def test_ft(): plot = Plot() vb = Verbose() # ============================================================================= # Parameters # ============================================================================= # # Flow # # permeability field phi = Constant(1) # porosity D = Constant(0.0252) # dispersivity K = Constant(1) # permeability # ============================================================================= # Mesh and Elements # ============================================================================= # Mesh mesh = QuadMesh(resolution=(30, 30)) # Mark left and right regions mesh.mark_region('left', lambda x, y: np.abs(x) < 1e-9, entity_type='half_edge') mesh.mark_region('right', lambda x, y: np.abs(x - 1) < 1e-9, entity_type='half_edge') # Elements p_element = QuadFE(2, 'Q1') # element for pressure c_element = QuadFE(2, 'Q1') # element for concentration # Dofhandlers p_dofhandler = DofHandler(mesh, p_element) c_dofhandler = DofHandler(mesh, c_element) p_dofhandler.distribute_dofs() c_dofhandler.distribute_dofs() # Basis functions p_ux = Basis(p_dofhandler, 'ux') p_uy = Basis(p_dofhandler, 'uy') p_u = Basis(p_dofhandler, 'u') p_inflow = lambda x, y: np.ones(shape=x.shape) p_outflow = lambda x, y: np.zeros(shape=x.shape) c_inflow = lambda x, y: np.zeros(shape=x.shape) # ============================================================================= # Solve the steady state flow equations # ============================================================================= vb.comment('Solving flow equations') # Define problem flow_problem = [ Form(1, test=p_ux, trial=p_ux), Form(1, test=p_uy, trial=p_uy), Form(0, test=p_u) ] # Assemble vb.tic('assembly') assembler = Assembler(flow_problem) assembler.add_dirichlet('left', 1) assembler.add_dirichlet('right', 0) assembler.assemble() vb.toc() # Solve linear system vb.tic('solve') A = assembler.get_matrix().tocsr() b = assembler.get_vector() x0 = assembler.assembled_bnd() # Interior nodes pa = np.zeros((p_u.n_dofs(), 1)) int_dofs = assembler.get_dofs('interior') pa[int_dofs, 0] = spla.spsolve(A, b - x0) # Resolve Dirichlet conditions dir_dofs, dir_vals = assembler.get_dirichlet(asdict=False) pa[dir_dofs] = dir_vals vb.toc() # Pressure function pfn = Nodal(data=pa, basis=p_u) px = pfn.differentiate((1, 0)) py = pfn.differentiate((1, 1)) #plot.contour(px) #plt.show() # ============================================================================= # Transport Equations # ============================================================================= # Specify initial condition c0 = Constant(1) dt = 1e-1 T = 6 N = int(np.ceil(T / dt)) c = Basis(c_dofhandler, 'c') cx = Basis(c_dofhandler, 'cx') cy = Basis(c_dofhandler, 'cy') print('assembling transport equations') k_phi = Kernel(f=phi) k_advx = Kernel(f=[K, px], F=lambda K, px: -K * px) k_advy = Kernel(f=[K, py], F=lambda K, py: -K * py) tht = 1 m = [Form(kernel=k_phi, test=c, trial=c)] s = [ Form(kernel=k_advx, test=c, trial=cx), Form(kernel=k_advy, test=c, trial=cy), Form(kernel=Kernel(D), test=cx, trial=cx), Form(kernel=Kernel(D), test=cy, trial=cy) ] problems = [m, s] assembler = Assembler(problems) assembler.add_dirichlet('left', 0, i_problem=0) assembler.add_dirichlet('left', 0, i_problem=1) assembler.assemble() x0 = assembler.assembled_bnd() # Interior nodes int_dofs = assembler.get_dofs('interior') # Dirichlet conditions dir_dofs, dir_vals = assembler.get_dirichlet(asdict=False) # System matrices M = assembler.get_matrix(i_problem=0) S = assembler.get_matrix(i_problem=1) # Initialize c0 and cp c0 = np.ones((c.n_dofs(), 1)) cp = np.zeros((c.n_dofs(), 1)) c_fn = Nodal(data=c0, basis=c) # # Compute solution # print('time stepping') for i in range(N): # Build system A = M + tht * dt * S b = M.dot(c0[int_dofs]) - (1 - tht) * dt * S.dot(c0[int_dofs]) # Solve linear system cp[int_dofs, 0] = spla.spsolve(A, b) # Add Dirichlet conditions cp[dir_dofs] = dir_vals # Record current iterate c_fn.add_samples(data=cp) # Update c0 c0 = cp.copy() #plot.contour(c_fn, n_sample=i) # # Quantity of interest # def F(c, px, py, entity=None): """ Compute c(x,y,t)*(grad p * n) """ n = entity.unit_normal() return c * (px * n[0] + py * n[1]) px.set_subsample(i=np.arange(41)) py.set_subsample(i=np.arange(41)) #kernel = Kernel(f=[c_fn,px,py], F=F) kernel = Kernel(c_fn) #print(kernel.n_subsample()) form = Form(kernel, flag='right', dmu='ds') assembler = Assembler(form, mesh=mesh) assembler.assemble() QQ = assembler.assembled_forms()[0].aggregate_data()['array'] Q = np.array([assembler.get_scalar(i_sample=i) for i in np.arange(N + 1)]) t = np.linspace(0, T, N + 1) plt.plot(t, Q) plt.show() print(Q)