Added a figure to illustrate the contributions to gravity

Former-commit-id: 9666e58c8ffadff83d316fec3c68600fe9b97477

theory/latex/Figures/Gravity/force.py

+import matplotlib
+matplotlib.use("Agg")
+from pylab import *
+# tex stuff
+#rc('font',**{'family':'serif','serif':['Palatino']})
+params = {'axes.labelsize': 14,
+          'text.fontsize': 14,
+          'legend.fontsize': 14,
+          'xtick.labelsize': 14,
+          'ytick.labelsize': 14,
+          'text.usetex': True,
+
+          'figure.figsize' : (9,6),
+          'figure.subplot.left'    : 0.08,  # the left side of the subplots of the figure
+          'figure.subplot.right'   : 0.95  ,  # the right side of the subplots of the figure
+          'figure.subplot.bottom'  : 0.1  ,  # the bottom of the subplots of the figure
+          'figure.subplot.top'     : 0.93  ,  # the top of the subplots of the figure
+          'figure.subplot.wspace'  : 0.2  ,  # the amount of width reserved for blank space between subplots
+          'figure.subplot.hspace'  : 0.2  ,  # the amount of height reserved for white space between subplots
+
+          'lines.markersize' : 4,
+
+          #'axes.formatter.limits' : (, 0),
+
+          'text.latex.unicode': True
+        }
+rcParams.update(params)
+rc('font', family='serif')
+import sys
+import math
+import os
+
+
+
+epsilon = 1.
+h = 2.8*epsilon
+r_s = 20.
+r_c = 4.5*r_s
+
+r=arange(0.1, 400, 1/1000.)
+
+numElem = size(r)
+f = zeros(numElem)
+fac = zeros(numElem)
+kernel = zeros(numElem)
+
+for i in range(numElem):
+    if ( r[i] >= r_c ):
+        f[i] = 0.
+    elif ( r[i] >= h ):
+        f[i] = (1. / r[i]**3)
+    else:
+        u = r[i] / h
+        if( u < 0.5 ):
+            f[i] = (10.666666666667 + u * u * (32.0 * u - 38.4)) / h**3 
+        else:
+            f[i] = (21.333333333333 - 48.0 * u + 38.4 * u * u - 10.666666666667 * u * u * u - 0.066666666667 / (u * u * u)) / h**3
+
+    fac[i] = math.erfc( r[i] / ( 2. * r_s ) ) + ( r[i] / ( r_s * sqrt( math.pi ) ) ) * exp( - r[i] * r[i] / ( 4 * r_s * r_s ) ) 
+    f[i] = f[i]*fac[i]
+
+for i in range(numElem):
+    u = r[i] / h
+    if( u < 0.5 ):
+        kernel[i] = (10.666666666667 + u * u * (32.0 * u - 38.4)) / h**3 
+    else:
+        kernel[i] = (21.333333333333 - 48.0 * u + 38.4 * u * u - 10.666666666667 * u * u * u - 0.066666666667 / (u * u * u)) / h**3
+
+figure()
+loglog(r/h, 1/r**3, 'b-', label="Newton's force")
+loglog(r/h, kernel, 'r-', label="Softening kernel")
+loglog(r/h, fac/r**3, 'g-', label="Tree force",)
+loglog(r/h, f, 'k--', label="Total short-range force", linewidth=3)
+
+
+plot([1, 1], [1e-10, 1e10], 'k--')
+text(0.85, 7e-9, "$h=2.8\epsilon$", rotation="vertical", fontsize=14, va="bottom")
+
+plot([epsilon/h, epsilon/h], [1e-10, 1e10], 'k--')
+text(0.85*epsilon/h, 7e-9, "$\epsilon$", rotation="vertical", fontsize=14, va="bottom")
+
+plot([r_s/h, r_s/h], [1e-10, 1e10], 'k--')
+text(0.85*r_s/h, 7e-9, "$r_s$", rotation="vertical", fontsize=14, va="bottom")
+
+plot([r_c/h, r_c/h], [1e-10, 1e10], 'k--')
+text(0.85*r_c/h, 7e-9, "$r_c$", rotation="vertical", fontsize=14, va="bottom")
+
+
+legend(loc="upper right")
+
+
+grid()
+
+xlim(1e-1, 200)
+xlabel("r/h")
+
+ylim(4e-10, 1e2)
+ylabel("Normalized acceleration")
+
+
+savefig("force.png")

theory/latex/swift.tex

@@ -197,39 +197,29 @@ time. In cosmological simulations, the softening length is a function of time.
 
