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Commit eec252f5 authored by Matthieu Schaller's avatar Matthieu Schaller
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Fixed the 1 < u softened derivatives. 

Move the (1/r)^n terms to the parent terms.
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1 merge request!1077Improved multipole acceptance criterion (MAC)
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theory/Multipoles/potential_derivatives.tex
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...@@ -74,6 +74,216 @@ truncated an softened gravity field $\varphi (\mathbf{r}, r_s, H) ...@@ -74,6 +74,216 @@ truncated an softened gravity field $\varphi (\mathbf{r}, r_s, H)
% \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 % \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} % \end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
\mathsf{\tilde{D}}_{1}(r, r_s, H) = % D_tilde_tilde_1 = D_tilde_1
\left\lbrace\begin{array}{rcl}
f(u)\times H^{-1} & \mbox{if} & u < 1,\\
%r^{-1} & \mbox{if} & u \geq 1,
\chi \times r^{-1} & \mbox{if} & u \geq 1~\mbox{and periodic}, \\
r^{-1} & \mbox{if} & u \geq 1~\mbox{and not periodic}.
\end{array}
\right.\nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
\mathsf{\tilde{D}}_{2}(r, r_s, H) = % D_tilde_tilde_3 = D_tilde_3 * r
\left\lbrace\begin{array}{rcl}
f'(u)\times H^{-2}& \mbox{if} & u < 1,\\
%-1 \times r^{-3} & \mbox{if} & u \geq 1,
\left(r\chi' - \chi\right) \times r^{-2} & \mbox{if} & u \geq 1~\mbox{and periodic}, \\
-1 \times r^{-2} & \mbox{if} & u \geq 1~\mbox{and not periodic}.
\end{array}
\right.\nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
\mathsf{\tilde{D}}_{3}(r, r_s, H) = % D_tilde_tilde_5 = D_tilde_5 * r^2
\left\lbrace\begin{array}{rcl}
\left(f''(u) - u^{-1}f'(u)\right)\times H^{-3}& \mbox{if} & u < 1,\\
%3\times r^{-5} & \mbox{if} & u \geq 1,
\left(r^2\chi'' - 3r\chi' + 3\chi \right)\times r^{-3} & \mbox{if} & u \geq 1~\mbox{and periodic}, \\
3 \times r^{-3} & \mbox{if} & u \geq 1~\mbox{and not periodic}.
\end{array}
\right.\nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
\mathsf{\tilde{D}}_{4}(r, r_s, H) = % D_tilde_tilde_7 = D_tilde_7 * r^3
\left\lbrace\begin{array}{rcl}
\left(f^{(3)}(u)-3u^{-1}f''(u)+3u^{-2}f'(u)\right)\times H^{-4} & \mbox{if} & u < 1,\\
%-15\times r^{-7} & \mbox{if} & u \geq 1,
\left(r^3\chi^{(3)} - 6r^2\chi''+15r\chi'-15\chi\right) \times r^{-4} & \mbox{if} & u \geq 1~\mbox{and periodic}, \\
-15 \times r^{-4} & \mbox{if} & u \geq 1~\mbox{and not periodic}.
\end{array}
\right.\nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
\mathsf{\tilde{D}}_{5}(r, r_s, H) = % D_tilde_tilde_9 = D_tilde_9 * r^4
\left\lbrace\begin{array}{rcl}
\left(f^{(4)}(u)-6u^{-1}f^{(3)}(u)+15u^{-2}f''(u)-15u^{-3}f'(u)\right)\times H^{-5}& \mbox{if} & u < 1,\\
%105\times r^{-9} & \mbox{if} & u \geq 1.
\left(r^4\chi^{(4)} - 10r^3\chi^{(3)} + 45r^2\chi'' - 105r\chi' + 105\chi \right) \times r^{-5} & \mbox{if} & u \geq 1~\mbox{and periodic}, \\
105 \times r^{-5} & \mbox{if} & u \geq 1~\mbox{and not periodic}.
\end{array}
\right.\nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align}
\mathsf{\tilde{D}}_{6}(r, r_s, H) = % D_tilde_tilde_11 = D_tilde_11 * r^4
\left\lbrace\begin{array}{rcl}
\left(f^{(5)}(u) -10u^{-1}f^{(4)}(u) +45u^{-2}f^{(3)} -105u^{-3}f''(u) + 105u^{-4}f'(u)\right)\times H^{-6}& \mbox{if} & u < 1,\\
%-945\times r^{-11} & \mbox{if} & u \geq 1.
