.. _demo_static_condensation: Static condensation of linear elasticity ======================================== Copyright (C) 2020 Michal Habera and Andreas Zilian This demo solves a Cook's plane stress elasticity test in a mixed space formulation. The test is a sloped cantilever under upward traction force at free end. Static condensation of internal (stress) degrees-of-freedom is demonstrated. :: import os import cffi import dolfinx import dolfinx.cpp import dolfinx.geometry import dolfinx.io import dolfinx.la import numba import numba.core.typing.cffi_utils as cffi_support import numpy import ufl from dolfinx.fem import locate_dofs_topological from dolfinx.mesh import locate_entities_boundary from mpi4py import MPI from petsc4py import PETSc filedir = os.path.dirname(__file__) infile = dolfinx.io.XDMFFile(MPI.COMM_WORLD, os.path.join(filedir, "cooks_tri_mesh.xdmf"), "r", encoding=dolfinx.cpp.io.XDMFFile.Encoding.ASCII) mesh = infile.read_mesh(name="Grid") infile.close() # Stress (Se) and displacement (Ue) elements Se = ufl.TensorElement("DG", mesh.ufl_cell(), 1, symmetry=True) Ue = ufl.VectorElement("CG", mesh.ufl_cell(), 2) S = dolfinx.FunctionSpace(mesh, Se) U = dolfinx.FunctionSpace(mesh, Ue) # Get local dofmap sizes for later local tensor tabulations Ssize = S.dolfin_element().space_dimension() Usize = U.dolfin_element().space_dimension() sigma, tau = ufl.TrialFunction(S), ufl.TestFunction(S) u, v = ufl.TrialFunction(U), ufl.TestFunction(U) def free_end(x): """Marks the leftmost points of the cantilever""" return numpy.isclose(x[0], 48.0) def left(x): """Marks left part of boundary, where cantilever is attached to wall""" return numpy.isclose(x[0], 0.0) # Locate all facets at the free end and assign them value 1 free_end_facets = locate_entities_boundary(mesh, 1, free_end) mt = dolfinx.mesh.MeshTags(mesh, 1, free_end_facets, 1) ds = ufl.Measure("ds", subdomain_data=mt) # Homogeneous boundary condition in displacement u_bc = dolfinx.Function(U) with u_bc.vector.localForm() as loc: loc.set(0.0) # Displacement BC is applied to the left side left_facets = locate_entities_boundary(mesh, 1, left) bdofs = locate_dofs_topological(U, 1, left_facets) bc = dolfinx.fem.DirichletBC(u_bc, bdofs) # Elastic stiffness tensor and Poisson ratio E, nu = 1.0, 1.0 / 3.0 def sigma_u(u): """Consitutive relation for stress-strain. Assuming plane-stress in XY""" eps = 0.5 * (ufl.grad(u) + ufl.grad(u).T) sigma = E / (1. - nu ** 2) * ((1. - nu) * eps + nu * ufl.Identity(2) * ufl.tr(eps)) return sigma a00 = ufl.inner(sigma, tau) * ufl.dx a10 = - ufl.inner(sigma, ufl.grad(v)) * ufl.dx a01 = - ufl.inner(sigma_u(u), tau) * ufl.dx f = ufl.as_vector([0.0, 1.0 / 16]) b1 = - ufl.inner(f, v) * ds(1) # JIT compile individual blocks tabulation kernels ufc_form00, _, _ = dolfinx.jit.ffcx_jit(mesh.mpi_comm(), a00) kernel00 = ufc_form00.integrals(0)[0].tabulate_tensor ufc_form01, _, _ = dolfinx.jit.ffcx_jit(mesh.mpi_comm(), a01) kernel01 = ufc_form01.integrals(0)[0].tabulate_tensor ufc_form10, _, _ = dolfinx.jit.ffcx_jit(mesh.mpi_comm(), a10) kernel10 = ufc_form10.integrals(0)[0].tabulate_tensor ffi = cffi.FFI() cffi_support.register_type(ffi.typeof('double _Complex'), numba.types.complex128) c_signature = numba.types.void( numba.types.CPointer(numba.typeof(PETSc.ScalarType())), numba.types.CPointer(numba.typeof(PETSc.ScalarType())), numba.types.CPointer(numba.typeof(PETSc.ScalarType())), numba.types.CPointer(numba.types.double), numba.types.CPointer(numba.types.int32), numba.types.CPointer(numba.types.uint8)) @numba.cfunc(c_signature, nopython=True) def tabulate_condensed_tensor_A(A_, w_, c_, coords_, entity_local_index, permutation=ffi.NULL): # Prepare target condensed local elem tensor A = numba.carray(A_, (Usize, Usize), dtype=PETSc.ScalarType) # Tabulate all sub blocks locally A00 = numpy.zeros((Ssize, Ssize), dtype=PETSc.ScalarType) kernel00(ffi.from_buffer(A00), w_, c_, coords_, entity_local_index, permutation) A01 = numpy.zeros((Ssize, Usize), dtype=PETSc.ScalarType) kernel01(ffi.from_buffer(A01), w_, c_, coords_, entity_local_index, permutation) A10 = numpy.zeros((Usize, Ssize), dtype=PETSc.ScalarType) kernel10(ffi.from_buffer(A10), w_, c_, coords_, entity_local_index, permutation) # A = - A10 * A00^{-1} * A01 A[:, :] = - A10 @ numpy.linalg.solve(A00, A01) # Prepare a Form with a condensed tabulation kernel integrals = {dolfinx.fem.IntegralType.cell: ([(-1, tabulate_condensed_tensor_A.address)], None)} a_cond = dolfinx.cpp.fem.Form([U._cpp_object, U._cpp_object], integrals, [], [], False, None) A_cond = dolfinx.fem.assemble_matrix(a_cond, [bc]) A_cond.assemble() b = dolfinx.fem.assemble_vector(b1) dolfinx.fem.apply_lifting(b, [a_cond], [[bc]]) b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE) dolfinx.fem.set_bc(b, [bc]) uc = dolfinx.Function(U) solver = PETSc.KSP().create(A_cond.getComm()) solver.setOperators(A_cond) solver.solve(b, uc.vector) # Pure displacement based formulation a = - ufl.inner(sigma_u(u), ufl.grad(v)) * ufl.dx A = dolfinx.fem.assemble_matrix(a, [bc]) A.assemble() # Create bounding box for function evaluation bb_tree = dolfinx.geometry.BoundingBoxTree(mesh, 2) # Check against standard table value p = numpy.array([48.0, 52.0, 0.0], dtype=numpy.float64) cell_candidates = dolfinx.geometry.compute_collisions_point(bb_tree, p) cell = dolfinx.cpp.geometry.select_colliding_cells(mesh, cell_candidates, p, 1) uc.vector.ghostUpdate(addv=PETSc.InsertMode.INSERT, mode=PETSc.ScatterMode.FORWARD) if len(cell) > 0: value = uc.eval(p, cell) print(value[1]) assert numpy.isclose(value[1], 23.95, rtol=1.e-2) # Check the equality of displacement based and mixed condensed global # matrices, i.e. check that condensation is exact assert numpy.isclose((A - A_cond).norm(), 0.0)