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SWIFT
SWIFTsim
Commits
1a8f00d5
Commit
1a8f00d5
authored
Sep 29, 2016
by
Matthieu Schaller
Browse files
Improved visual quality of kernel function plots
parent
3e463812
Changes
1
Hide whitespace changes
Inline
Side-by-side
theory/SPH/Kernels/kernels.py
View file @
1a8f00d5
...
...
@@ -26,31 +26,53 @@ from scipy.optimize import fsolve
from
matplotlib.font_manager
import
FontProperties
import
numpy
params
=
{
'axes.labelsize'
:
10
,
'axes.titlesize'
:
8
,
'font.size'
:
10
,
'legend.fontsize'
:
9
,
'xtick.labelsize'
:
10
,
'ytick.labelsize'
:
10
,
'xtick.major.pad'
:
2.5
,
'ytick.major.pad'
:
2.5
,
'text.usetex'
:
True
,
'figure.figsize'
:
(
4.15
,
4.15
),
'figure.subplot.left'
:
0.14
,
'figure.subplot.right'
:
0.99
,
'figure.subplot.bottom'
:
0.06
,
'figure.subplot.top'
:
0.99
,
params
=
{
'axes.labelsize'
:
9
,
'axes.titlesize'
:
10
,
'font.size'
:
12
,
'legend.fontsize'
:
12
,
'xtick.labelsize'
:
9
,
'ytick.labelsize'
:
9
,
'text.usetex'
:
True
,
'figure.figsize'
:
(
3.15
,
3.15
),
'figure.subplot.left'
:
0.17
,
'figure.subplot.right'
:
0.99
,
'figure.subplot.bottom'
:
0.08
,
'figure.subplot.top'
:
0.99
,
'figure.subplot.wspace'
:
0.
,
'figure.subplot.hspace'
:
0.
,
'lines.markersize'
:
6
,
'lines.linewidth'
:
1.5
,
'lines.linewidth'
:
3.
,
'text.latex.unicode'
:
True
}
rcParams
.
update
(
params
)
rc
(
'font'
,
**
{
'family'
:
'sans-serif'
,
'sans-serif'
:[
'Times'
]})
# params = {
# 'axes.labelsize': 10,
# 'axes.titlesize': 8,
# 'font.size': 10,
# 'legend.fontsize': 9,
# 'xtick.labelsize': 10,
# 'ytick.labelsize': 10,
# 'xtick.major.pad': 2.5,
# 'ytick.major.pad': 2.5,
# 'text.usetex': True,
# 'figure.figsize' : (4.15,4.15),
# 'figure.subplot.left' : 0.14,
# 'figure.subplot.right' : 0.99,
# 'figure.subplot.bottom' : 0.06,
# 'figure.subplot.top' : 0.99,
# 'figure.subplot.wspace' : 0. ,
# 'figure.subplot.hspace' : 0. ,
# 'lines.markersize' : 6,
# 'lines.linewidth' : 1.5,
# 'text.latex.unicode': True
# }
# rcParams.update(params)
# rc('font',**{'family':'sans-serif','sans-serif':['Times']})
#Parameters
eta
=
1.2348422195325
# Resolution (Gives 48 neighbours for a cubic spline kernel)
dx
=
1.5
#4 #2.7162 # Mean inter-particle separation
...
...
