def taylor_green(n):
mesh = UnitSquareMesh(n, n)
nu = 1 / 100
U_e = ('-(cos(pi*(x[0]))*sin(pi*(x[1]))) * exp(-2.0*nu*pi*pi*t)',
' (cos(pi*(x[1]))*sin(pi*(x[0]))) * exp(-2.0*nu*pi*pi*t)')
P_e = '-0.25*(cos(2*pi*(x[0])) + cos(2*pi*(x[1]))) * exp(-4.0*nu*pi*pi*t)'
u_e = Expression(U_e, degree=3, t=0, nu=float(nu))
K_e = 0.5 * assemble(inner(u_e, u_e) * dx(mesh))
###############CHORIN###############
(u1, p1, time1) = chorin_method(mesh, c_d, U_e, P_e)
K1 = 0.5 * assemble(inner(u1, u1) * dx(mesh))
er1 = K_e - K1
###############IPCS###############
(u2, p2, time2) = ipcs_method(mesh, c_d, U_e, P_e)
K2 = 0.5 * assemble(inner(u2, u2) * dx(mesh))
er2 = K_e - K2
###############GALERKIN###############
(u3, p3, time3) = galerkin_method(mesh, c_d, U_e, P_e)
K3 = 0.5 * assemble(inner(u3, u3) * dx(mesh))
er3 = K_e - K3
degrees1 = u1.vector().size() + p1.vector().size()
degrees2 = u3.vector().size() + p3.vector().size()
return mesh, er1, er2, er3, time1, time2, time3, degrees1, degrees2
def cylinder_flow(n):
channel = Rectangle(Point(0, 0), Point(2.2, 0.41))
cylinder = Circle(Point(0.2, 0.2), 0.05)
domain = channel - cylinder
mesh = generate_mesh(domain, n)
p_e = -0.111444953719
###############CHORIN###############
(u1, p1, time1) = chorin_method(mesh)
p1_a = p1(0.45, 0.2)
p1_b = p1(0.55, 0.2)
p_1 = p1_b - p1_a
er1 = p_e - p_1
###############IPCS###############
(u2, p2, time2) = ipcs_method(mesh)
p2_a = p2(0.45, 0.2)
p2_b = p2(0.55, 0.2)
p_2 = p2_b - p2_a
er2 = p_e - p_2
###############GALERKIN###############
(u3, p3, time3) = galerkin_method(mesh)
p3_a = p3(0.45, 0.2)
p3_b = p3(0.55, 0.2)
p_3 = p3_b - p3_a
er3 = p_e - p_3
degrees1 = u1.vector().size() + p1.vector().size()
degrees2 = u3.vector().size() + p3.vector().size()
return mesh, er1, er2, er3, time1, time2, time3, degrees1, degrees2
# Set parameter values
T = 0.5 # final time
dt = 0.2 * h / 1. # time step CFL with 1 = max. velocity
nu = 1 / 8 # kinematic viscosity
rho = 1
k = dt
u0 = Constant((0.0, 0.0))
p0 = Constant(0.0)
f = Constant((0.0, 0.0))
u_e = 0.44321183655
inicio1 = time.time()
(u1, p1) = chorin_method(rho, n, k, u, v, nu, f, p, q, bcu, bcp, dt, V, Q,
teste, u0, p0, T)
fim1 = time.time()
u1_a = u1(1.0, 0.5)[0]
print('Tempo chorin: ', fim1 - inicio1)
print('velocidade x : ', u1_a)
errorChorin = 1 - (u_e - u1_a) / u_e
print('error chorin: ', errorChorin)
print('degrees of freedom: ', u1.vector().size() + p1.vector().size())
plot(u1)
plt.savefig('VelocityChorin' + str(i) + '.png')
plot(p1)
plt.savefig('PressureChorin' + str(i) + '.png')
