}} -\def\bq{\mbox{\boldmath$ q$}} -\def\br{\mbox{\boldmath$ r$}} -\def\bs{\mbox{\boldmath$ s$}} -\def\bt{\mbox{\boldmath$ t$}} -\def\bu{\mbox{\boldmath$ u$}} -\def\bv{\mbox{\boldmath$ v$}} -\def\bw{\mbox{\boldmath$ w$}} -\def\bx{\mbox{\boldmath$ x$}} -\def\by{\mbox{\boldmath$ y$}} -\def\bz{\mbox{\boldmath$ z$}} -%********************************* -%Start main paper -%********************************* -\centerline{\Large{\bf PRISMS PhaseField}} -\smallskip -\centerline{\Large{\bf Anisotropy (with Coupled CH-AC Dynamics)}} -\bigskip - -Consider a free energy expression of the form: -\begin{equation} - \Pi(c, \eta, \grad \eta) = \int_{\Omega} \left( f_{\alpha}(1-H) + f_{\beta}H \right) + \frac{1}{2} | \gamma( \mathbf{n} ) \grad \eta |^2 ~dV -\end{equation} -where $f_{\alpha}$ and $f_{\beta}$ are the free energy densities corresponding to $\alpha$ and $\beta$ phases, respectively, and are functions of composition $c$. $H$ is a function of the structural order parameter $\eta$. The interface normal vector $\mathbf{n}$ is given by -\begin{equation} -\mathbf{n} = \frac{\grad \eta}{|\grad \eta|} -\end{equation} -for $\grad \eta \ne \mathbf{0}$, and $\mathbf{n} = \mathbf{0}$ when $\grad \eta = \mathbf{0}$. - -\section{Variational treatment} -Following standard variational arguments (see Cahn-Hilliard formulation), we obtain the chemical potentials: -\begin{align} - \mu_{c} &= (f_{\alpha,c}(1-H)+f_{\beta,c}H) \\ - \mu_{\eta} &= (f_{\beta,c}-f_{\alpha,c})H_{,\eta} - \grad \cdot \mathbf{m} -\end{align} -The components of the anisotropic gradient $\mathbf{m}$ are given by -\begin{equation} -m_i = \gamma(\mathbf{n}) \left( \grad \eta + |\grad \eta| (\delta_{ij}-n_i n_j) \frac{\partial \gamma (\mathbf{n})}{n_j} \right), -\end{equation} -where $\delta_{ij}$ is the Kronecker delta. - -\section{Kinetics} -Now the PDE for Cahn-Hilliard dynamics is given by: -\begin{align} - \frac{\partial c}{\partial t} &= -~\grad \cdot (-M_c\grad \mu_c)\\ - &=M_c~\grad \cdot (\grad (f_{\alpha,c}(1-H)+f_{\beta,c}H)) - \end{align} - and the PDE for Allen-Cahn dynamics is given by: - \begin{align} - \frac{\partial \eta}{\partial t} &= -M_\eta \mu_\eta \\ - &=-M_\eta ~ ((f_{\beta,c}-f_{\alpha,c})H_{,\eta} - \grad \cdot \mathbf{m}) -\end{align} -where $M_c$ and $M_\eta$ are the constant mobilities. - -\section{Time discretization} -Considering forward Euler explicit time stepping, we have the time discretized kinetics equation: -\begin{align} - \eta^{n+1} &= \eta^{n} - \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n - \grad \cdot \mathbf{m}^n) \\ -c^{n+1} &= c^{n} + \Delta t M_{\eta}~\grad \cdot (\grad (f_{\alpha,c}^n(1-H^{n})+f_{\beta,c}^n H^{n})) -\end{align} - -\section{Weak formulation} -In the weak formulation, considering an arbitrary variation $w$, the above equations can be expressed as residual equations: -\begin{align} - \int_{\Omega} w \eta^{n+1} ~dV &= \int_{\Omega} w \eta^{n} - w \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n - \kappa \Delta \eta^n) ~dV\\ - &=\int_{\Omega} w \left( \underbrace{\eta^{n} - \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n)}_{r_{\eta}} \right)+ \grad w \cdot \underbrace{(- \Delta t M_{\eta}) \mathbf{m}^n}_{r_{\eta x}} ~dV -\end{align} -and -\begin{align} - \int_{\Omega} w c^{n+1} ~dV &= \int_{\Omega} w c^{n} + w \Delta t M_{c}~ \grad \cdot (\grad (f_{\alpha,c}^n(1-H^{n})+f_{\beta,c}^n H^{n})) ~dV\\ - &= \int_{\Omega} w \underbrace{c^{n}}_{r_c} + \grad w \underbrace{(-\Delta t M_{c})~ [~(f_{\alpha,cc}^n(1-H^{n})+f_{\beta,cc}^n H^{n}) \grad c + ~((f_{\beta,c}^n-f_{\alpha,c}^n)H^{n}_{,\eta} \grad \eta) ] }_{r_{cx}} ~dV -\end{align} - -\vskip 0.25in -The above values of $r_{\eta}$, $r_{\eta x}$, $r_{c}$ and $r_{cx}$ are used to define the residuals in the following equations file: \\ -\textit{applications/CHAC\_anisotropy/equations.h} - - -\end{document} \ No newline at end of file diff --git a/applications/CHAC_anisotropyRegularized/CHAC_anisotropyRegularized.md b/applications/CHAC_anisotropyRegularized/CHAC_anisotropyRegularized.md new file mode 100644 index 000000000..e03525c51 --- /dev/null +++ b/applications/CHAC_anisotropyRegularized/CHAC_anisotropyRegularized.md @@ -0,0 +1,154 @@ +## PRISMS PhaseField: Regularized Anisotropy (with Coupled CH-AC Dynamics) + +Consider a free energy expression of the form: + +$$ +\begin{equation} + \Pi(c, \eta, \nabla \eta) = \int_{\Omega} \left( f_{\alpha}(1-H) + f_{\beta}H \right) + \frac{1}{2} | \gamma( \mathbf{n} ) \nabla \eta |^2 + \frac{\delta^2}{2} (\Delta \eta)^2 ~dV +\end{equation} +$$ + +where $f_{\alpha}$ and $f_{\beta}$ are the free energy densities corresponding to $\alpha$ and $\beta$ phases, respectively, and are functions of composition $c$. $H$ is a function of the structural order parameter $\eta$. $\delta$ is a scalar regularization parameter. The interface normal vector $\mathbf{n}$ is given by + +$$ +\begin{equation} +\mathbf{n} = \frac{\nabla \eta}{|\nabla \eta|} +\end{equation} +$$ + +for $\nabla \eta \ne \mathbf{0}$, and $\mathbf{n} = \mathbf{0}$ when $\nabla \eta = \mathbf{0}$. + +## Variational Treatment +Following standard variational arguments (see Cahn-Hilliard formulation), we obtain the chemical potentials: + +$$ +\begin{align} + \mu_{c} &= (f_{\alpha,c}(1-H)+f_{\beta,c}H) \\ + \mu_{\eta} &= (f_{\beta,c}-f_{\alpha,c})H_{,\eta} - \nabla \cdot \mathbf{m} + \delta^2 \Delta(\Delta \eta) +\end{align} +$$ + +The components of the anisotropic gradient $\mathbf{m}$ are given by + +$$ +\begin{equation} +m_i = \gamma(\mathbf{n}) \left( \nabla \eta + |\nabla \eta| (\delta_{ij}-n_i n_j) \frac{\partial \gamma (\mathbf{n})}{n_j} \right), +\end{equation} +$$ + +where $\delta_{ij}$ is the Kronecker delta. + +## Kinetics +Now the PDE for Cahn-Hilliard dynamics is given by: + +$$ +\begin{align} +\frac{\partial c}{\partial t} &= -~\nabla \cdot (-M_c\nabla \mu_c)\\ +&=M_c~\nabla \cdot (\nabla (f_{\alpha,c}(1-H)+f_{\beta,c}H)) +\end{align} +$$ + +and the PDE for Allen-Cahn dynamics is given by: + +$$ +\begin{align} +\frac{\partial \eta}{\partial t} &= -M_\eta \mu_\eta \\ +&=-M_\eta ~ [(f_{\beta,c}-f_{\alpha,c})H_{,\eta} - \nabla \cdot \mathbf{m} + \delta^2 \Delta(\Delta \eta)] +\end{align} +$$ + +where $M_c$ and $M_\eta$ are the constant mobilities. In order that the formulation only includes second order derivatives, an auxiliary field $\phi$ is introduced to break up the biharmonic term: + +$$ +\begin{align} +\phi = \Delta \eta +\end{align} +$$ + +and the PDE for Allen-Cahn dynamics becomes + +$$ +\begin{align} + \frac{\partial \eta}{\partial t} =-M_\eta ~ ((f_{\beta,c}-f_{\alpha,c})H_{,\eta} - \nabla \cdot \mathbf{m}) + \delta^2 \Delta \phi . +\end{align} +$$ + +## Time discretization +Considering forward Euler explicit time stepping, we have the time discretized kinetics equation: + +$$ +\begin{align} + \phi^{n+1} &= \Delta \eta^n \\ + \eta^{n+1} &= \eta^{n} - \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n - \nabla \cdot \mathbf{m}^n + \delta^2 \Delta \phi^n) \\ +c^{n+1} &= c^{n} + \Delta t M_{\eta}~\nabla \cdot (\nabla (f_{\alpha,c}^n(1-H^{n})+f_{\beta,c}^n H^{n})) +\end{align} +$$ + +## Weak formulation +In the weak formulation, considering an arbitrary variation $w$, the above equations can be expressed as residual equations. + +$$ +\begin{align} + \int_{\Omega} w \phi^{n+1} ~dV &=\int_{\Omega} \nabla w \cdot \nabla \eta^n ~dV +\end{align} +$$ + +$$ +\begin{align} +r_{\phi x} &= \nabla \eta^n +\end{align} +$$ + +$$ +\begin{align} + \int_{\Omega} w \eta^{n+1} ~dV &= \int_{\Omega} w \eta^{n} - w \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n - \kappa \Delta \eta^n) ~dV +\end{align} +$$ + +$$ +\begin{align} + &=\int_{\Omega} w \left( \eta^{n} - \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n) \right)+ \nabla w \cdot (- \Delta t M_{\eta}) ( \mathbf{m}^n - \delta^2 + \phi^n) ~dV +\end{align} +$$ + +$$ +\begin{align} +r_{\eta} &= \eta^{n} - \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n) +\end{align} +$$ + + +$$ +\begin{align} +r_{\eta x} &= (- \Delta t M_{\eta}) ( \mathbf{m}^n - \delta^2 \phi^n) +\end{align} +$$ + +and + +$$ +\begin{align} + \int_{\Omega} w c^{n+1} ~dV &= \int_{\Omega} w c^{n} + w \Delta t M_{c}~ \nabla \cdot (\nabla (f_{\alpha,c}^n(1-H^{n})+f_{\beta,c}^n H^{n})) ~dV +\end{align} +$$ + +$$ +\begin{align} +&= \int_{\Omega} w c^{n} + \nabla w (-\Delta t M_{c})~ [~(f_{\alpha,cc}^n(1-H^{n})+f_{\beta,cc}^n H^{n}) \nabla c + ~((f_{\beta,c}^n-f_{\alpha,c}^n)H^{n}_{,\eta} \nabla \eta^n) ] ~dV +\end{align} +$$ + +$$ +\begin{align} +r_c &= c^{n} +\end{align} +$$ + +$$ +\begin{align} +r_{cx} &= (-\Delta t M_{c})~ [~(f_{\alpha,cc}^n(1-H^{n})+f_{\beta,cc}^n H^{n}) \nabla c + ~((f_{\beta,c}^n-f_{\alpha,c}^n)H^{n}_{,\eta} \nabla \eta^n) ] +\end{align} +$$ + +The above values of $r_{\phi x}$, $r_{\eta}$, $r_{\eta x}$, $r_{c}$ and $r_{cx}$ are used to define the residuals in the following equations file: `applications/CHAC\_anisotropyRegularized/equations.h` diff --git a/applications/CHAC_anisotropyRegularized/CHAC_anisotropyRegularized.pdf b/applications/CHAC_anisotropyRegularized/CHAC_anisotropyRegularized.pdf deleted file mode 100644 index 8dcbe0543..000000000 Binary files a/applications/CHAC_anisotropyRegularized/CHAC_anisotropyRegularized.pdf and /dev/null differ diff --git a/applications/CHAC_anisotropyRegularized/tex_files/CHAC_anisotropyRegularized.tex b/applications/CHAC_anisotropyRegularized/tex_files/CHAC_anisotropyRegularized.tex deleted file mode 100644 index 21ee47b92..000000000 --- a/applications/CHAC_anisotropyRegularized/tex_files/CHAC_anisotropyRegularized.tex +++ /dev/null @@ -1,253 +0,0 @@ -\documentclass[10pt]{article} -\usepackage{amsmath} -\usepackage{bm} -\usepackage{bbm} -\usepackage{mathrsfs} -\usepackage{graphicx} -\usepackage{wrapfig} -\usepackage{subcaption} -\usepackage{epsfig} -\usepackage{amsfonts} -\usepackage{amssymb} -\usepackage{amsmath} -\usepackage{wrapfig} -\usepackage{graphicx} -\usepackage{psfrag} -\newcommand{\sun}{\ensuremath{\odot}} % sun symbol is \sun -\let\vaccent=\v % rename builtin command \v{} to \vaccent{} -\renewcommand{\v}[1]{\ensuremath{\mathbf{#1}}} % for vectors -\newcommand{\gv}[1]{\ensuremath{\mbox{\boldmath$ #1 $}}} -\newcommand{\grad}[1]{\gv{\nabla} #1} -\renewcommand{\baselinestretch}{1.2} -\jot 5mm -\graphicspath{{./figures/}} -%text dimensions -\textwidth 6.5 in -\oddsidemargin .2 in -\topmargin -0.2 in -\textheight 8.5 in -\headheight 0.2in -\overfullrule = 0pt -\pagestyle{plain} -\def\newpar{\par\vskip 0.5cm} -\begin{document} -% -%---------------------------------------------------------------------- -% Define symbols -%---------------------------------------------------------------------- -% -\def\iso{\mathbbm{1}} -\def\half{{\textstyle{1 \over 2}}} -\def\third{{\textstyle{1 \over 3}}} -\def\fourth{{\textstyle{{1 \over 4}}}} -\def\twothird{{\textstyle {{2 \over 3}}}} -\def\ndim{{n_{\rm dim}}} -\def\nint{n_{\rm int}} -\def\lint{l_{\rm int}} -\def\nel{n_{\rm el}} -\def\nf{n_{\rm f}} -\def\DIV {\hbox{\af div}} -\def\GRAD{\hbox{\af Grad}} -\def\sym{\mathop{\rm sym}\nolimits} -\def\tr{\mathop{\rm tr}\nolimits} -\def\dev{\mathop{\rm dev}\nolimits} -\def\Dev{\mathop{\rm Dev}\nolimits} -\def\DEV{\mathop {\rm DEV}\nolimits} -\def\bfb {{\bi b}} -\def\Bnabla{\nabla} -\def\bG{{\bi G}} -\def\jmpdelu{{\lbrack\!\lbrack \Delta u\rbrack\!\rbrack}} -\def\jmpudot{{\lbrack\!\lbrack\dot u\rbrack\!\rbrack}} -\def\jmpu{{\lbrack\!\lbrack u\rbrack\!\rbrack}} -\def\jmphi{{\lbrack\!\lbrack\varphi\rbrack\!\rbrack}} -\def\ljmp{{\lbrack\!\lbrack}} -\def\rjmp{{\rbrack\!\rbrack}} -\def\sign{{\rm sign}} -\def\nn{{n+1}} -\def\na{{n+\vartheta}} -\def\nna{{n+(1-\vartheta)}} -\def\nt{{n+{1\over 2}}} -\def\nb{{n+\beta}} -\def\nbb{{n+(1-\beta)}} -%--------------------------------------------------------- -% Bold Face Math Characters: -% All In Format: \B***** . -%--------------------------------------------------------- -\def\bOne{\mbox{\boldmath$1$}} -\def\BGamma{\mbox{\boldmath$\Gamma$}} -\def\BDelta{\mbox{\boldmath$\Delta$}} -\def\BTheta{\mbox{\boldmath$\Theta$}} -\def\BLambda{\mbox{\boldmath$\Lambda$}} -\def\BXi{\mbox{\boldmath$\Xi$}} -\def\BPi{\mbox{\boldmath$\Pi$}} -\def\BSigma{\mbox{\boldmath$\Sigma$}} -\def\BUpsilon{\mbox{\boldmath$\Upsilon$}} -\def\BPhi{\mbox{\boldmath$\Phi$}} -\def\BPsi{\mbox{\boldmath$\Psi$}} -\def\BOmega{\mbox{\boldmath$\Omega$}} -\def\Balpha{\mbox{\boldmath$\alpha$}} -\def\Bbeta{\mbox{\boldmath$\beta$}} -\def\Bgamma{\mbox{\boldmath$\gamma$}} -\def\Bdelta{\mbox{\boldmath$\delta$}} -\def\Bepsilon{\mbox{\boldmath$\epsilon$}} -\def\Bzeta{\mbox{\boldmath$\zeta$}} -\def\Beta{\mbox{\boldmath$\eta$}} -\def\Btheta{\mbox{\boldmath$\theta$}} -\def\Biota{\mbox{\boldmath$\iota$}} -\def\Bkappa{\mbox{\boldmath$\kappa$}} -\def\Blambda{\mbox{\boldmath$\lambda$}} -\def\Bmu{\mbox{\boldmath$\mu$}} -\def\Bnu{\mbox{\boldmath$\nu$}} -\def\Bxi{\mbox{\boldmath$\xi$}} -\def\Bpi{\mbox{\boldmath$\pi$}} -\def\Brho{\mbox{\boldmath$\rho$}} -\def\Bsigma{\mbox{\boldmath$\sigma$}} -\def\Btau{\mbox{\boldmath$\tau$}} -\def\Bupsilon{\mbox{\boldmath$\upsilon$}} -\def\Bphi{\mbox{\boldmath$\phi$}} -\def\Bchi{\mbox{\boldmath$\chi$}} -\def\Bpsi{\mbox{\boldmath$\psi$}} -\def\Bomega{\mbox{\boldmath$\omega$}} -\def\Bvarepsilon{\mbox{\boldmath$\varepsilon$}} -\def\Bvartheta{\mbox{\boldmath$\vartheta$}} -\def\Bvarpi{\mbox{\boldmath$\varpi$}} -\def\Bvarrho{\mbox{\boldmath$\varrho$}} -\def\Bvarsigma{\mbox{\boldmath$\varsigma$}} -\def\Bvarphi{\mbox{\boldmath$\varphi$}} -\def\bone{\mathbf{1}} -\def\bzero{\mathbf{0}} -%--------------------------------------------------------- -% Bold Face Math Italic: -% All In Format: \b* . -%--------------------------------------------------------- -\def\bA{\mbox{\boldmath$ A$}} -\def\bB{\mbox{\boldmath$ B$}} -\def\bC{\mbox{\boldmath$ C$}} -\def\bD{\mbox{\boldmath$ D$}} -\def\bE{\mbox{\boldmath$ E$}} -\def\bF{\mbox{\boldmath$ F$}} -\def\bG{\mbox{\boldmath$ G$}} -\def\bH{\mbox{\boldmath$ H$}} -\def\bI{\mbox{\boldmath$ I$}} -\def\bJ{\mbox{\boldmath$ J$}} -\def\bK{\mbox{\boldmath$ K$}} -\def\bL{\mbox{\boldmath$ L$}} -\def\bM{\mbox{\boldmath$ M$}} -\def\bN{\mbox{\boldmath$ N$}} -\def\bO{\mbox{\boldmath$ O$}} -\def\bP{\mbox{\boldmath$ P$}} -\def\bQ{\mbox{\boldmath$ Q$}} -\def\bR{\mbox{\boldmath$ R$}} -\def\bS{\mbox{\boldmath$ S$}} -\def\bT{\mbox{\boldmath$ T$}} -\def\bU{\mbox{\boldmath$ U$}} -\def\bV{\mbox{\boldmath$ V$}} -\def\bW{\mbox{\boldmath$ W$}} -\def\bX{\mbox{\boldmath$ X$}} -\def\bY{\mbox{\boldmath$ Y$}} -\def\bZ{\mbox{\boldmath$ Z$}} -\def\ba{\mbox{\boldmath$ a$}} -\def\bb{\mbox{\boldmath$ b$}} -\def\bc{\mbox{\boldmath$ c$}} -\def\bd{\mbox{\boldmath$ d$}} -\def\be{\mbox{\boldmath$ e$}} -\def\bff{\mbox{\boldmath$ f$}} -\def\bg{\mbox{\boldmath$ g$}} -\def\bh{\mbox{\boldmath$ h$}} -\def\bi{\mbox{\boldmath$ i$}} -\def\bj{\mbox{\boldmath$ j$}} -\def\bk{\mbox{\boldmath$ k$}} -\def\bl{\mbox{\boldmath$ l$}} -\def\bm{\mbox{\boldmath$ m$}} -\def\bn{\mbox{\boldmath$ n$}} -\def\bo{\mbox{\boldmath$ o$}} -\def\bp{\mbox{\boldmath$ p$}} -\def\bq{\mbox{\boldmath$ q$}} -\def\br{\mbox{\boldmath$ r$}} -\def\bs{\mbox{\boldmath$ s$}} -\def\bt{\mbox{\boldmath$ t$}} -\def\bu{\mbox{\boldmath$ u$}} -\def\bv{\mbox{\boldmath$ v$}} -\def\bw{\mbox{\boldmath$ w$}} -\def\bx{\mbox{\boldmath$ x$}} -\def\by{\mbox{\boldmath$ y$}} -\def\bz{\mbox{\boldmath$ z$}} -%********************************* -%Start main paper -%********************************* -\centerline{\Large{\bf PRISMS PhaseField}} -\smallskip -\centerline{\Large{\bf Regularized Anisotropy (with Coupled CH-AC Dynamics)}} -\bigskip - -Consider a free energy expression of the form: -\begin{equation} - \Pi(c, \eta, \grad \eta) = \int_{\Omega} \left( f_{\alpha}(1-H) + f_{\beta}H \right) + \frac{1}{2} | \gamma( \mathbf{n} ) \grad \eta |^2 + \frac{\delta^2}{2} (\Delta \eta)^2 ~dV -\end{equation} -where $f_{\alpha}$ and $f_{\beta}$ are the free energy densities corresponding to $\alpha$ and $\beta$ phases, respectively, and are functions of composition $c$. $H$ is a function of the structural order parameter $\eta$. $\delta$ is a scalar regularization parameter. The interface normal vector $\mathbf{n}$ is given by -\begin{equation} -\mathbf{n} = \frac{\grad \eta}{|\grad \eta|} -\end{equation} -for $\grad \eta \ne \mathbf{0}$, and $\mathbf{n} = \mathbf{0}$ when $\grad \eta = \mathbf{0}$. - -\section{Variational treatment} -Following standard variational arguments (see Cahn-Hilliard formulation), we obtain the chemical potentials: -\begin{align} - \mu_{c} &= (f_{\alpha,c}(1-H)+f_{\beta,c}H) \\ - \mu_{\eta} &= (f_{\beta,c}-f_{\alpha,c})H_{,\eta} - \grad \cdot \mathbf{m} + \delta^2 \Delta(\Delta \eta) -\end{align} -The components of the anisotropic gradient $\mathbf{m}$ are given by -\begin{equation} -m_i = \gamma(\mathbf{n}) \left( \grad \eta + |\grad \eta| (\delta_{ij}-n_i n_j) \frac{\partial \gamma (\mathbf{n})}{n_j} \right), -\end{equation} -where $\delta_{ij}$ is the Kronecker delta. - -\section{Kinetics} -Now the PDE for Cahn-Hilliard dynamics is given by: -\begin{align} - \frac{\partial c}{\partial t} &= -~\grad \cdot (-M_c\grad \mu_c)\\ - &=M_c~\grad \cdot (\grad (f_{\alpha,c}(1-H)+f_{\beta,c}H)) - \end{align} - and the PDE for Allen-Cahn dynamics is given by: - \begin{align} - \frac{\partial \eta}{\partial t} &= -M_\eta \mu_\eta \\ - &=-M_\eta ~ [(f_{\beta,c}-f_{\alpha,c})H_{,\eta} - \grad \cdot \mathbf{m} + \delta^2 \Delta(\Delta \eta)] -\end{align} -where $M_c$ and $M_\eta$ are the constant mobilities. In order that the formulation only includes second order derivatives, an auxiliary field $\phi$ is introduced to break up the biharmonic term: -\begin{align} -\phi = \Delta \eta -\end{align} -and the PDE for Allen-Cahn dynamics becomes -\begin{align} - \frac{\partial \eta}{\partial t} =-M_\eta ~ ((f_{\beta,c}-f_{\alpha,c})H_{,\eta} - \grad \cdot \mathbf{m}) + \delta^2 \Delta \phi . -\end{align} - -\section{Time discretization} -Considering forward Euler explicit time stepping, we have the time discretized kinetics equation: -\begin{align} - \phi^{n+1} &= \Delta \eta^n \\ - \eta^{n+1} &= \eta^{n} - \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n - \grad \cdot \mathbf{m}^n + \delta^2 \Delta \phi^n) \\ -c^{n+1} &= c^{n} + \Delta t M_{\eta}~\grad \cdot (\grad (f_{\alpha,c}^n(1-H^{n})+f_{\beta,c}^n H^{n})) -\end{align} - -\section{Weak formulation} -In the weak formulation, considering an arbitrary variation $w$, the above equations can be expressed as residual equations. -\begin{align} - \int_{\Omega} w \phi^{n+1} ~dV &=\int_{\Omega} \grad w \cdot \underbrace{ \grad \eta^n }_{r_{\phi x}} ~dV -\end{align} - -\begin{align} - \int_{\Omega} w \eta^{n+1} ~dV &= \int_{\Omega} w \eta^{n} - w \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n - \kappa \Delta \eta^n) ~dV\\ - &=\int_{\Omega} w \left( \underbrace{\eta^{n} - \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n)}_{r_{\eta}} \right)+ \grad w \cdot \underbrace{(- \Delta t M_{\eta}) ( \mathbf{m}^n - \delta^2 \phi^n)}_{r_{\eta x}} ~dV -\end{align} -and -\begin{align} - \int_{\Omega} w c^{n+1} ~dV &= \int_{\Omega} w c^{n} + w \Delta t M_{c}~ \grad \cdot (\grad (f_{\alpha,c}^n(1-H^{n})+f_{\beta,c}^n H^{n})) ~dV\\ - &= \int_{\Omega} w \underbrace{c^{n}}_{r_c} + \grad w \underbrace{(-\Delta t M_{c})~ [~(f_{\alpha,cc}^n(1-H^{n})+f_{\beta,cc}^n H^{n}) \grad c + ~((f_{\beta,c}^n-f_{\alpha,c}^n)H^{n}_{,\eta} \grad \eta^n) ] }_{r_{cx}} ~dV -\end{align} - -\vskip 0.25in -The above values of $r_{\phi x}$, $r_{\eta}$, $r_{\eta x}$, $r_{c}$ and $r_{cx}$ are used to define the residuals in the following equations file: \textit{applications/CHAC\_anisotropyRegularized/equations.h} - - -\end{document} \ No newline at end of file diff --git a/applications/CHAC_performance_test/formulation_coupledCHAC.md b/applications/CHAC_performance_test/formulation_coupledCHAC.md new file mode 100644 index 000000000..93d40e594 --- /dev/null +++ b/applications/CHAC_performance_test/formulation_coupledCHAC.md @@ -0,0 +1,113 @@ +## PRISMS PhaseField: Coupled Cahn-Hilliard and Allen-Cahn Dynamics + +Consider a free energy expression of the form: + +$$ +\begin{equation} + \Pi(c, \eta, \nabla \eta) = \int_{\Omega} \left( f_{\alpha}(1-H) + f_{\beta}H \right) + \frac{\kappa}{2} \nabla \eta \cdot \nabla \eta ~dV +\end{equation} +$$ + +where $f_{\alpha}$ and $f_{\beta}$ are the free energy densities corresponding to $\alpha$ and $\beta$ phases, respectively, and are functions of composition $c$. $H$ is a function of the structural order parameter $\eta$. Note that we don't have the nablaient terms for the composition, i.e, $\nabla c$ terms, unlike classical Cahn-Hillard formulation. + +## Variational treatment +Following standard variational arguments (see Cahn-Hilliard formulation), we obtain the chemical potentials: + +$$ +\begin{align} +\mu_{c} &= (f_{\alpha,c}(1-H)+f_{\beta,c}H) +\end{align} +$$ + +$$ +\begin{align} +\mu_{\eta} &= (f_{\beta}-f_{\alpha})H_{,\eta} - \kappa \Delta \eta +\end{align} +$$ + +## Kinetics +Now the PDE for Cahn-Hilliard dynamics is given by: + +$$ +\begin{align} +\frac{\partial c}{\partial t} &= -~\nabla \cdot (-M_c\nabla \mu_c)\\ +&=M_c~\nabla \cdot (\nabla (f_{\alpha,c}(1-H)+f_{\beta,c}H)) +\end{align} +$$ + +and the PDE for Allen-Cahn dynamics is given by: + +$$ +\begin{align} +\frac{\partial \eta}{\partial t} &= -M_\eta \mu_\eta \\ +&=-M_\eta ~ ((f_{\beta}-f_{\alpha})H_{,\eta} - \kappa \Delta \eta) +\end{align} +$$ + +where $M_c$ and $M_\eta$ are the constant mobilities. + +## Time discretization +Considering forward Euler explicit time stepping, we have the time discretized kinetics equation: + +$$ +\begin{align} + \eta^{n+1} &= \eta^{n} - \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n - \kappa \Delta \eta^n) \\ +c^{n+1} &= c^{n} + \Delta t M_{\eta}~\nabla \cdot (\nabla (f_{\alpha,c}^n(1-H^{n})+f_{\beta,c}^n H^{n})) +\end{align} +$$ + +## Weak formulation +In the weak formulation, considering an arbitrary variation $w$, the above equations can be expressed as residual equations: + +$$ +\begin{align} +\int_{\Omega} w \eta^{n+1} ~dV &= \int_{\Omega} w \eta^{n} - w \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n - \kappa \Delta \eta^n) ~dV +\end{align} +$$ + +$$ +\begin{align} +&=\int_{\Omega} w \left( \eta^{n} - \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n) \right)+ \nabla w \cdot (- \Delta t M_{\eta}\kappa) \nabla \eta^{n} ~dV +\end{align} +$$ + +$$ +\begin{align} +r_{\eta} &= \eta^{n} - \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n) +\end{align} +$$ + +$$ +\begin{align} +r_{\eta x} &= (- \Delta t M_{\eta}\kappa) \nabla \eta^{n} +\end{align} +$$ + +and + +$$ +\begin{align} +\int_{\Omega} w c^{n+1} ~dV &= \int_{\Omega} w c^{n} + w \Delta t M_{c}~ \nabla \cdot (\nabla (f_{\alpha,c}^n(1-H^{n})+f_{\beta,c}^n H^{n})) ~dV +\end{align} +$$ + +$$ +\begin{align} +&= \int_{\Omega} w c^{n} + \nabla w (-\Delta t M_{c})~ [~(f_{\alpha,cc}^n(1-H^{n})+f_{\beta,cc}^n H^{n}) \nabla c + ~((f_{\beta,c}^n-f_{\alpha,c}^n)H^{n}_{,\eta} \nabla \eta) ] ~dV +\end{align} +$$ + +$$ +\begin{align} +r_c &= c^{n} +\end{align} +$$ + +$$ +\begin{align} +r_{cx} &= (-\Delta t M_{c})~ [~(f_{\alpha,cc}^n(1-H^{n})+f_{\beta,cc}^n H^{n}) \nabla c + ~((f_{\beta,c}^n-f_{\alpha,c}^n)H^{n}_{,\eta} \nabla \eta) ] +\end{align} +$$ + +The above values of $r_{\eta}$, $r_{\eta x}$, $r_{c}$ and $r_{cx}$ are used to define the residuals in the following parameters file: `applications/coupledCahnHilliardAllenCahn/parameters.h` + diff --git a/applications/CHAC_performance_test/formulation_coupledCHAC.pdf b/applications/CHAC_performance_test/formulation_coupledCHAC.pdf deleted file mode 100644 index e37e836e7..000000000 Binary files a/applications/CHAC_performance_test/formulation_coupledCHAC.pdf and /dev/null differ diff --git a/applications/CHAC_performance_test/tex_files/coupledCHAC.tex b/applications/CHAC_performance_test/tex_files/coupledCHAC.tex deleted file mode 100644 index 0eecb018c..000000000 --- a/applications/CHAC_performance_test/tex_files/coupledCHAC.tex +++ /dev/null @@ -1,233 +0,0 @@ -\documentclass[10pt]{article} -\usepackage{amsmath} -\usepackage{bm} -\usepackage{bbm} -\usepackage{mathrsfs} -\usepackage{graphicx} -\usepackage{wrapfig} -\usepackage{subcaption} -\usepackage{epsfig} -\usepackage{amsfonts} -\usepackage{amssymb} -\usepackage{amsmath} -\usepackage{wrapfig} -\usepackage{graphicx} -\usepackage{psfrag} -\newcommand{\sun}{\ensuremath{\odot}} % sun symbol is \sun -\let\vaccent=\v % rename builtin command \v{} to \vaccent{} -\renewcommand{\v}[1]{\ensuremath{\mathbf{#1}}} % for vectors -\newcommand{\gv}[1]{\ensuremath{\mbox{\boldmath$ #1 $}}} -\newcommand{\grad}[1]{\gv{\nabla} #1} -\renewcommand{\baselinestretch}{1.2} -\jot 5mm -\graphicspath{{./figures/}} -%text dimensions -\textwidth 6.5 in -\oddsidemargin .2 in -\topmargin -0.2 in -\textheight 8.5 in -\headheight 0.2in -\overfullrule = 0pt -\pagestyle{plain} -\def\newpar{\par\vskip 0.5cm} -\begin{document} -% -%---------------------------------------------------------------------- -% Define symbols -%---------------------------------------------------------------------- -% -\def\iso{\mathbbm{1}} -\def\half{{\textstyle{1 \over 2}}} -\def\third{{\textstyle{1 \over 3}}} -\def\fourth{{\textstyle{{1 \over 4}}}} -\def\twothird{{\textstyle {{2 \over 3}}}} -\def\ndim{{n_{\rm dim}}} -\def\nint{n_{\rm int}} -\def\lint{l_{\rm int}} -\def\nel{n_{\rm el}} -\def\nf{n_{\rm f}} -\def\DIV {\hbox{\af div}} -\def\GRAD{\hbox{\af Grad}} -\def\sym{\mathop{\rm sym}\nolimits} -\def\tr{\mathop{\rm tr}\nolimits} -\def\dev{\mathop{\rm dev}\nolimits} -\def\Dev{\mathop{\rm Dev}\nolimits} -\def\DEV{\mathop {\rm DEV}\nolimits} -\def\bfb {{\bi b}} -\def\Bnabla{\nabla} -\def\bG{{\bi G}} -\def\jmpdelu{{\lbrack\!\lbrack \Delta u\rbrack\!\rbrack}} -\def\jmpudot{{\lbrack\!\lbrack\dot u\rbrack\!\rbrack}} -\def\jmpu{{\lbrack\!\lbrack u\rbrack\!\rbrack}} -\def\jmphi{{\lbrack\!\lbrack\varphi\rbrack\!\rbrack}} -\def\ljmp{{\lbrack\!\lbrack}} -\def\rjmp{{\rbrack\!\rbrack}} -\def\sign{{\rm sign}} -\def\nn{{n+1}} -\def\na{{n+\vartheta}} -\def\nna{{n+(1-\vartheta)}} -\def\nt{{n+{1\over 2}}} -\def\nb{{n+\beta}} -\def\nbb{{n+(1-\beta)}} -%--------------------------------------------------------- -% Bold Face Math Characters: -% All In Format: \B***** . -%--------------------------------------------------------- -\def\bOne{\mbox{\boldmath$1$}} -\def\BGamma{\mbox{\boldmath$\Gamma$}} -\def\BDelta{\mbox{\boldmath$\Delta$}} -\def\BTheta{\mbox{\boldmath$\Theta$}} -\def\BLambda{\mbox{\boldmath$\Lambda$}} -\def\BXi{\mbox{\boldmath$\Xi$}} -\def\BPi{\mbox{\boldmath$\Pi$}} -\def\BSigma{\mbox{\boldmath$\Sigma$}} -\def\BUpsilon{\mbox{\boldmath$\Upsilon$}} -\def\BPhi{\mbox{\boldmath$\Phi$}} -\def\BPsi{\mbox{\boldmath$\Psi$}} -\def\BOmega{\mbox{\boldmath$\Omega$}} -\def\Balpha{\mbox{\boldmath$\alpha$}} -\def\Bbeta{\mbox{\boldmath$\beta$}} -\def\Bgamma{\mbox{\boldmath$\gamma$}} -\def\Bdelta{\mbox{\boldmath$\delta$}} -\def\Bepsilon{\mbox{\boldmath$\epsilon$}} -\def\Bzeta{\mbox{\boldmath$\zeta$}} -\def\Beta{\mbox{\boldmath$\eta$}} -\def\Btheta{\mbox{\boldmath$\theta$}} -\def\Biota{\mbox{\boldmath$\iota$}} -\def\Bkappa{\mbox{\boldmath$\kappa$}} -\def\Blambda{\mbox{\boldmath$\lambda$}} -\def\Bmu{\mbox{\boldmath$\mu$}} -\def\Bnu{\mbox{\boldmath$\nu$}} -\def\Bxi{\mbox{\boldmath$\xi$}} -\def\Bpi{\mbox{\boldmath$\pi$}} -\def\Brho{\mbox{\boldmath$\rho$}} -\def\Bsigma{\mbox{\boldmath$\sigma$}} -\def\Btau{\mbox{\boldmath$\tau$}} -\def\Bupsilon{\mbox{\boldmath$\upsilon$}} -\def\Bphi{\mbox{\boldmath$\phi$}} -\def\Bchi{\mbox{\boldmath$\chi$}} -\def\Bpsi{\mbox{\boldmath$\psi$}} -\def\Bomega{\mbox{\boldmath$\omega$}} -\def\Bvarepsilon{\mbox{\boldmath$\varepsilon$}} -\def\Bvartheta{\mbox{\boldmath$\vartheta$}} -\def\Bvarpi{\mbox{\boldmath$\varpi$}} -\def\Bvarrho{\mbox{\boldmath$\varrho$}} -\def\Bvarsigma{\mbox{\boldmath$\varsigma$}} -\def\Bvarphi{\mbox{\boldmath$\varphi$}} -\def\bone{\mathbf{1}} -\def\bzero{\mathbf{0}} -%--------------------------------------------------------- -% Bold Face Math Italic: -% All In Format: \b* . -%--------------------------------------------------------- -\def\bA{\mbox{\boldmath$ A$}} -\def\bB{\mbox{\boldmath$ B$}} -\def\bC{\mbox{\boldmath$ C$}} -\def\bD{\mbox{\boldmath$ D$}} -\def\bE{\mbox{\boldmath$ E$}} -\def\bF{\mbox{\boldmath$ F$}} -\def\bG{\mbox{\boldmath$ G$}} -\def\bH{\mbox{\boldmath$ H$}} -\def\bI{\mbox{\boldmath$ I$}} -\def\bJ{\mbox{\boldmath$ J$}} -\def\bK{\mbox{\boldmath$ K$}} -\def\bL{\mbox{\boldmath$ L$}} -\def\bM{\mbox{\boldmath$ M$}} -\def\bN{\mbox{\boldmath$ N$}} -\def\bO{\mbox{\boldmath$ O$}} -\def\bP{\mbox{\boldmath$ P$}} -\def\bQ{\mbox{\boldmath$ Q$}} -\def\bR{\mbox{\boldmath$ R$}} -\def\bS{\mbox{\boldmath$ S$}} -\def\bT{\mbox{\boldmath$ T$}} -\def\bU{\mbox{\boldmath$ U$}} -\def\bV{\mbox{\boldmath$ V$}} -\def\bW{\mbox{\boldmath$ W$}} -\def\bX{\mbox{\boldmath$ X$}} -\def\bY{\mbox{\boldmath$ Y$}} -\def\bZ{\mbox{\boldmath$ Z$}} -\def\ba{\mbox{\boldmath$ a$}} -\def\bb{\mbox{\boldmath$ b$}} -\def\bc{\mbox{\boldmath$ c$}} -\def\bd{\mbox{\boldmath$ d$}} -\def\be{\mbox{\boldmath$ e$}} -\def\bff{\mbox{\boldmath$ f$}} -\def\bg{\mbox{\boldmath$ g$}} -\def\bh{\mbox{\boldmath$ h$}} -\def\bi{\mbox{\boldmath$ i$}} -\def\bj{\mbox{\boldmath$ j$}} -\def\bk{\mbox{\boldmath$ k$}} -\def\bl{\mbox{\boldmath$ l$}} -\def\bm{\mbox{\boldmath$ m$}} -\def\bn{\mbox{\boldmath$ n$}} -\def\bo{\mbox{\boldmath$ o$}} -\def\bp{\mbox{\boldmath$ p$}} -\def\bq{\mbox{\boldmath$ q$}} -\def\br{\mbox{\boldmath$ r$}} -\def\bs{\mbox{\boldmath$ s$}} -\def\bt{\mbox{\boldmath$ t$}} -\def\bu{\mbox{\boldmath$ u$}} -\def\bv{\mbox{\boldmath$ v$}} -\def\bw{\mbox{\boldmath$ w$}} -\def\bx{\mbox{\boldmath$ x$}} -\def\by{\mbox{\boldmath$ y$}} -\def\bz{\mbox{\boldmath$ z$}} -%********************************* -%Start main paper -%********************************* -\centerline{\Large{\bf PRISMS PhaseField}} -\smallskip -\centerline{\Large{\bf Coupled Cahn-Hilliard and Allen-Cahn Dynamics}} -\bigskip - -Consider a free energy expression of the form: -\begin{equation} - \Pi(c, \eta, \grad \eta) = \int_{\Omega} \left( f_{\alpha}(1-H) + f_{\beta}H \right) + \frac{\kappa}{2} \grad \eta \cdot \grad \eta ~dV -\end{equation} -where $f_{\alpha}$ and $f_{\beta}$ are the free energy densities corresponding to $\alpha$ and $\beta$ phases, respectively, and are functions of composition $c$. $H$ is a function of the structural order parameter $\eta$. Note that we don't have the gradient terms for the composition, i.e, $\grad c$ terms, unlike classical Cahn-Hillard formulation. - -\section{Variational treatment} -Following standard variational arguments (see Cahn-Hilliard formulation), we obtain the chemical potentials: -\begin{align} - \mu_{c} &= (f_{\alpha,c}(1-H)+f_{\beta,c}H) \\ - \mu_{\eta} &= (f_{\beta}-f_{\alpha})H_{,\eta} - \kappa \Delta \eta -\end{align} - -\section{Kinetics} -Now the PDE for Cahn-Hilliard dynamics is given by: -\begin{align} - \frac{\partial c}{\partial t} &= -~\grad \cdot (-M_c\grad \mu_c)\\ - &=M_c~\grad \cdot (\grad (f_{\alpha,c}(1-H)+f_{\beta,c}H)) - \end{align} - and the PDE for Allen-Cahn dynamics is given by: - \begin{align} - \frac{\partial \eta}{\partial t} &= -M_\eta \mu_\eta \\ - &=-M_\eta ~ ((f_{\beta}-f_{\alpha})H_{,\eta} - \kappa \Delta \eta) -\end{align} -where $M_c$ and $M_\eta$ are the constant mobilities. - -\section{Time discretization} -Considering forward Euler explicit time stepping, we have the time discretized kinetics equation: -\begin{align} - \eta^{n+1} &= \eta^{n} - \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n - \kappa \Delta \eta^n) \\ -c^{n+1} &= c^{n} + \Delta t M_{\eta}~\grad \cdot (\grad (f_{\alpha,c}^n(1-H^{n})+f_{\beta,c}^n H^{n})) -\end{align} - -\section{Weak formulation} -In the weak formulation, considering an arbitrary variation $w$, the above equations can be expressed as residual equations: -\begin{align} - \int_{\Omega} w \eta^{n+1} ~dV &= \int_{\Omega} w \eta^{n} - w \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n - \kappa \Delta \eta^n) ~dV\\ - &=\int_{\Omega} w \left( \underbrace{\eta^{n} - \Delta t M_{\eta}~ ((f_{\beta,c}^n-f_{\alpha,c}^n)H_{,\eta}^n)}_{r_{\eta}} \right)+ \grad w \cdot \underbrace{(- \Delta t M_{\eta}\kappa) \grad \eta^{n}}_{r_{\eta x}} ~dV -\end{align} -and -\begin{align} - \int_{\Omega} w c^{n+1} ~dV &= \int_{\Omega} w c^{n} + w \Delta t M_{c}~ \grad \cdot (\grad (f_{\alpha,c}^n(1-H^{n})+f_{\beta,c}^n H^{n})) ~dV\\ - &= \int_{\Omega} w \underbrace{c^{n}}_{r_c} + \grad w \underbrace{(-\Delta t M_{c})~ [~(f_{\alpha,cc}^n(1-H^{n})+f_{\beta,cc}^n H^{n}) \grad c + ~((f_{\beta,c}^n-f_{\alpha,c}^n)H^{n}_{,\eta} \grad \eta) ] }_{r_{cx}} ~dV -\end{align} - -\vskip 0.25in -The above values of $r_{\eta}$, $r_{\eta x}$, $r_{c}$ and $r_{cx}$ are used to define the residuals in the following parameters file: \\ -\textit{applications/coupledCahnHilliardAllenCahn/parameters.h} - - -\end{document} \ No newline at end of file diff --git a/applications/MgNd_precipitate_single_Bppp/precipitateEvolution_KKS.md b/applications/MgNd_precipitate_single_Bppp/precipitateEvolution_KKS.md new file mode 100644 index 000000000..c5154ae25 --- /dev/null +++ b/applications/MgNd_precipitate_single_Bppp/precipitateEvolution_KKS.md @@ -0,0 +1,321 @@ +# KKS Phase Field Model of Precipitate Evolution (September 14th, 2023) + +The model employed in this application is described in detail in the article: +DeWitt et al., Misfit-driven $\beta'''$ precipitate composition and morphology in Mg-Nd alloys, +Acta Materialia **137**, 378-389 (2017). + +## Variational formulation +The total free energy of the system (neglecting boundary terms) is of the form, + + +$$ +\begin{equation} +\Pi(c, \eta_1, \eta_2, \eta_3, \epsilon) = \int_{\Omega} f(c, \eta_1, \eta_2, \eta_3, \epsilon) ~dV +\end{equation} +$$ + +where $c$ is the concentration of the $\beta$ phase, $\eta_p$ are the structural order parameters and $\varepsilon$ is the small strain tensor. $f$, the free energy density is given by + +$$ +\begin{equation} + f(c, \eta_1, \eta_2, \eta_3, \epsilon) = f_{chem}(c, \eta_1, \eta_2, \eta_3) + f_{grad}(\eta_1, \eta_2, \eta_3) + f_{elastic}(c,\eta_1, \eta_2, \eta_3,\epsilon) +\end{equation} +$$ + +where + +$$ +\begin{equation} +f_{chem}(c, \eta_1, \eta_2, \eta_3) = f_{\alpha}(c,\eta_1, \eta_2, \eta_3) \left( 1- \sum_{p=1}^3 H(\eta_p)\right) + f_{\beta}(c,\eta_1, \eta_2, \eta_3) \sum_{p=1}^3 H(\eta_p)+ W f_{Landau}(\eta_1, \eta_2, \eta_3) +\end{equation} +$$ + +$$ +\begin{equation} +f_{grad}(\eta_1, \eta_2, \eta_3) = \frac{1}{2} \sum_{p=1}^3 \kappa_{ij}^{\eta_p} \eta_{p,i} \eta_{p,j} +\end{equation} +$$ + +$$ +\begin{gather} +f_{elastic}(c,\eta_1, \eta_2, \eta_3,\epsilon) = \frac{1}{2} C_{ijkl}(\eta_1, \eta_2, \eta_3) \left( \varepsilon_{ij} - \varepsilon ^0_{ij}(c, \eta_1, \eta_2, \eta_3) \right)\left( \varepsilon_{kl} - \varepsilon^0_{kl}(c, \eta_1, \eta_2, \eta_3)\right) +\end{gather} +$$ + +$$ +\begin{gather} +\varepsilon^0(c, \eta_1, \eta_2, \eta_3) = H(\eta_1) \varepsilon^0_{\eta_1} (c_{\beta})+ H(\eta_2) \varepsilon^0_{\eta_2} (c_{\beta}) + H(\eta_3) \varepsilon^0_{\eta_3} (c_{\beta}) +C(\eta_1, \eta_2, \eta_3) = H(\eta_1) C_{\eta_1}+ H(\eta_2) C_{\eta_2} + H(\eta_3) C_{\eta_3} + \left( 1- H(\eta_1)-H(\eta_2)-H(\eta_3)\right) C_{\alpha} +\end{gather} +$$ + +Here $\varepsilon^0_{\eta_p}$ are the composition dependent stress free strain transformation tensor corresponding to each structural order parameter, which is a function of the $\beta$ phase concentration, $c_{\beta}$, defined below. + +In the KKS model (Kim 1999), the interfacial region is modeled as a mixture of the $\alpha$ and $\beta$ phases with concentrations $c_{alpha}$ and $c_{beta}$, respectively. The homogenous free energies for each phase, $f_{\alpha}$ and $f_{\beta}$ in this case, are typically given as functions of $c_{\alpha}$ and $c_{\beta}$, rather than directly as functions of $c$ and $\eta_p$. Thus, $f_{chem}(c, \eta_1, \eta_2, \eta_3)$ can be rewritten as + +$$ +\begin{equation} +f_{chem}(c, \eta_1, \eta_2, \eta_3) = f_{\alpha}(c_{\alpha}) \left( 1- \sum_{p=1}^3 H(\eta_p)\right) + f_{\beta}(c_{\beta}) \sum_{p=1}^3 H(\eta_p)+ W f_{Landau}(\eta_1, \eta_2, \eta_3) +\end{equation} +$$ + +The concentration in each phase is determined by the following system of equations: + +$$ +\begin{gather} +c = c_{\alpha} \left( 1- \sum_{p=1}^3 H(\eta_p)\right) + c_{\beta} \sum_{p=1}^3 H(\eta_p) \\ +\frac{\partial f_{\alpha}(c_{\alpha})}{\partial c_{\alpha}} = \frac{\partial f_{\beta}(c_{\beta})}{\partial c_{\beta}} +\end{gather} +$$ + +Given the following parabolic functions for the single-phase homogenous free energies: + +$$ +\begin{gather} +f_{\alpha}(c_{\alpha}) = A_{2} c_{\alpha}^2 + A_{1} c_{\alpha} + A_{0} \\ +f_{\beta}(c_{\beta}) = B_{2} c_{\beta}^2 + B_{1} c_{\beta} + B_{0} +\end{gather} +$$ + +the single-phase concentrations are: + +$$ +\begin{gather} +c_{\alpha} = \frac{ B_2 c + \frac{1}{2} (B_1 - A_1) \sum_{p=1}^3 H(\eta_p) }{A_2 \sum_{p=1}^3 H(\eta_p) + B_2 \left( 1- \sum_{p=1}^3 H(\eta_p)\right) } \\ +c_{\beta} = \frac{ A_2 c + \frac{1}{2} (A_1 - B_1) \left[1-\sum_{p=1}^3 H(\eta_p)\right] }{A_2 \sum_{p=1}^3 H(\eta_p) + B_2 \left[ 1- \sum_{p=1}^3 H(\eta_p)\right] } +\end{gather} +$$ + +## Required inputs + +- $f_{\alpha}(c_{\alpha}), f_{\beta}(c_{\beta})$ - Homogeneous chemical free energy of the components of the binary system, example form given above +- $f_{Landau}(\eta_1, \eta_2, \eta_3)$ - Landau free energy term that controls the interfacial energy and prevents precipitates with different orientation varients from overlapping, example form given in Appendix I +- \$W$ - Barrier height for the Landau free energy term, used to control the thickness of the interface +- $H(\eta_p)$ - Interpolation function for connecting the $\alpha$ phase and the $p^{th}$ orientation variant of the $\beta$ phase, example form given in Appendix I +- $\kappa^{\eta_p}$ - gradient penalty tensor for the $p^{th}$ orientation variant of the $\beta$ phase +- $C_{\eta_p}$ - fourth order elasticity tensor (or its equivalent second order Voigt representation) for the $p^{th}$ orientation variant of the $\beta$ phase +- $C_{\alpha}$ - fourth order elasticity tensor (or its equivalent second order Voigt representation) for the $\alpha$ phase +- $\varepsilon^0_{\eta_p}$ - stress free strain transformation tensor for the $p^{th}$ orientation variant of the $\beta$ phase + + +In addition, to drive the kinetics, we need: +- $M$ - mobility value for the concentration field +- $L$ - mobility value for the structural order parameter field + + +## Variational treatment +We obtain chemical potentials for the chemical potentials for the concentration and the structural order parameters by taking variational derivatives of $\Pi$: + +$$ +\begin{align} + \mu_{c} &= f_{\alpha,c} \left( 1- H(\eta_1)-H(\eta_2)-H(\eta_3)\right) +f_{\beta,c} \left( H(\eta_1) + H(\eta_2) + H(\eta_3) \right) + C_{ijkl} (- \varepsilon^0_{ij,c}) \left( \varepsilon_{kl} - \varepsilon^0_{kl}\right) +\end{align} +$$ + +$$ +\begin{align} +\mu_{\eta_p} &= [ f_{\beta}-f_{\alpha} -(c_{\beta}-c_{\alpha}) f_{\beta,c_{\beta}}] H(\eta_p){,\eta_p} + W f_{Landau,\eta_p}- +\kappa_{ij}^{\eta_p} \eta_{p,ij} + C_{ijkl} (- \varepsilon^0_{ij,\eta_p}) \left( \varepsilon_{kl} - \varepsilon^0_{kl}\right) + \frac{1}{2} C_{ijkl,\eta_p} \left( \varepsilon_{ij} - \varepsilon_{ij}^0 \right) \left( \varepsilon_{kl} - \varepsilon_{kl}^0\right) +\end{align} +$$ + +## Kinetics +Now the PDE for Cahn-Hilliard dynamics is given by: + +$$ +\begin{align} +\frac{\partial c}{\partial t} &= ~\nabla \cdot \left( \frac{1}{f_{,cc}}M \nabla \mu_c \right) +\end{align} +$$ + +where $M$ is a constant mobility and the factor of $\frac{1}{f_{,cc}}$ is added to guarentee constant diffusivity in the two phases. The PDE for Allen-Cahn dynamics is given by: + +$$ +\begin{align} +\frac{\partial \eta_p}{\partial t} &= - L \mu_{\eta_p} +\end{align} +$$ + +where $L$ is a constant mobility. + +## Mechanics +Considering variations on the displacement $u$ of the from $u+\epsilon w$, we have + +$$ +\begin{align} +\delta_u \Pi &= \int_{\Omega} \nabla w : C(\eta_1, \eta_2, \eta_3) : \left( \varepsilon - \varepsilon^0(c,\eta_1, \eta_2, \eta_3)\right) ~dV = 0 +\end{align} +$$ + +where $\sigma = C(\eta_1, \eta_2, \eta_3) : \left( \varepsilon - \varepsilon^0(c,\eta_1, \eta_2, \eta_3)\right)$ is the stress tensor. + +## Time discretization +Using forward Euler explicit time stepping, equations + +$$ +\begin{align} +\frac{\partial c}{\partial t} &= ~\nabla \cdot \left( \frac{1}{f_{,cc}}M \nabla \mu_c \right) +\end{align} +$$ + +and + +$$ +\begin{align} +\frac{\partial \eta_p}{\partial t} &= - L \mu_{\eta_p} +\end{align} +$$ + +become: + +$$ +\begin{align} +c^{n+1} = c^{n}+\Delta t \left[\nabla \cdot \left(\frac{1}{f_{,cc}} M \nabla \mu_c \right) \right] +\end{align} +$$ + +$$ +\begin{align} +\eta_p^{n+1} = \eta_p^n -\Delta t L \mu_{\eta_p} +\end{align} +$$ + +## Weak formulation +Writing equations + +$$ +\begin{align} +\frac{\partial c}{\partial t} &= ~\nabla \cdot \left( \frac{1}{f_{,cc}}M \nabla \mu_c \right) +\end{align} +$$ + +and + +$$ +\begin{align} +\frac{\partial \eta_p}{\partial t} &= - L \mu_{\eta_p} +\end{align} +$$ + +in the weak form, with the arbirary variation given by $w$ yields: + +$$ +\begin{align} +\int_\Omega w c^{n+1} dV &= \int_\Omega wc^{n}+w \Delta t \left[\nabla \cdot \left(\frac{1}{f_{,cc}} M \nabla \mu_c \right) \right] dV +\end{align} +$$ + +$$ +\begin{align} +%&= \int_\Omega wc^{n}+\nabla w \cdot (\Delta t M \nabla \mu_c ) dV +\end{align} +$$ + +$$ +\begin{align} +\int_\Omega w \eta_p^{n+1} dV &= \int_\Omega w \eta_p^{n}-w \Delta t L \mu_{\eta_p} dV +\end{align} +$$ + +The expression of $\frac{1}{f_{,cc}} \mu_c$ can be written as: + +$$ +\begin{align} +\frac{1}{f_{,cc}} \nabla \mu_c = & \nabla c + (c_{\alpha}-c_{\beta}) \sum_{p=1}^3 H(\eta_p)_{,\eta_p} \nabla \eta_p +\end{align} +$$ + +$$ +\begin{align} +&+ \frac{1}{f_{,cc}} \left(\sum_{p=1}^3 (C_{ijkl}^{\eta_p} - C_{ijkl}^{\alpha} )\nabla \eta_p H_{,\eta_p}(\eta_p) \right)(-\epsilon_{ij,c}^0)(\epsilon_{ij} - \epsilon_{ij}^0) +\end{align} +$$ + +$$ +\begin{align} +&- \frac{1}{f_{,cc}} C_{ijkl} \left( \sum_{p=1}^3 \left( H_{,\eta_p}(\eta_p) \epsilon_{ij,c}^{0\eta_p} + \sum_{q=1}^3 \left( H(\eta_p) \epsilon_{ij,c\eta_q}^{0\eta_p} \right) \right) \nabla \eta_p + H(\eta_p) \epsilon_{ij,cc}^{0\eta_p} \nabla c \right)(\epsilon_{kl}-\epsilon_{kl}^0) +\end{align} +$$ + +$$ +\begin{align} +&+ \frac{1}{f_{,cc}} C_{ijkl} (-\epsilon_{ij,c}^0) \left( \nabla \epsilon_{kl} - \left( \sum_{p=1}^3 \left(H_{,\eta_p}(\eta_p) \epsilon_{kl}^{0\eta_p} -\sum_{q=1}^3 \epsilon_{kl,\eta_q}^{\eta_q} H(\eta_q) \right)\nabla \eta_p + H(\eta_p) \epsilon_{kl,c}^{0\eta_p} \nabla c \right) \right) +\end{align} +$$ + +Applying the divergence theorem to equation + +$$ +\begin{align} +\int_\Omega w c^{n+1} dV &= \int_\Omega wc^{n}+w \Delta t \left[\nabla \cdot \left(\frac{1}{f_{,cc}} M \nabla \mu_c \right) \right] dV +\end{align} +$$ + +one can derive the residual terms $r_c$ and $r_{cx}$: + +$$ +\begin{equation} +\int_\Omega w c^{n+1} dV = \int_\Omega wc^{n} +\nabla w \cdot (-\Delta t M \frac{1}{f_{,cc}} \nabla \mu_c) dV +\end{equation} +$$ + +$$ +\begin{align} +r_c &= c^{n} +\end{align} +$$ + +$$ +\begin{align} +r_{cx} &= -\Delta t M \frac{1}{f_{,cc}} \nabla \mu_c +\end{align} +$$ + +Expanding $\mu_{\eta_p}$ in equation + +$$ +\begin{align} +\int_\Omega w \eta_p^{n+1} dV &= \int_\Omega w \eta_p^{n}-w \Delta t L \mu_{\eta_p} dV +\end{align} +$$ + +and applying the divergence theorem yields the residual terms $r_{\eta_p}$ and $r_{\eta_p x}$: + +$$ +\begin{align} +\int_\Omega w \eta_p^{n+1} dV &= +\end{align} +$$ + +$$ +\begin{align} +&\int_\Omega w \left(\eta_p^{n}-\Delta t L \left( (f_{\beta}-f_{\alpha})H_{,\eta_p}(\eta_p^n) -(c_{\beta}-c_{\alpha}) f_{\beta,c_{\beta}}H_{,\eta_p}(\eta_p^n) + W f_{Landau,\eta_p} +-C_{ijkl} (H_{,\eta_p}(\eta_p) \epsilon_{ij}^{0 \eta_p}) (\epsilon_{kl} - \epsilon_{kl}^{0}) + \frac{1}{2} \left((C_{ijkl}^{\eta_p} - C_{ijkl}^{\alpha}) H_{,\eta_p}(\eta_p) \right) (\epsilon_{ij} - \epsilon_{ij}^{0}) (\epsilon_{kl} - \epsilon_{kl}^{0}) \right) \right) &+ \nabla w \cdot \left(-\Delta t L \kappa_{ij}^{\eta_p} \eta_{p,i}^n \right) dV +\end{align} +$$ + +where + +$$ +\begin{align} +r_{\eta_p} &= \eta_p^{n}-\Delta t L \left( (f_{\beta}-f_{\alpha})H_{,\eta_p}(\eta_p^n) -(c_{\beta}-c_{\alpha}) f_{\beta,c_{\beta}}H_{,\eta_p}(\eta_p^n) + W f_{Landau,\eta_p} +-C_{ijkl} (H_{,\eta_p}(\eta_p) \epsilon_{ij}^{0 \eta_p}) (\epsilon_{kl} - \epsilon_{kl}^{0}) + \frac{1}{2} \left((C_{ijkl}^{\eta_p} - C_{ijkl}^{\alpha}) H_{,\eta_p}(\eta_p) \right) (\epsilon_{ij} - \epsilon_{ij}^{0}) (\epsilon_{kl} - \epsilon_{kl}^{0}) \right) +\end{align} +$$ + +$$ +\begin{align} +r_{\eta_p x} &= -\Delta t L \kappa_{ij}^{\eta_p} \eta_{p,i}^n +\end{align} +$$ + +## Appendix I: Example functions for $f_{\alpha}$, $f_{\beta}$, $f_{Landau}$, $H(\eta_p)$ + +$$ +\begin{gather} +f_{\alpha}(c_{\alpha}) = A_{2} c_{\alpha}^2 + A_{1} c_{\alpha} + A_{0} \\ +f_{\beta}(c_{\beta}) = B_{2} c_{\beta}^2 + B_{1} c_{\beta} + B_{0} \\ +f_{Landau}(\eta_1, \eta_2, \eta_3) = (\eta_1^2 + \eta_2^2 + \eta_3^2) - 2(\eta_1^3 + \eta_2^3 + \eta_3^3) + (\eta_1^4 + \eta_2^4 + \eta_3^4) + 5 (\eta_1^2 \eta_2^2 + \eta_2^2 \eta_3^2 + \eta_1^2 \eta_3^2) + 5(\eta_1^2 \eta_2^2 \eta_3^2) \\ +H(\eta_p) = 3 \eta_p^2 - 2 \eta_p^3 +\end{gather} +$$ diff --git a/applications/MgNd_precipitate_single_Bppp/precipitateEvolution_KKS.pdf b/applications/MgNd_precipitate_single_Bppp/precipitateEvolution_KKS.pdf deleted file mode 100644 index 10beabd27..000000000 Binary files a/applications/MgNd_precipitate_single_Bppp/precipitateEvolution_KKS.pdf and /dev/null differ diff --git a/applications/MgNd_precipitate_single_Bppp/tex_files/precipitateEvolution_KKS.pdf b/applications/MgNd_precipitate_single_Bppp/tex_files/precipitateEvolution_KKS.pdf deleted file mode 100644 index 10beabd27..000000000 Binary files a/applications/MgNd_precipitate_single_Bppp/tex_files/precipitateEvolution_KKS.pdf and /dev/null differ diff --git a/applications/MgNd_precipitate_single_Bppp/tex_files/precipitateEvolution_KKS.tex b/applications/MgNd_precipitate_single_Bppp/tex_files/precipitateEvolution_KKS.tex deleted file mode 100644 index 0d3f68921..000000000 --- a/applications/MgNd_precipitate_single_Bppp/tex_files/precipitateEvolution_KKS.tex +++ /dev/null @@ -1,331 +0,0 @@ -% *********************************************************** -% ******************* PHYSICS HEADER ************************ -% *********************************************************** -% Version 2 -\documentclass[11pt]{article} -\usepackage{amsmath} % AMS Math Package -\usepackage{amsthm} % Theorem Formatting -\usepackage{amssymb} % Math symbols such as \mathbb -\usepackage{graphicx} % Allows for eps images -\usepackage{multicol} % Allows for multiple columns -\usepackage[dvips,letterpaper,margin=0.75in,bottom=0.5in]{geometry} - 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- -\makenomenclature -\makeindex - -\title{KKS Phase Field Model of Precipitate Evolution} -\begin{document} -\maketitle -\nomenclature[a]{$c$}{Concentration (Cahn-Hilliard order parameter)} -\nomenclature[b]{$\eta$}{Structural order parameter (Allen-Cahn order parameter)} -\nomenclature[c]{$\bE$}{Lagrange strain tensor (Mechanics order parameter)} -\nomenclature[d]{$\Pi$}{Total free energy of the system} -\nomenclature[e]{$F$}{Local free energy density} -\nomenclature[f]{$\mathcal{J}$}{Concentration flux} -\nomenclature[g]{$\mu$}{Chemical potential} -\nomenclature[h]{$\kappa^c$}{Cahn-Hilliard gradient coefficient} -\nomenclature[i]{$\kappa^{\eta}$}{Allen-Cahn gradient coefficient} -\nomenclature[j]{$L^{c}$}{Concentration mobility} -\nomenclature[k]{$L^{\eta}$}{Structural order parameter mobility} -\nomenclature[l]{$\omega$}{Variations over primal field} -\nomenclature[m]{$\mathcal{M}$}{Boundary chemical potential like term} -\nomenclature[n]{$\bn$}{Nomal vector in the current configuration} -\nomenclature[o]{$(\theta,~\phi)$}{Polar angles of the interface normal, $\bn$} -\centerline{\today} -\printnomenclature[1cm] -\vspace{.5in} - -\noindent The model employed in this application is described in detail in the article:\\ -\\ -DeWitt et al., Misfit-driven $\beta'''$ precipitate composition and morphology in Mg-Nd alloys,\\ -Acta Materialia {\bf 137}, 378-389 (2017). - -\section{Variational formulation} -The total free energy of the system (neglecting boundary terms) is of the form, -\begin{equation} -\Pi(c, \eta_1, \eta_2, \eta_3, \Bepsilon) = \int_{\Omega} f(c, \eta_1, \eta_2, \eta_3, \Bepsilon) ~dV -\end{equation} -where $c$ is the concentration of the $\beta$ phase, $\eta_p$ are the structural order parameters and $\Bvarepsilon$ is the small strain tensor. $f$, the free energy density is given by -\begin{equation} - f(c, \eta_1, \eta_2, \eta_3, \Bepsilon) = f_{chem}(c, \eta_1, \eta_2, \eta_3) + f_{grad}(\eta_1, \eta_2, \eta_3) + f_{elastic}(c,\eta_1, \eta_2, \eta_3,\Bepsilon) -\end{equation} -where -\begin{gather} -f_{chem}(c, \eta_1, \eta_2, \eta_3) = f_{\alpha}(c,\eta_1, \eta_2, \eta_3) \left( 1- \sum_{p=1}^3 H(\eta_p)\right) + f_{\beta}(c,\eta_1, \eta_2, \eta_3) \sum_{p=1}^3 H(\eta_p)+ W f_{Landau}(\eta_1, \eta_2, \eta_3) \\ -f_{grad}(\eta_1, \eta_2, \eta_3) = \frac{1}{2} \sum_{p=1}^3 \Bkappa^{\eta_p}_{ij} \eta_{p,i} \eta_{p,j} \\ -f_{elastic}(c,\eta_1, \eta_2, \eta_3,\Bepsilon) = \frac{1}{2} \bC_{ijkl}(\eta_1, \eta_2, \eta_3) \left( \Bvarepsilon_{ij} - \Bvarepsilon ^0_{ij}(c, \eta_1, \eta_2, \eta_3) \right)\left( \Bvarepsilon_{kl} - \Bvarepsilon^0_{kl}(c, \eta_1, \eta_2, \eta_3)\right) \\ -\Bvarepsilon^0(c, \eta_1, \eta_2, \eta_3) = H(\eta_1) \Bvarepsilon^0_{\eta_1} (c_{\beta})+ H(\eta_2) \Bvarepsilon^0_{\eta_2} (c_{\beta}) + H(\eta_3) \Bvarepsilon^0_{\eta_3} (c_{\beta}) \\ -\bC(\eta_1, \eta_2, \eta_3) = H(\eta_1) \bC_{\eta_1}+ H(\eta_2) \bC_{\eta_2} + H(\eta_3) \bC_{\eta_3} + \left( 1- H(\eta_1)-H(\eta_2)-H(\eta_3)\right) \bC_{\alpha} -\end{gather} -Here $\Bvarepsilon^0_{\eta_p}$ are the composition dependent stress free strain transformation tensor corresponding to each structural order parameter, which is a function of the $\beta$ phase concentration, $c_{\beta}$, defined below. - -In the KKS model (Kim 1999), the interfacial region is modeled as a mixture of the $\alpha$ and $\beta$ phases with concentrations $c_{alpha}$ and $c_{beta}$, respectively. The homogenous free energies for each phase, $f_{\alpha}$ and $f_{\beta}$ in this case, are typically given as functions of $c_{\alpha}$ and $c_{\beta}$, rather than directly as functions of $c$ and $\eta_p$. Thus, $f_{chem}(c, \eta_1, \eta_2, \eta_3)$ can be rewritten as -\begin{equation} -f_{chem}(c, \eta_1, \eta_2, \eta_3) = f_{\alpha}(c_{\alpha}) \left( 1- \sum_{p=1}^3 H(\eta_p)\right) + f_{\beta}(c_{\beta}) \sum_{p=1}^3 H(\eta_p)+ W f_{Landau}(\eta_1, \eta_2, \eta_3) -\end{equation} - -The concentration in each phase is determined by the following system of equations: -\begin{gather} -c = c_{\alpha} \left( 1- \sum_{p=1}^3 H(\eta_p)\right) + c_{\beta} \sum_{p=1}^3 H(\eta_p) \\ -\frac{\partial f_{\alpha}(c_{\alpha})}{\partial c_{\alpha}} = \frac{\partial f_{\beta}(c_{\beta})}{\partial c_{\beta}} -\end{gather} - -Given the following parabolic functions for the single-phase homogenous free energies: -\begin{gather} -f_{\alpha}(c_{\alpha}) = A_{2} c_{\alpha}^2 + A_{1} c_{\alpha} + A_{0} \\ -f_{\beta}(c_{\beta}) = B_{2} c_{\beta}^2 + B_{1} c_{\beta} + B_{0} -\end{gather} -the single-phase concentrations are: -\begin{gather} -c_{\alpha} = \frac{ B_2 c + \frac{1}{2} (B_1 - A_1) \sum_{p=1}^3 H(\eta_p) }{A_2 \sum_{p=1}^3 H(\eta_p) + B_2 \left( 1- \sum_{p=1}^3 H(\eta_p)\right) } \\ -c_{\beta} = \frac{ A_2 c + \frac{1}{2} (A_1 - B_1) \left[1-\sum_{p=1}^3 H(\eta_p)\right] }{A_2 \sum_{p=1}^3 H(\eta_p) + B_2 \left[ 1- \sum_{p=1}^3 H(\eta_p)\right] } -\end{gather} - -\section{Required inputs} -\begin{itemize} -\item $f_{\alpha}(c_{\alpha}), f_{\beta}(c_{\beta})$ - Homogeneous chemical free energy of the components of the binary system, example form given above -\item $f_{Landau}(\eta_1, \eta_2, \eta_3)$ - Landau free energy term that controls the interfacial energy and prevents precipitates with different orientation varients from overlapping, example form given in Appendix I -\item $W$ - Barrier height for the Landau free energy term, used to control the thickness of the interface -\item $H(\eta_p)$ - Interpolation function for connecting the $\alpha$ phase and the $p^{th}$ orientation variant of the $\beta$ phase, example form given in Appendix I -\item $\Bkappa^{\eta_p}$ - gradient penalty tensor for the $p^{th}$ orientation variant of the $\beta$ phase -\item $\bC_{\eta_p}$ - fourth order elasticity tensor (or its equivalent second order Voigt representation) for the $p^{th}$ orientation variant of the $\beta$ phase -\item $\bC_{\alpha}$ - fourth order elasticity tensor (or its equivalent second order Voigt representation) for the $\alpha$ phase -\item $\Bvarepsilon^0_{\eta_p}$ - stress free strain transformation tensor for the $p^{th}$ orientation variant of the $\beta$ phase -\end{itemize} -In addition, to drive the kinetics, we need: -\begin{itemize} -\item $M$ - mobility value for the concentration field -\item $L$ - mobility value for the structural order parameter field -\end{itemize} - -\section{Variational treatment} -%From the variational derivatives given in Appendix II, we obtain the chemical potentials for the concentration and the structural order parameters: -We obtain chemical potentials for the chemical potentials for the concentration and the structural order parameters by taking variational derivatives of $\Pi$: -\begin{align} - \mu_{c} &= f_{\alpha,c} \left( 1- H(\eta_1)-H(\eta_2)-H(\eta_3)\right) +f_{\beta,c} \left( H(\eta_1) + H(\eta_2) + H(\eta_3) \right) + \bC_{ijkl} (- \Bvarepsilon^0_{ij,c}) \left( \Bvarepsilon_{kl} - \Bvarepsilon^0_{kl}\right) \\ - \mu_{\eta_p} &= [ f_{\beta}-f_{\alpha} -(c_{\beta}-c_{\alpha}) f_{\beta,c_{\beta}}] H(\eta_p)_{,\eta_p} + W f_{Landau,\eta_p}- \Bkappa^{\eta_p}_{ij} \eta_{p,ij} + \bC_{ijkl} (- \Bvarepsilon^0_{ij,\eta_p}) \left( \Bvarepsilon_{kl} - \Bvarepsilon^0_{kl}\right) + \frac{1}{2} \bC_{ijkl,\eta_p} \left( \Bvarepsilon_{ij} - \Bvarepsilon ^0_{ij} \right) \left( \Bvarepsilon_{kl} - \Bvarepsilon^0_{kl}\right) -\end{align} - -\section{Kinetics} -Now the PDE for Cahn-Hilliard dynamics is given by: -\begin{align} - \frac{\partial c}{\partial t} &= ~\grad \cdot \left( \frac{1}{f_{,cc}}M \grad \mu_c \right) \label{CH_eqn} - \end{align} - where $M$ is a constant mobility and the factor of $\frac{1}{f_{,cc}}$ is added to guarentee constant diffusivity in the two phases. The PDE for Allen-Cahn dynamics is given by: - \begin{align} - \frac{\partial \eta_p}{\partial t} &= - L \mu_{\eta_p} \label{AC_eqn} -\end{align} -where $L$ is a constant mobility. - -\section{Mechanics} -Considering variations on the displacement $u$ of the from $u+\epsilon w$, we have -\begin{align} -\delta_u \Pi &= \int_{\Omega} \grad w : \bC(\eta_1, \eta_2, \eta_3) : \left( \Bvarepsilon - \Bvarepsilon^0(c,\eta_1, \eta_2, \eta_3)\right) ~dV = 0 \\ -\end{align} -where $\Bsigma = \bC(\eta_1, \eta_2, \eta_3) : \left( \Bvarepsilon - \Bvarepsilon^0(c,\eta_1, \eta_2, \eta_3)\right)$ is the stress tensor. \\ - -\section{Time discretization} -Using forward Euler explicit time stepping, equations \ref{CH_eqn} and \ref{AC_eqn} become: -\begin{align} -c^{n+1} = c^{n}+\Delta t \left[\nabla \cdot \left(\frac{1}{f_{,cc}} M \nabla \mu_c \right) \right]\\ -\eta_p^{n+1} = \eta_p^n -\Delta t L \mu_{\eta_p} -\end{align} - -\section{Weak formulation} -Writing equations \ref{CH_eqn} and \ref{AC_eqn} in the weak form, with the arbirary variation given by $w$ yields: -\begin{align} -\int_\Omega w c^{n+1} dV &= \int_\Omega wc^{n}+w \Delta t \left[\nabla \cdot \left(\frac{1}{f_{,cc}} M \nabla \mu_c \right) \right] dV \label{CH_weak} \\ -%&= \int_\Omega w\underbrace{c^{n}}_{r_c}+\nabla w \cdot (\underbrace{\Delta t M \nabla \mu_c}_{r_{cx}} ) dV \\ -\int_\Omega w \eta_p^{n+1} dV &= \int_\Omega w \eta_p^{n}-w \Delta t L \mu_{\eta_p} dV \label{AC_weak} -%&= \int_\Omega w\underbrace{c^{n}}_{r_c}+\nabla w \cdot (\underbrace{\Delta t M \nabla \mu_c}_{r_{cx}} ) dV -\end{align} - -The expression of $\frac{1}{f_{,cc}} \mu_c$ can be written as: -\begin{equation} -\begin{split} -\frac{1}{f_{,cc}} \nabla \mu_c = & \nabla c + (c_{\alpha}-c_{\beta}) \sum_{p=1}^3 H(\eta_p)_{,\eta_p} \nabla \eta_p \\ -&+ \frac{1}{f_{,cc}} \left[ \sum_{p=1}^3 (C_{ijkl}^{\eta_p} - C_{ijkl}^{\alpha} )\nabla \eta_p H(\eta_p)_{,\eta_p} \right](-\epsilon_{ij,c}^0)(\epsilon_{ij} - \epsilon_{ij}^0) \\ -&- \frac{1}{f_{,cc}} C_{ijkl} \left[ \sum_{p=1}^3 \left( H(\eta_p)_{,\eta_p} \epsilon_{ij,c}^{0\eta_p} + \sum_{q=1}^3 \left( H(\eta_p) \epsilon_{ij,c\eta_q}^{0\eta_p} \right) \right) \nabla \eta_p + H(\eta_p) \epsilon_{ij,cc}^{0\eta_p} \nabla c \right](\epsilon_{kl}-\epsilon_{kl}^0)\\ -&+ \frac{1}{f_{,cc}} C_{ijkl} (-\epsilon_{ij,c}^0) \left[ \nabla \epsilon_{kl} - \left( \sum_{p=1}^3 \left(H(\eta_p)_{,\eta_p} \epsilon_{kl}^{0\eta_p} -\sum_{q=1}^3 \epsilon_{kl,\eta_q}^{\eta_q} H(\eta_q) \right)\nabla \eta_p + H(\eta_p) \epsilon_{kl,c}^{0\eta_p} \nabla c \right) \right] -\end{split} -\end{equation} - -Applying the divergence theorem to equation \ref{CH_weak}, one can derive the residual terms $r_c$ and $r_{cx}$: -\begin{equation} -\int_\Omega w c^{n+1} dV = \int_\Omega w\underbrace{c^{n}}_{r_c}+\nabla w \cdot (\underbrace{-\Delta t M \frac{1}{f_{,cc}} \nabla \mu_c}_{r_{cx}} ) dV -\end{equation} - -Expanding $\mu_{\eta_p}$ in equation \ref{AC_weak} and applying the divergence theorem yields the residual terms $r_{\eta_p}$ and $r_{\eta_p x}$: -\begin{equation} -\begin{split} -\int_\Omega w \eta_p^{n+1} dV &= \\ -&\int_\Omega w \Bigg\{\underbrace{\eta_p^{n}-\Delta t L \bigg[(f_{\beta}-f_{\alpha})H(\eta_p^n)_{,\eta_p} -(c_{\beta}-c_{\alpha}) f_{\beta,c_{\beta}}H(\eta_p^n)_{,\eta_p} + W f_{Landau,\eta_p}}_{r_{\eta_p}} \\ -&\underbrace{ -C_{ijkl} \left( H(\eta_p)_{,\eta_p} \epsilon_{ij}^{0 \eta_p}\right)\left(\epsilon_{kl} - \epsilon_{kl}^{0} \right) + \frac{1}{2} \left[ (C_{ijkl}^{\eta_p} - C_{ijkl}^{\alpha}) H(\eta_p)_{,\eta_p} \right] \left(\epsilon_{ij} - \epsilon_{ij}^{0} \right) \left(\epsilon_{kl} - \epsilon_{kl}^{0} \right) \bigg] }_{r_{\eta_p}~cont.} \Bigg\} \\ -&+ \nabla w \cdot (\underbrace{-\Delta t L \Bkappa^{\eta_p}_{ij} \eta_{p,i}^n}_{r_{\eta_p x}} ) dV -\end{split} -\end{equation} - -\section{Appendix I: Example functions for $f_{\alpha}$, $f_{\beta}$, $f_{Landau}$, $H(\eta_p)$ } -\begin{gather} -f_{\alpha}(c_{\alpha}) = A_{2} c_{\alpha}^2 + A_{1} c_{\alpha} + A_{0} \\ -f_{\beta}(c_{\beta}) = B_{2} c_{\beta}^2 + B_{1} c_{\beta} + B_{0} \\ -f_{Landau}(\eta_1, \eta_2, \eta_3) = (\eta_1^2 + \eta_2^2 + \eta_3^2) - 2(\eta_1^3 + \eta_2^3 + \eta_3^3) + (\eta_1^4 + \eta_2^4 + \eta_3^4) + 5 (\eta_1^2 \eta_2^2 + \eta_2^2 \eta_3^2 + \eta_1^2 \eta_3^2) + 5(\eta_1^2 \eta_2^2 \eta_3^2) \\ -H(\eta_p) = 3 \eta_p^2 - 2 \eta_p^3 -\end{gather} - -\end{document} \ No newline at end of file