@@ -135,13 +157,13 @@ xx = linspace(0., 5*h, 1000)
maxY
=
1.2
*
Gaussian
(
0
,
h
)
# Plot the kernels
plot
(
xx
,
Gaussian
(
xx
,
h
),
'k-'
,
linewidth
=
0.7
,
label
=
"${
\\
rm %14s
\\
quad
H=
\\
infty}$"
%
(
"Gaussian~~~~~~"
))
plot
(
xx
,
W_cubic_spline
(
xx
),
'b-'
,
label
=
"${
\\
rm %14s
\\
quad
H=%4.3f}$"
%
(
"Cubic~spline~~"
,
H_cubic
))
plot
(
xx
,
W_quartic_spline
(
xx
),
'c-'
,
label
=
"${
\\
rm %14s
\\
quad
H=%4.3f}$"
%
(
"Quartic~spline"
,
H_quartic
))
plot
(
xx
,
W_quintic_spline
(
xx
),
'g-'
,
label
=
"${
\\
rm %14s
\\
quad
H=%4.3f}$"
%
(
"Quintic~spline"
,
H_quintic
))
plot
(
xx
,
W_WendlandC2
(
xx
),
'r-'
,
label
=
"${
\\
rm %14s
\\
quad
H=%4.3f}$"
%
(
"Wendland~C2~"
,
H_WendlandC2
))
plot
(
xx
,
W_WendlandC4
(
xx
),
'm-'
,
label
=
"${
\\
rm %14s
\\
quad
H=%4.3f}$"
%
(
"Wendland~C4~"
,
H_WendlandC4
))
plot
(
xx
,
W_WendlandC6
(
xx
),
'y-'
,
label
=
"${
\\
rm %14s
\\
quad
H=%4.3f}$"
%
(
"Wendland~C6~"
,
H_WendlandC6
))
plot
(
xx
,
Gaussian
(
xx
,
h
),
'k-'
,
linewidth
=
0.7
,
label
=
"${
\\
rm %14s
~
H=
\\
infty}$"
%
(
"Gaussian~~~~~~"
))
plot
(
xx
,
W_cubic_spline
(
xx
),
'b-'
,
label
=
"${
\\
rm %14s
~
H=%4.3f}$"
%
(
"Cubic~spline~~"
,
H_cubic
)
,
lw
=
1.5
)
plot
(
xx
,
W_quartic_spline
(
xx
),
'c-'
,
label
=
"${
\\
rm %14s
~
H=%4.3f}$"
%
(
"Quartic~spline"
,
H_quartic
)
,
lw
=
1.5
)
plot
(
xx
,
W_quintic_spline
(
xx
),
'g-'
,
label
=
"${
\\
rm %14s
~
H=%4.3f}$"
%
(
"Quintic~spline"
,
H_quintic
)
,
lw
=
1.5
)
plot
(
xx
,
W_WendlandC2
(
xx
),
'r-'
,
label
=
"${
\\
rm %14s
~
H=%4.3f}$"
%
(
"Wendland~C2~"
,
H_WendlandC2
)
,
lw
=
1.5
)
plot
(
xx
,
W_WendlandC4
(
xx
),
'm-'
,
label
=
"${
\\
rm %14s
~
H=%4.3f}$"
%
(
"Wendland~C4~"
,
H_WendlandC4
)
,
lw
=
1.5
)
plot
(
xx
,
W_WendlandC6
(
xx
),
'y-'
,
label
=
"${
\\
rm %14s
~
H=%4.3f}$"
%
(
"Wendland~C6~"
,
H_WendlandC6
)
,
lw
=
1.5
)
# Indicate the position of H
arrow
(
H_cubic
,
0.12
*
maxY
,
0.
,
-
0.12
*
maxY
*
0.9
,
fc
=
'b'
,
ec
=
'b'
,
length_includes_head
=
True
,
head_width
=
0.03
,
head_length
=
0.12
*
maxY
*
0.3
)
...
...
@@ -152,33 +174,33 @@ arrow(H_WendlandC4, 0.12*maxY , 0., -0.12*maxY*0.9, fc='m', ec='m', length_inclu
arrow
(
H_WendlandC6
,
0.12
*
maxY
,
0.
,
-
0.12
*
maxY
*
0.9
,
fc
=
'y'
,
ec
=
'y'
,
length_includes_head
=
True
,
head_width
=
0.03
,
head_length
=
0.12
*
maxY
*
0.3
)
# Show h
plot
([
h
,
h
],
[
0.
,
maxY
],
'k:'
,
linewidth
=
0.5
)
text
(
h
,
maxY
*
0.
3
5
,
"$h
\\
equiv
\\
eta
\\
langle x
\\
rangle
= %.4f$"
%
h
,
rotation
=
90
,
backgroundcolor
=
'w'
,
ha
=
'center'
,
va
=
'bottom'
)
plot
([
h
,
h
],
[
0.
,
0.63
*
maxY
],
'k:'
,
linewidth
=
0.5
)
text
(
h
,
maxY
*
0.
6
5
,
"$h
\\
equiv
\\
eta
\\
langle x
\\
rangle
$"
,
rotation
=
90
,
ha
=
'center'
,
va
=
'bottom'
,
fontsize
=
9
)
# Show sigma
plot
([
h
/
2
,
h
/
2
],
[
0.
