# Copyright (C) 2022-2025 Nathan Sime and Jørgen S. Dokken # # This file is part of DOLFINX_MPC # # SPDX-License-Identifier: MIT # # This demo illustrates how to apply a slip condition on an # interface not aligned with the coordiante axis. # The demos solves the Stokes problem using the nest functionality to # avoid using mixed function spaces and as a non-linear problem. from __future__ import annotations from pathlib import Path from mpi4py import MPI from petsc4py import PETSc import basix import dolfinx.io import gmsh import numpy as np import scipy.sparse.linalg import ufl from dolfinx import default_real_type, default_scalar_type from dolfinx.io import gmsh as gmshio from ufl.core.expr import Expr import dolfinx_mpc import dolfinx_mpc.utils def create_mesh_gmsh( L: int = 2, H: int = 1, res: float = 0.1, theta: float = np.pi / 5, wall_marker: int = 1, outlet_marker: int = 2, inlet_marker: int = 3, ): """ Create a channel of length L, height H, rotated theta degrees around origin, with facet markers for inlet, outlet and walls. Parameters ---------- L The length of the channel H Width of the channel res Mesh resolution (uniform) theta Rotation angle wall_marker Integer used to mark the walls of the channel outlet_marker Integer used to mark the outlet of the channel inlet_marker Integer used to mark the inlet of the channel """ gmsh.initialize() if MPI.COMM_WORLD.rank == 0: gmsh.model.add("Square duct") # Create rectangular channel channel = gmsh.model.occ.addRectangle(0, 0, 0, L, H) gmsh.model.occ.synchronize() # Find entity markers before rotation surfaces = gmsh.model.occ.getEntities(dim=1) walls = [] inlets = [] outlets = [] for surface in surfaces: com = gmsh.model.occ.getCenterOfMass(surface[0], surface[1]) if np.allclose(com, [0, H / 2, 0]): inlets.append(surface[1]) elif np.allclose(com, [L, H / 2, 0]): outlets.append(surface[1]) elif np.isclose(com[1], 0) or np.isclose(com[1], H): walls.append(surface[1]) # Rotate channel theta degrees in the xy-plane gmsh.model.occ.rotate([(2, channel)], 0, 0, 0, 0, 0, 1, theta) gmsh.model.occ.synchronize() # Add physical markers gmsh.model.addPhysicalGroup(2, [channel], 1) gmsh.model.setPhysicalName(2, 1, "Fluid volume") gmsh.model.addPhysicalGroup(1, walls, wall_marker) gmsh.model.setPhysicalName(1, wall_marker, "Walls") gmsh.model.addPhysicalGroup(1, inlets, inlet_marker) gmsh.model.setPhysicalName(1, inlet_marker, "Fluid inlet") gmsh.model.addPhysicalGroup(1, outlets, outlet_marker) gmsh.model.setPhysicalName(1, outlet_marker, "Fluid outlet") # Set uniform mesh size gmsh.option.setNumber("Mesh.CharacteristicLengthMin", res) gmsh.option.setNumber("Mesh.CharacteristicLengthMax", res) # Generate mesh gmsh.model.mesh.generate(2) # Convert gmsh model to DOLFINx Mesh and meshtags mesh_data = gmshio.model_to_mesh(gmsh.model, MPI.COMM_WORLD, 0, gdim=2) assert mesh_data.cell_tags is not None ft = mesh_data.facet_tags gmsh.finalize() return mesh_data.mesh, ft # ------------------- Mesh and function space creation ------------------------ mesh, mt = create_mesh_gmsh(res=0.1) fdim = mesh.topology.dim - 1 # Create the function space cellname = mesh.ufl_cell().cellname() Ve = basix.ufl.element(basix.ElementFamily.P, cellname, 2, shape=(mesh.geometry.dim,), dtype=default_real_type) Qe = basix.ufl.element(basix.ElementFamily.P, cellname, 1, dtype=default_real_type) V = dolfinx.fem.functionspace(mesh, Ve) Q = dolfinx.fem.functionspace(mesh, Qe) W = ufl.MixedFunctionSpace(V, Q) def inlet_velocity_expression(x): return np.stack( ( np.sin(np.pi * np.sqrt(x[0] ** 2 + x[1] ** 2)), 5 * x[1] * np.sin(np.pi * np.sqrt(x[0] ** 2 + x[1] ** 2)), ) ) # ----------------------Defining boundary conditions---------------------- # Inlet velocity Dirichlet BC inlet_velocity = dolfinx.fem.Function(V) inlet_velocity.interpolate(inlet_velocity_expression) inlet_velocity.x.scatter_forward() dofs = dolfinx.fem.locate_dofs_topological(V, 1, mt.find(3)) bc1 = dolfinx.fem.dirichletbc(inlet_velocity, dofs) # Collect Dirichlet boundary conditions bcs = [bc1] # Slip conditions for walls n = dolfinx_mpc.utils.create_normal_approximation(V, mt, 1) with dolfinx.common.Timer("~Stokes: Create slip constraint"): mpc = dolfinx_mpc.MultiPointConstraint(V) mpc.create_slip_constraint(V, (mt, 1), n, bcs=bcs) mpc.finalize() mpc_q = dolfinx_mpc.MultiPointConstraint(Q) mpc_q.finalize() def tangential_proj(u: Expr, n: Expr): """ See for instance: https://link.springer.com/content/pdf/10.1023/A:1022235512626.pdf """ return (ufl.Identity(u.ufl_shape[0]) - ufl.outer(n, n)) * u def sym_grad(u: Expr): return ufl.sym(ufl.grad(u)) def T(u: Expr, p: Expr, mu: Expr): return 2 * mu * sym_grad(u) - p * ufl.Identity(u.ufl_shape[0]) # --------------------------Variational problem--------------------------- # Traditional terms mu = 1 f = dolfinx.fem.Constant(mesh, default_scalar_type((0, 0))) Wh = ufl.MixedFunctionSpace(mpc.function_space, mpc_q.function_space) uh = dolfinx.fem.Function(mpc.function_space) ph = dolfinx.fem.Function(mpc_q.function_space) (v, q) = ufl.TestFunctions(W) F = 2 * mu * ufl.inner(sym_grad(uh), sym_grad(v)) * ufl.dx F -= ufl.inner(ph, ufl.div(v)) * ufl.dx F -= ufl.inner(ufl.div(uh), q) * ufl.dx F -= ufl.inner(f, v) * ufl.dx F -= ufl.inner(dolfinx.fem.Constant(mesh, default_scalar_type(0.0)), q) * ufl.dx # No prescribed shear stress n = ufl.FacetNormal(mesh) g_tau = tangential_proj(dolfinx.fem.Constant(mesh, default_scalar_type(((0, 0), (0, 0)))) * n, n) ds = ufl.Measure("ds", domain=mesh, subdomain_data=mt, subdomain_id=1) # Terms due to slip condition # Explained in for instance: https://arxiv.org/pdf/2001.10639.pdf F -= ufl.inner(ufl.outer(n, n) * ufl.dot(2 * mu * sym_grad(uh), n), v) * ds F -= ufl.inner(ufl.outer(n, n) * ufl.dot(-ph * ufl.Identity(uh.ufl_shape[0]), n), v) * ds F -= ufl.inner(g_tau, v) * ds u, p = ufl.TrialFunctions(W) P = 2 * mu * ufl.inner(sym_grad(u), sym_grad(v)) * ufl.dx P -= ufl.inner(ufl.outer(n, n) * ufl.dot(2 * mu * sym_grad(u), n), v) * ds P += ufl.inner(p, q) * ufl.dx tol = 1e-7 problem = dolfinx_mpc.NonlinearProblem( ufl.extract_blocks(F), [uh, ph], mpc=[mpc, mpc_q], bcs=bcs, kind="nest", petsc_options={ "snes_type": "newtonls", "snes_atol": tol, "snes_rtol": tol, "snes_error_if_not_converged": True, "snes_monitor": None, "snes_linesearch_type": "none", "ksp_error_if_not_converged": True, "ksp_type": "minres", "ksp_rtol": 1e-8, "pc_type": "fieldsplit", "pc_fieldsplit_type": "additive", }, ) nested_IS = problem.A.getNestISs() ksp = problem.solver.getKSP() pc = ksp.getPC() pc.setFieldSplitIS(("u", nested_IS[0][0]), ("p", nested_IS[0][1])) ksp_u, ksp_p = pc.getFieldSplitSubKSP() ksp_u.setType("preonly") ksp_u.getPC().setType("gamg") ksp_p.setType("preonly") ksp_p.getPC().setType("jacobi") ksp.setMonitor( lambda