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- """
- FEniCS tutorial demo program: Incompressible Navier-Stokes equations
- for channel flow (Poisseuille) on the unit square using the
- Incremental Pressure Correction Scheme (IPCS).
-
- u' + u . nabla(u)) - div(sigma(u, p)) = f
- div(u) = 0
- """
-
- from __future__ import print_function
- from fenics import *
- import numpy as np
-
- T = 10.0 # final time
- num_steps = 500 # number of time steps
- dt = T / num_steps # time step size
- mu = 1 # kinematic viscosity
- rho = 1 # density
-
- # Create mesh and define function spaces
- mesh = UnitSquareMesh(16, 16)
- V = VectorFunctionSpace(mesh, 'P', 2)
- Q = FunctionSpace(mesh, 'P', 1)
-
- # Define boundaries
- inflow = 'near(x[0], 0)'
- outflow = 'near(x[0], 1)'
- walls = 'near(x[1], 0) || near(x[1], 1)'
-
- # Define boundary conditions
- bcu_noslip = DirichletBC(V, Constant((0, 0)), walls)
- bcp_inflow = DirichletBC(Q, Constant(8), inflow)
- bcp_outflow = DirichletBC(Q, Constant(0), outflow)
- bcu = [bcu_noslip]
- bcp = [bcp_inflow, bcp_outflow]
-
- # Define trial and test functions
- u = TrialFunction(V)
- v = TestFunction(V)
- p = TrialFunction(Q)
- q = TestFunction(Q)
-
- # Define functions for solutions at previous and current time steps
- u_n = Function(V)
- u_ = Function(V)
- p_n = Function(Q)
- p_ = Function(Q)
-
- # Define expressions used in variational forms
- U = 0.5*(u_n + u)
- n = FacetNormal(mesh)
- f = Constant((0, 0))
- k = Constant(dt)
- mu = Constant(mu)
- rho = Constant(rho)
-
- # Define strain-rate tensor
- def epsilon(u):
- return sym(nabla_grad(u))
-
- # Define stress tensor
- def sigma(u, p):
- return 2*mu*epsilon(u) - p*Identity(len(u))
-
- # Define variational problem for step 1
- F1 = rho*dot((u - u_n) / k, v)*dx + \
- rho*dot(dot(u_n, nabla_grad(u_n)), v)*dx \
- + inner(sigma(U, p_n), epsilon(v))*dx \
- + dot(p_n*n, v)*ds - dot(mu*nabla_grad(U)*n, v)*ds \
- - dot(f, v)*dx
- a1 = lhs(F1)
- L1 = rhs(F1)
-
- # Define variational problem for step 2
- a2 = dot(nabla_grad(p), nabla_grad(q))*dx
- L2 = dot(nabla_grad(p_n), nabla_grad(q))*dx - (1/k)*div(u_)*q*dx
-
- # Define variational problem for step 3
- a3 = dot(u, v)*dx
- L3 = dot(u_, v)*dx - k*dot(nabla_grad(p_ - p_n), v)*dx
-
- # Assemble matrices
- A1 = assemble(a1)
- A2 = assemble(a2)
- A3 = assemble(a3)
-
- # Apply boundary conditions to matrices
- [bc.apply(A1) for bc in bcu]
- [bc.apply(A2) for bc in bcp]
-
- # Time-stepping
- t = 0
- for n in range(num_steps):
-
- # Update current time
- t += dt
-
- # Step 1: Tentative velocity step
- b1 = assemble(L1)
- [bc.apply(b1) for bc in bcu]
- solve(A1, u_.vector(), b1)
-
- # Step 2: Pressure correction step
- b2 = assemble(L2)
- [bc.apply(b2) for bc in bcp]
- solve(A2, p_.vector(), b2)
-
- # Step 3: Velocity correction step
- b3 = assemble(L3)
- solve(A3, u_.vector(), b3)
-
- # Plot solution
- plot(u_)
-
- # Compute error
- u_e = Expression(('4*x[1]*(1.0 - x[1])', '0'), degree=2)
- u_e = interpolate(u_e, V)
- error = np.abs(u_e.vector().array() - u_.vector().array()).max()
- print('t = %.2f: error = %.3g' % (t, error))
- print('max u:', u_.vector().array().max())
-
- # Update previous solution
- u_n.assign(u_)
- p_n.assign(p_)
-
- # Hold plot
- interactive()
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