,
maxY
],
'k:'
,
linewidth
=
0.5
)
text
(
h
/
2
,
maxY
*
0.
0
5
,
"$
\\
sigma
\\
equiv h/2$"
,
rotation
=
90
,
backgroundcolor
=
'w'
,
ha
=
'center'
,
va
=
'bottom'
)
plot
([
h
/
2
,
h
/
2
],
[
0.
,
0.63
*
maxY
],
'k:'
,
linewidth
=
0.5
)
text
(
h
/
2
,
maxY
*
0.
6
5
,
"$
\\
sigma
\\
equiv h/2$"
,
rotation
=
90
,
ha
=
'center'
,
va
=
'bottom'
,
fontsize
=
9
)
# Show <x>
plot
([
dx
,
dx
],
[
0.
,
maxY
],
'k:'
,
linewidth
=
0.5
)
text
(
dx
,
maxY
*
0.
3
5
,
"$
\\
langle x
\\
rangle = %.1f$"
%
dx
,
rotation
=
90
,
backgroundcolor
=
'w'
,
ha
=
'center'
,
va
=
'bottom'
)
plot
([
dx
,
dx
],
[
0.
,
0.63
*
maxY
],
'k:'
,
linewidth
=
0.5
)
text
(
dx
,
maxY
*
0.
6
5
,
"$
\\
langle x
\\
rangle = %.1f$"
%
dx
,
rotation
=
90
,
ha
=
'center'
,
va
=
'bottom'
,
fontsize
=
9
)
xlim
(
0.
,
2.5
*
h
)
ylim
(
0.
,
maxY
)
gca
().
xaxis
.
set_ticklabels
([])
ylabel
(
"$W(r,h)$"
,
labelpad
=
1.5
)
legend
(
loc
=
"upper right"
,
handlelength
=
1.2
,
handletextpad
=
0.2
)
legend
(
loc
=
"upper right"
,
frameon
=
False
,
handletextpad
=
0.1
,
handlelength
=
1.2
,
fontsize
=
8
)
# Same but now in log space
subplot
(
212
,
yscale
=
"log"
)
plot
(
xx
,
Gaussian
(
xx
,
h
),
'k-'
,
linewidth
=
0.7
,
label
=
"${
\\
rm Gaussian}$"
)
plot
(
xx
,
W_cubic_spline
(
xx
),
'b-'
,
label
=
"${
\\
rm Cubic~spline}$"
)
plot
(
xx
,
W_quartic_spline
(
xx
),
'c-'
,
label
=
"${
\\
rm Quartic~spline}$"
)
plot
(
xx
,
W_quintic_spline
(
xx
),
'g-'
,
label
=
"${
\\
rm Quintic~spline}$"
)
plot
(
xx
,
W_WendlandC2
(
xx
),
'r-'
,
label
=
"${
\\
rm Wendland~C2}$"
)
plot
(
xx
,
W_WendlandC4
(
xx
),
'm-'
,
label
=
"${
\\
rm Wendland~C4}$"
)
plot
(
xx
,
W_WendlandC6
(
xx
),
'y-'
,
label
=
"${
\\
rm Wendland~C6}$"
)
plot
(
xx
,
Gaussian
(
xx
,
h
),
'k-'
,
linewidth
=
0.7
,
label
=
"${
\\
rm Gaussian}$"
,
lw
=
1.5
)
plot
(
xx
,
W_cubic_spline
(
xx
),
'b-'
,
label
=
"${
\\
rm Cubic~spline}$"
,
lw
=
1.5
)
plot
(
xx
,
W_quartic_spline
(
xx
),
'c-'
,
label
=
"${
\\
rm Quartic~spline}$"
,
lw
=
1.5
)
plot
(
xx
,
W_quintic_spline
(
xx
),
'g-'
,
label
=
"${
\\
rm Quintic~spline}$"
,
lw
=
1.5
)
plot
(
xx
,
W_WendlandC2
(
xx
),
'r-'
,
label
=
"${
\\
rm Wendland~C2}$"
,
lw
=
1.5
)
plot
(
xx
,
W_WendlandC4
(
xx
),
'm-'
,
label
=
"${
\\
rm Wendland~C4}$"
,
lw
=
1.5
)
plot
(
xx
,
W_WendlandC6
(
xx
),
'y-'
,
label
=
"${
\\
rm Wendland~C6}$"
,
lw
=
1.5
)
# Show h
plot
([
h
,
h
],
[
0.
