Commit bbed7285 authored by Matthieu Schaller's avatar Matthieu Schaller
Browse files

Updated documentation to contain all the truncated derivatives

parent d50a7bb1
......@@ -161,4 +161,34 @@ keywords = "adaptive algorithms"
}
@ARTICLE{Springel2005,
author = {{Springel}, V.},
title = "{The cosmological simulation code GADGET-2}",
journal = {\mnras},
eprint = {astro-ph/0505010},
keywords = {methods: numerical, galaxies: interactions, dark matter},
year = 2005,
month = dec,
volume = 364,
pages = {1105-1134},
doi = {10.1111/j.1365-2966.2005.09655.x},
adsurl = {http://adsabs.harvard.edu/abs/2005MNRAS.364.1105S},
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}
@ARTICLE{Bagla2003,
author = {{Bagla}, J.~S. and {Ray}, S.},
title = "{Performance characteristics of TreePM codes}",
journal = {\na},
eprint = {astro-ph/0212129},
year = 2003,
month = sep,
volume = 8,
pages = {665-677},
doi = {10.1016/S1384-1076(03)00056-3},
adsurl = {http://adsabs.harvard.edu/abs/2003NewA....8..665B},
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}
......@@ -4,6 +4,7 @@
\usepackage{times}
\usepackage{comment}
\newcommand{\gadget}{{\sc Gadget}\xspace}
\newcommand{\swift}{{\sc Swift}\xspace}
\newcommand{\nbody}{$N$-body\xspace}
......@@ -32,7 +33,7 @@ Making gravity great again.
\input{potential_softening}
\input{fmm_summary}
\input{gravity_derivatives}
%\input{gravity_derivatives}
\input{mesh_summary}
\bibliographystyle{mnras}
......
......@@ -162,21 +162,21 @@ read:
All the kernels (Eqs.~\ref{eq:fmm:P2M}-\ref{eq:fmm:L2L}) are rather
straightforward to evaluate as they are only made of additions and
multiplications (provided $\mathsf{D}$ can be evaluated quickly, see
Sec.~\ref{ssec:grav_derivatives}), which are extremely efficient
instructions on modern architectures. However, the fully expanded sums
can lead to rather large and prone to typo expressions. To avoid any
mishaps, we use a \texttt{python} script to generate C code in which
all the sums are unrolled and correct by construction. In \swift, we
implemented the kernels up to order $p=5$, as it proved to be accurate
enough for our purpose, but this could be extended to higher order
easily. This implies storing $56$ numbers per cell for each
$\textsf{M}$ and $\textsf{F}$ plus three numbers for the location of
the centre of mass. For leaf-cells with large numbers of particles, as
in \swift, this is a small memory overhead. One further small
improvement consists in choosing $\mathbf{z}_A$ to be the centre of
mass of cell $A$ rather than its geometrical centre. The first order
multipoles ($\mathsf{M}_{100},\mathsf{M}_{010},\mathsf{M}_{001}$) then
vanish by construction. This allows us to simplify some of the
expressions and helps reduce, albeit by a small fraction, the memory
footprint of the tree structure.
multiplications (provided $\mathsf{D}$ can be evaluated quickly),
which are extremely efficient instructions on modern
architectures. However, the fully expanded sums can lead to rather
large and prone to typo expressions. To avoid any mishaps, we use a
\texttt{python} script to generate C code in which all the sums are
unrolled and correct by construction. In \swift, we implemented the
kernels up to order $p=5$, as it proved to be accurate enough for our
purpose, but this could be extended to higher order easily. This
implies storing $56$ numbers per cell for each $\textsf{M}$ and
$\textsf{F}$ plus three numbers for the location of the centre of
mass. For leaf-cells with large numbers of particles, as in \swift,
this is a small memory overhead. One further small improvement
consists in choosing $\mathbf{z}_A$ to be the centre of mass of cell
$A$ rather than its geometrical centre. The first order multipoles
($\mathsf{M}_{100},\mathsf{M}_{010},\mathsf{M}_{001}$) then vanish by
construction. This allows us to simplify some of the expressions and
helps reduce, albeit by a small fraction, the memory footprint of the
tree structure.
