Z = np.zeros((n+2, n+2), [('U', np.double), ('V', np.double)])
U, V = Z['U'], Z['V']
u, v = U[1:-1, 1:-1], V[1:-1, 1:-1]
r = 20
u[...] = 1.0
U[n//2-r:n//2+r, n//2-r:n//2+r] = 0.50
V[n//2-r:n//2+r, n//2-r:n//2+r] = 0.25
u += 0.05*np.random.uniform(-1, 1, (n, n))
v += 0.05*np.random.uniform(-1, 1, (n, n))
sy, sx = V.shape
grd = Grid(sx=sx, sy=sy, resx=sx, resy=sy)
grd.lineWidth(0).wireframe(False).lighting(ambient=0.5)
formula = r'(u,v)=(D_u\cdot\Delta u -u v v+F(1-u), D_v\cdot\Delta v +u v v -(F+k)v)'
ltx = Latex(formula, s=15, pos=(0,-sy/1.9,0))
print('Du, Dv, F, k, name =', Du, Dv, F, k, name)
settings.useDepthPeeling = False
for step in range(Nsteps):
for i in range(25):
Lu = ( U[0:-2, 1:-1] +
U[1:-1, 0:-2] - 4*U[1:-1, 1:-1] + U[1:-1, 2:] +
U[2: , 1:-1])
Lv = ( V[0:-2, 1:-1] +
V[1:-1, 0:-2] - 4*V[1:-1, 1:-1] + V[1:-1, 2:] +
V[2: , 1:-1])
uvv = u*v*v
u += Du*Lu - uvv + F*(1-u)
v += Dv*Lv + uvv - (F+k)*v
from vedo import Latex
# https://matplotlib.org/tutorials/text/mathtext.html
latex1 = r'x= \frac{ - b \pm \sqrt {b^2 - 4ac} }{2a}'
latex2 = r'\mathcal{A}\mathrm{sin}(2 \omega t)'
latex3 = r'I(Y | X)=\sum_{x \in \mathcal{X}, y \in \mathcal{Y}} p(x, y) \log \left(\frac{p(x)}{p(x, y)}\right)'
latex4 = r'\Gamma_{\epsilon}(x)=\left[1-e^{-2 \pi \epsilon}\right]^{1-x} \prod_{n=0}^{\infty} \frac{1-\exp (-2 \pi \epsilon(n+1))}{1-\exp (-2 \pi \epsilon(x+n))}'
latex5 = r'\left( \begin{array}{l}{c t^{\prime}} \\ {x^{\prime}} \\ {y^{\prime}} \\ {z^{\prime}}\end{array}\right)=\left( \begin{array}{cccc}{\gamma} & {-\gamma \beta} & {0} & {0} \\ {-\gamma \beta} & {\gamma} & {0} & {0} \\ {0} & {0} & {1} & {0} \\ {0} & {0} & {0} & {1}\end{array}\right) \left( \begin{array}{l}{c t} \\ {x} \\ {y} \\ {z}\end{array}\right)'
latex6 = r'\mathrm{CO}_{2}+6 \mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}+6 \mathrm{O}_{2}'
latex7 = r'x \mathrm{(arb. units)}'
ltx = Latex(latex4, s=1, c='darkblue', bg='', alpha=0.9, usetex=False, fromweb=False)
ltx.crop(0.3, 0.3) # crop top and bottom 30%
ltx.pos(2,0,0)
ltx.show(axes=1)
v = TestFunction(V)
a = dot(grad(w), grad(v)) * dx
L = p * v * dx
# Compute solution
w = Function(V)
solve(a == L, w, bc)
p = interpolate(p, V)
# Curve plot along x = 0 comparing p and w
import numpy as np
tol = 0.001 # avoid hitting points outside the domain
y = np.linspace(-1 + tol, 1 - tol, 101)
points = [(0, y_) for y_ in y] # 2D points
w_line = np.array([w(point) for point in points])
p_line = np.array([p(point) for point in points])
#######################################################################
from vedo.dolfin import plot
from vedo import Line, Latex
pde = r'-T \nabla^{2} D=p, ~\Omega=\left\{(x, y) | x^{2}+y^{2} \leq R\right\}'
tex = Latex(pde, pos=(0, 1.1, .1), s=0.2, c='w')
wline = Line(y, w_line * 10, c='white', lw=4)
pline = Line(y, p_line / 4, c='lightgreen', lw=4)
plot(w, wline, tex, at=0, N=2, bg='bb', text='Deflection')
plot(p, pline, at=1, bg='bb', text='Load')
from vedo import Latex, show
from vedo.pyplot import histogram
import numpy as np
N = 2000
x = np.random.randn(N) * 1.0
y = np.random.randn(N) * 1.5
# hexagonal binned histogram:
histo = histogram(
x,
y,
bins=10,
mode='hexbin',
xtitle="\sigma_x =1.0",
ytitle="\sigma_y =1.5",
ztitle="counts",
fill=True,
cmap='terrain',
)
# add a formula:
f = r'f(x, y)=A \exp \left(-\left(\frac{\left(x-x_{o}\right)^{2}}'
f += r'{2 \sigma_{x}^{2}}+\frac{\left(y-y_{o}\right)^{2}}'
f += r'{2 \sigma_{y}^{2}}\right)\right)'
formula = Latex(f, c='k', s=1.5).rotateX(90).rotateZ(90).pos(1.5, -2, 1)
show(histo, formula, axes=1, viewup='z')