\left(r^5\chi^{(5)} - 15r^4\chi^{(4)} + 105r^3\chi^{(3)} - 420r^2\chi'' + 945r \chi' - 945\chi\right) \times r^{-6} & \mbox{if} & u \geq 1~\mbox{and periodic}, \\
-945\times r^{-6} & \mbox{if} & u \geq 1~\mbox{and not periodic}.
\end{array}
\right.\nonumber
\end{align}
In the case $u<1$ and using $f(u)$ given by \ref{eq:fmm:potential}, we can simplify the expressions to get:
\begin{align}
\mathsf{\tilde{D}}_{1} &= (-3u^7 + 15u^6 - 28u^5 + 21u^4 - 7u^2 + 3) \times H^{-1}, \nonumber \\
\mathsf{\tilde{D}}_{2} &= (-21u^6 + 90u^5 - 140u^4 + 84u^3 - 14u) \times H^{-2}, \nonumber \\
\mathsf{\tilde{D}}_{3} &= (-105u^5 + 360u^4 - 420u^3 + 168u^2) \times H^{-3}, \nonumber \\
\mathsf{\tilde{D}}_{4} &= (-315u^4 + 720u^3 - 420u^2) \times H^{-4}, \nonumber \\
\mathsf{\tilde{D}}_{5} &= (-315u^3 + 420u) \times H^{-5}, \nonumber \\
\mathsf{\tilde{D}}_{6} &= (315u^2 - 1260) \times H^{-6}. \nonumber
\end{align}
We can now write out all the derivatives used in the M2L and M2P kernels:
\begin{align}
\mathsf{D}_{000}(\mathbf{r}) = \varphi (\mathbf{r}, r_s, H) =
\mathsf{\tilde{D}}_{1} \nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\rule{6cm}{0.4pt}
\begin{align}
\mathsf{D}_{100}(\mathbf{r}) = \frac{\partial}{\partial r_x} \varphi (\mathbf{r}, r_s, H) =
\left(\frac{r_x}{r}\right) \mathsf{\tilde{D}}_{2} \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}, r_s, H) =
\left(\frac{r_x}{r}\right)^2 \mathsf{\tilde{D}}_{3} + \left(\frac{1}{r}\right)\mathsf{\tilde{D}}_{2}\nonumber
\end{align}
\begin{align}
\mathsf{D}_{110}(\mathbf{r}) = \frac{\partial^2}{\partial r_x\partial r_y} \varphi (\mathbf{r}, r_s, H) =
\left(\frac{r_x}{r}\right) \left(\frac{r_y}{r}\right) \mathsf{\tilde{D}}_{3} \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}, r_s, H) =
\left(\frac{r_x}{r}\right)^3 \mathsf{\tilde{D}}_{4}
+ 3 \left(\frac{r_x}{r}\right) \left(\frac{1}{r}\right) \mathsf{\tilde{D}}_{3} \nonumber
\end{align}
\begin{align}
\mathsf{D}_{210}(\mathbf{r}) = \frac{\partial^3}{\partial r_x^2 r_y} \varphi (\mathbf{r}, r_s, H) =
\left(\frac{r_x}{r}\right)^2 \left(\frac{r_y}{r}\right) \mathsf{\tilde{D}}_{4} + \left(\frac{r_y}{r}\right) \left(\frac{1}{r}\right) \mathsf{\tilde{D}}_{3} \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}, r_s, H) =
\left(\frac{r_x}{r}\right)\left(\frac{r_y}{r}\right)\left(\frac{r_z}{r}\right) \mathsf{\tilde{D}}_{4} \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}, r_s, H) =
\left(\frac{r_x}{r}\right)^4 \mathsf{\tilde{D}}_{5}+
6\left(\frac{r_x}{r}\right)^2 \left(\frac{1}{r}\right) \mathsf{\tilde{D}}_{4} +
3 \left(\frac{1}{r}\right)^2 \mathsf{\tilde{D}}_{3}
\nonumber
\end{align}
\begin{align}
\mathsf{D}_{310}(\mathbf{r}) &= \frac{\partial^4}{\partial r_x^3
\partial r_y} \varphi (\mathbf{r}, r_s, H) =
\left(\frac{r_x}{r}\right)^3 \left(\frac{r_y}{r}\right) \mathsf{\tilde{D}}_{4} +
