def jump_condition(x,y,u):
u_desired = desired_func(x,y)
u_n_desired = hf.norm_grad(u_desired,(x,y),lvl_func)
#for Robin boundary condition only
def sigma(x,y):
return -0.25 * (hf.XYtoR(x-x0, y-y0)+10**-17)
sigma_Robin = sigma(x, y)
g_Robin = u_desired + sigma_Robin * u_n_desired
#################################
if(bnd_type == "Dirichlet"):
a_mesh = - u_desired
b_mesh = - beta_m * hf.grad_frame(u,(x,y),lvl_func)
elif(bnd_type == "Neumann"):
a_mesh = - u
b_mesh = - beta_m * u_n_desired
elif(bnd_type == "Robin_u"):
a_mesh = - u
b_mesh = - beta_m * (g_Robin - u)/sigma_Robin
elif(bnd_type == "Robin_u_n"):
a_mesh = -(g_Robin - sigma_Robin*u_n_desired)
b_mesh = - beta_m * hf.grad_frame(u,(x,y),lvl_func)
elif(bnd_type == "exact"):
a_mesh = - u_desired
b_mesh = - beta_m * u_n_desired
else:
raise Exception("error: invalid boundary format")
return (a_mesh, b_mesh)
def get_source(a, b, mesh, lvl_func_, f_mat_):
xmesh, ymesh = mesh
h = xmesh[0, 1] - xmesh[0, 0]
phi = lvl_func_(xmesh, ymesh)
#in the soruce term paper, the phi they use are inverted
phi_inv = -phi
#Discretization of the source term - formula (prev paper)
# S_mat = np.zeros_like(xmesh)
# H_h_mat = H(phi_inv,h)
# H_mat = np.heaviside(phi_inv,1)
# delta_mat = delta(phi_inv,h)
# S_mat = a*hf.laplace(H_mat,h,h) - (hf.norm_grad(a,(xmesh,ymesh),lvl_func_) + b)*delta_mat*hf.abs_grad(phi_inv,h,h) + H_h_mat * f_mat_
#Discretization of the source term - formula (8)
# a_eff = (a + (hf.norm_grad(a,(xmesh,ymesh),lvl_func_) - b) * phi_inv / (hf.abs_grad(phi_inv,h,h)+singular_null))
# H_h_mat = H(phi_inv,h)
# H_mat = np.heaviside(phi_inv,1)
# S_mat = hf.laplace(a_eff * H_mat,h,h) - H_h_mat * hf.laplace(a_eff,h,h) + H_h_mat * f_mat_
#Discretization of the source term - formula (7)
H_h_mat = H(phi_inv, h)
H_mat = np.heaviside(phi_inv, 1)
term1 = hf.laplace(a * H_mat, h, h)
term2 = -H_h_mat * hf.laplace(a, h, h)
term3 = -(b - hf.norm_grad(a, (xmesh, ymesh), lvl_func_)) * delta(
phi_inv, h) * hf.abs_grad(phi_inv, h, h)
term4 = H_h_mat * f_mat_
S_mat = term1 + term2 + term3 + term4
return S_mat
def source_test():
setup_grid(64)
setup_equations("exact")
phi = lvl_func(xmesh, ymesh)
a_mesh = desired_func(xmesh, ymesh)
b_mesh = hf.norm_grad(a_mesh, (xmesh, ymesh), lvl_func)
source = get_source(a_mesh, b_mesh, (xmesh, ymesh), lvl_func,
rhs_func(xmesh, ymesh))
plt.pcolor(xmesh, ymesh, source)
plt.xlabel("x")
plt.ylabel("y")
plt.colorbar()
return H(-phi, h)