,
1.
],
'k:'
,
linewidth
=
0.5
)
...
...
@@ -191,21 +213,22 @@ plot([dx, dx], [0., 1.], 'k:', linewidth=0.5)
# Show plot properties
text
(
h
/
5.
,
1e-3
,
"$
\\
langle x
\\
rangle = %3.1f$"
%
(
dx
),
va
=
"top"
,
backgroundcolor
=
'w'
)
text
(
h
/
5.
+
0.06
,
3e-4
,
"$
\\
eta = %5.4f$"
%
(
eta
),
va
=
"top"
,
backgroundcolor
=
'w'
)
text
(
h
/
5.
,
2e-3
,
"$
\\
langle x
\\
rangle = %3.1f$"
%
(
dx
),
va
=
"top"
,
backgroundcolor
=
'w'
,
fontsize
=
9
)
text
(
h
/
5.
+
0.06
,
4e-4
,
"$
\\
eta = %5.4f$"
%
(
eta
),
va
=
"top"
,
backgroundcolor
=
'w'
,
fontsize
=
9
)
text
(
h
/
5.
+
0.06
,
8e-5
,
"$h =
\\
eta
\\
langle x
\\
rangle = %5.4f$"
%
(
h
),
va
=
"top"
,
backgroundcolor
=
'w'
,
fontsize
=
9
)
# Show number of neighbours
text
(
1.
9
*
h
,
2e-1
/
2.9
**
0
,
"$N_{
\\
rm ngb}=
\\
infty$"
,
fontsize
=
10
)
text
(
1.
9
*
h
,
2e-1
/
2.9
**
1
,
"$N_{
\\
rm ngb}=%3.1f$"
%
(
N_H_cubic
),
color
=
'b'
,
fontsize
=
9
)
text
(
1.
9
*
h
,
2e-1
/
2.9
**
2
,
"$N_{
\\
rm ngb}=%3.1f$"
%
(
N_H_quartic
),
color
=
'c'
,
fontsize
=
9
)
text
(
1.
9
*
h
,
2e-1
/
2.9
**
3
,
"$N_{
\\
rm ngb}=%3.1f$"
%
(
N_H_quintic
),
color
=
'g'
,
fontsize
=
9
)
text
(
1.
9
*
h
,
2e-1
/
2.9
**
4
,
"$N_{
\\
rm ngb}=%3.1f$"
%
(
N_H_WendlandC2
),
color
=
'r'
,
fontsize
=
9
)
text
(
1.
9
*
h
,
2e-1
/
2.9
**
5
,
"$N_{
\\
rm ngb}=%3.1f$"
%
(
N_H_WendlandC4
),
color
=
'm'
,
fontsize
=
9
)
text
(
1.
9
*
h
,
2e-1
/
2.9
**
6
,
"$N_{
\\
rm ngb}=%3.0f$"
%
(
N_H_WendlandC6
),
color
=
'y'
,
fontsize
=
9
)
text
(
1.
75
*
h
,
2e-1
/
2.9
**
0
,
"$N_{
\\
rm ngb}=
\\
infty$"
,
fontsize
=
10
)
text
(
1.
75
*
h
,
2e-1
/
2.9
**
1
,
"$N_{
\\
rm ngb}=%3.1f$"
%
(
N_H_cubic
),
color
=
'b'
,
fontsize
=
9
)
text
(
1.
75
*
h
,
2e-1
/
2.9
**
2
,
"$N_{
\\
rm ngb}=%3.1f$"
%
(
N_H_quartic
),
color
=
'c'
,
fontsize
=
9
)
text
(
1.
75
*
h
,
2e-1
/
2.9
**
3
,
"$N_{
\\
rm ngb}=%3.1f$"
%
(
N_H_quintic
),
color
=
'g'
,
fontsize
=
9
)
text
(
1.
75
*
h
,
2e-1
/
2.9
**
4
,
"$N_{
\\
rm ngb}=%3.1f$"
%
(
N_H_WendlandC2
),
color
=
'r'
,
fontsize
=
9
)
text
(
1.
75
*
h
,
2e-1
/
2.9
**
5
,
"$N_{
\\
rm ngb}=%3.1f$"
%
(
N_H_WendlandC4
),
color
=
'm'
,
fontsize
=
9
)
text
(
1.
