def test03_solve_2d(self): """ Test problem with Neumann conditions """ # # Define Mesh # mesh = QuadMesh(resolution=(2, 1)) mesh.cells.get_child(1).mark(1) mesh.cells.refine(refinement_flag=1) # Mark left and right boundaries bm_left = lambda x, dummy: np.abs(x) < 1e-9 bm_right = lambda x, dummy: np.abs(1 - x) < 1e-9 mesh.mark_region('left', bm_left, entity_type='half_edge') mesh.mark_region('right', bm_right, entity_type='half_edge') for etype in ['Q1', 'Q2', 'Q3']: # # Define element and basis type # element = QuadFE(2, etype) dofhandler = DofHandler(mesh, element) dofhandler.distribute_dofs() u = Basis(dofhandler, 'u') ux = Basis(dofhandler, 'ux') uy = Basis(dofhandler, 'uy') # # Exact solution # ue = Nodal(f=lambda x: x[:, 0], basis=u) # # Set up forms # one = Constant(1) ax = Form(kernel=Kernel(one), trial=ux, test=ux) ay = Form(kernel=Kernel(one), trial=uy, test=uy) L = Form(kernel=Kernel(Constant(0)), test=u) Ln = Form(kernel=Kernel(one), test=u, dmu='ds', flag='right') problem = [ax, ay, L, Ln] assembler = Assembler(problem, mesh) assembler.add_dirichlet('left', dir_fn=0) assembler.add_hanging_nodes() assembler.assemble() # # Automatic solve # ya = assembler.solve() self.assertTrue(np.allclose(ue.data()[:, 0], ya)) # # Explicit solve # # System Matrices A = assembler.get_matrix().toarray() b = assembler.get_vector() x0 = assembler.assembled_bnd() # Solve linear system xa = np.zeros(u.n_dofs()) int_dofs = assembler.get_dofs('interior') xa[int_dofs] = np.linalg.solve(A, b - x0) # Resolve Dirichlet conditions dir_dofs, dir_vals = assembler.get_dirichlet(asdict=False) xa[dir_dofs] = dir_vals[:, 0] # Resolve hanging nodes C = assembler.hanging_node_matrix() xa += C.dot(xa) self.assertTrue(np.allclose(ue.data()[:, 0], xa))
def test02_solve_2d(self): """ Solve 2D problem with hanging nodes """ # Mesh mesh = QuadMesh(resolution=(2, 2)) mesh.cells.get_leaves()[0].mark(0) mesh.cells.refine(refinement_flag=0) mesh.mark_region('left', lambda x, y: abs(x) < 1e-9, entity_type='half_edge') mesh.mark_region('right', lambda x, y: abs(x - 1) < 1e-9, entity_type='half_edge') # Element Q1 = QuadFE(2, 'Q1') dofhandler = DofHandler(mesh, Q1) dofhandler.distribute_dofs() dofhandler.set_hanging_nodes() # Basis functions phi = Basis(dofhandler, 'u') phi_x = Basis(dofhandler, 'ux') phi_y = Basis(dofhandler, 'uy') # # Define problem # problem = [ Form(1, trial=phi_x, test=phi_x), Form(1, trial=phi_y, test=phi_y), Form(0, test=phi) ] ue = Nodal(f=lambda x: x[:, 0], basis=phi) xe = ue.data().ravel() # # Assemble without Dirichlet and without Hanging Nodes # assembler = Assembler(problem, mesh) assembler.add_dirichlet('left', dir_fn=0) assembler.add_dirichlet('right', dir_fn=1) assembler.add_hanging_nodes() assembler.assemble() # Get dofs for different regions int_dofs = assembler.get_dofs('interior') # Get matrix and vector A = assembler.get_matrix().toarray() b = assembler.get_vector() x0 = assembler.assembled_bnd() # Solve linear system xa = np.zeros(phi.n_dofs()) xa[int_dofs] = np.linalg.solve(A, b - x0) # Resolve Dirichlet conditions dir_dofs, dir_vals = assembler.get_dirichlet(asdict=False) xa[dir_dofs] = dir_vals[:, 0] # Resolve hanging nodes C = assembler.hanging_node_matrix() xa += C.dot(xa) self.assertTrue(np.allclose(xa, xe))