From fea6c5c4f204c0ba4591c66f619a28a721a632a1 Mon Sep 17 00:00:00 2001 From: Umberto Zerbinati Date: Thu, 25 Jun 2026 14:16:12 +0100 Subject: [PATCH 01/13] Build periodic Firedrake meshes from periodic Netgen meshes When a Netgen mesh carries periodic identifications, ngsPETSc now produces a vertex-merged (periodic) DMPlex. This finishes the job on the Firedrake side: the Mesh() netgen branch detects periodicity, builds the intermediate topology without reordering, and attaches a discontinuous DG1 coordinate field (MeshGeometry._periodic_coordinates) carrying the un-wrapped geometry via _postprocess_periodic_mesh, exactly as Firedrake's own periodic meshes. Each Firedrake cell is matched to its Netgen element by geometry (hashing the wrapped per-cell coordinates), which is robust to reordering. The DG coordinate space uses the equispaced variant required by _set_dg_coordinates. High-order curving of periodic meshes raises NotImplementedError for now. Adds regression tests for a periodic square (one and two directions), a periodic cylinder, and the too-coarse error path. Co-Authored-By: Claude Opus 4.8 (1M context) --- firedrake/mesh.py | 118 +++++++++++++++++++++- tests/firedrake/regression/test_netgen.py | 89 ++++++++++++++++ 2 files changed, 204 insertions(+), 3 deletions(-) diff --git a/firedrake/mesh.py b/firedrake/mesh.py index b6aa2e3f15..7b32ced67f 100644 --- a/firedrake/mesh.py +++ b/firedrake/mesh.py @@ -3055,6 +3055,80 @@ def curve_field(self, order, permutation_tol=1e-8, cg_field=None): new_coordinates.dat.data_wo_with_halos[broken_indices] = own_curved_points return new_coordinates + @PETSc.Log.EventDecorator() + def _periodic_coordinates(self, permutation_tol=1e-8): + '''Return a discontinuous coordinate field for a periodic netgen mesh. + + A periodic netgen mesh is converted by ngsPETSc into a DMPlex whose + topology is periodic (the identified vertices are merged) but whose + continuous coordinates are "wrapped" at the periodic seam. This method + builds the discontinuous (DG1) coordinate field carrying each cell's + true, un-wrapped corner coordinates, which is then attached to the mesh + (see :func:`~.utility_meshes._postprocess_periodic_mesh`). + + This method requires that the mesh has been constructed from a netgen + mesh that carries periodic identifications. + + :arg permutation_tol: tolerance used to match the reference element nodes. + ''' + utils.check_netgen_installed() + from firedrake.netgen import find_permutation + from firedrake.function import Function + from firedrake.functionspace import VectorFunctionSpace + from ngsPETSc.plex import buildPeriodicVertexMap + from ngsPETSc.utils.utils import trim_util + + if not hasattr(self, "netgen_mesh"): + raise ValueError("Cannot build periodic coordinates for a mesh that " + "has not been generated by netgen.") + + # Netgen element -> vertex connectivity (0-based). + if self.topological_dimension == 2: + ng_element = self.netgen_mesh.Elements2D() + else: + ng_element = self.netgen_mesh.Elements3D() + conn = trim_util(ng_element.NumPy()["nodes"]) + + # The same vertex merging ngsPETSc applied when it built the periodic plex. + old_to_new, survivors, _ = buildPeriodicVertexMap(self.netgen_mesh) + coords = self.netgen_mesh.Coordinates() + # `unwrapped` is the true geometry (what we want to store); `wrapped` is the + # merged/representative geometry, which coincides with Firedrake's continuous + # coordinates on the periodic plex. + unwrapped = coords[conn] + wrapped = coords[survivors][old_to_new[conn]] + + # Index netgen cells by their (rounded) set of wrapped vertex coordinates, + # so each Firedrake cell can be matched to its netgen element by geometry + # alone. This is robust to mesh reordering and parallel redistribution. + def cell_key(pts): + return tuple(sorted(tuple(np.round(p, 8)) for p in pts)) + lookup = {cell_key(wrapped[e]): e for e in range(wrapped.shape[0])} + + # Build the DG1 coordinate field, initialised from the (wrapped) continuous + # coordinates, then overwrite every cell with its un-wrapped geometry. The + # equispaced DG element matches the layout expected by _set_dg_coordinates + # (as used by Firedrake's own periodic meshes). + broken_space = VectorFunctionSpace( + self, finat.ufl.FiniteElement("DG", self.ufl_cell(), 1, variant="equispaced") + ) + new_coordinates = Function(broken_space).interpolate(self.coordinates) + cell_nodes = new_coordinates.cell_node_map().values + data = new_coordinates.dat.data + # Snapshot the wrapped coordinates (in Firedrake node order) before + # overwriting, so the per-cell matching always sees the interpolated values. + wrapped_fd = data[cell_nodes].real.copy() + + for i in range(cell_nodes.shape[0]): + fd_nodes = wrapped_fd[i] + e = lookup[cell_key(fd_nodes)] + # permutation taking the netgen node order to this cell's node order + permutation = find_permutation( + wrapped[e][np.newaxis], fd_nodes[np.newaxis], tol=permutation_tol + )[0] + data[cell_nodes[i]] = unwrapped[e][permutation] + return new_coordinates + @PETSc.Log.EventDecorator() def make_mesh_from_coordinates(coordinates, name, tolerance=0.5): @@ -3303,6 +3377,13 @@ def Mesh(meshfile, **kwargs): :param netgen_flags: The dictionary of flags to be passed to ngsPETSc. + If the Netgen mesh carries periodic identifications (e.g. created with + ``shape.Identify(..., IdentificationType.PERIODIC, ...)``) the resulting + Firedrake mesh is periodic: the identified vertices are merged and a + discontinuous coordinate field carries the un-wrapped geometry. The mesh + must be fine enough that no cell spans a full period, and high-order curving + (``degree != 1``) of periodic meshes is not currently supported. + When the mesh is read from a file the following mesh formats are supported (determined, case insensitively, from the filename extension): @@ -3360,6 +3441,7 @@ def Mesh(meshfile, **kwargs): # they all immediately call a petsc4py which in turn uses a PETSc # internal comm geometric_dim = kwargs.get("dim", None) + netgen_periodic = False if isinstance(meshfile, PETSc.DMPlex): plex = meshfile if MPI.Comm.Compare(user_comm, plex.comm.tompi4py()) not in {MPI.CONGRUENT, MPI.IDENT}: @@ -3372,6 +3454,11 @@ def Mesh(meshfile, **kwargs): netgen_firedrake_mesh = FiredrakeMesh(meshfile, netgen_flags, user_comm) plex = netgen_firedrake_mesh.meshMap.petscPlex plex.setName(_generate_default_mesh_topology_name(name)) + # A periodic netgen mesh produces a vertex-merged (periodic) topology that + # is finished off with a discontinuous coordinate field below. That field + # is built on the un-reordered topology (as for Firedrake's own periodic + # meshes), so suppress reordering here and reapply it in postprocessing. + netgen_periodic = len(netgen_firedrake_mesh.meshMap.ngMesh.GetIdentifications()) > 0 else: basename, ext = os.path.splitext(meshfile) @@ -3395,7 +3482,11 @@ def Mesh(meshfile, **kwargs): plex.setName(_generate_default_mesh_topology_name(name)) # Create mesh topology submesh_parent = kwargs.get("submesh_parent", None) - topology = MeshTopology(plex, name=plex.getName(), reorder=reorder, + # A periodic netgen mesh is finished off with a discontinuous coordinate field + # built on the un-reordered topology; the requested reordering is reapplied in + # _postprocess_periodic_mesh. + topology_reorder = False if netgen_periodic else reorder + topology = MeshTopology(plex, name=plex.getName(), reorder=topology_reorder, distribution_parameters=distribution_parameters, distribution_name=kwargs.get("distribution_name"), permutation_name=kwargs.get("permutation_name"), @@ -3407,9 +3498,30 @@ def Mesh(meshfile, **kwargs): mesh.netgen_mesh = netgen_firedrake_mesh.meshMap.ngMesh mesh.netgen_flags = netgen_flags - # Curve the mesh, if requested degree = netgen_flags.get("degree", 1) - if degree != 1: + periodic = len(mesh.netgen_mesh.GetIdentifications()) > 0 + if periodic: + # ngsPETSc produced a periodic (vertex-merged) topology; attach the + # discontinuous coordinate field carrying the un-wrapped geometry. + if degree != 1: + raise NotImplementedError( + "High-order curving of periodic netgen meshes is not supported yet." + ) + from firedrake.utility_meshes import _postprocess_periodic_mesh + permutation_tol = netgen_flags.get("permutation_tol", 1e-8) + coordinates = mesh._periodic_coordinates(permutation_tol=permutation_tol) + temp = _postprocess_periodic_mesh(coordinates, + mesh.comm, + distribution_parameters, + reorder, + name, + kwargs.get("distribution_name"), + kwargs.get("permutation_name")) + temp.netgen_mesh = mesh.netgen_mesh + temp.netgen_flags = mesh.netgen_flags + mesh = temp + # Curve the mesh, if requested + elif degree != 1: permutation_tol = netgen_flags.get("permutation_tol", 1e-8) cg = netgen_flags.get("cg", None) coordinates = mesh.curve_field( diff --git a/tests/firedrake/regression/test_netgen.py b/tests/firedrake/regression/test_netgen.py index 8524b52748..e3a0b2e9b3 100644 --- a/tests/firedrake/regression/test_netgen.py +++ b/tests/firedrake/regression/test_netgen.py @@ -342,3 +342,92 @@ def adapt(mesh, eta): break mesh = adapt(mesh, eta) assert error_estimators[-1] < 0.06 + + +def _occ_periodic_square(maxh, directions="x"): + from netgen.occ import Rectangle, OCCGeometry, X, Y, gp_Trsf, gp_Vec + from netgen.meshing import IdentificationType + shape = Rectangle(1, 1).Face() + shape.edges.Min(X).name, shape.edges.Max(X).name = "left", "right" + shape.edges.Min(Y).name, shape.edges.Max(Y).name = "bottom", "top" + if "x" in directions: + shape.edges.Min(X).Identify(shape.edges.Max(X), "px", + IdentificationType.PERIODIC, + gp_Trsf.Translation(gp_Vec(1, 0, 0))) + if "y" in directions: + shape.edges.Min(Y).Identify(shape.edges.Max(Y), "py", + IdentificationType.PERIODIC, + gp_Trsf.Translation(gp_Vec(0, 1, 0))) + return OCCGeometry(shape, dim=2).GenerateMesh(maxh=maxh) + + +def _solve_periodic_helmholtz(mesh, uex): + # Solve (I - div grad) u = f, where f is manufactured so the exact solution + # is the periodic field uex; return the L2 error. + V = FunctionSpace(mesh, "CG", 1) + u, v = TrialFunction(V), TestFunction(V) + f = uex - div(grad(uex)) + a = (inner(u, v) + inner(grad(u), grad(v))) * dx + L = inner(f, v) * dx + uh = Function(V) + solve(a == L, uh) + return sqrt(assemble(inner(uh - uex, uh - uex) * dx)) + + +@pytest.mark.skipnetgen +def test_netgen_periodic_square(): + # A periodic netgen mesh identifies the seam DOFs, so the periodic boundary + # markers are absent and the geometry is recovered exactly. + mesh = Mesh(_occ_periodic_square(0.1, directions="x")) + assert abs(assemble(Constant(1.0) * dx(domain=mesh)) - 1.0) < 1e-10 + + x, y = SpatialCoordinate(mesh) + # exact solution periodic in x; the seam continuity is essential to recover it. + err = _solve_periodic_helmholtz(mesh, sin(2 * pi * x)) + assert err < 5e-2 + + +@pytest.mark.skipnetgen +def test_netgen_periodic_square_both_directions(): + mesh = Mesh(_occ_periodic_square(0.1, directions="xy")) + assert abs(assemble(Constant(1.0) * dx(domain=mesh)) - 1.0) < 1e-10 + # both pairs identified: no exterior facets remain (a torus). + assert mesh.exterior_facets.set.total_size == 0 + x, y = SpatialCoordinate(mesh) + err = _solve_periodic_helmholtz(mesh, sin(2 * pi * x) * cos(2 * pi * y)) + assert err < 5e-2 + + +@pytest.mark.skipnetgen +def test_netgen_periodic_cylinder(): + # Periodic along the axis of a cylinder: a curved-boundary periodic mesh. + from netgen.occ import Cylinder, OCCGeometry, Pnt, gp_Trsf, gp_Vec, Z + from netgen.meshing import IdentificationType + cyl = Cylinder(Pnt(0, 0, 0), Z, r=1.0, h=1.0) + cyl.faces.Min(Z).Identify(cyl.faces.Max(Z), "pz", + IdentificationType.PERIODIC, + gp_Trsf.Translation(gp_Vec(0, 0, 1))) + mesh = Mesh(OCCGeometry(cyl).GenerateMesh(maxh=0.15)) + # Volume approaches pi as the polygonal boundary is refined. + assert abs(assemble(Constant(1.0) * dx(domain=mesh)) - pi) < 2e-2 + x, y, z = SpatialCoordinate(mesh) + err = _solve_periodic_helmholtz(mesh, sin(2 * pi * z)) + assert err < 2e-1 + + +@pytest.mark.skipnetgen +def test_netgen_periodic_too_coarse(): + # A mesh too coarse along the periodic direction produces seam-spanning cells + # that collapse on merging; this must raise a clear error rather than build a + # broken mesh. + from netgen.occ import Box, OCCGeometry, X, gp_Trsf, gp_Vec, Pnt + from netgen.meshing import IdentificationType + box = Box(Pnt(0, 0, 0), Pnt(1, 1, 1)) + box.faces.Min(X).Identify(box.faces.Max(X), "px", + IdentificationType.PERIODIC, + gp_Trsf.Translation(gp_Vec(1, 0, 0))) + ngmesh = OCCGeometry(box).GenerateMesh(maxh=0.4) + with pytest.raises(ValueError, match="degenerate"): + Mesh(ngmesh) + + From e327c74d05935b180ed57830d180df67719e5962 Mon Sep 17 00:00:00 2001 From: Umberto Zerbinati Date: Sun, 5 Jul 2026 14:59:56 +0100 Subject: [PATCH 02/13] Document periodic Netgen meshes in the demo (periodic-cylinder tokamak) Add a "Periodic Meshes" section to the Netgen integration demo showing how to declare OCC periodic identifications and import the resulting mesh into Firedrake. The worked example is the periodic-cylinder ("screw pinch") reduced model of a tokamak: a cylinder whose two circular ends are identified, making the axial (toroidal) direction periodic. It labels the lateral wall, checks the geometry survives the vertex merge (volume 2*pi^2), and solves a manufactured Helmholtz problem with Dirichlet data on the wall and periodicity on the ends, demonstrating seam continuity. Also documents the too-coarse ValueError and the degree==1 limitation. Co-Authored-By: Claude Opus 4.8 (1M context) --- demos/netgen/netgen_mesh.py.rst | 96 +++++++++++++++++++++++++++++++++ 1 file changed, 96 insertions(+) diff --git a/demos/netgen/netgen_mesh.py.rst b/demos/netgen/netgen_mesh.py.rst index 6d0f488c83..ecf8e67091 100755 --- a/demos/netgen/netgen_mesh.py.rst +++ b/demos/netgen/netgen_mesh.py.rst @@ -411,3 +411,99 @@ It is also possible to construct high-order meshes using the ``SplineGeometry``, .. figure:: Example7.png :align: center :alt: Example of a curved mesh of order 2 generated from a geometry described using Netgen CSG2d. + +Periodic Meshes +--------------- +Netgen can identify pairs of vertices lying on opposite boundaries of a geometry as being *the same* point. +When such a mesh is imported into Firedrake, the identified vertices are merged in the mesh topology, so that +a continuous (CG) function space automatically shares its degrees of freedom across the seam: the mesh is +genuinely **periodic**. This is exactly the representation Firedrake uses for its built-in +``PeriodicRectangleMesh``/``PeriodicBoxMesh``, and it is now available for any Netgen geometry carrying +periodic identifications. + +Identifications are declared on the geometry, before meshing, with the OCC ``Identify`` method:: + + shape_a.Identify(shape_b, name, IdentificationType.PERIODIC, transformation) + +where ``transformation`` is the rigid motion (typically a translation) that maps ``shape_a`` onto ``shape_b``. +Netgen then meshes the two boundaries compatibly and records the vertex pairs; Firedrake consumes them +automatically -- no extra flag on the ``Mesh`` constructor is required. + +As a physically motivated example we build the *periodic cylinder*, the classic reduced ("screw pinch") model +of a tokamak plasma column. A tokamak is a torus, so the plasma is periodic in the toroidal direction; in the +large-aspect-ratio limit one straightens a toroidal section into a cylinder and identifies its two circular +ends, recovering periodicity along the axis. We take the axial (toroidal) coordinate to run over :math:`[0, 2\pi)` +and identify the two end caps by a translation of :math:`2\pi` along ``Z``:: + + from netgen.occ import Cylinder, OCCGeometry, Pnt, Z, gp_Trsf, gp_Vec + from netgen.meshing import IdentificationType + from math import pi as PI + + cyl = Cylinder(Pnt(0, 0, 0), Z, r=1.0, h=2 * PI) + # Label the lateral wall, then the two end caps that we will identify. + for face in cyl.faces: + face.name = "wall" + cyl.faces.Min(Z).name = "bottom" + cyl.faces.Max(Z).name = "top" + # Identify the bottom cap with the top cap: a translation of 2*pi along Z + # maps one onto the other, making the axial direction periodic. + cyl.faces.Min(Z).Identify(cyl.faces.Max(Z), "toroidal", + IdentificationType.PERIODIC, + gp_Trsf.Translation(gp_Vec(0, 0, 2 * PI))) + ngmsh = OCCGeometry(cyl).GenerateMesh(maxh=0.4) + msh = Mesh(ngmsh) + VTKFile("output/Tokamak.pvd").write(msh) + +.. warning:: + + The mesh must contain at least a handful of cells along each periodic direction. If a single cell spans a + whole period, its two ends are identified and the cell collapses; Firedrake then raises a ``ValueError`` + asking you to refine along the periodic direction. Here the axis has length :math:`2\pi` and ``maxh=0.4`` + gives roughly sixteen cells along it, which is ample. Only ``degree == 1`` periodic meshes are supported + for now. + +Because the two end caps have been identified, no boundary markers survive on them: the seam has become an +*interior* set of facets, and the only labelled boundary that remains is the lateral wall. This is what makes +a continuous field wrap around continuously in the axial direction. We can verify the geometry survived the +merge intact -- the volume of the cylinder is :math:`\pi r^2 h = 2\pi^2` -- while the ends carry no exterior +facets:: + + volume = assemble(Constant(1.0) * dx(domain=msh)) + PETSc.Sys.Print(f"cylinder volume: {volume:.4f} (exact 2*pi**2 = {2 * PI**2:.4f})") + +To show that the periodicity is doing real work, we solve a Helmholtz problem whose exact solution is periodic +in the axial coordinate and vanishes on the lateral wall, + +.. math:: + + u_{\text{ex}}(x, y, z) = \cos(z)\,\bigl(1 - x^2 - y^2\bigr), + +so that we can impose a homogeneous Dirichlet condition on the wall while relying on the identified ends for +continuity along the axis. We look up the id of the ``"wall"`` boundary with ``GetRegionNames`` (as in the +Poisson example above) and manufacture the right-hand side :math:`f = u_{\text{ex}} - \Delta u_{\text{ex}}` for +:math:`(I - \Delta)u = f`:: + + V = FunctionSpace(msh, "CG", 2) + x, y, z = SpatialCoordinate(msh) + uex = cos(z) * (1 - x**2 - y**2) + f = uex - div(grad(uex)) + + u = TrialFunction(V) + v = TestFunction(V) + a = (inner(u, v) + inner(grad(u), grad(v))) * dx + L = inner(f, v) * dx + + labels = [i + 1 for i, name in enumerate(ngmsh.GetRegionNames(codim=1)) if name == "wall"] + bc = DirichletBC(V, 0, labels) + + sol = Function(V) + solve(a == L, sol, bcs=bc) + VTKFile("output/TokamakSolution.pvd").write(sol) + + error = sqrt(assemble(inner(sol - uex, sol - uex) * dx)) + PETSc.Sys.Print(f"L2 error: {error:.2e}") + +The recovered solution is continuous across the identified ends: opening ``output/TokamakSolution.pvd`` in +ParaView, the field wraps seamlessly from the top cap back to the bottom, exactly as a toroidal mode should. +Had the ends *not* been identified, the same computation would leave an artificial jump at the seam and the +manufactured solution would not be recovered. From 1f5925cb76273fe69decae1996f8d0f650181682 Mon Sep 17 00:00:00 2001 From: Umberto Zerbinati Date: Fri, 10 Jul 2026 16:07:18 +0100 Subject: [PATCH 03/13] Destroy object the garbage collector would have issue with --- firedrake/mesh.py | 11 +++++++---- firedrake/netgen.py | 10 ++++++++-- 2 files changed, 15 insertions(+), 6 deletions(-) diff --git a/firedrake/mesh.py b/firedrake/mesh.py index 8e6ebed99e..4560185c87 100644 --- a/firedrake/mesh.py +++ b/firedrake/mesh.py @@ -2944,10 +2944,13 @@ def refine_marked_elements(self, mark, netgen_flags=None): mark_np = mvec.getArray()[cellNum] else: sfBCInv = self.sfBC_orig.createInverse() - _, mvec0 = self.topology_dm.distributeField(sfBCInv, - self._cell_numbering, - mvec) - mark_np = mvec0.getArray() + section0, mvec0 = self.topology_dm.distributeField(sfBCInv, + self._cell_numbering, + mvec) + mark_np = mvec0.getArray().copy() + sfBCInv.destroy() + section0.destroy() + mvec0.destroy() max_refs = 0 if mark_np.size == 0 else int(mark_np.max()) # Create a copy of the netgen mesh netgen_mesh = self.netgen_mesh.Copy() diff --git a/firedrake/netgen.py b/firedrake/netgen.py index 57b442b83a..6d255a1219 100644 --- a/firedrake/netgen.py +++ b/firedrake/netgen.py @@ -70,12 +70,18 @@ def netgen_distribute(V: firedrake.functionspaceimpl.WithGeometryBase, for i in np.ndindex(V.shape): di = netgen_data[(..., *i)].flatten() vec0[:len(di)] = di - _, vec = plex.distributeField(sf, section0, vec0) - arr = vec.getArray() + section_i, vec_i = plex.distributeField(sf, section0, vec0) + arr = vec_i.getArray() if plex_data is None: plex_data = np.empty(arr.shape + V.shape, dtype=dtype) plex_data[(..., *i)] = arr + section_i.destroy() + vec_i.destroy() plex_data = plex_data.reshape(-1, *nshape[1:]) + # Destroy the remaining transients created in this call. + sfBCInv.destroy() + section0.destroy() + vec0.destroy() return plex_data From 4407328d1773f3d474d083c7406eee5b5b908bc3 Mon Sep 17 00:00:00 2001 From: ThighamOxford Date: Fri, 31 Jul 2026 11:56:57 +0100 Subject: [PATCH 04/13] merge conflict --- demos/netgen/netgen_mesh.py.rst | 187 ++++++++++++++++++++++++++++++++ 1 file changed, 187 insertions(+) diff --git a/demos/netgen/netgen_mesh.py.rst b/demos/netgen/netgen_mesh.py.rst index 6d0f488c83..903fe94c94 100755 --- a/demos/netgen/netgen_mesh.py.rst +++ b/demos/netgen/netgen_mesh.py.rst @@ -411,3 +411,190 @@ It is also possible to construct high-order meshes using the ``SplineGeometry``, .. figure:: Example7.png :align: center :alt: Example of a curved mesh of order 2 generated from a geometry described using Netgen CSG2d. + +Periodic Meshes +--------------- +Netgen can identify pairs of vertices lying on opposite boundaries of a geometry as being *the same* point. +When such a mesh is imported into Firedrake, the identified vertices are merged in the mesh topology, so that +a continuous (CG) function space automatically shares its degrees of freedom across the seam: the mesh is +genuinely **periodic**. This is exactly the representation Firedrake uses for its built-in +``PeriodicRectangleMesh``/``PeriodicBoxMesh``, and it is now available for any Netgen geometry carrying +periodic identifications. + +Identifications are declared on the geometry, before meshing, with the OCC ``Identify`` method:: + + shape_a.Identify(shape_b, name, IdentificationType.PERIODIC, transformation) + +where ``transformation`` is the rigid motion (typically a translation) that maps ``shape_a`` onto ``shape_b``. +Netgen then meshes the two boundaries compatibly and records the vertex pairs; Firedrake consumes them +automatically -- no extra flag on the ``Mesh`` constructor is required. + +As a physically motivated example we build the *periodic cylinder*, the classic reduced ("screw pinch") model +of a tokamak plasma column. A tokamak is a torus, so the plasma is periodic in the toroidal direction; in the +large-aspect-ratio limit one straightens a toroidal section into a cylinder and identifies its two circular +ends, recovering periodicity along the axis. We take the axial (toroidal) coordinate to run over :math:`[0, 2\pi)` +and identify the two end caps by a translation of :math:`2\pi` along ``Z``:: + + from netgen.occ import Cylinder, OCCGeometry, Pnt, Z, gp_Trsf, gp_Vec + from netgen.meshing import IdentificationType + from math import pi as PI + + cyl = Cylinder(Pnt(0, 0, 0), Z, r=1.0, h=2 * PI) + # Label the lateral wall, then the two end caps that we will identify. + for face in cyl.faces: + face.name = "wall" + cyl.faces.Min(Z).name = "bottom" + cyl.faces.Max(Z).name = "top" + # Identify the bottom cap with the top cap: a translation of 2*pi along Z + # maps one onto the other, making the axial