\subsection{Coupling the FMM to a mesh for periodic long-range forces}
\label{ssec:mesh_summary}
\begin{equation}
S(x) = \frac{e^x}{1 + e^x}
\end{equation}
We truncate the potential and forces computed via the FMM using a
smooth function that drops quickly to zero at some scale $r_s$ set by
the top-level mesh. Traditionally, implementations have used
expressions which are cheap to evaluate in Fourier space
\citep[e.g.][]{Bagla2003,
Springel2005}. This, however, implies a large cost for each
interaction computed within the tree as the real-space truncation
function won't have a simple analytic form that can be evaluated
efficiently by computers. Since the FMM scheme involves to not only
evaluate the forces but higher-order derivatives, a more appropiate
choice is necessary. We use the sigmoid $S(x) \equiv \frac{e^x}{1 + e^x}$
as the basis of our truncation function write for the potential:
\begin{align}
\varphi_s(r) &= \frac{1}{r}\left[2 - 2S\left(\frac{2r}{r_s}\right)\right] \nonumber\\
&= \frac{1}{r}\left[2 - \frac{2e^{\frac{2r}{r_s}}}{1+e^{\frac{2r}{r_s}}}\right]
\varphi_s(r) &= \frac{1}{r} \chi(r, r_s) = \frac{1}{r}\times\left[2 - 2S\left(\frac{2r}{r_s}\right)\right].% \nonumber\\
%&= \frac{1}{r}\left[2 - \frac{2e^{\frac{2r}{r_s}}}{1+e^{\frac{2r}{r_s}}}.\right]
\end{align}
This function alongside the trunctation function used in \gadget is
shown on Fig.~\ref{fig:fmm:potential_short}. This choice of $S(x)$ can
seem rather cumbersome at first but writing $\alpha(x) \equiv (1+e^x)^{-1}$,
one can expressed all derivatives of $S(x)$ as simple polynomials in
$\alpha(x)$, which are easy to evaluate. For instance, in the case of
the direct force evaluation between two particles, we obtain
\begin{align}
|\mathbf{f}_s(r)| &= \frac{1}{r^2}\left[\frac{4r}{r_s}S'\left(\frac{2r}{r_s}\right) - 2S\left(\frac{2r}{r_s}\right) + 2\right] \nonumber \\
&= \frac{1}{r^2}\left[\frac{4r}{r_s}\frac{e^{\frac{2r}{r_s}}}{(1+e^{\frac{2r}{r_s}})^2} - \frac{2e^{\frac{2r}{r_s}}}{1+e^{\frac{2r}{r_s}}} + 2\right]
|\mathbf{f}_s(r)| &=
\frac{1}{r^2}\times\left[\frac{4r}{r_s}S'\left(\frac{2r}{r_s}\right) -
2S\left(\frac{2r}{r_s}\right) + 2\right] \nonumber \\
%&=
%\frac{1}{r^2}\left[\frac{4r}{r_s}\frac{e^{\frac{2r}{r_s}}}{(1+e^{\frac{2r}{r_s}})^2}
%- \frac{2e^{\frac{2r}{r_s}}}{1+e^{\frac{2r}{r_s}}} + 2\right]
&=
\frac{1}{r^2}\times 2 \left[x\alpha(x) - x\alpha(x)^2 - e^x\alpha(x) + 1\right],
\end{align}
with $x\equiv2r/r_s$. The truncated force is compared to the Newtonian
force on Fig.~\ref{fig:fmm:force_short}. At distance $r<r_s/10$, the
truncation term is negligibly close to one and the truncated forces
can be replaced by their Newtonian equivalent. We use this
optimization in \swift and only compute truncated forces between pairs
of particles that are in tree-leaves larger than $1/10$ of the mesh
size or between two tree-leaves distant by more than that amount.
