Hermes2D: Example 06 (04/28/2010)

91 days ago by solin

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The Code:
# Hermes2D: Example 06 (bc-newton) # This example explains how to use Newton boundary conditions. Again, # a Filter is used to visualize the solution gradient. # # PDE: Laplace equation -Laplace u = 0 (this equation describes, among # many other things, also stationary heat transfer in a homogeneous linear # material). # # BC: u = T1 ... fixed temperature on Gamma_3 (Dirichlet) # du/dn = 0 ... insulated wall on Gamma_2 and Gamma_4 (Neumann) # du/dn = H*(u - T0) ... heat flux on Gamma_1 (Newton) # # Note that the last BC can be written in the form du/dn - H*u = -h*T0. # The following parameters can be changed P_INIT = 3 # uniform polynomial degree in the mesh UNIFORM_REF_LEVEL = 2 # number of initial uniform mesh refinements CORNER_REF_LEVEL = 12 # number of mesh refinements towards the re-entrant corner # Import modules from hermes2d import Mesh, MeshView, H1Shapeset, PrecalcShapeset, H1Space, \ LinSystem, WeakForm, DummySolver, Solution, ScalarView from hermes2d.examples.c06 import set_bc, set_forms # Initialize the mesh mesh = Mesh() # Create a mesh from a list of nodes, elements, boundary and nurbs. mesh.create([ [0, -1], [1, -1], [-1, 0], [0, 0], [1, 0], [-1, 1], [0, 1], [0.707106781, 0.707106781] ], [ [0, 1, 4, 3, 0], [3, 4, 7, 0], [3, 7, 6, 0], [2, 3, 6, 5, 0] ], [ [0, 1, 1], [1, 4, 2], [3, 0, 4], [4, 7, 2], [7, 6, 2], [2, 3, 4], [6, 5, 2], [5, 2, 3] ], [ [4, 7, 45], [7, 6, 45] ]) # Mesh refinements for i in range(UNIFORM_REF_LEVEL): mesh.refine_all_elements() mesh.refine_towards_vertex(3, CORNER_REF_LEVEL) # Initialize shapeset and cache shapeset = H1Shapeset() pss = PrecalcShapeset(shapeset) # Create an H1 space space = H1Space(mesh, shapeset) space.set_uniform_order(P_INIT) # Set boundary conditions set_bc(space) # Enumerate degrees of freedom space.assign_dofs() # Initialize the discrete problem wf = WeakForm(1) set_forms(wf) # Initialize the linear system and solver solver = DummySolver() sys = LinSystem(wf, solver) sys.set_spaces(space) sys.set_pss(pss) # Assemble the stiffness matrix and solve the system sln = Solution() sys.assemble() sys.solve_system(sln) # Display the solution sln.plot(filename="a.png") # Display the mesh mesh.plot(space=space, filename="b.png") # Positioning the images print """<html><table border=1><tr><td><img src="cell://a.png"></span></td><td><img src="cell://b.png" width="540" height="405"></td></tr></table></html>"""