def _fluxes(h, q, g):
r"""Calculate the flux of mass and momentum for the shallow water
equations"""
I = firedrake.Identity(2)
F_h = q
F_q = outer(q, q) / h + 0.5 * g * h**2 * I
return F_h, F_q
def initialize_solvers(self):
# Kinematics # Right Cauchy-Green tensor
if self.nonlin:
d = self.X.geometric_dimension()
I = fd.Identity(d) # Identity tensor
F = I + fd.grad(self.X) # Deformation gradient
C = F.T * F
E = (C - I) / 2. # Green-Lagrangian strain
# E = 1./2.*( fd.grad(self.X).T + fd.grad(self.X) + fd.grad(self.X).T * fd.grad(self.X) ) # alternative equivalent definition
else:
E = 1. / 2. * (fd.grad(self.X).T + fd.grad(self.X)
) # linear strain
self.W = (self.lam / 2.) * (fd.tr(E))**2 + self.mu * fd.tr(E * E)
# f = fd.Constant((0, 0, -self.g)) # body force / rho
# T = self.surface_force()
# Total potential energy
Pi = self.W * fd.dx
# Compute first variation of Pi (directional derivative about X in the direction of v)
F_expr = fd.derivative(Pi, self.X, self.v)
self.DBC = fd.DirichletBC(self.V, fd.as_vector([0., 0., 0.]),
self.bottom_id)
# delX = fd.nabla_grad(self.X)
# delv_B = fd.nabla_grad(self.v)
# T_x_dv = self.lam * fd.div(self.X) * fd.div(self.v) \
# + self.mu * ( fd.inner( delX, delv_B + fd.transpose(delv_B) ) )
self.a_U = fd.dot(self.trial, self.v) * fd.dx
# self.L_U = ( fd.dot( self.U, self.v ) - self.dt/2./self.rho * T_x_dv ) * fd.dx
self.L_U = fd.dot(
self.U, self.v) * fd.dx - self.dt / 2. / self.rho * F_expr #\
# + self.dt/2./self.rho*fd.dot(T,self.v)*fd.ds(1) # surface force at x==0 plane
# + self.dt/2.*fd.dot(f,self.v)*fd.dx # body force
self.a_X = fd.dot(self.trial, self.v) * fd.dx
# self.L_interface = fd.dot(self.phi_vect, self.v) * fd.ds(self.interface_id)
self.L_X = fd.dot((self.X + self.dt * self.U), self.v) * fd.dx #\
# - self.dt/self.rho * self.L_interface
self.LVP_U = fd.LinearVariationalProblem(self.a_U,
self.L_U,
self.U,
bcs=[self.DBC])
self.LVS_U = fd.LinearVariationalSolver(self.LVP_U)
self.LVP_X = fd.LinearVariationalProblem(self.a_X,
self.L_X,
self.X,
bcs=[self.DBC])
self.LVS_X = fd.LinearVariationalSolver(self.LVP_X)
dt = np.sqrt(tol)
dtc = fd.Constant(dt)
n = fd.FacetNormal(mesh)
h = fd.sqrt(2) * fd.CellVolume(mesh) / fd.CellDiameter(mesh)
# advective velocity
velocity = fd.Function(vDG1)
velocity.interpolate(v_inlet)
vnorm = fd.sqrt(fd.dot(velocity, velocity))
# upwind term
vn = 0.5 * (fd.dot(velocity, n) + abs(fd.dot(velocity, n)))
cupw = fd.conditional(fd.dot(velocity, n) > 0, ch, lmbd_h)
# Diffusion tensor
Diff = fd.Identity(mesh.geometric_dimension()) * Dm
# Diff = fd.Identity(mesh.geometric_dimension())*(Dm + d_t*vnorm) + \
# fd.Constant(d_l-d_t)*fd.outer(velocity, velocity)/vnorm
# stability
# tau = fd.Constant(1.0) / h + abs(fd.dot(velocity, n))
tau = fd.Constant(5) / h + vn
# numerical flux
chat = lmbd_h