The truncation function in Fourier space reads
\begin{equation}
\tilde\varphi_l(k) = \frac{1}{k^2}\left[\frac{\upi}{2}kr_s\textrm{csch}\left(\frac{\upi}{2}kr_s\right) \right]
\tilde\varphi_l(k) =
\frac{1}{k^2}\left[\frac{\upi}{2}kr_s\textrm{csch}\left(\frac{\upi}{2}kr_s\right)
\right]
\end{equation}
\begin{figure}
\includegraphics[width=\columnwidth]{potential_short.pdf}
\caption{aa}
\caption{Truncated potential used in \swift (green line) and \gadget
(yellow line) alongside the full Newtonian potential (blue dasheed
line). The green dash-dotted line corresponds to the same
trunctation function where the exponential in the sigmoid is
replaced by a sixth order Taylor expansion. At $r>4r_s$, the
truncated potential becomes negligible.}
\label{fig:fmm:potential_short}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{force_short.pdf}
\caption{bb}
\caption{used in \swift (green line) and \gadget
(yellow line) alongside the full Newtonian force term (blue dasheed
line). The green dash-dotted line corresponds to the same
trunctation function where the exponential in the sigmoid is
replaced by a sixth order Taylor expansion. At $r<r_s/10$, the
truncated forces becomes almost equal to the Newtonian ones and can
safely be replaced by their simpler form. The deviation between the
exact expression and the one obtained from Taylor expansion at
$r>r_s$ has a small impact since no pairs of particles should
interact directly over distances of order the mesh size. }
\label{fig:fmm:force_short}
\end{figure}
......
......@@ -3,136 +3,165 @@
For completeness, we give here the full expression for the first few
derivatives of the potential that are used in our FMM scheme. We use
the notation $\mathbf{r}=(r_x, r_y, r_z)$, $r = |\mathbf{r}|$ and
$u=r/H$. We can construct the higher order derivatives by successively
applying the "chain rule". We show representative examples of the
first few relevant ones here split by order. We start by constructing
common quantities that appear in derivatives of multiple orders.
the notation $\mathbf{r}=(r_x, r_y, r_z)$, $r = |\mathbf{r}|$, $u=r/H$
and $x=2r/r_s$. We also assume $H \ll r_s$. We can construct the higher order derivatives by
successively applying the "chain rule". We show representative
examples of the first few relevant ones here split by order. We start
by constructing derivatives of the truncated potentials:
\begin{align}
\mathsf{\tilde{D}}_{1}(r, u, H) =
\alpha(x) &= \left(1 + e^x\right)^{-1} \nonumber \\
\chi(r, r_s) &= 2\left(1 - e^{2r/r_s}\alpha(2r/r_s) \right) \nonumber \\
\chi'(r, r_s) &= \frac{2}{r_s}\left(2\alpha(x)^2 - 2\alpha(x)\right) \nonumber \\
\chi''(r, r_s) &= \frac{4}{r_s^2}\left(4\alpha(x)^3 - 6\alpha(x)^2 + 2\alpha(x)\right) \nonumber \\
\chi^{(3)}(r, r_s) &= \frac{8}{r_s^3} \left(12\alpha(x)^4 - 24\alpha(x)^3 + 14\alpha(x)^2 -2 \alpha(x)\right) \nonumber \\