 \section{Gravity interactions}
 
-The interaction is computed between all pairs of particles. The acceleration of particle $i$ due to one other particle
-$j$ is given by:
+The gravity interactions are split in two parts, the sort range interactions computed via the FMM method and the long 
+range interactions computed through a PM scheme. Short range interactions are also soften on a scale $\epsilon$ to 
+reduce the effect of spurious two-body relaxation. For implementation purposes, we define the quantity $h_c = 
+2.8\epsilon$, the radius at which the force becomes Newtonian and is not soften any more.
+
+The short range force is computed thanks to FMM. The expression for the acceleration of a particle $i$ due to a body 
+(particle or monopole) $j$ is given by:
 
 \begin{equation}
- \vec{a_i} = \left\lbrace \begin{array}{rcl}
-	   \Bigg. G_N\dfrac{m_j}{|\vec{r}_{ij}|^3}\vec{r}_{ij}& \mbox{if} & \Big.\dfrac{|\vec{r}_{ij}|}{\epsilon}
-\geq 1 \\
-         \Bigg. G_N\dfrac{m_j}{\epsilon^3}\phi\left(\frac{|\vec{r}_{ij}|}{\epsilon}\right)\vec{r}_{ij} & \mbox{if} &
-\Big.\dfrac{|\vec{r}_{ij}|}{\epsilon} < 1 \\
-                     \end{array}
-\right.
+ \vec{a} = f \cdot \vec{r}_{ij},
 \end{equation}
-
-where the softened potential $\phi(u)$ is given by:
+ where $f$ is the norm of the acceleration given by
 
 \begin{equation}
-\phi(u) =  \left\lbrace \begin{array}{rcl}
-	     \frac{32}{3} - \frac{192}{5}u^2 + 32u^3& \mbox{if} & u < \Big.\frac{1}{2}\\
-	     \frac{64}{3} - 48u + \frac{192}{5} u^2 - \frac{32}{3}u^3  - \frac{2}{30}\dfrac{1}{u^3}& \mbox{if} & \Big.u
-\geq \frac{1}{2}\\
-                        \end{array}
-\right.
+ f = G\left\lbrace \begin{array}{rcl}
+                    \frac{1}{|\vec{r}_{ij}|^3}  & \mbox{if} & |\vec{r}_{ij}| > h_c \\
+                    (\frac{32}{3} + \frac{|\vec{r}_{ij}|^2}{h_c^2} (32 \frac{|\vec{r}_{ij}|}{h_c} - 38.4)) / h_c^3  & 
+\mbox{if} & h_c \geq |\vec{r}_{ij}| > \frac{h_c}{2} \\	
+                    \frac{1}{|\vec{r}_{ij}|^3}  & \mbox{if} & |\vec{r}_{ij}| \leq \frac{h_c}{2} \\		
+		    \end{array}\right.
 \end{equation}
 
-and $G_N$ is Newton's constant for gravity. The softening length used in this equation is the maximum of the softening
-length of both particles, i.e. $\epsilon = \max(\epsilon_i, \epsilon_j)$. Notice that this softened potential is
-derived from the smoothing kernel used in the SPH part of the calculation with a smoothing length $h=1.4\epsilon$. This
-corresponds to a Plummer sphere of size $\epsilon$. \\
-
-
-\textcolor{red}{This is the softening potential used in GADGET. There is something weird about this function that I want
-to check but for now it should be sufficient.}\\
-
+ 
 The maximal time step a particle is allowed to have is related to the acceleration by:
 
 \begin{equation}