3 \left(\frac{r_x}{r}\right) \left(\frac{r_y}{r}\right) \left(\frac{1}{r}\right) \mathsf{\tilde{D}}_{3}
\nonumber
\end{align}
\begin{align}
\mathsf{D}_{220}(\mathbf{r}) &= \frac{\partial^4}{\partial r_x^2
\partial r_y^2} \varphi (\mathbf{r}, r_s, H) =
\left(\frac{r_x}{r}\right)^2 \left(\frac{r_y}{r}\right)^2 \mathsf{\tilde{D}}_{5} +
\left(\frac{r_x}{r}\right)^2 \left(\frac{1}{r}\right) \mathsf{\tilde{D}}_{4} +
\left(\frac{r_y}{r}\right)^2 \left(\frac{1}{r}\right) \mathsf{\tilde{D}}_{4} +
\left(\frac{1}{r}\right)^2 \mathsf{\tilde{D}}_{3}
\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}, r_s, H) =
\left(\frac{r_x}{r}\right)^2\left(\frac{r_y}{r}\right)\left(\frac{r_z}{r}\right) \mathsf{\tilde{D}}_{5} +
\left(\frac{r_y}{r}\right)\left(\frac{r_z}{r}\right)\left(\frac{1}{r}\right) \mathsf{\tilde{D}}_{4}
\nonumber
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent\rule{6cm}{0.4pt}
\begin{align}
\mathsf{D}_{500}(\mathbf{r}) &= \frac{\partial^5}{\partial r_x^5}
\varphi (\mathbf{r}, r_s, H) =
\left(\frac{r_x}{r}\right)^5 \mathsf{\tilde{D}}_{6} +
10\left(\frac{r_x}{r}\right)^3\left(\frac{1}{r}\right)\mathsf{\tilde{D}}_{5} +
15\left(\frac{r_x}{r}\right)\left(\frac{1}{r}\right)^2\mathsf{\tilde{D}}_{4}
\nonumber
\end{align}
\begin{align}
\mathsf{D}_{410}(\mathbf{r}) &= \frac{\partial^5}{\partial r_x^4
\partial r_y} \varphi (\mathbf{r}, r_s, H) =
\left(\frac{r_x}{r}\right)^4 \left(\frac{r_y}{r}\right) \mathsf{\tilde{D}}_{6} +
6 \left(\frac{r_x}{r}\right)^2 \left(\frac{r_y}{r}\right)\left(\frac{1}{r}\right) \mathsf{\tilde{D}}_{5} +
3 \left(\frac{r_y}{r}\right) \left(\frac{1}{r}\right)^2\mathsf{\tilde{D}}_{4}
\nonumber
\end{align}
\begin{align}
\mathsf{D}_{320}(\mathbf{r}) &= \frac{\partial^5}{\partial r_x^3
\partial r_y^2} \varphi (\mathbf{r}, r_s, H) =
\left(\frac{r_x}{r}\right)^3 \left(\frac{r_y}{r}\right)^2 \mathsf{\tilde{D}}_{6} +
\left(\frac{r_x}{r}\right)^3 \left(\frac{1}{r}\right)\mathsf{\tilde{D}}_{5} +
3 \left(\frac{r_x}{r}\right) \left(\frac{r_y}{r}\right)^2 \left(\frac{1}{r}\right)\mathsf{\tilde{D}}_{5} +
3 \left(\frac{r_x}{r}\right) \left(\frac{1}{r}\right)^2\mathsf{\tilde{D}}_{4}
\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}, r_s, H) =
\left(\frac{r_x}{r}\right)^3 \left(\frac{r_y}{r}\right) \left(\frac{r_z}{r}\right) \mathsf{\tilde{D}}_{6} +
3 \left(\frac{r_x}{r}\right) \left(\frac{r_y}{r}\right) \left(\frac{r_z}{r}\right) \left(\frac{1}{r}\right)\mathsf{\tilde{D}}_{5}
\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}, r_s, H) =
\left(\frac{r_x}{r}\right)^2 \left(\frac{r_y}{r}\right)^2 \left(\frac{r_z}{r}\right) \mathsf{\tilde{D}}_{6} +
\left(\frac{r_x}{r}\right)^2 \left(\frac{r_z}{r}\right) \left(\frac{1}{r}\right)\mathsf{\tilde{D}}_{5} +
\left(\frac{r_y}{r}\right)^2 \left(\frac{r_z}{r}\right) \left(\frac{1}{r}\right)\mathsf{\tilde{D}}_{5} +
\left(\frac{r_z}{r}\right) \left(\frac{1}{r}\right)^2\mathsf{\tilde{D}}_{4}
\nonumber
\end{align}