\chi^{(4)}(r, r_s) &= \frac{16}{r_s^4} \left(48\alpha(x)^5 - 120\alpha(x)^4 + 100\alpha(x)^3 -30 \alpha(x)^2 + 2\alpha(x)\right) \nonumber \\
\chi^{(5)}(r, r_s) &= \frac{32}{r_s^5} \left(240\alpha(x)^6 - 720\alpha(x)^5 + 780\alpha(x)^4 - 360\alpha(x)^3 + 62\alpha(x)^2 - 2\alpha(x) \right) \nonumber
\end{align}
We can now construct common quantities that appear in derivatives of
multiple orders:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
\mathsf{\tilde{D}}_{1}(r, r_s, H) =
\left\lbrace\begin{array}{rcl}
\left(-3u^7 + 15u^6 - 28u^5 + 21u^4 - 7u^2 + 3\right)\times H^{-1} & \mbox{if} & u < 1,\\
r^{-1} & \mbox{if} & u \geq 1,
%r^{-1} & \mbox{if} & u \geq 1,
\chi(r, r_s) \times r^{-1} & \mbox{if} & u \geq 1,
\end{array}
\right.\nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
\mathsf{\tilde{D}}_{3}(r, u, H) =
\mathsf{\tilde{D}}_{3}(r, r_s, H) =
\left\lbrace\begin{array}{rcl}
-\left(21u^5 - 90u^4 + 140u^3 -84u^2 +14\right)\times H^{-3}& \mbox{if} & u < 1,\\
-1 \times r^{-3} & \mbox{if} & u \geq 1,
%-1 \times r^{-3} & \mbox{if} & u \geq 1,
\left(r\chi'(r, r_s) - \chi(r, r_s)\right) \times r^{-3} & \mbox{if} & u \geq 1,
\end{array}
\right.\nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
\mathsf{\tilde{D}}_{5}(r, u, H) =
\mathsf{\tilde{D}}_{5}(r, r_s, H) =
\left\lbrace\begin{array}{rcl}
\left(-105u^3 + 360u^2 - 420u + 168\right)\times H^{-5}& \mbox{if} & u < 1,\\
3\times r^{-5} & \mbox{if} & u \geq 1,
%3\times r^{-5} & \mbox{if} & u \geq 1,
\left(r^2\chi''(r, r_s) - 3r\chi'(r, r_s) + 3\chi(r, r_s) \right)\times r^{-5} & \mbox{if} & u \geq 1,
\end{array}
\right.\nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
\mathsf{\tilde{D}}_{7}(r, u, H) =
\mathsf{\tilde{D}}_{7}(r, r_s, H) =
\left\lbrace\begin{array}{rcl}
-\left(315u - 720 + 420u^{-1}\right)\times H^{-7} & \mbox{if} & u < 1,\\
-15\times r^{-7} & \mbox{if} & u \geq 1,
%-15\times r^{-7} & \mbox{if} & u \geq 1,
\left(r^3\chi^{(3)}(r, r_s) - 6r^2\chi''(r, r_s)+15r\chi'(r, r_s)-15\chi(r, r_s)\right) \times r^{-7} & \mbox{if} & u \geq 1,
\end{array}
\right.\nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
\mathsf{\tilde{D}}_{9}(r, u, H) =
\mathsf{\tilde{D}}_{9}(r, r_s, H) =
\left\lbrace\begin{array}{rcl}
\left(-315u^{-1} + 420u^{-3}\right)\times H^{-9}& \mbox{if} & u < 1,\\
105\times r^{-9} & \mbox{if} & u \geq 1.
%105\times r^{-9} & \mbox{if} & u \geq 1.
\left(r^4\chi^{(4)}(r, r_s) - 10r^3\chi^{(3)} + 45r^2\chi''(r, r_s) - 105r\chi'(r, r_s) + 105\chi(r, r_s) \right) \times r^{-9} & \mbox{if} & u \geq 1
\end{array}
\right.\nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
\mathsf{\tilde{D}}_{11}(r, r_s, H) =
\left\lbrace\begin{array}{rcl}
-\left(315u^{-3} - 1260u^{-5}\right)\times H^{-11}& \mbox{if} & u < 1,\\
%-945\times r^{-11} & \mbox{if} & u \geq 1.
\left(r^5\chi^{(5)}(r, r_s) - 15r^4\chi^{(4)}(r, r_s) + 105r^3\chi^{(3)}(r, r_s) - 420r^2\chi''(r, r_s) + 945r \chi'(r, r_s) - 945\chi(r, r_s)\right) \times r^{-11} & \mbox{if} & u \geq 1.