qhat = qh + tau * (ch - chat) * n + velocity * chat
# qhat_n = fd.dot(qh, n) + tau*(ch - chat) + chat*vn
a_u = (
fd.inner(fd.inv(Diff) * qh, vh) * fd.dx - ch * fd.div(vh) * fd.dx +
# internal faces
L = p_out*fd.inner(v, n)*fd.ds(outlet) + \
fd.inner(f, v)*fd.dx + idt*c_t*p0*q*fd.dx
# -------------------------------------------------------------------------
# transport
# coefficients
K = fd.Constant(K)
Dt = fd.Constant(dt)
c_mid = 0.5 * (c + c0) # Crank-Nicolson timestepping
# advective velocity
vel = fd.Function(V, name='velocity')
vnorm = fd.sqrt(fd.dot(vel, vel))
# Diffusion tensor
Diff = fd.Identity(mesh.geometric_dimension())*(Dm + alphaT*vnorm) + \
(alphaL-alphaT)*fd.outer(vel, vel)/vnorm
# source term
fc = fd.Constant(s)
# weak form (transport)
F1 = w * (c - c0) * fd.dx + Dt * (w * fd.dot(vel, fd.grad(c_mid)) + fd.dot(
fd.grad(w), Diff * fd.grad(c_mid)) + w * K * c_mid -
fc * w) * fd.dx # - Dt*h_n*w*fd.ds(outlet)
# strong form
R = (c - c0) + Dt * (fd.dot(vel, fd.grad(c_mid)) -
fd.div(Diff * fd.grad(c_mid)) + K * c_mid - fc)
# *** Adding SUPG stabilizing and shock cap. terms ***
# 3.4) Variational Form
# ----------------------
# coefficients
dt = np.sqrt(tol)
dtc = fd.Constant(dt)
n = fd.FacetNormal(mesh)
h = fd.sqrt(2) * fd.CellVolume(mesh) / fd.CellDiameter(mesh)
# advective velocity
velocity = fd.Function(vDG1)
velocity.interpolate(fd.Constant(v_inlet))
vnorm = fd.sqrt(fd.dot(velocity, velocity))
# Diffusion tensor
if d_t < 0:
Diff = fd.Identity(mesh.geometric_dimension()) * Dm
else:
Diff = fd.Identity(mesh.geometric_dimension())*(Dm + d_t*vnorm) + \
fd.Constant(d_l-d_t)*fd.outer(velocity, velocity)/vnorm
# upwind ter
# vn = 0.5*(fd.dot(velocity, n) + abs(fd.dot(velocity, n)))
# cupw = fd.conditional(vn < 0, ch, lmbd_h)
# stability
tau_d = fd.Constant(max([Dm, tau_e])) / h
tau_a = abs(fd.dot(velocity, n))
# tau_a = vn
# numerical flux
chat = lmbd_h
def sigma(u):
d = u.geometric_dimension() # space dimension
return self.lam * fd.nabla_div(u) * fd.Identity(
d) + 2 * self.mu * epsilon(u)
# way #1: natural bdr condition
# src = None
# nat_bdr = fd.conditional(x**2 + y**2 < h_mesh**2,
# -0.1, 0.) # 10 m3/d
# way #2: source term
nat_bdr = None
src = fd.conditional(x**2 + y**2 < h_mesh**2, 0.1 * n, 0.) # 10 m3/d/m3
# %%
# physical parameters
beta = 4.5e-8 # 1/Pa
rho = lambda x: 1000 * (1 + beta * (x - 1e5))
mu = 1e-3
phi = 0.1
k = 1e-14 * fd.Identity(dim)
g_ = -10 # None (without gravity)
g = fd.as_vector(
tuple([0 for d in range(dim - 1)]) + (0 if g_ is None else -g_, ))
# %%
# set forms
# ---------
W = fd.FunctionSpace(mesh, "CG", 1)
q = fd.TestFunction(W)
p = fd.Function(W, name="p")
p0 = fd.Function(W, name="p0")
def Identity(dim):
r"""Return the unit tensor of a given dimension"""
if dim == 1:
return firedrake.Constant(1.0)
return firedrake.Identity(dim)