\begin{comment}
\noindent\rule{12cm}{1pt}\\
Old version \\
\noindent\rule{12cm}{1pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align} \begin{align}
\mathsf{\tilde{D}}_{1}(r, r_s, H) = \mathsf{\tilde{D}}_{1}(r, r_s, H) =
...@@ -266,6 +476,7 @@ r_y \mathsf{\tilde{D}}_{5}(r, r_s, H) \nonumber ...@@ -266,6 +476,7 @@ r_y \mathsf{\tilde{D}}_{5}(r, r_s, H) \nonumber
\nonumber \nonumber
\end{align} \end{align}
\end{comment}
...@@ -285,94 +496,3 @@ r_y \mathsf{\tilde{D}}_{5}(r, r_s, H) \nonumber ...@@ -285,94 +496,3 @@ r_y \mathsf{\tilde{D}}_{5}(r, r_s, H) \nonumber
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{comment}
\noindent\rule{6cm}{0.4pt}
\begin{align}
\mathsf{D}_{100}(\mathbf{r}) = \frac{\partial}{\partial r_x} \varphi (\mathbf{r},H) =
\left\lbrace\begin{array}{rcl}
-\frac{r_x}{H^3} \left(21u^5 - 90u^4 + 140u^3 - 84u^2 + 14\right) & \mbox{if} & u < 1,\\
-\frac{r_x}{r^3} & \mbox{if} & u \geq 1,
\end{array}
\right.\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) =
\left\lbrace\begin{array}{rcl}
\frac{r_x^2}{H^5}\left(-105u^3+360u^2-420u+168\right) -
\frac{1}{H^3} \left(21u^5 - 90u^4 + 140u^3 - 84u^2 + 14\right) & \mbox{if} & u < 1,\\
3\frac{r_x^2}{r^5} - \frac{1}{r^3} & \mbox{if} & u \geq 1,
\end{array}
\right.\nonumber
\end{align}
\begin{align}
\mathsf{D}_{110}(\mathbf{r}) = \frac{\partial^2}{\partial r_x\partial r_y} \varphi (\mathbf{r},H) =
\left\lbrace\begin{array}{rcl}
\frac{r_xr_y}{H^5}\left(-105u^3+360u^2-420u+168\right) & \mbox{if} & u < 1,\\
3\frac{r_xr_y}{r^5} & \mbox{if} & u \geq 1,
\end{array}
\right.\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) =
\left\lbrace\begin{array}{rcl}
-\frac{r_x^3}{H^7} \left(315u - 720 + 420u^{-1}\right) +
\frac{3r_x}{H^5}\left(-105u^3+360u^2-420u+168\right) & \mbox{if} & u < 1,\\
-15\frac{r_x^3}{r^7} + 9 \frac{r_x}{r^5} & \mbox{if} & u \geq 1,
\end{array}
\right.\nonumber
\end{align}
\begin{align}
\mathsf{D}_{210}(\mathbf{r}) = \frac{\partial^3}{\partial r_x^3} \varphi (\mathbf{r},H) =
\left\lbrace\begin{array}{rcl}
-\frac{r_x^2r_y}{H^7} \left(315u - 720 + 420u^{-1}\right) +
\frac{r_y}{H^5}\left(-105u^3+360u^2-420u+168\right) & \mbox{if} & u < 1,\\
-15\frac{r_x^2r_y}{r^7} + 3 \frac{r_y}{r^5} & \mbox{if} & u \geq 1,
\end{array}
\right.\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) =
\left\lbrace\begin{array}{rcl}
-\frac{r_xr_yr_z}{H^7} \left(315u - 720 + 420u^{-1}\right) & \mbox{if} & u < 1,\\
-15\frac{r_xr_yr_z}{r^7} & \mbox{if} & u \geq 1,
\end{array}
\right.\nonumber
\end{align}
\noindent\rule{6cm}{0.4pt}
\begin{align}
\mathsf{D}_{400}(\mathbf{r}) &=
\nonumber
\end{align}
\begin{align}
\mathsf{D}_{310}(\mathbf{r}) &=
\nonumber
\end{align}
\begin{align}
\mathsf{D}_{220}(\mathbf{r}) &=
\nonumber
\end{align}
\begin{align}
\mathsf{D}_{211}(\mathbf{r}) &=
\nonumber
\end{align}
\end{comment}
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