def display_velocity( q , p ,mu , mu2 = None):
W = 5*SIGMA
res = 30
N_nodes = res**2
store = np.outer( np.linspace(-W,W , res), np.ones(res) )
nodes = np.zeros( [N_nodes , DIM] )
nodes[:,0] = np.reshape( store , N_nodes )
nodes[:,1] = np.reshape( store.T , N_nodes )
K,DK,D2K,D3K = jpf.derivatives_of_kernel(nodes , q)
vel_field = np.einsum('ijab,jb->ia',K,p) - np.einsum('ijabc,jbc->ia',DK,mu)
if mu2 != None:
vel_field = vel_field + np.einsum('ijabcd,jbcd->ia',D2K,mu2)
U = vel_field[:,0]
V = vel_field[:,1]
plt.figure()
plt.quiver( nodes[:,0] , nodes[:,1] , U , V , scale=10 )
plt.plot(q[:,0],q[:,1],'ro')
for i in range(0,N):
if np.linalg.norm(p[i]) < 1e-4: continue
plt.arrow(q[i,0], q[i,1], p[i,0], p[i,1], head_width=0.1, head_length=0.2, lw = 4.0, fc='b', ec='b')
plt.arrow(q[i,0], q[i,1], p[i,0], p[i,1], head_width=0.1, head_length=0.2, lw = 2.0, fc='w', ec='w')
plt.axis('equal')
plt.axis([- W, W,- W, W ])
return plt.gcf()
def display_streamlines(q, p, mu):
W = 5 * SIGMA
res = 30
N_nodes = res**DIM
store = np.outer(np.linspace(-W, W, res), np.ones(res))
nodes = np.zeros([N_nodes, DIM])
nodes[:, 0] = np.reshape(store, N_nodes)
nodes[:, 1] = np.reshape(store.T, N_nodes)
K, DK, D2K, D3K = jpf.derivatives_of_kernel(nodes, q)
vel_field = np.einsum('ijab,jb->ia', K, p) - np.einsum(
'ijabc,jbc->ia', DK, mu)
U = vel_field[:, 0]
V = vel_field[:, 1]
U2 = U.reshape(res, res)
V2 = V.reshape(res, res)
Y, X = np.mgrid[-W:W:30j, -W:W:30j]
speed = np.sqrt(U2 * U2 + V2 * V2)
lw = 5 * speed / speed.max()
plt.figure()
plt.streamplot(X, Y, U2.T, V2.T, color='k', linewidth=lw)
plt.plot(q[:, 0], q[:, 1], 'ro')
for i in range(0, N):
plt.arrow(q[i, 0],
q[i, 1],
0.2 * p[i, 0],
0.2 * p[i, 1],
head_width=0.05,
head_length=0.1,
fc='b',
ec='b')
plt.axis([-W, W, -W, W])
return plt.gcf()
def display_vorticity(q, p, mu, quiver=None):
W = 5 * SIGMA
res = 100
N_nodes = res**DIM
store = np.outer(np.linspace(-W, W, res), np.ones(res))
nodes = np.zeros([N_nodes, DIM])
nodes[:, 0] = np.reshape(store, N_nodes)
nodes[:, 1] = np.reshape(store.T, N_nodes)
K, DK, D2K, D3K = jpf.derivatives_of_kernel(nodes, q)
J = np.zeros([2, 2])
J[0, 1] = -1.
J[1, 0] = 1.
vorticity = np.einsum('da,ijabd,jb->i', J, DK, p) - np.einsum(
'da,ijabcd,jbc->i', J, D2K, mu)
Z = np.reshape(vorticity[:, ], [res, res])
X = np.reshape(nodes[:, 0], [res, res])
Y = np.reshape(nodes[:, 1], [res, res])
plt.figure()
plt.contourf(X, Y, Z, 10, cmap=plt.cm.rainbow)
# plt.contour( X,Y, Z , 10, color='k' )
plt.plot(q[:, 0], q[:, 1], 'wo')
for i in range(0, N):
plt.arrow(q[i, 0],
q[i, 1],
0.2 * p[i, 0],
0.2 * p[i, 1],
head_width=0.05,
head_length=0.1,
fc='b',
ec='b')
if quiver == "on":
res = 30
N_nodes = res**DIM
store = np.outer(np.linspace(-W, W, res), np.ones(res))
nodes = np.zeros([N_nodes, DIM])
nodes[:, 0] = np.reshape(store, N_nodes)
nodes[:, 1] = np.reshape(store.T, N_nodes)
K, DK, D2K, D3K = jpf.derivatives_of_kernel(nodes, q)
vel_field = np.einsum('ijab,jb->ia', K, p) - np.einsum(
'ijabc,jbc->ia', DK, mu)
U = vel_field[:, 0]
V = vel_field[:, 1]
plt.quiver(nodes[:, 0], nodes[:, 1], U, V, scale=20)
plt.axis([-W, W, -W, W])
plt.axis('equal')
return plt.gcf()
def display_vorticity( q , p ,mu , quiver = None ):
W = 5*SIGMA
res = 100
N_nodes = res**DIM
store = np.outer( np.linspace(-W,W , res), np.ones(res) )
nodes = np.zeros( [N_nodes , DIM] )
nodes[:,0] = np.reshape( store , N_nodes )
nodes[:,1] = np.reshape( store.T , N_nodes )
K,DK,D2K,D3K = jpf.derivatives_of_kernel(nodes,q)
J = np.zeros([2,2])
J[0,1] = -1.
