diff options
Diffstat (limited to 'Docs/source/latex_theory')
| -rw-r--r-- | Docs/source/latex_theory/AMR/AMR.tex | 4 | ||||
| -rw-r--r-- | Docs/source/latex_theory/Boosted_frame/Boosted_frame.tex | 4 | ||||
| -rw-r--r-- | Docs/source/latex_theory/Makefile | 8 | ||||
| -rw-r--r-- | Docs/source/latex_theory/PML/PML.tex | 24 | ||||
| -rw-r--r-- | Docs/source/latex_theory/input_output/input_output.tex | 4 | ||||
| -rw-r--r-- | Docs/source/latex_theory/theory.tex | 12 |
6 files changed, 31 insertions, 25 deletions
diff --git a/Docs/source/latex_theory/AMR/AMR.tex b/Docs/source/latex_theory/AMR/AMR.tex index bd3f3bd6e..23e32f6a1 100644 --- a/Docs/source/latex_theory/AMR/AMR.tex +++ b/Docs/source/latex_theory/AMR/AMR.tex @@ -1,5 +1,9 @@ \input{newcommands} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Mesh refinement} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \begin{figure}[htb] \centering \includegraphics[width=15cm]{ICNSP_2011_Vay_fig1.png} diff --git a/Docs/source/latex_theory/Boosted_frame/Boosted_frame.tex b/Docs/source/latex_theory/Boosted_frame/Boosted_frame.tex index c3cb08ca2..c0e7f41f8 100644 --- a/Docs/source/latex_theory/Boosted_frame/Boosted_frame.tex +++ b/Docs/source/latex_theory/Boosted_frame/Boosted_frame.tex @@ -1,5 +1,9 @@ \input{newcommands} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Moving window and optimal Lorentz boosted frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + The simulations of plasma accelerators from first principles are extremely computationally intensive, due to the need to resolve the evolution of a driver (laser or particle beam) and an accelerated particle beam into a plasma structure that is orders of magnitude longer and wider than the accelerated beam. As is customary in the modeling of particle beam dynamics in standard particle accelerators, a moving window is commonly used to follow the driver, the wake and the accelerated beam. This results in huge savings, by avoiding the meshing of the entire plasma that is orders of magnitude longer than the other length scales of interest. \begin{figure} diff --git a/Docs/source/latex_theory/Makefile b/Docs/source/latex_theory/Makefile index 30ae06964..ee6aefb88 100644 --- a/Docs/source/latex_theory/Makefile +++ b/Docs/source/latex_theory/Makefile @@ -4,9 +4,11 @@ SRC_FILES = theory.tex \ all: $(SRC_FILES) clean pandoc intro.tex --mathjax --wrap=preserve --bibliography allbibs.bib -o intro.rst - mv intro.rst ../theory - pandoc theory.tex --mathjax --wrap=preserve --bibliography allbibs.bib -o warpx_theory.rst - mv warpx_theory.rst ../theory + pandoc AMR/AMR.tex --mathjax --wrap=preserve --bibliography allbibs.bib -o amr.rst + pandoc Boosted_frame/Boosted_frame.tex --mathjax --wrap=preserve --bibliography allbibs.bib -o boosted_frame.rst + pandoc input_output/input_output.tex --mathjax --wrap=preserve --bibliography allbibs.bib -o input_output.rst + pandoc PML/PML.tex --mathjax --wrap=preserve --bibliography allbibs.bib -o PML.rst + mv *.rst ../theory cd ../../../../picsar/Doxygen/pages/latex_theory/; pandoc theory.tex --mathjax --wrap=preserve --bibliography allbibs.bib -o picsar_theory.rst mv ../../../../picsar/Doxygen/pages/latex_theory/picsar_theory.rst ../theory cp ../../../../picsar/Doxygen/images/PIC.png ../theory diff --git a/Docs/source/latex_theory/PML/PML.tex b/Docs/source/latex_theory/PML/PML.tex index 6956b5182..7c5f09619 100644 --- a/Docs/source/latex_theory/PML/PML.tex +++ b/Docs/source/latex_theory/PML/PML.tex @@ -1,5 +1,9 @@ \input{newcommands} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Boundary conditions} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \subsection{Open boundary condition for electromagnetic waves} For the TE case, the original Berenger's Perfectly Matched Layer (PML) writes @@ -10,7 +14,7 @@ For the TE case, the original Berenger's Perfectly Matched Layer (PML) writes \varepsilon _{0}\frac{\partial E_{y}}{\partial t}+\sigma _{x}E_{y} = & -\frac{\partial H_{z}}{\partial x}\label{PML_def_2} \\ \mu _{0}\frac{\partial H_{zx}}{\partial t}+\sigma ^{*}_{x}H_{zx} = & -\frac{\partial E_{y}}{\partial x}\label{PML_def_3} \\ \mu _{0}\frac{\partial H_{zy}}{\partial t}+\sigma ^{*}_{y}H_{zy} = & \frac{\partial E_{x}}{\partial y}\label{PML_def_4} \\ -H_{z} = & H_{zx}+H_{zy}\label{PML_def_5} +H_{z} = & H_{zx}+H_{zy}\label{PML_def_5} \end{eqnarray} This can be generalized to @@ -21,7 +25,7 @@ This can be generalized to \varepsilon _{0}\frac{\partial E_{y}}{\partial t}+\sigma _{x}E_{y} = & -\frac{c_{x}}{c}\frac{\partial H_{z}}{\partial x}+\overline{\sigma }_{x}H_{z}\label{APML_def_2} \\ \mu _{0}\frac{\partial H_{zx}}{\partial t}+\sigma ^{*}_{x}H_{zx} = & -\frac{c^{*}_{x}}{c}\frac{\partial E_{y}}{\partial x}+\overline{\sigma }_{x}^{*}E_{y}\label{APML_def_3} \\ \mu _{0}\frac{\partial H_{zy}}{\partial t}+\sigma ^{*}_{y}H_{zy} = & \frac{c^{*}_{y}}{c}\frac{\partial E_{x}}{\partial y}+\overline{\sigma }_{y}^{*}E_{x}\label{APML_def_4} \\ -H_{z} = & H_{zx}+H_{zy}\label{APML_def_5} +H_{z} = & H_{zx}+H_{zy}\label{APML_def_5} \end{eqnarray} For $c_{x}=c_{y}=c^{*}_{x}=c^{*}_{y}=c$ and $\overline{\sigma }_{x}=\overline{\sigma }_{y}=\overline{\sigma }_{x}^{*}=\overline{\sigma }_{y}^{*}=0$, @@ -39,7 +43,7 @@ angle $\varphi$ relative to the x axis E_{x} = & -E_{0}\sin \varphi e^{i\omega \left( t-\alpha x-\beta y\right) }\label{Plane_wave_APML_def_1} \\ E_{y} = & E_{0}\cos \varphi e^{i\omega \left( t-\alpha x-\beta y\right) }\label{Plane_wave_APML_def_2} \\ H_{zx} = & H_{zx0}e^{i\omega \left( t-\alpha x-\beta y\right) }\label{Plane_wave_AMPL_def_3} \\ -H_{zy} = & H_{zy0}e^{i\omega \left( t-\alpha x-\beta y\right) }\label{Plane_wave_APML_def_4} +H_{zy} = & H_{zy0}e^{i\omega \left( t-\alpha x-\beta y\right) }\label{Plane_wave_APML_def_4} \end{eqnarray} @@ -55,7 +59,7 @@ and (\ref{APML_def_4}) gives \varepsilon _{0}E_{0}\sin \varphi -i\frac{\sigma _{y}}{\omega }E_{0}\sin \varphi = & \beta \frac{c_{y}}{c}\left( H_{zx0}+H_{zy0}\right) +i\frac{\overline{\sigma }_{y}}{\omega }\left( H_{zx0}+H_{zy0}\right) \label{Plane_wave_APML_1_1} \\ \varepsilon _{0}E_{0}\cos \varphi -i\frac{\sigma _{x}}{\omega }E_{0}\cos \varphi = & \alpha \frac{c_{x}}{c}\left( H_{zx0}+H_{zy0}\right) -i\frac{\overline{\sigma }_{x}}{\omega }\left( H_{zx0}+H_{zy0}\right) \label{Plane_wave_APML_1_2} \\ \mu _{0}H_{zx0}-i\frac{\sigma ^{*}_{x}}{\omega }H_{zx0} = & \alpha \frac{c^{*}_{x}}{c}E_{0}\cos \varphi -i\frac{\overline{\sigma }^{*}_{x}}{\omega }E_{0}\cos \varphi \label{Plane_wave_APML_1_3} \\ -\mu _{0}H_{zy0}-i\frac{\sigma ^{*}_{y}}{\omega }H_{zy0} = & \beta \frac{c^{*}_{y}}{c}E_{0}\sin \varphi +i\frac{\overline{\sigma }^{*}_{y}}{\omega }E_{0}\sin \varphi \label{Plane_wave_APML_1_4} +\mu _{0}H_{zy0}-i\frac{\sigma ^{*}_{y}}{\omega }H_{zy0} = & \beta \frac{c^{*}_{y}}{c}E_{0}\sin \varphi +i\frac{\overline{\sigma }^{*}_{y}}{\omega }E_{0}\sin \varphi \label{Plane_wave_APML_1_4} \end{eqnarray} @@ -64,7 +68,7 @@ and (\ref{Plane_wave_APML_1_2}), we get \begin{eqnarray} \beta = & \left[ Z\left( \varepsilon _{0}-i\frac{\sigma _{y}}{\omega }\right) \sin \varphi -i\frac{\overline{\sigma }_{y}}{\omega }\right] \frac{c}{c_{y}}\label{Plane_wave_APML_beta_of_g} \\ -\alpha = & \left[ Z\left( \varepsilon _{0}-i\frac{\sigma _{x}}{\omega }\right) \cos \varphi +i\frac{\overline{\sigma }_{x}}{\omega }\right] \frac{c}{c_{x}}\label{Plane_wave_APML_alpha_of_g} +\alpha = & \left[ Z\left( \varepsilon _{0}-i\frac{\sigma _{x}}{\omega }\right) \cos \varphi +i\frac{\overline{\sigma }_{x}}{\omega }\right] \frac{c}{c_{x}}\label{Plane_wave_APML_alpha_of_g} \end{eqnarray} @@ -75,14 +79,14 @@ and (\ref{Plane_wave_APML_alpha_of_g}) yields \begin{eqnarray} \frac{1}{Z} = & \frac{Z\left( \varepsilon _{0}-i\frac{\sigma _{x}}{\omega }\right) \cos \varphi \frac{c^{*}_{x}}{c_{x}}+i\frac{\overline{\sigma }_{x}}{\omega }\frac{c^{*}_{x}}{c_{x}}-i\frac{\overline{\sigma }^{*}_{x}}{\omega }}{\mu _{0}-i\frac{\sigma ^{*}_{x}}{\omega }}\cos \varphi \nonumber \\ - + & \frac{Z\left( \varepsilon _{0}-i\frac{\sigma _{y}}{\omega }\right) \sin \varphi \frac{c^{*}_{y}}{c_{y}}-i\frac{\overline{\sigma }_{y}}{\omega }\frac{c^{*}_{y}}{c_{y}}+i\frac{\overline{\sigma }^{*}_{y}}{\omega }}{\mu _{0}-i\frac{\sigma ^{*}_{y}}{\omega }}\sin \varphi + + & \frac{Z\left( \varepsilon _{0}-i\frac{\sigma _{y}}{\omega }\right) \sin \varphi \frac{c^{*}_{y}}{c_{y}}-i\frac{\overline{\sigma }_{y}}{\omega }\frac{c^{*}_{y}}{c_{y}}+i\frac{\overline{\sigma }^{*}_{y}}{\omega }}{\mu _{0}-i\frac{\sigma ^{*}_{y}}{\omega }}\sin \varphi \end{eqnarray} If $c_{x}=c^{*}_{x}$, $c_{y}=c^{*}_{y}$, $\overline{\sigma }_{x}=\overline{\sigma }^{*}_{x}$, $\overline{\sigma }_{y}=\overline{\sigma }^{*}_{y}$, $\frac{\sigma _{x}}{\varepsilon _{0}}=\frac{\sigma ^{*}_{x}}{\mu _{0}}$ and $\frac{\sigma _{y}}{\varepsilon _{0}}=\frac{\sigma ^{*}_{y}}{\mu _{0}}$ then \begin{eqnarray} -Z = & \pm \sqrt{\frac{\mu _{0}}{\varepsilon _{0}}}\label{APML_impedance} +Z = & \pm \sqrt{\frac{\mu _{0}}{\varepsilon _{0}}}\label{APML_impedance} \end{eqnarray} @@ -93,7 +97,7 @@ retained after discretization, as shown subsequently in this paper. Calling $\psi$ any component of the field and $\psi _{0}$ its magnitude, we get from (\ref{Plane_wave_APML_def_1}), (\ref{Plane_wave_APML_beta_of_g}), -(\ref{Plane_wave_APML_alpha_of_g}) and (\ref{APML_impedance}) that +(\ref{Plane_wave_APML_alpha_of_g}) and (\ref{APML_impedance}) that \begin{equation} \label{Plane_wave_absorption} @@ -104,7 +108,7 @@ its magnitude, we get from (\ref{Plane_wave_APML_def_1}), (\ref{Plane_wave_APML_ We assume that we have an APML layer of thickness $\delta$ (measured along $x$) and that $\sigma _{y}=\overline{\sigma }_{y}=0$ and $c_{y}=c.