75
*
h
,
2e-1
/
2.9
**
6
,
"$N_{
\\
rm ngb}=%3.0f$"
%
(
N_H_WendlandC6
),
color
=
'y'
,
fontsize
=
9
)
xlim
(
0.
,
2.5
*
h
)
ylim
(
1e-5
,
0.7
)
xlabel
(
"$r$"
,
labelpad
=
0
)
xlabel
(
"$r$"
,
labelpad
=
-
1
)
ylabel
(
"$W(r,h)$"
,
labelpad
=
0.5
)
savefig
(
"kernels.pdf"
)
...
...
@@ -282,13 +305,13 @@ figure()
subplot
(
211
)
plot
([
0
,
2.5
*
h
],
[
0.
,
0.
],
'k--'
,
linewidth
=
0.7
)
plot
(
xx
,
d_Gaussian
(
xx
,
h
),
'k-'
,
linewidth
=
0.7
,
label
=
"${
\\
rm Gaussian}$"
)
plot
(
xx
,
dW_cubic_spline
(
xx
),
'b-'
,
label
=
"${
\\
rm Cubic~spline}$"
)
plot
(
xx
,
dW_quartic_spline
(
xx
),
'c-'
,
label
=
"${
\\
rm Quartic~spline}$"
)
plot
(
xx
,
dW_quintic_spline
(
xx
),
'g-'
,
label
=
"${
\\
rm Quintic~spline}$"
)
plot
(
xx
,
dW_WendlandC2
(
xx
),
'r-'
,
label
=
"${
\\
rm Wendland~C2}$"
)
plot
(
xx
,
dW_WendlandC4
(
xx
),
'm-'
,
label
=
"${
\\
rm Wendland~C4}$"
)
plot
(
xx
,
dW_WendlandC6
(
xx
),
'y-'
,
label
=
"${
\\
rm Wendland~C6}$"
)
plot
(
xx
,
d_Gaussian
(
xx
,
h
),
'k-'
,
linewidth
=
0.7
,
label
=
"${
\\
rm Gaussian}$"
,
lw
=
1.5
)
plot
(
xx
,
dW_cubic_spline
(
xx
),
'b-'
,
label
=
"${
\\
rm Cubic~spline}$"
,
lw
=
1.5
)
plot
(
xx
,
dW_quartic_spline
(
xx
),
'c-'
,
label
=
"${
\\
rm Quartic~spline}$"
,
lw
=
1.5
)
plot
(
xx
,
dW_quintic_spline
(
xx
),
'g-'
,
label
=
"${
\\
rm Quintic~spline}$"
,
lw
=
1.5
)
plot
(
xx
,
dW_WendlandC2
(
xx
),
'r-'
,
label
=
"${
\\
rm Wendland~C2}$"
,
lw
=
1.5
)
plot
(
xx
,
dW_WendlandC4
(
xx
),
'm-'
,
label
=
"${
\\
rm Wendland~C4}$"
,
lw
=
1.5
)
plot
(
xx
,
dW_WendlandC6
(
xx
),
'y-'
,
label
=
"${
\\
rm Wendland~C6}$"
,
lw
=
1.5
)
maxY
=
d_Gaussian
(
h
/
2
,
h
)
...
...
@@ -307,7 +330,7 @@ gca().xaxis.set_ticklabels([])
ylim
(
1.1
*
maxY
,
-
0.1
*
maxY
)
xlabel
(
"$r$"
,
labelpad
=
0
)
ylabel
(
"$
\\
partial W(r,h)/
\\
partial r$"
,
labelpad
=
0.5
)
legend
(
loc
=
"lower right"
)
legend
(
loc
=
"lower right"
,
frameon
=
False
,
handletextpad
=
0.1
,
handlelength
=
1.2
,
fontsize
=
8
)
...
...
@@ -317,23 +340,23 @@ maxY = d2_Gaussian(h,h)
plot
([
h
/
2
,
h
/
2
],
[
-
4
*
maxY
,
1.4
*
maxY
],
'k:'
,
linewidth
=
0.5
)
plot
([
h
,
h
],
[
-
4
*
maxY
,
1.4
*
maxY
],
'k:'
,
linewidth
=
0.5
)
plot
([
dx
,
dx
],
[
-
4
*
maxY
,
1.4
*
maxY
],
'k:'
,
linewidth
=
0.5
)
text
(
h
/
2
,
-
3.