J[1,0] = 1.
vorticity = np.einsum('da,ijabd,jb->i',J,DK,p) - np.einsum('da,ijabcd,jbc->i',J,D2K,mu)
Z = np.reshape(vorticity[:,], [res,res])
X = np.reshape( nodes[:,0] , [res,res] )
Y = np.reshape( nodes[:,1] , [res,res] )
plt.figure()
plt.contourf( X,Y, Z , 10, cmap=plt.cm.rainbow )
# plt.contour( X,Y, Z , 10, color='k' )
plt.plot(q[:,0],q[:,1],'wo')
for i in range(0,N):
plt.arrow(q[i,0], q[i,1], 0.2*p[i,0], 0.2*p[i,1], head_width=0.05, head_length=0.1, fc='b', ec='b')
if quiver=="on":
res = 30
N_nodes = res**DIM
store = np.outer( np.linspace(-W,W , res), np.ones(res) )
nodes = np.zeros( [N_nodes , DIM] )
nodes[:,0] = np.reshape( store , N_nodes )
nodes[:,1] = np.reshape( store.T , N_nodes )
K,DK,D2K,D3K = jpf.derivatives_of_kernel(nodes , q)
vel_field = np.einsum('ijab,jb->ia',K,p) - np.einsum('ijabc,jbc->ia',DK,mu)
U = vel_field[:,0]
V = vel_field[:,1]
plt.quiver( nodes[:,0] , nodes[:,1] , U , V , scale = 20 )
plt.axis([- W, W,- W, W ])
plt.axis('equal')
return plt.gcf()
def display_velocity(q, p, mu, mu2=None):
W = 5 * SIGMA
res = 30
N_nodes = res**2
store = np.outer(np.linspace(-W, W, res), np.ones(res))
nodes = np.zeros([N_nodes, DIM])
nodes[:, 0] = np.reshape(store, N_nodes)
nodes[:, 1] = np.reshape(store.T, N_nodes)
K, DK, D2K, D3K = jpf.derivatives_of_kernel(nodes, q)
vel_field = np.einsum('ijab,jb->ia', K, p) - np.einsum(
'ijabc,jbc->ia', DK, mu)
if mu2 != None:
vel_field = vel_field + np.einsum('ijabcd,jbcd->ia', D2K, mu2)
U = vel_field[:, 0]
V = vel_field[:, 1]
plt.figure()
plt.quiver(nodes[:, 0], nodes[:, 1], U, V, scale=10)
plt.plot(q[:, 0], q[:, 1], 'ro')
for i in range(0, N):
if np.linalg.norm(p[i]) < 1e-4: continue
plt.arrow(q[i, 0],
q[i, 1],
p[i, 0],
p[i, 1],
head_width=0.1,
head_length=0.2,
lw=4.0,
fc='b',
ec='b')
plt.arrow(q[i, 0],
q[i, 1],
p[i, 0],
p[i, 1],
head_width=0.1,
head_length=0.2,
lw=2.0,
fc='w',
ec='w')
plt.axis('equal')
plt.axis([-W, W, -W, W])
return plt.gcf()
def display_streamlines( q , p ,mu ):
W = 5*SIGMA
res = 30
N_nodes = res**DIM
store = np.outer( np.linspace(-W,W , res), np.ones(res) )
nodes = np.zeros( [N_nodes , DIM] )
nodes[:,0] = np.reshape( store , N_nodes )
nodes[:,1] = np.reshape( store.T , N_nodes )
K,DK,D2K,D3K = jpf.derivatives_of_kernel(nodes , q)
vel_field = np.einsum('ijab,jb->ia',K,p) - np.einsum('ijabc,jbc->ia',DK,mu)
U = vel_field[:,0]
V = vel_field[:,1]
U2 = U.reshape(res,res)
V2 = V.reshape(res,res)
Y,X = np.mgrid[-W:W:30j, -W:W:30j]
speed = np.sqrt(U2*U2 + V2*V2)
lw = 5*speed / speed.max()
plt.figure()
plt.streamplot( X , Y , U2.T , V2.T , color='k', linewidth=lw )
plt.plot(q[:,0],q[:,1],'ro')
for i in range(0,N):
plt.arrow(q[i,0], q[i,1], 0.2*p[i,0], 0.2*p[i,1], head_width=0.05, head_length=0.1, fc='b', ec='b')
plt.axis([- W, W,- W, W ])
return plt.gcf()