def pressure_driven(n): mesh = UnitSquareMesh(n, n) h = mesh.hmin() V = VectorFunctionSpace(mesh, 'P', 2) Q = FunctionSpace(mesh, 'P', 1) # Define Boundaries and Boundary Condictions inflow = 'near(x[0], 0)' outflow = 'near(x[0], 1)' walls = 'near(x[1], 0) || near(x[1], 1)' bcu_noslip = DirichletBC(V, Constant((0.0, 0.0)), walls) bcp_in = DirichletBC(Q, Constant(1.0), inflow) bcp_out = DirichletBC(Q, Constant(0.0), outflow) bcu = [bcu_noslip] bcp = [bcp_in, bcp_out] # Define variacional parameters u = TrialFunction(V) p = TrialFunction(Q) v = TestFunction(V) q = TestFunction(Q) # Set parameter values T = 0.5 # final time dt = 0.2*h/1. # time step CFL with 1 = max. velocity # Set parameter values T = 0.5 # final time dt = 0.2*h/1. # time step CFL with 1 = max. velocity nu = 1/8 # kinematic viscosity rho = 1 k = dt u0 = Constant((0.0, 0.0)) p0 = Constant(0.0) f = Constant((0.0, 0.0)) teste = 0 u_e = 0.44321183655 ###############CHORIN############### inicio1 = time.time() (u1,p1) = chorin_method(k, u, v, nu, f, p, q, bcu, bcp, dt, V, Q, teste, u0, p0, T) fim1 = time.time() time1 = (fim1 - inicio1) u1_a = u1(1.0,0.5)[0] er1 = u_e - u1_a ###############IPCS############### (u2,p2,time2) = ipcs_method(mesh) u2_a = u2(1.0,0.5)[0] er2 = u_e - u2_a ###############IPCS############### (u3, p3, time3) = galerkin_method(mesh) u3_a = u3(1.0,0.5)[0] er3 = u_e - u3_a degrees1 = u1.vector().size() + p1.vector().size() degrees2 = u3.vector().size() + p3.vector().size() return mesh, er1, er2, er3, time1, time2, time3, degrees1, degrees2
def driven_cavity(n):
mesh = UnitSquareMesh(n, n)
h = mesh.hmin()
V = VectorFunctionSpace(mesh, 'P', 2)
Q = FunctionSpace(mesh, 'P', 1)
# Define Boundaries and Boundary Condictions
upperedge = 'near(x[1], 1)'
noslip = 'near(x[1], 0) || near(x[0], 1) || near(x[0], 0)'
bcu_noslip = DirichletBC(V, Constant((0.0, 0.0)), noslip)
bcu_top = DirichletBC(V, Constant((1.0, 0.0)), upperedge)
bcu = [bcu_noslip, bcu_top]
bcp = []
# Define variacional parameters
u = TrialFunction(V)
p = TrialFunction(Q)
v = TestFunction(V)
q = TestFunction(Q)
# Set parameter values
T = 2.5 # final time
dt = 0.2 * h / 1. # time step CFL with 1 = max. velocity
nu = 1 / 1000 # kinematic viscosity
k = dt
u0 = Constant((0.0, 0.0))
p0 = Constant(0.0)
f = Constant((0.0, 0.0))
rho = 1
n = FacetNormal(mesh)
teste = 0
psi_e = -0.0585236
###############CHORIN###############
inicio1 = time.time()
(u1, p1) = chorin_method(k, u, v, nu, f, p, q, bcu, bcp, dt, V, Q, teste,
u0, p0, T)
fim1 = time.time()
time1 = (fim1 - inicio1)
psi1 = stream_function(u1)
stream_min1 = np.array(psi1.vector()).min()
er1 = psi_e - stream_min1
###############IPCS###############
inicio2 = time.time()
(u2, p2) = ipcs_method(rho, n, k, u, v, nu, f, p, q, bcu, bcp, dt, V, Q,
teste, u0, p0, T)
fim2 = time.time()
time2 = (fim2 - inicio2)
psi2 = stream_function(u2)
stream_min2 = np.array(psi2.vector()).min()
er2 = psi_e - stream_min2
###############IPCS###############
(u3, p3, time3) = galerkin_method(mesh)
psi3 = stream_function(u3)
stream_min3 = np.array(psi3.vector()).min()
er3 = psi_e - stream_min3
degrees1 = u1.vector().size() + p1.vector().size()
degrees2 = u3.vector().size() + p3.vector().size()
return mesh, er1, er2, er3, time1, time2, time3, degrees1, degrees2