direction periodic. + cyl.faces.Min(Z).Identify(cyl.faces.Max(Z), "toroidal", + IdentificationType.PERIODIC, + gp_Trsf.Translation(gp_Vec(0, 0, 2 * PI))) + ngmsh = OCCGeometry(cyl).GenerateMesh(maxh=0.4) + msh = Mesh(ngmsh) + VTKFile("output/PeriodicCylinder.pvd").write(msh) + +.. warning:: + + The mesh must contain at least a handful of cells along each periodic direction. If a single cell spans a + whole period, its two ends are identified and the cell collapses; Firedrake then raises a ``ValueError`` + asking you to refine along the periodic direction. Here the axis has length :math:`2\pi` and ``maxh=0.4`` + gives roughly sixteen cells along it, which is ample. Only ``degree == 1`` periodic meshes are supported + for now. + +Because the two end caps have been identified, no boundary markers survive on them: the seam has become an +*interior* set of facets, and the only labelled boundary that remains is the lateral wall. This is what makes +a continuous field wrap around continuously in the axial direction. We can verify the geometry survived the +merge intact -- the volume of the cylinder is :math:`\pi r^2 h = 2\pi^2` -- while the ends carry no exterior +facets:: + + volume = assemble(Constant(1.0) * dx(domain=msh)) + PETSc.Sys.Print(f"cylinder volume: {volume:.4f} (exact 2*pi**2 = {2 * PI**2:.4f})") + +To show that the periodicity is doing real work, we solve a Helmholtz problem whose exact solution is periodic +in the axial coordinate and vanishes on the lateral wall, + +.. math:: + + u_{\text{ex}}(x, y, z) = \cos(z)\,\bigl(1 - x^2 - y^2\bigr), + +so that we can impose a homogeneous Dirichlet condition on the wall while relying on the identified ends for +continuity along the axis. We look up the id of the ``"wall"`` boundary with ``GetRegionNames`` (as in the +Poisson example above) and manufacture the right-hand side :math:`f = u_{\text{ex}} - \Delta u_{\text{ex}}` for +:math:`(I - \Delta)u = f`:: + + V = FunctionSpace(msh, "CG", 2) + x, y, z = SpatialCoordinate(msh) + uex = cos(z) * (1 - x**2 - y**2) + f = uex - div(grad(uex)) + + u = TrialFunction(V) + v = TestFunction(V) + a = (inner(u, v) + inner(grad(u), grad(v))) * dx + L = inner(f, v) * dx + + labels = [i + 1 for i, name in enumerate(ngmsh.GetRegionNames(codim=1)) if name == "wall"] + bc = DirichletBC(V, 0, labels) + + sol = Function(V) + solve(a == L, sol, bcs=bc) + VTKFile("output/PeriodicCylinderSolution.pvd").write(sol) + + error = sqrt(assemble(inner(sol - uex, sol - uex) * dx)) + PETSc.Sys.Print(f"L2 error: {error:.2e}") + +The recovered solution is continuous across the identified ends: opening ``output/PeriodicCylinderSolution.pvd`` in +ParaView, the field wraps seamlessly from the top cap back to the bottom, exactly as a toroidal mode should. +Had the ends *not* been identified, the same computation would leave an artificial jump at the seam and the +manufactured solution would not be recovered. + +We can also solve a Helmholtz problem on a realistic tokamak geometry, although we no longer have an analytical solution to test against. +We now work in cylindrical coordinate space (R, phi, Z) where we construct a cross-section in the (R,Z) plane with the appropriate tokamak shape and then extrude in phi by :math:`2\pi`. + + from firedrake import * + from netgen.occ import ( + OCCGeometry, WorkPlane, Axes, Pnt, Z, X, + gp_Trsf, gp_Vec + ) + from netgen.meshing import IdentificationType + from math import pi as PI + import math + + # Geometry parameters - large aspect ratio tokamak + R0 = 3.0 + a = 1.0 + kappa = 2.0 + delta = 0.3 + + n_boundary = 40 + mesh_size = 0.5 + + # Build a tokamak cross section and extrude periodically + + alpha = math.asin(delta) + + boundary_points = [] + for i in range(n_boundary): + theta = 2.0 * math.pi * i / n_boundary + R = R0 + a * math.cos(theta + alpha * math.sin(theta)) + Zc = kappa * a * math.sin(theta) + boundary_points.append((R, Zc)) + + wp = WorkPlane(Axes((0, 0, 0), n=Z, h=X)) + + # Start at the first boundary point, then draw a closed polyline + R0p, Z0p = boundary_points[0] + wp.MoveTo(R0p, Z0p) + for R, Zc in boundary_points[1:]: + wp.LineTo(R, Zc) + wp.Close() + + face = wp.Face() + + # Extrude in periodic phi direction and identify end caps. + + phi_length = 2 * PI + solid = face.Extrude(phi_length * Z) + + # Side wall(s) + for f in solid.faces: + f.name = "wall" + + bottom = solid.faces.Min(Z) + top = solid.faces.Max(Z) + bottom.name = "bottom" + top.name = "top" + + # Periodic identification of the end caps + bottom.Identify( + top, + "periodic_phi", + IdentificationType.PERIODIC, + gp_Trsf.Translation(gp_Vec(0, 0, phi_length)), + ) + + # Mesh and convert to Firedrake + ngmsh = OCCGeometry(solid).GenerateMesh(maxh=mesh_size) + msh = Mesh(ngmsh, name="TokamakPeriodic") + VTKFile("output/Tokamak_3d_Mesh.pvd").write(msh) + +Now we solve a Helmholtz problem again with source :math:`f = \cos(z)xy`. + + V = FunctionSpace(msh, "CG", 2) + x, y, z = SpatialCoordinate(msh) + + f = cos(z)* x*y + + u = TrialFunction(V) + v = TestFunction(V) + a = (inner(u, v) + inner(grad(u), grad(v))) * dx + L = inner(f, v) * dx + + labels = [i + 1 for i, name in enumerate(ngmsh.GetRegionNames(codim=1)) + if name == "wall"] + bc = DirichletBC(V, 0, labels) + + sol = Function(V) + solve(a == L, sol, bcs=bc) + + VTKFile("output/TokamakSolution.pvd").write(sol) From 7557a736a61b205a9626bd2b250508368c761c90 Mon Sep 17 00:00:00 2001 From: ThighamOxford Date: Mon, 3 Aug 2026 11:32:12 +0100 Subject: [PATCH 05/13] fix permutation_tol --- firedrake/mesh.py | 9 +++------ 1 file changed, 3 insertions(+), 6 deletions(-) diff --git a/firedrake/mesh.py b/firedrake/mesh.py index b4a04792df..b2034f6634 100644 --- a/firedrake/mesh.py +++ b/firedrake/mesh.py @@ -3077,7 +3077,7 @@ def curve_field(self, order, permutation_tol=None, cg_field=None): return new_coordinates @PETSc.Log.EventDecorator() - def _periodic_coordinates(self, permutation_tol=1e-8): + def _periodic_coordinates(self): '''Return a discontinuous coordinate field for a periodic netgen mesh. A periodic netgen mesh is converted by ngsPETSc into a DMPlex whose @@ -3089,8 +3089,6 @@ def _periodic_coordinates(self, permutation_tol=1e-8): This method requires that the mesh has been constructed from a netgen mesh that carries periodic identifications. - - :arg permutation_tol: tolerance used to match the reference element nodes. ''' utils.check_netgen_installed() from firedrake.netgen import find_permutation @@ -3145,7 +3143,7 @@ def cell_key(pts): e = lookup[cell_key(fd_nodes)] # permutation taking the netgen node order to this cell's node order permutation = find_permutation( - wrapped[e][np.newaxis], fd_nodes[np.newaxis], tol=permutation_tol + wrapped[e][np.newaxis], fd_nodes[np.newaxis] )[0] data[cell_nodes[i]] = unwrapped[e][permutation] return new_coordinates @@ -3530,7 +3528,7 @@ def Mesh(meshfile, **kwargs): ) from firedrake.utility_meshes import _postprocess_periodic_mesh permutation_tol = netgen_flags.get("permutation_tol", None) - coordinates = mesh._periodic_coordinates(permutation_tol=permutation_tol) + coordinates = mesh._periodic_coordinates() temp = _postprocess_periodic_mesh(coordinates, mesh.comm, distribution_parameters, @@ -3546,7 +3544,6 @@ def Mesh(meshfile, **kwargs): cg = netgen_flags.get("cg", None) coordinates = mesh.curve_field( order=degree, - permutation_tol=permutation_tol, cg_field=cg, ) # Do not redistribute the mesh From 6a46b633a53bfd6f646bdb3fcc5bc00045ec124d Mon Sep 17 00:00:00 2001 From: ThighamOxford Date: Mon, 3 Aug 2026 13:16:24 +0100 Subject: [PATCH 06/13] Written a periodic_meshes demo for Firedrake and placed in introductory tutorials. This takes some of the demo from netgen_meshes to make it less lengthy. --- demos/periodic_meshes/Example1.png | Bin 0 -> 23187 bytes demos/periodic_meshes/Example2.png | Bin 0 -> 477115 bytes demos/periodic_meshes/Example3.png | Bin 0 -> 548855 bytes demos/periodic_meshes/Example4.png | Bin 0 -> 534936 bytes demos/periodic_meshes/Example5.png | Bin 0 -> 505494 bytes demos/periodic_meshes/Example6.png | Bin 0 -> 602914 bytes demos/periodic_meshes/periodic_meshes.py.rst | 430 +++++++++++++++++++ docs/source/intro_tut.rst | 1 + 8 files changed, 431 insertions(+) create mode 100644 demos/periodic_meshes/Example1.png create mode 100644 demos/periodic_meshes/Example2.png create mode 100644 demos/periodic_meshes/Example3.png create mode 100644 demos/periodic_meshes/Example4.png create mode 100644 demos/periodic_meshes/Example5.png create mode 100644 demos/periodic_meshes/Example6.png create mode 100644 demos/periodic_meshes/periodic_meshes.py.rst diff --git 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of this demo is to summarise the support for periodic meshes in Firedrake. +Firedrake can build periodic meshes for simple one- and two-dimensional geometries and can also periodically extrude two-dimensional meshes into three-dimensional prism meshes. +Firedrake also has support for periodic tetrahedral meshes generated in Netgen. + +1D Periodic Poisson Problem +--------------------------- +We solve the 1D periodic Poisson problem + +.. math:: + + - \Delta u = \sin(x), \quad u(0) = u(2 \pi), + +on :math:`\Omega = [0, 2 \pi]`. This problem has a trivial nullspace of constants; if we fix the constant to be zero then this problem has an analytical solution of :math:`u(x) = \sin(x)`. +We use the Firedrake function ``PeriodicIntervalMesh`` to build the mesh and implement the usual weak formulation of the Poisson equation: :: + + from firedrake import * + from firedrake.petsc import PETSc + from math import pi + + mesh = PeriodicIntervalMesh(32, 2*pi) + x, = SpatialCoordinate(mesh) + + V = FunctionSpace(mesh, "CG", 1) + u = TrialFunction(V); v = TestFunction(V) + + uh = Function(V, name="Numerical"); uex = Function(V, name="Exact") + uex.interpolate(sin(x)) + + a = dot(grad(u), grad(v)) * dx + L = sin(x) * v * dx + + problem = LinearVariationalProblem(a, L, uh) + +We use the ``VectorSpaceBasis`` function to tell PETSc that the problem has a +constant nullspace:: + + nullspace = VectorSpaceBasis(constant=True) + + solver = LinearVariationalSolver( + problem, + nullspace=nullspace, + transpose_nullspace=nullspace, + ) + + solver.solve() + + l2 = errornorm(uex, uh, norm_type="L2") + PETSc.Sys.Print(f"L2 error = {l2:.16e}") + +We plot the solution below. + + +.. figure:: Example1.png + :align: center + :alt: Finite element solution to 1D periodic Poisson problem. + + Finite element solution to 1D periodic Poisson problem. + +2D Periodic Poisson Problem +--------------------------- + +Now we solve the 2D :math:`x`-periodic Poisson problem + +.. math:: + + - \Delta u = (1 + \pi^2) \sin(x) \sin(\pi y), \quad u(0,y) = u(2 \pi, y), \quad u(x, 0) = u(x, 1) = 0, + +on :math:`\Omega = [0, 2 \pi] \times [0,1]`. The nullspace is fixed for this problem. The exact solution is :math:`u(x,y) = \sin(x)\sin(\pi y)`. +We use the Firedrake function ``PeriodicRectangleMesh`` to build the mesh: if you call this function without calling the argument ``PeriodicRectangleMesh`` then the rectangle will be periodic in both :math:`x` and :math:`y`. +In our case we only want periodicity in :math:`x`. :: + + from firedrake import * + from firedrake.petsc import PETSc + from math import pi + + nx = 32 + ny = 16 + + mesh = PeriodicRectangleMesh(nx, ny, 2*pi, 1.0, direction="x") + x, y = SpatialCoordinate(mesh) + + V = FunctionSpace(mesh, "CG", 1) + u = TrialFunction(V); v = TestFunction(V) + uh = Function(V, name="Numerical"); u_exact = Function(V, name="Exact") + u_exact.interpolate(sin(x)*sin(pi*y)) + + f = (1 + pi**2)*sin(x)*sin(pi*y) + a = inner(grad(u), grad(v))*dx + L = f*v*dx + + # Homogeneous Dirichlet BCs on y = 0 and y = 1. + bcs = [ + DirichletBC(V, 0.0, 3), + DirichletBC(V, 0.0, 4), + ] + + solve(a == L, uh, bcs=bcs) + + l2 = errornorm(u_exact, uh, norm_type="L2") + PETSc.Sys.Print(f"L2 error = {l2:.3e}") + +We plot the solution below. + + +.. figure:: Example2.png + :align: center + :alt: Finite element solution to 2D periodic Poisson problem. + + Finite element solution to 2D periodic Poisson problem. + +3D Periodic Poisson Problem +--------------------------- + +Now we solve the 3D :math:`z`-periodic Poisson problem + +.. math:: + + - \Delta u = (5 - x^2 - y^2)\sin(z), \quad u(x,y,0) = u(x, y, 2 \pi), \quad u\vert_{\text{walls}} = 0, + +on the domain + +.. math:: + + \Omega = + \left\{ + (x,y,z)\in\mathbb{R}^3 : + x^2+y^2\le 1,\; + 0\le z\le 2\pi + \right\}. + +The exact solution is +:math:`u(x,y,z) = (1-x^2-y^2)\sin(z)`. +In the case of cube or cuboid 3D periodic domains, Firedrake is able to make tetrahedral periodic meshes using the commands ``PeriodicUnitCubeMesh`` and ``PeriodicBoxMesh`` respectively. +For this problem we need a periodic cylinder: we first build a 2D mesh for the desired cross section of the cylinder and then we use the command ``ExtrudedMesh`` with flag ``periodic=True`` to generate a mesh of prisms. :: + + from firedrake import * + from firedrake.petsc import PETSc + from math import pi + + Refinement = 3 + base = UnitDiskMesh(Refinement) + + mesh = ExtrudedMesh( + base, + layers=32, + layer_height=2*pi/32, + periodic=True, + ) + +The two important arguments are ``layers`` and ``layer_height``, which tell Firedrake how far to extrude the mesh. By default ``layer_height`` is 1/``layers``. :: + + x, y, z = SpatialCoordinate(mesh) + + V = FunctionSpace(mesh, "CG", 1) + + u = TrialFunction(V); v = TestFunction(V) + + uh = Function(V, name="Numerical"); u_exact = Function(V, name="Exact") + u_exact.interpolate((1 - x*x - y*y)*sin(z)) + + f = (5 - x*x - y*y)*sin(z) + + a = inner(grad(u), grad(v))*dx + L = f*v*dx + + bc = DirichletBC(V, 0.0, "on_boundary") + + solve( + a == L, + uh, + bcs=bc, + ) + + PETSc.Sys.Print(f"L2 error = {errornorm(u_exact, uh, norm_type='L2'):.3e}") + +We plot the solution below. + + +.. figure:: Example3.png + :align: center + :alt: Cross-section of the finite element solution to a 3D periodic Poisson problem. + + Cross-section of the finite element solution to a 3D periodic Poisson problem using prismatic elements. + +Periodic Meshes From Netgen +--------------------------- + +Netgen can identify pairs of vertices lying on opposite boundaries of a geometry as being *the same* point. +When such a mesh is imported into Firedrake, the identified vertices are merged in the mesh topology, so that +a continuous (CG) function space automatically shares its degrees of freedom across the seam: the mesh is +genuinely **periodic**. This is exactly the representation Firedrake uses for its built-in ``PeriodicBoxMesh``, and it is now available for any Netgen geometry carrying +periodic identifications. + +Identifications are declared on the geometry, before meshing, with the OCC +``Identify`` method:: + + shape_a.Identify( + shape_b, + name, + IdentificationType.PERIODIC, + transformation, + ) + +where ``transformation`` is the rigid motion (typically a translation) that maps ``shape_a`` onto ``shape_b``. +Netgen then meshes the two boundaries compatibly and records the vertex pairs; Firedrake consumes them +automatically -- no extra flag on the ``Mesh`` constructor is required. + +As a physically motivated example we build the *periodic cylinder*, the classic reduced ("screw pinch") model +of a tokamak plasma column. A tokamak is a torus, so the plasma is periodic in the toroidal direction; in the +large-aspect-ratio limit one straightens a toroidal section into a cylinder and identifies its two circular +ends, recovering periodicity along the axis. We take the axial (toroidal) coordinate to run over :math:`[0, 2\pi)` +and identify the two end caps by a translation of :math:`2\pi` along ``Z``:: + + from netgen.occ import Cylinder, OCCGeometry, Pnt, Z, gp_Trsf, gp_Vec + from netgen.meshing import IdentificationType + from math import pi as PI + + cyl = Cylinder(Pnt(0, 0, 0), Z, r=1.0, h=2 * PI) + # Label the lateral wall, then the two end caps that we will identify. + for face in cyl.faces: + face.name = "wall" + cyl.faces.Min(Z).name = "bottom" + cyl.faces.Max(Z).name = "top" + # Identify the bottom cap with the top cap: a translation of 2*pi along Z + # maps one onto the other, making the axial direction periodic. + cyl.faces.Min(Z).Identify(cyl.faces.Max(Z), "toroidal", + IdentificationType.PERIODIC, + gp_Trsf.Translation(gp_Vec(0, 0, 2 * PI))) + ngmsh = OCCGeometry(cyl).GenerateMesh(maxh=0.4) + msh = Mesh(ngmsh) + +.. warning:: + + The mesh must contain at least a handful of cells along each periodic direction. If a single cell spans a + whole period, its two ends are identified and the cell collapses; Firedrake then raises a ``ValueError`` + asking you to refine along the periodic direction. Here the axis has length :math:`2\pi` and ``maxh=0.4`` + gives roughly sixteen cells along it, which is ample. Only ``degree == 1`` periodic meshes are supported + for now. + +Because the two end caps have been identified, no boundary markers survive on them: the seam has become an +*interior* set of facets, and the only labelled boundary that remains is the lateral wall. This is what makes +a continuous field wrap around continuously in the axial direction. We can verify the geometry survived the +merge intact -- the volume of the cylinder is :math:`\pi r^2 h = 2\pi^2` -- while the ends carry no exterior +facets:: + + volume = assemble(Constant(1.0) * dx(domain=msh)) + PETSc.Sys.Print(f"cylinder volume: {volume:.4f} (exact 2*pi**2 = {2 * PI**2:.4f})") + +To show that the periodicity is doing real work, we solve a Helmholtz problem whose exact solution is periodic +in the axial coordinate and vanishes on the lateral wall, + +.. math:: + + u_{\text{ex}}(x, y, z) = \cos(z)\,\bigl(1 - x^2 - y^2\bigr), + +so that we can impose a homogeneous Dirichlet condition on the wall while relying on the identified ends for +continuity along the axis. We look up the id of the ``"wall"`` boundary with ``GetRegionNames`` (as in the +Poisson example above) and manufacture the right-hand side :math:`f = u_{\text{ex}} - \Delta u_{\text{ex}}` for +:math:`(I - \Delta)u = f`:: + + V = FunctionSpace(msh, "CG", 2) + x, y, z = SpatialCoordinate(msh) + uex = cos(z) * (1 - x**2 - y**2) + f = uex - div(grad(uex)) + + u = TrialFunction(V); v = TestFunction(V) + a = (inner(u, v) + inner(grad(u), grad(v))) * dx + L = inner(f, v) * dx + + labels = [i + 1 for i, name in enumerate(ngmsh.GetRegionNames(codim=1)) if name == "wall"] + bc = DirichletBC(V, 0, labels) + + sol = Function(V) + solve(a == L, sol, bcs=bc) + + error = sqrt(assemble(inner(sol - uex, sol - uex) * dx)) + PETSc.Sys.Print(f"L2 error: {error:.2e}") + +We plot the solution below. + + +.. figure:: Example4.png + :align: center + :alt: Finite element solution to 3D periodic Helmholtz problem using tetrahedral elements. + + Finite element solution to 3D periodic Helmholtz problem. + +We can also solve a Helmholtz problem on a realistic tokamak geometry, although we no longer have an analytical solution to test against. +We now work with the physical cylindrical coordinates :math:`(R,\phi,Z)`. +The cross-section is constructed in the :math:`(R,Z)` plane and extruded +through one period in :math:`\phi`:: + + from firedrake import * + from netgen.occ import ( + OCCGeometry, WorkPlane, Axes, Pnt, Z, X, + gp_Trsf, gp_Vec + ) + from netgen.meshing import IdentificationType + from math import pi as PI + import math + + # Geometry parameters - large aspect ratio tokamak + R0 = 3.0 + a = 1.0 + kappa = 2.0 + delta = 0.3 + + n_boundary = 40 + mesh_size = 0.5 + + # Build a tokamak cross section and extrude periodically + + alpha = math.asin(delta) + + boundary_points = [] + for i in range(n_boundary): + theta = 2.0 * math.pi * i / n_boundary + R = R0 + a * math.cos(theta + alpha * math.sin(theta)) + Zc = kappa * a * math.sin(theta) + boundary_points.append((R, Zc)) + + wp = WorkPlane(Axes((0, 0, 0), n=Z, h=X)) + + # Start at the first boundary point, then draw a closed polyline + R0p, Z0p = boundary_points[0] + wp.MoveTo(R0p, Z0p) + for R, Zc in boundary_points[1:]: + wp.LineTo(R, Zc) + wp.Close() + + face = wp.Face() + + # Extrude in periodic phi direction and identify end caps. + + phi_length = 2 * PI + solid = face.Extrude(phi_length * Z) + + # Side wall(s) + for f in solid.faces: + f.name = "wall" + + bottom = solid.faces.Min(Z) + top = solid.faces.Max(Z) + bottom.name = "bottom" + top.name = "top" + + # Periodic identification of the end caps + bottom.Identify( + top, + "periodic_phi", + IdentificationType.PERIODIC, + gp_Trsf.Translation(gp_Vec(0, 0, phi_length)), + ) + + # Mesh and convert to Firedrake + ngmsh = OCCGeometry(solid).GenerateMesh(maxh=mesh_size) + msh = Mesh(ngmsh, name="TokamakPeriodic") + +The physical cylindrical coordinate convention is :math:`(R,\phi,Z)`. +However, the Netgen mesh stores its coordinate components in the order +:math:`(R,Z,\phi)`: the first two components are the coordinates of the +cross-section and the third is the extrusion coordinate. We therefore unpack +``SpatialCoordinate`` as ``R, Zc, phi``. + +We solve the Helmholtz problem :math:`(I-\Delta)u=f` with source term +:math:`f(R,\phi,Z)=RZ\cos(\phi)`:: + + V = FunctionSpace(msh, "CG", 2) + R, Zc, phi = SpatialCoordinate(msh) + + f = R*Zc*cos(phi) + + u = TrialFunction(V) + v = TestFunction(V) + a_form = (inner(u, v) + inner(grad(u), grad(v))) * dx + L = inner(f, v) * dx + + labels = [ + i + 1 + for i, name in enumerate(ngmsh.GetRegionNames(codim=1)) + if name == "wall" + ] + bc = DirichletBC(V, 0, labels) + + sol = Function(V, name="Solution") + solve(a_form == L, sol, bcs=bc) + +We plot the output in :math:`(R, \phi, Z)` cylindrical coordinates: + + +.. figure:: Example5.png + :align: center + :width: 85% + :alt: Finite element solution to 3D periodic Helmholtz problem in cylindrical tokamak geometry with tetrahedral elements. + + Finite element solution to 3D periodic Helmholtz problem in cylindrical tokamak geometry. + +Since the solution is represented as a function on the mesh, we can obtain a +Cartesian visualisation by interpolating the cylindrical-to-Cartesian map + +.. math:: + + (R,\phi,Z) \longmapsto + \bigl(R\cos(\phi),R\sin(\phi),Z\bigr) + +into the mesh coordinate field. The mesh components must again be unpacked in +their stored order, :math:`(R,Z,\phi)`:: + + R, Zc, phi = SpatialCoordinate(msh) + msh.coordinates.interpolate(as_vector(( + R*cos(phi), + R*sin(phi), + Zc, + ))) + VTKFile("output/TokamakCartesianSolution.pvd").write(sol) + +We plot a cross-section of the solution in Cartesian coordinates: + +.. figure:: Example6.png + :align: center + :width: 95% + :alt: Finite element solution to 3D periodic Helmholtz problem in Cartesian tokamak geometry. + + Finite element solution to 3D periodic Helmholtz problem in Cartesian tokamak geometry with tetrahedral elements. + diff --git a/docs/source/intro_tut.rst b/docs/source/intro_tut.rst index 93b3d3afeb..fe9b46e3b8 100644 --- a/docs/source/intro_tut.rst +++ b/docs/source/intro_tut.rst @@ -17,3 +17,4 @@ Introductory Tutorials A linear shallow water equations example using a Strang timestepping scheme. A linear wave equation with optional mass lumping. Creating Firedrake-compatible meshes in Gmsh. + Creating periodic meshes. From 3426bd6a4e39ec353513b58c903fb60f6781dd9f Mon Sep 17 00:00:00 2001 From: ThighamOxford Date: Mon, 3 Aug 2026 13:26:18 +0100 Subject: [PATCH 07/13] Added periodic_meshes to test_demos_run.py --- tests/firedrake/demos/test_demos_run.py | 1 + 1 file changed, 1 insertion(+) diff --git a/tests/firedrake/demos/test_demos_run.py b/tests/firedrake/demos/test_demos_run.py index f53a611c2b..0b1a9321b2 100644 --- a/tests/firedrake/demos/test_demos_run.py +++ b/tests/firedrake/demos/test_demos_run.py @@ -43,6 +43,7 @@ Demo(("netgen", "netgen_mesh"), ["mumps", "netgen", "slepc", "vtk"]), Demo(("nonlinear_QG_winddrivengyre", "qg_winddrivengyre"), ["vtk"]), Demo(("parallel-printing", "parprint"), []), + Demo(("periodic_meshes", "periodic_meshes"), ["netgen", "vtk"]), Demo(("poisson", "poisson_mixed"), ["vtk"]), Demo(("patch", "poisson_mg_patches"), []), Demo(("patch", "stokes_vanka_patches"), []), From 68bf2bab81ada4137cec9013a7d41c5b625d5974 Mon Sep 17 00:00:00 2001 From: ThighamOxford Date: Mon, 3 Aug 2026 15:15:52 +0100 Subject: [PATCH 08/13] Fixed code block that was not intedned to be run in test --- demos/periodic_meshes/periodic_meshes.py.rst | 18 ++++++++++-------- 1 file changed, 10 insertions(+), 8 deletions(-) diff --git a/demos/periodic_meshes/periodic_meshes.py.rst b/demos/periodic_meshes/periodic_meshes.py.rst index 03a4647137..f51826fef1 100644 --- a/demos/periodic_meshes/periodic_meshes.py.rst +++ b/demos/periodic_meshes/periodic_meshes.py.rst @@ -197,14 +197,16 @@ genuinely **periodic**. This is exactly the representation Firedrake uses for it periodic identifications. Identifications are declared on the geometry, before meshing, with the OCC -``Identify`` method:: +``Identify`` method: - shape_a.Identify( - shape_b, - name, - IdentificationType.PERIODIC, - transformation, - ) +.. code-block:: python + + shape_a.Identify( + shape_b, + name, + IdentificationType.PERIODIC, + transformation, + ) where ``transformation`` is the rigid motion (typically a translation) that maps ``shape_a`` onto ``shape_b``. Netgen then meshes the two boundaries compatibly and records the vertex pairs; Firedrake consumes them @@ -417,7 +419,7 @@ their stored order, :math:`(R,Z,\phi)`:: R*sin(phi), Zc, ))) - VTKFile("output/TokamakCartesianSolution.pvd").write(sol) + VTKFile("TokamakCartesianSolution.pvd").write(sol) We plot a cross-section of the solution in Cartesian coordinates: From 0ffaaa3575785f728e4f928e45a505931af19b3b Mon Sep 17 00:00:00 2001 From: ThighamOxford Date: Mon, 3 Aug 2026 18:55:07 +0100 Subject: [PATCH 09/13] I have edited the demo to match the Firedrake code style, and I have updated figures. --- demos/periodic_meshes/Example2.png | Bin 477115 -> 374602 bytes demos/periodic_meshes/Example3.png | Bin 548855 -> 317405 bytes demos/periodic_meshes/Example4.png | Bin 534936 -> 459297 bytes demos/periodic_meshes/Example5.png | Bin 505494 -> 441200 bytes demos/periodic_meshes/Example6.png | Bin 602914 -> 621626 bytes demos/periodic_meshes/periodic_meshes.py.rst | 180 +++++++++++-------- 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geometries and can also periodically extrude two-dimensional meshes into three-dimensional prism meshes. Firedrake also has support for periodic tetrahedral meshes generated in Netgen. +We begin by importing the necessary libraries: :: + + from firedrake import * + 1D Periodic Poisson Problem --------------------------- We solve the 1D periodic Poisson problem @@ -18,28 +22,28 @@ We solve the 1D periodic Poisson problem on :math:`\Omega = [0, 2 \pi]`. This problem has a trivial nullspace of constants; if we fix the constant to be zero then this problem has an analytical solution of :math:`u(x) = \sin(x)`. We use the Firedrake function ``PeriodicIntervalMesh`` to build the mesh and implement the usual weak formulation of the Poisson equation: :: - from firedrake import * - from firedrake.petsc import PETSc - from math import pi - mesh = PeriodicIntervalMesh(32, 2*pi) x, = SpatialCoordinate(mesh) V = FunctionSpace(mesh, "CG", 1) - u = TrialFunction(V); v = TestFunction(V) + u = TrialFunction(V) + v = TestFunction(V) - uh = Function(V, name="Numerical"); uex = Function(V, name="Exact") - uex.interpolate(sin(x)) + uh = Function(V, name="Numerical") + u_exact = sin(x) a = dot(grad(u), grad(v)) * dx - L = sin(x) * v * dx + L = inner(sin(x), v) * dx problem = LinearVariationalProblem(a, L, uh) We use the ``VectorSpaceBasis`` function to tell PETSc that the problem has a constant nullspace:: - nullspace = VectorSpaceBasis(constant=True) + nullspace = VectorSpaceBasis( + constant=True, + comm=mesh.comm, + ) solver = LinearVariationalSolver( problem, @@ -49,8 +53,7 @@ constant nullspace:: solver.solve() - l2 = errornorm(uex, uh, norm_type="L2") - PETSc.Sys.Print(f"L2 error = {l2:.16e}") + print(f"L2 error = {errornorm(u_exact, uh, norm_type='L2'):.3e}") We plot the solution below. @@ -64,7 +67,8 @@ We plot the solution below. 2D Periodic Poisson Problem --------------------------- -Now we solve the 2D :math:`x`-periodic Poisson problem +In two dimensions, we can choose which boundaries are periodic. We consider +an :math:`x`-periodic Poisson problem, .. math:: @@ -74,10 +78,6 @@ on :math:`\Omega = [0, 2 \pi] \times [0,1]`. The nullspace is fixed for this pro We use the Firedrake function ``PeriodicRectangleMesh`` to build the mesh: if you call this function without calling the argument ``PeriodicRectangleMesh`` then the rectangle will be periodic in both :math:`x` and :math:`y`. In our case we only want periodicity in :math:`x`. :: - from firedrake import * - from firedrake.petsc import PETSc - from math import pi - nx = 32 ny = 16 @@ -85,24 +85,23 @@ In our case we only want periodicity in :math:`x`. :: x, y = SpatialCoordinate(mesh) V = FunctionSpace(mesh, "CG", 1) - u = TrialFunction(V); v = TestFunction(V) - uh = Function(V, name="Numerical"); u_exact = Function(V, name="Exact") - u_exact.interpolate(sin(x)*sin(pi*y)) + u = TrialFunction(V) + v = TestFunction(V) + uh = Function(V, name="Numerical") + u_exact = sin(x) * sin(pi*y) f = (1 + pi**2)*sin(x)*sin(pi*y) a = inner(grad(u), grad(v))*dx L = f*v*dx # Homogeneous Dirichlet BCs on y = 0 and y = 1. - bcs = [ - DirichletBC(V, 0.0, 3), - DirichletBC(V, 0.0, 4), - ] + bcs = DirichletBC(V, 0.0, (3, 4)) solve(a == L, uh, bcs=bcs) - l2 = errornorm(u_exact, uh, norm_type="L2") - PETSc.Sys.Print(f"L2 error = {l2:.3e}") + print(f"L2 error = {errornorm(u_exact, uh, norm_type='L2'):.3e}") + VTKFile("output/rectangle_poisson.pvd").write(uh) + We plot the solution below. @@ -138,17 +137,14 @@ The exact solution is In the case of cube or cuboid 3D periodic domains, Firedrake is able to make tetrahedral periodic meshes using the commands ``PeriodicUnitCubeMesh`` and ``PeriodicBoxMesh`` respectively. For this problem we need a periodic cylinder: we first build a 2D mesh for the desired cross section of the cylinder and then we use the command ``ExtrudedMesh`` with flag ``periodic=True`` to generate a mesh of prisms. :: - from firedrake import * - from firedrake.petsc import PETSc - from math import pi - - Refinement = 3 - base = UnitDiskMesh(Refinement) + R = 2.0 + refinements = 2 + base = UnitDiskMesh(refinements) mesh = ExtrudedMesh( base, layers=32, - layer_height=2*pi/32, + layer_height=2*pi*R/32, periodic=True, ) @@ -158,15 +154,16 @@ The two important arguments are ``layers`` and ``layer_height``, which tell Fire V = FunctionSpace(mesh, "CG", 1) - u = TrialFunction(V); v = TestFunction(V) + u = TrialFunction(V) + v = TestFunction(V) - uh = Function(V, name="Numerical"); u_exact = Function(V, name="Exact") - u_exact.interpolate((1 - x*x - y*y)*sin(z)) + uh = Function(V, name="Numerical") + u_exact = (1 - x*x - y*y) * sin(z) - f = (5 - x*x - y*y)*sin(z) + f = (5 - x*x - y*y) * sin(z) a = inner(grad(u), grad(v))*dx - L = f*v*dx + L = inner(f, v) * dx bc = DirichletBC(V, 0.0, "on_boundary") @@ -176,7 +173,9 @@ The two important arguments are ``layers`` and ``layer_height``, which tell Fire bcs=bc, ) - PETSc.Sys.Print(f"L2 error = {errornorm(u_exact, uh, norm_type='L2'):.3e}") + print(f"L2 error = {errornorm(u_exact, uh, norm_type='L2'):.3e}") + VTKFile("output/cylinder_poisson.pvd").write(uh) + We plot the solution below. @@ -215,14 +214,18 @@ automatically -- no extra flag on the ``Mesh`` constructor is required. As a physically motivated example we build the *periodic cylinder*, the classic reduced ("screw pinch") model of a tokamak plasma column. A tokamak