*
maxY
,
"$
\\
sigma
\\
equiv h/2$"
,
rotation
=
90
,
backgroundcolor
=
'w'
,
ha
=
'center'
,
va
=
'bottom'
)
text
(
h
,
-
3.
*
maxY
,
"$h
\\
equiv
\\
eta
\\
langle x
\\
rangle
= %.4f$"
%
h
,
rotation
=
90
,
backgroundcolor
=
'w'
,
ha
=
'center'
,
va
=
'bottom'
)
text
(
dx
,
-
3.
*
maxY
,
"$
\\
langle x
\\
rangle = %.1f$"
%
dx
,
rotation
=
90
,
backgroundcolor
=
'w'
,
ha
=
'center'
,
va
=
'bottom'
)
text
(
h
/
2
,
-
2.9
*
maxY
,
"$
\\
sigma
\\
equiv h/2$"
,
rotation
=
90
,
backgroundcolor
=
'w'
,
ha
=
'center'
,
va
=
'bottom'
,
fontsize
=
9
)
text
(
h
,
-
2.9
*
maxY
,
"$h
\\
equiv
\\
eta
\\
langle x
\\
rangle
$"
,
rotation
=
90
,
backgroundcolor
=
'w'
,
ha
=
'center'
,
va
=
'bottom'
,
fontsize
=
9
)
text
(
dx
,
-
2.9
*
maxY
,
"$
\\
langle x
\\
rangle = %.1f$"
%
dx
,
rotation
=
90
,
backgroundcolor
=
'w'
,
ha
=
'center'
,
va
=
'bottom'
,
fontsize
=
9
)
plot
([
0
,
2.5
*
h
],
[
0.
,
0.
],
'k--'
,
linewidth
=
0.7
)
plot
(
xx
,
d2_Gaussian
(
xx
,
h
),
'k-'
,
linewidth
=
0.7
,
label
=
"${
\\
rm Gaussian}$"
)
plot
(
xx
,
d2W_cubic_spline
(
xx
),
'b-'
,
label
=
"${
\\
rm Cubic~spline}$"
)
plot
(
xx
,
d2W_quartic_spline
(
xx
),
'c-'
,
label
=
"${
\\
rm Quartic~spline}$"
)
plot
(
xx
,
d2W_quintic_spline
(
xx
),
'g-'
,
label
=
"${
\\
rm Quintic~spline}$"
)
plot
(
xx
,
d2W_WendlandC2
(
xx
),
'r-'
,
label
=
"${
\\
rm Wendland~C2}$"
)
plot
(
xx
,
d2W_WendlandC4
(
xx
),
'm-'
,
label
=
"${
\\
rm Wendland~C4}$"
)
plot
(
xx
,
d2W_WendlandC6
(
xx
),
'y-'
,
label
=
"${
\\
rm Wendland~C6}$"
)
plot
(
xx
,
d2W_cubic_spline
(
xx
),
'b-'
,
label
=
"${
\\
rm Cubic~spline}$"
,
lw
=
1.5
)
plot
(
xx
,
d2W_quartic_spline
(
xx
),
'c-'
,
label
=
"${
\\
rm Quartic~spline}$"
,
lw
=
1.5
)
plot
(
xx
,
d2W_quintic_spline
(
xx
),
'g-'
,
label
=
"${
\\
rm Quintic~spline}$"
,
lw
=
1.5
)
plot
(
xx
,
d2W_WendlandC2
(
xx
),
'r-'
,
label
=
"${
\\
rm Wendland~C2}$"
,
lw
=
1.5
)
plot
(
xx
,
d2W_WendlandC4
(
xx
),
'm-'
,
label
=
"${
\\
rm Wendland~C4}$"
,
lw
=
1.5
)
plot
(
xx
,
d2W_WendlandC6
(
xx
),
'y-'
,
label
=
"${
\\
rm Wendland~C6}$"
,
lw
=
1.5
)
xlim
(
0.
,
2.5
*
h
)
ylim
(
-
3.2
*
maxY
,
1.3
*
maxY
)
xlabel
(
"$r$"
,
labelpad
=
0
)
xlabel
(
"$r$"
,
labelpad
=
-
1
)
ylabel
(
"$
\\
partial^2 W(r,h)/
\\
partial r^2$"
,
labelpad
=
0.5
)
...
...
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