$ Using (\ref{Plane_wave_absorption}), we determine -that the coefficient of reflection given by this layer is +that the coefficient of reflection given by this layer is \begin{eqnarray} R_{APML}\left( \theta \right) = & e^{-\left( \sigma _{x}\cos \varphi /\varepsilon _{0}c_{x}+\overline{\sigma }_{x}c/c_{x}\right) \delta }e^{-\left( \sigma _{x}\cos \varphi /\varepsilon _{0}c_{x}-\overline{\sigma }_{x}c/c_{x}\right) \delta }\nonumber \\ @@ -187,7 +191,7 @@ H_z = & H_{zx}+H_{zy} c_x = & c e^{-\sigma_x\Delta t} \frac{\sigma_x \Delta x}{1-e^{-\sigma_x\Delta t}} \\ c_y = & c e^{-\sigma_y\Delta t} \frac{\sigma_y \Delta y}{1-e^{-\sigma_y\Delta t}} \\ c^*_x = & c e^{-\sigma^*_x\Delta t} \frac{\sigma^*_x \Delta x}{1-e^{-\sigma^*_x\Delta t}} \\ -c^*_y = & c e^{-\sigma^*_y\Delta t} \frac{\sigma^*_y \Delta y}{1-e^{-\sigma^*_y\Delta t}} +c^*_y = & c e^{-\sigma^*_y\Delta t} \frac{\sigma^*_y \Delta y}{1-e^{-\sigma^*_y\Delta t}} \end{align} \end{subequations} diff --git a/Docs/source/latex_theory/input_output/input_output.tex b/Docs/source/latex_theory/input_output/input_output.tex index 117af9cd0..870ce2e44 100644 --- a/Docs/source/latex_theory/input_output/input_output.tex +++ b/Docs/source/latex_theory/input_output/input_output.tex @@ -1,5 +1,9 @@ \input{newcommands} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Inputs and outputs} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + Initialization of the plasma columns and drivers (laser or particle beam) is performed via the specification of multidimensional functions that describe the initial state with, if needed, a time dependence, or from reconstruction of distributions based on experimental data. Care is needed when initializing quantities in parallel to avoid double counting and ensure smoothness of the distributions at the interface of computational domains. When the sum of the initial distributions of charged particles is not charge neutral, initial fields are computed using generally a static approximation with Poisson solves accompanied by proper relativistic scalings \cite{Vaypop2008, CowanPRSTAB13}. Outputs include dumps of particle and field quantities at regular intervals, histories of particle distributions moments, spectra, etc, and plots of the various quantities. In parallel simulations, the diagnostic subroutines need to handle additional complexity from the domain decomposition, as well as large amount of data that may necessitate data reduction in some form before saving to disk. diff --git a/Docs/source/latex_theory/theory.tex b/Docs/source/latex_theory/theory.tex index 643c53786..cee24d8ff 100644 --- a/Docs/source/latex_theory/theory.tex +++ b/Docs/source/latex_theory/theory.tex @@ -23,25 +23,13 @@ \begin{document} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Mesh refinement} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input{AMR/AMR.tex} %%%%%\input{Particle_pushers/Vay_pusher.tex} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Boundary conditions} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input{PML/PML.tex} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Moving window and optimal Lorentz boosted frame} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input{Boosted_frame/Boosted_frame.tex} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Inputs and outputs} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input{input_output/input_output.tex} |