def norm_der(x, y):
u_x = A * np.cos(A * x) * np.cos(B * y)
u_y = -B * np.sin(A * x) * np.sin(B * y)
n_x = alpha**2 * x / (hf.XYtoR(alpha * x, beta * y) + singular_null)
n_y = beta**2 * y / (hf.XYtoR(alpha * x, beta * y) + singular_null)
u_n = (u_x * n_x + u_y * n_y) / (hf.XYtoR(n_x, n_y) + singular_null)
return u_n
def norm_der(x, y):
u_x = -0.5 * (x - x0)
u_y = -0.5 * (y - y0)
n_x = x / (hf.XYtoR(x, y) + singular_null)
n_y = y / (hf.XYtoR(x, y) + singular_null)
u_n = (u_x * n_x + u_y * n_y) / (hf.XYtoR(n_x, n_y) + singular_null)
return u_n
def norm_der(x, y):
theta = hf.XYtoTheta(x, y)
r = hf.XYtoR(x, y)
u_x = A * np.cos(A * x) * np.cos(B * y)
u_y = -B * np.sin(A * x) * np.sin(B * y)
dphi_dr = 1.
dphi_dtheta = -coef * theta_coef * np.cos(theta_coef * theta +
theta_const)
n_x = np.cos(theta) * dphi_dr - np.sin(theta) * dphi_dtheta / (
r + singular_null)
n_y = np.sin(theta) * dphi_dr + np.cos(theta) * dphi_dtheta / (
r + singular_null)
u_n = (u_x * n_x + u_y * n_y) / (hf.XYtoR(n_x, n_y) + singular_null)
return u_n
def sigma(x,y):
return -0.25 * (hf.XYtoR(x-x0, y-y0)+10**-17)
def level(x,y):
return np.array(hf.XYtoR(x,y) - r0)
def poisson_jacobi_solver(u_init_, maxIterNum_, mesh_, beta_, rhs_func_, lvl_func_, jmp_func_):
xmesh, ymesh = mesh_
beta_p, beta_m = beta_
#level grid
phi = lvl_func_(xmesh,ymesh)
h = xmesh[0,1]-xmesh[0,0]
# u = u_init_
u=np.zeros_like(xmesh)
# normal
r_temp = hf.XYtoR(xmesh, ymesh) + 10**-17
n1, n2 = xmesh/r_temp, ymesh/r_temp
# the source on the rhs
#source = np.zeros_like(u)
source = np.zeros_like(xmesh)
isOut = np.array(np.greater(phi,0.0),dtype = int)
source += (1.0 - isOut) * rhs_func_(xmesh, ymesh)
a_mesh, b_mesh = jmp_func_(xmesh, ymesh, u_init_)
def get_theta(u_cur,u_next):
return np.abs(u_cur)/(np.abs(u_cur)+np.abs(u_next))
def get_effective_beta(beta_cur,beta_next,theta):
ret = beta_cur * beta_next / (beta_next * theta + beta_cur * (1-theta))
return ret
beta_x = np.zeros_like(xmesh[:,1:])
beta_y = np.zeros_like(ymesh[1:,:])
isOutx1 = np.array(np.greater(phi[:,1:],0.0),dtype = int)
isOutx2 = np.array(np.greater(phi[:,:-1],0.0),dtype = int)
isOuty1 = np.array(np.greater(phi[1:,:],0.0),dtype = int)
isOuty2 = np.array(np.greater(phi[:-1,:],0.0),dtype = int)
# step is 1 if k+1 is out and k is in
# step is -1 if k is out and k+1 is in
# step is 0 if both out or in
xstep = isOutx1 - isOutx2
ystep = isOuty1 - isOuty2
xstep_p = np.array(np.greater( xstep,0.0),dtype = int)
xstep_m = np.array(np.greater(-xstep,0.0),dtype = int)
ystep_p = np.array(np.greater( ystep,0.0),dtype = int)
ystep_m = np.array(np.greater(-ystep,0.0),dtype = int)
theta_x = get_theta(phi[:,:-1],phi[:,1:])
theta_y = get_theta(phi[:-1,:],phi[1:,:])
beta_eff_x = np.zeros_like(beta_x)
beta_eff_y = np.zeros_like(beta_y)
beta_eff_x += xstep_p * get_effective_beta(beta_m,beta_p,theta_x)
beta_eff_x += xstep_m * get_effective_beta(beta_p,beta_m,theta_x)
beta_eff_y += ystep_p * get_effective_beta(beta_m,beta_p,theta_y)
beta_eff_y += ystep_m * get_effective_beta(beta_p,beta_m,theta_y)
a_jump_x = a_mesh[:,:-1] * (1-theta_x) + a_mesh[:,1:] * theta_x
a_jump_y = a_mesh[:-1,:] * (1-theta_y) + a_mesh[1:,:] * theta_y
b_jump_x = b_mesh[:,:-1]*(1-theta_x)*n1[:,:-1] + b_mesh[:,1:]*theta_x*n1[:,1:]
b_jump_y = b_mesh[:-1,:]*(1-theta_y)*n2[:-1,:] + b_mesh[1:,:]*theta_y*n2[1:,:]
# first additional term
source[:,:-1] += a_jump_x * beta_eff_x*xstep/h**2
source[:, 1:] += -a_jump_x * beta_eff_x*xstep/h**2