\end{array}
\right.\nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Starting from the potential (Eq. \ref{eq:fmm:potential},
reproduced here for completeness), we can now build all the relevent derivatives
\begin{align}
\mathsf{D}_{000}(\mathbf{r}) = \varphi (\mathbf{r},H) =
\mathsf{\tilde{D}}_{1}(r, u, H) \nonumber
\mathsf{D}_{000}(\mathbf{r}) = \varphi (\mathbf{r}, r_s, H) =
\mathsf{\tilde{D}}_{1}(r, r_s, H) \nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\rule{6cm}{0.4pt}
\begin{align}
\mathsf{D}_{100}(\mathbf{r}) = \frac{\partial}{\partial r_x} \varphi (\mathbf{r},H) =
r_x \mathsf{\tilde{D}}_{3}(r, u, H) \nonumber
\mathsf{D}_{100}(\mathbf{r}) = \frac{\partial}{\partial r_x} \varphi (\mathbf{r}, r_s, H) =
r_x \mathsf{\tilde{D}}_{3}(r, r_s, H) \nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\rule{6cm}{0.4pt}
\begin{align}
\mathsf{D}_{200}(\mathbf{r}) = \frac{\partial^2}{\partial r_x^2} \varphi (\mathbf{r},H) =
r_x^2 \mathsf{\tilde{D}}_{5}(r, u, H) +
\mathsf{\tilde{D}}_{3}(r, u, H)\nonumber
\mathsf{D}_{200}(\mathbf{r}) = \frac{\partial^2}{\partial r_x^2} \varphi (\mathbf{r}, r_s, H) =
r_x^2 \mathsf{\tilde{D}}_{5}(r, r_s, H) +
\mathsf{\tilde{D}}_{3}(r, r_s, H)\nonumber
\end{align}
\begin{align}
\mathsf{D}_{110}(\mathbf{r}) = \frac{\partial^2}{\partial r_x\partial r_y} \varphi (\mathbf{r},H) =
r_x r_y \mathsf{\tilde{D}}_{5}(r, u, H) \nonumber
\mathsf{D}_{110}(\mathbf{r}) = \frac{\partial^2}{\partial r_x\partial r_y} \varphi (\mathbf{r}, r_s, H) =
r_x r_y \mathsf{\tilde{D}}_{5}(r, r_s, H) \nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\rule{6cm}{0.4pt}
\begin{align}
\mathsf{D}_{300}(\mathbf{r}) = \frac{\partial^3}{\partial r_x^3} \varphi (\mathbf{r},H) =
r_x^3 \mathsf{\tilde{D}}_{7}(r, u, H)
+ 3 r_x \mathsf{\tilde{D}}_{5}(r, u, H) \nonumber
\mathsf{D}_{300}(\mathbf{r}) = \frac{\partial^3}{\partial r_x^3} \varphi (\mathbf{r}, r_s, H) =
r_x^3 \mathsf{\tilde{D}}_{7}(r, r_s, H)
+ 3 r_x \mathsf{\tilde{D}}_{5}(r, r_s, H) \nonumber
\end{align}
\begin{align}
\mathsf{D}_{210}(\mathbf{r}) = \frac{\partial^3}{\partial r_x^2 r_y} \varphi (\mathbf{r},H) =
r_x^2 r_y \mathsf{\tilde{D}}_{7}(r, u, H) +
r_y \mathsf{\tilde{D}}_{5}(r, u, H) \nonumber
\mathsf{D}_{210}(\mathbf{r}) = \frac{\partial^3}{\partial r_x^2 r_y} \varphi (\mathbf{r}, r_s, H) =
r_x^2 r_y \mathsf{\tilde{D}}_{7}(r, r_s, H) +