ctx, it, r: PETSc.Sys.Print( # type: ignore f"Iteration: {it:>4d}, |r| = {r:.3e}" ) ) _, converged, num_iterations = problem.solve() # ------------------------------ Output ---------------------------------- uh.name = "u" ph.name = "p" outdir = Path("results") outdir.mkdir(exist_ok=True, parents=True) with dolfinx.io.XDMFFile(mesh.comm, outdir / "demo_stokes_nest_nonlinear.xdmf", "w") as outfile: outfile.write_mesh(mesh) outfile.write_meshtags(mt, mesh.geometry) outfile.write_function(ph) with dolfinx.io.VTXWriter(mesh.comm, outdir / "stokes_nest_uh_nonlinear.bp", uh, engine="BP4") as vtx: vtx.write(0.0) # -------------------- Verification -------------------------------- # Transfer data from the MPC problem to numpy arrays for comparison with dolfinx.common.Timer("~Stokes: Verification of problem by global matrix reduction"): W = dolfinx.fem.functionspace(mesh, basix.ufl.mixed_element([Ve, Qe])) V, V_to_W = W.sub(0).collapse() _, Q_to_W = W.sub(1).collapse() # Inlet velocity Dirichlet BC inlet_velocity = dolfinx.fem.Function(V) inlet_velocity.interpolate(inlet_velocity_expression) inlet_velocity.x.scatter_forward() W0 = W.sub(0) dofs = dolfinx.fem.locate_dofs_topological((W0, V), 1, mt.find(3)) bc1 = dolfinx.fem.dirichletbc(inlet_velocity, dofs, W0) # Collect Dirichlet boundary conditions bcs = [bc1] # Slip conditions for walls n = dolfinx_mpc.utils.create_normal_approximation(V, mt, 1) with dolfinx.common.Timer("~Stokes: Create slip constraint"): mpc = dolfinx_mpc.MultiPointConstraint(W) mpc.create_slip_constraint(W.sub(0), (mt, 1), n, bcs=bcs) mpc.finalize() (u, p) = ufl.TrialFunctions(W) (v, q) = ufl.TestFunctions(W) a = (2 * mu * ufl.inner(sym_grad(u), sym_grad(v)) - ufl.inner(p, ufl.div(v)) - ufl.inner(ufl.div(u), q)) * ufl.dx L = ufl.inner(f, v) * ufl.dx # Terms due to slip condition # Explained in for instance: https://arxiv.org/pdf/2001.10639.pdf a -= ufl.inner(ufl.outer(n, n) * ufl.dot(T(u, p, mu), n), v) * ds L += ufl.inner(g_tau, v) * ds af = dolfinx.fem.form(a) Lf = dolfinx.fem.form(L) # Solve the MPC problem using a global transformation matrix # and numpy solvers to get reference values # Generate reference matrices and unconstrained solution A_org = dolfinx.fem.petsc.assemble_matrix(af, bcs) A_org.assemble() L_org = dolfinx.fem.petsc.assemble_vector(Lf) dolfinx.fem.petsc.apply_lifting(L_org, [af], [bcs]) L_org.ghostUpdate(addv=PETSc.InsertMode.ADD_VALUES, mode=PETSc.ScatterMode.REVERSE) # type: ignore dolfinx.fem.petsc.set_bc(L_org, bcs) root = 0 # Gather LHS, RHS and solution on one process A_csr = dolfinx_mpc.utils.gather_PETScMatrix(A_org, root=root) K = dolfinx_mpc.utils.gather_transformation_matrix(mpc, root=root) L_np = dolfinx_mpc.utils.gather_PETScVector(L_org, root=root) u_mpc = dolfinx_mpc.utils.gather_PETScVector(uh.x.petsc_vec, root=root) p_mpc = dolfinx_mpc.utils.gather_PETScVector(ph.x.petsc_vec, root=root) up_mpc = np.hstack([u_mpc, p_mpc]) if MPI.COMM_WORLD.rank == root: KTAK = K.T * A_csr * K reduced_L = K.T @ L_np # Solve linear system d = scipy.sparse.linalg.spsolve(KTAK, reduced_L) # Back substitution to full solution vector uh_numpy = K @ d assert np.allclose(np.linalg.norm(uh_numpy, 2), np.linalg.norm(up_mpc, 2)) # -------------------- List timings -------------------------- dolfinx.common.list_timings(MPI.COMM_WORLD)