source[:-1,:] += a_jump_y * beta_eff_y*ystep/h**2
source[ 1:,:] += -a_jump_y * beta_eff_y*ystep/h**2
beta_x_k = ((beta_p + beta_m) + (beta_p - beta_m)*xstep)/2
beta_x_kp1 = ((beta_p + beta_m) - (beta_p - beta_m)*xstep)/2
beta_y_k = ((beta_p + beta_m) + (beta_p - beta_m)*ystep)/2
beta_y_kp1 = ((beta_p + beta_m) - (beta_p - beta_m)*ystep)/2
# second additional term
source[:,:-1] += (b_jump_x * beta_eff_x * xstep / h) * ((1-theta_x)/beta_x_k)
source[:, 1:] += (b_jump_x * beta_eff_x * xstep / h) * (theta_x/beta_x_kp1)
source[:-1,:] += (b_jump_y * beta_eff_y * ystep / h) * ((1-theta_y)/beta_y_k)
source[ 1:,:] += (b_jump_y * beta_eff_y * ystep / h) * (theta_y/beta_y_kp1)
isBoundary_x = np.abs(xstep)
isBoundary_y = np.abs(ystep)
beta_x += beta_eff_x * isBoundary_x
beta_x += isOutx2 * (1-isBoundary_x) * beta_p
beta_x += (1-isOutx2) * (1-isBoundary_x) * beta_m
beta_y += beta_eff_y * isBoundary_y
beta_y += isOuty2 * (1-isBoundary_y) * beta_p
beta_y += (1-isOuty2) * (1-isBoundary_y) * beta_m
beta_sum = beta_x[1:-1,:-1] + beta_x[1:-1,1:] + beta_y[:-1,1:-1] + beta_y[1:,1:-1]
# source_bcc = np.copy(source)
#
# a_jump_x = a_mesh[:,:-1] * theta_x + a_mesh[:,1:] * (1 - theta_x)
# a_jump_y = a_mesh[:-1,:] * theta_y + a_mesh[1:,:] * (1 - theta_y)
#
# b_jump_x = b_mesh[:,:-1]*theta_x*n1[:,:-1] + b_mesh[:,1:]*(1-theta_x)*n1[:,1:]
# b_jump_y = b_mesh[:-1,:]*theta_y*n2[:-1,:] + b_mesh[1:,:]*(1-theta_y)*n2[1:,:]
#
# # first additional term
# source_bcc[:,:-1] += a_jump_x * beta_eff_x*xstep/h**2
# source_bcc[:,1:] += -a_jump_x * beta_eff_x*xstep/h**2
# source_bcc[:-1,:] += a_jump_y * beta_eff_y*ystep/h**2
# source_bcc[1:,:] += -a_jump_y * beta_eff_y*ystep/h**2
# # second additional term
# source_bcc[:,:-1] += (b_jump_x * beta_eff_x * xstep / h) * ((1-theta_x)/beta_x_k)
# source_bcc[:,1:] += (b_jump_x * beta_eff_x * xstep / h) * (theta_x/beta_x_kp1)
# source_bcc[:-1,:] += (b_jump_y * beta_eff_y * ystep / h) * ((1-theta_y)/beta_y_k)
# source_bcc[1:,:] += (b_jump_y * beta_eff_y * ystep / h) * (theta_y/beta_y_kp1)
# plt.pcolor(xmesh,ymesh,source)
# plt.xlabel("x")
# plt.ylabel("y")
# plt.colorbar()
#record data
u_prev = u*(1-isOut)
global iterNum_record
iterNum_record = 0
for i in range(maxIterNum_):
iterNum_record += 1
# enforce boundary condition
# sol = solution1(xmesh, ymesh)
u[ 0, :] = np.zeros_like(u[ 0, :])
u[-1, :] = np.zeros_like(u[-1, :])
u[ :, 0] = np.zeros_like(u[ :, 0])
u[ :,-1] = np.zeros_like(u[ :,-1])
u_new = np.copy(u)
# update u according to Jacobi method formula
# https://en.wikipedia.org/wiki/Jacobi_method
del_u = u[1:-1,2:]*beta_x[1:-1,1:] + u[1:-1,0:-2]*beta_x[1:-1,0:-1] +\
u[2:,1:-1]*beta_y[1:,1:-1] + u[0:-2,1:-1]*beta_y[0:-1,1:-1]
u_new[1:-1,1:-1] = -h**2/beta_sum * (source[1:-1,1:-1] - del_u/h**2)
u = u_new
# check convergence and print process
check_convergence_rate = 10**-11
if(i % int(maxIterNum_/100) < 0.1):
u_cur = u* (1-isOut)
maxDif = np.max(np.abs(u_cur - u_prev)) / np.max(np.abs(u_cur))
L2Dif = hf.L_n_norm(np.abs(u_cur - u_prev)) / hf.L_n_norm(u_cur)
if(L2Dif < check_convergence_rate):
break;
else:
u_prev = u_cur
sys.stdout.write("\rProgress: %4g%%" % (i *100.0/maxIterNum_))
sys.stdout.flush()
print("")
return u
def level(x, y):
theta = np.arctan(y / (x + singular_null))
r = hf.XYtoR(x, y)
return -(const + coef * np.sin(theta_coef * theta + theta_const) - r)
def level(x, y):
return -0.9 + hf.XYtoR(alpha * x, beta * y)