r_y \mathsf{\tilde{D}}_{5}(r, r_s, H) \nonumber
\end{align}
\begin{align}
\mathsf{D}_{111}(\mathbf{r}) = \frac{\partial^3}{\partial r_x\partial r_y\partial r_z} \varphi (\mathbf{r},H) =
r_x r_y r_z \mathsf{\tilde{D}}_{7}(r, u, H) \nonumber
\mathsf{D}_{111}(\mathbf{r}) = \frac{\partial^3}{\partial r_x\partial r_y\partial r_z} \varphi (\mathbf{r}, r_s, H) =
r_x r_y r_z \mathsf{\tilde{D}}_{7}(r, r_s, H) \nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\rule{6cm}{0.4pt}
\begin{align}
\mathsf{D}_{400}(\mathbf{r}) &= \frac{\partial^4}{\partial r_x^4}
\varphi (\mathbf{r},H) =
r_x^4 \mathsf{\tilde{D}}_{9}(r, u, H)+
6r_x^2 \mathsf{\tilde{D}}_{7}(r, u, H) +
3 \mathsf{\tilde{D}}_{5}(r, u, H)
\varphi (\mathbf{r}, r_s, H) =
r_x^4 \mathsf{\tilde{D}}_{9}(r, r_s, H)+
6r_x^2 \mathsf{\tilde{D}}_{7}(r, r_s, H) +
3 \mathsf{\tilde{D}}_{5}(r, r_s, H)
\nonumber
\end{align}
\begin{align}
\mathsf{D}_{310}(\mathbf{r}) &= \frac{\partial^4}{\partial r_x^3
\partial r_y} \varphi (\mathbf{r},H) =
r_x^3 r_y \mathsf{\tilde{D}}_{9}(r, u, H) +
3 r_x r_y \mathsf{\tilde{D}}_{7}(r, u, H)
\partial r_y} \varphi (\mathbf{r}, r_s, H) =
r_x^3 r_y \mathsf{\tilde{D}}_{9}(r, r_s, H) +
3 r_x r_y \mathsf{\tilde{D}}_{7}(r, r_s, H)
\nonumber
\end{align}
\begin{align}
\mathsf{D}_{220}(\mathbf{r}) &= \frac{\partial^4}{\partial r_x^2
\partial r_y^2} \varphi (\mathbf{r},H) =
r_x^2 r_y^2 \mathsf{\tilde{D}}_{9}(r, u, H) +
r_x^2 \mathsf{\tilde{D}}_{7}(r, u, H) +
r_y^2 \mathsf{\tilde{D}}_{7}(r, u, H) +
\mathsf{\tilde{D}}_{5}(r, u, H)
\partial r_y^2} \varphi (\mathbf{r}, r_s, H) =
r_x^2 r_y^2 \mathsf{\tilde{D}}_{9}(r, r_s, H) +
r_x^2 \mathsf{\tilde{D}}_{7}(r, r_s, H) +
r_y^2 \mathsf{\tilde{D}}_{7}(r, r_s, H) +
\mathsf{\tilde{D}}_{5}(r, r_s, H)
\nonumber
\end{align}
\begin{align}
\mathsf{D}_{211}(\mathbf{r}) &= \frac{\partial^4}{\partial r_x^2
\partial r_y \partial r_z} \varphi (\mathbf{r},H) =
r_x^2 r_y r_z \mathsf{\tilde{D}}_{9}(r, u, H) +
r_y r_z \mathsf{\tilde{D}}_{7}(r, u, H)
\partial r_y \partial r_z} \varphi (\mathbf{r}, r_s, H) =
r_x^2 r_y r_z \mathsf{\tilde{D}}_{9}(r, r_s, H) +
r_y r_z \mathsf{\tilde{D}}_{7}(r, r_s, H)
\nonumber
\end{align}
......@@ -140,47 +169,47 @@ r_y \mathsf{\tilde{D}}_{5}(r, u, H) \nonumber
\noindent\rule{6cm}{0.4pt}
\begin{align}
\mathsf{D}_{500}(\mathbf{r}) &= \frac{\partial^5}{\partial r_x^5}
\varphi (\mathbf{r},H) =
r_x^5 \mathsf{\tilde{D}}_{11}(r, u, H) +
10r_x^3\mathsf{\tilde{D}}_{9}(r, u, H) +
15r_x\mathsf{\tilde{D}}_{7}(r, u, H)
\varphi (\mathbf{r}, r_s, H) =
r_x^5 \mathsf{\tilde{D}}_{11}(r, r_s, H) +
10r_x^3\mathsf{\tilde{D}}_{9}(r, r_s, H) +
15r_x\mathsf{\tilde{D}}_{7}(r, r_s, H)
\nonumber
\end{align}
\begin{align}
\mathsf{D}_{410}(\mathbf{r}) &= \frac{\partial^5}{\partial r_x^4
\partial r_y} \varphi (\mathbf{r},H) =
r_x^4 r_y \mathsf{\tilde{D}}_{11}(r, u, H) +
6 r_x^2 r_y \mathsf{\tilde{D}}_{9}(r, u, H) +
3 r_y \mathsf{\tilde{D}}_{7}(r, u, H)
\partial r_y} \varphi (\mathbf{r}, r_s, H) =
r_x^4 r_y \mathsf{\tilde{D}}_{11}(r, r_s, H) +
6 r_x^2 r_y \mathsf{\tilde{D}}_{9}(r, r_s, H) +
3 r_y \mathsf{\tilde{D}}_{7}(r, r_s, H)
\nonumber
\end{align}
\begin{align}
\mathsf{D}_{320}(\mathbf{r}) &= \frac{\partial^5}{\partial r_x^3
\partial r_y^2} \varphi (\mathbf{r},H) =
r_x^3 r_y^2 \mathsf{\tilde{D}}_{11}(r, u, H) +
r_x^3 \mathsf{\tilde{D}}_{9}(r, u, H) +
3 r_x r_y^2 \mathsf{\tilde{D}}_{9}(r, u, H) +
3 r_x \mathsf{\tilde{D}}_{7}(r, u, H)
\partial r_y^2} \varphi (\mathbf{r}, r_s, H) =
r_x^3 r_y^2 \mathsf{\tilde{D}}_{11}(r, r_s, H) +
r_x^3 \mathsf{\tilde{D}}_{9}(r, r_s, H) +
3 r_x r_y^2 \mathsf{\tilde{D}}_{9}(r, r_s, H) +
3 r_x \mathsf{\tilde{D}}_{7}(r, r_s, H)
\nonumber
\end{align}
\begin{align}
\mathsf{D}_{311}(\mathbf{r}) &= \frac{\partial^5}{\partial r_x^3
\partial r_y \partial r_z} \varphi (\mathbf{r},H) =
r_x^3 r_y r_z \mathsf{\tilde{D}}_{11}(r, u, H) +
3 r_x r_y r_z \mathsf{\tilde{D}}_{9}(r, u, H)
\partial r_y \partial r_z} \varphi (\mathbf{r}, r_s, H) =
r_x^3 r_y r_z \mathsf{\tilde{D}}_{11}(r, r_s, H) +
3 r_x r_y r_z \mathsf{\tilde{D}}_{9}(r, r_s, H)
\nonumber
\end{align}
\begin{align}
\mathsf{D}_{221}(\mathbf{r}) &= \frac{\partial^5}{\partial r_x^2
\partial r_y^2 \partial r_z} \varphi (\mathbf{r},H) =
r_x^2 r_y^2 r_z \mathsf{\tilde{D}}_{11}(r, u, H) +
r_x^2 r_z \mathsf{\tilde{D}}_{9}(r, u, H) +
r_y^2 r_z \mathsf{\tilde{D}}_{9}(r, u, H) +
r_z \mathsf{\tilde{D}}_{y}(r, u, H)
\partial r_y^2 \partial r_z} \varphi (\mathbf{r}, r_s, H) =
r_x^2 r_y^2 r_z \mathsf{\tilde{D}}_{11}(r, r_s, H) +
r_x^2 r_z \mathsf{\tilde{D}}_{9}(r, r_s, H) +
r_y^2 r_z \mathsf{\tilde{D}}_{9}(r, r_s, H) +
r_z \mathsf{\tilde{D}}_{y}(r, r_s, H)
\nonumber
\end{align}
......
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