is a torus, so the plasma is periodic in the toroidal direction; in the large-aspect-ratio limit one straightens a toroidal section into a cylinder and identifies its two circular -ends, recovering periodicity along the axis. We take the axial (toroidal) coordinate to run over :math:`[0, 2\pi)` -and identify the two end caps by a translation of :math:`2\pi` along ``Z``:: +ends, recovering periodicity along the axis. + +We take the axial (toroidal) coordinate to run over :math:`[0, 2\pi)` +and identify the two end caps by a translation of :math:`2\pi R_0` along ``Z``. +:math:`R_0` is called the major radius and ensures that the torus that the cylinder defines is the correct length. :: from netgen.occ import Cylinder, OCCGeometry, Pnt, Z, gp_Trsf, gp_Vec from netgen.meshing import IdentificationType - from math import pi as PI - cyl = Cylinder(Pnt(0, 0, 0), Z, r=1.0, h=2 * PI) + R0 = 1 + height = 2 * pi * R0 + cyl = Cylinder(Pnt(0, 0, 0), Z, r=1.0, h=height) # Label the lateral wall, then the two end caps that we will identify. for face in cyl.faces: face.name = "wall" @@ -232,7 +235,7 @@ and identify the two end caps by a translation of :math:`2\pi` along ``Z``:: # maps one onto the other, making the axial direction periodic. cyl.faces.Min(Z).Identify(cyl.faces.Max(Z), "toroidal", IdentificationType.PERIODIC, - gp_Trsf.Translation(gp_Vec(0, 0, 2 * PI))) + gp_Trsf.Translation(gp_Vec(0, 0, height))) ngmsh = OCCGeometry(cyl).GenerateMesh(maxh=0.4) msh = Mesh(ngmsh) @@ -247,18 +250,21 @@ and identify the two end caps by a translation of :math:`2\pi` along ``Z``:: Because the two end caps have been identified, no boundary markers survive on them: the seam has become an *interior* set of facets, and the only labelled boundary that remains is the lateral wall. This is what makes a continuous field wrap around continuously in the axial direction. We can verify the geometry survived the -merge intact -- the volume of the cylinder is :math:`\pi r^2 h = 2\pi^2` -- while the ends carry no exterior +merge intact -- the volume of the cylinder is :math:`\pi r^2 h = 2\pi^2 R0` -- while the ends carry no exterior facets:: volume = assemble(Constant(1.0) * dx(domain=msh)) - PETSc.Sys.Print(f"cylinder volume: {volume:.4f} (exact 2*pi**2 = {2 * PI**2:.4f})") + print( + f"cylinder volume: {volume:.4f} " + f"(exact 2*pi**2*R0 = {2*pi**2*R0:.4f})" + ) To show that the periodicity is doing real work, we solve a Helmholtz problem whose exact solution is periodic in the axial coordinate and vanishes on the lateral wall, .. math:: - u_{\text{ex}}(x, y, z) = \cos(z)\,\bigl(1 - x^2 - y^2\bigr), + u_{\text{ex}}(x, y, z) = \cos(z/R_0) \bigl(1 - x^2 - y^2\bigr), so that we can impose a homogeneous Dirichlet condition on the wall while relying on the identified ends for continuity along the axis. We look up the id of the ``"wall"`` boundary with ``GetRegionNames`` (as in the @@ -267,23 +273,25 @@ Poisson example above) and manufacture the right-hand side :math:`f = u_{\text{e V = FunctionSpace(msh, "CG", 2) x, y, z = SpatialCoordinate(msh) - uex = cos(z) * (1 - x**2 - y**2) - f = uex - div(grad(uex)) + u_exact = cos(z/R0) * (1 - x**2 - y**2) + f = u_exact - div(grad(u_exact)) - u = TrialFunction(V); v = TestFunction(V) + u = TrialFunction(V) + v = TestFunction(V) a = (inner(u, v) + inner(grad(u), grad(v))) * dx L = inner(f, v) * dx labels = [i + 1 for i, name in enumerate(ngmsh.GetRegionNames(codim=1)) if name == "wall"] bc = DirichletBC(V, 0, labels) - sol = Function(V) - solve(a == L, sol, bcs=bc) + uh = Function(V) + solve(a == L, uh, bcs=bc) - error = sqrt(assemble(inner(sol - uex, sol - uex) * dx)) - PETSc.Sys.Print(f"L2 error: {error:.2e}") + print(f"L2 error = {errornorm(u_exact, uh, norm_type='L2'):.3e}") + VTKFile("output/cylinder_helmholtz.pvd").write(uh) -We plot the solution below. + +We plot the solution below with a "crinkle cut" cross-section to inspect the solution. .. figure:: Example4.png @@ -292,18 +300,27 @@ We plot the solution below. Finite element solution to 3D periodic Helmholtz problem. +Netgen Tokamak Example +--------------------------- + + We can also solve a Helmholtz problem on a realistic tokamak geometry, although we no longer have an analytical solution to test against. We now work with the physical cylindrical coordinates :math:`(R,\phi,Z)`. The cross-section is constructed in the :math:`(R,Z)` plane and extruded -through one period in :math:`\phi`:: +through one period in :math:`\phi`. +We parametrize the tokamak cross-section using the formula + +.. math:: + + R(\theta) = R0 + a \cos(\theta + \sin^{-1}(\delta) \sin(\theta)), \quad Z(\theta) = a \kappa \sin(\theta). + +where :math:`\delta` is the triangularity, :math:`\kappa` is the elongation, :math:`a` is the tokamak minor radius, and :math:`\theta \in [0, 2\pi]`. :: - from firedrake import * from netgen.occ import ( OCCGeometry, WorkPlane, Axes, Pnt, Z, X, gp_Trsf, gp_Vec ) from netgen.meshing import IdentificationType - from math import pi as PI import math # Geometry parameters - large aspect ratio tokamak @@ -313,7 +330,7 @@ through one period in :math:`\phi`:: delta = 0.3 n_boundary = 40 - mesh_size = 0.5 + maxh = 0.5 # Build a tokamak cross section and extrude periodically @@ -321,7 +338,7 @@ through one period in :math:`\phi`:: boundary_points = [] for i in range(n_boundary): - theta = 2.0 * math.pi * i / n_boundary + theta = 2.0 * pi * i / n_boundary R = R0 + a * math.cos(theta + alpha * math.sin(theta)) Zc = kappa * a * math.sin(theta) boundary_points.append((R, Zc)) @@ -339,8 +356,8 @@ through one period in :math:`\phi`:: # Extrude in periodic phi direction and identify end caps. - phi_length = 2 * PI - solid = face.Extrude(phi_length * Z) + height = 2 * pi * R0 + solid = face.Extrude(height * Z) # Side wall(s) for f in solid.faces: @@ -356,11 +373,11 @@ through one period in :math:`\phi`:: top, "periodic_phi", IdentificationType.PERIODIC, - gp_Trsf.Translation(gp_Vec(0, 0, phi_length)), + gp_Trsf.Translation(gp_Vec(0, 0, height)), ) # Mesh and convert to Firedrake - ngmsh = OCCGeometry(solid).GenerateMesh(maxh=mesh_size) + ngmsh = OCCGeometry(solid).GenerateMesh(maxh=maxh) msh = Mesh(ngmsh, name="TokamakPeriodic") The physical cylindrical coordinate convention is :math:`(R,\phi,Z)`. @@ -369,17 +386,23 @@ However, the Netgen mesh stores its coordinate components in the order cross-section and the third is the extrusion coordinate. We therefore unpack ``SpatialCoordinate`` as ``R, Zc, phi``. -We solve the Helmholtz problem :math:`(I-\Delta)u=f` with source term -:math:`f(R,\phi,Z)=RZ\cos(\phi)`:: +We solve the Helmholtz problem + +.. math:: + + (I-\Delta)u=f + +with source term +:math:`f(R,\phi,Z)=RZ\cos(\phi)`. :: V = FunctionSpace(msh, "CG", 2) R, Zc, phi = SpatialCoordinate(msh) - f = R*Zc*cos(phi) + f = R * Zc * cos(phi) u = TrialFunction(V) v = TestFunction(V) - a_form = (inner(u, v) + inner(grad(u), grad(v))) * dx + a = (inner(u, v) + inner(grad(u), grad(v))) * dx L = inner(f, v) * dx labels = [ @@ -389,10 +412,11 @@ We solve the Helmholtz problem :math:`(I-\Delta)u=f` with source term ] bc = DirichletBC(V, 0, labels) - sol = Function(V, name="Solution") - solve(a_form == L, sol, bcs=bc) + uh = Function(V, name="Solution") + solve(a == L, uh, bcs=bc) + VTKFile("output/tokamak_helmholtz.pvd").write(uh) -We plot the output in :math:`(R, \phi, Z)` cylindrical coordinates: +We plot the output in :math:`(R, \phi, Z)`, a cylindrical representation: .. figure:: Example5.png @@ -408,25 +432,25 @@ Cartesian visualisation by interpolating the cylindrical-to-Cartesian map .. math:: (R,\phi,Z) \longmapsto - \bigl(R\cos(\phi),R\sin(\phi),Z\bigr) + \bigl( R\cos(\phi / R_0), R\sin(\phi / R_0), Z \bigr) into the mesh coordinate field. The mesh components must again be unpacked in their stored order, :math:`(R,Z,\phi)`:: - R, Zc, phi = SpatialCoordinate(msh) + R, Zc, phiR0 = SpatialCoordinate(msh) msh.coordinates.interpolate(as_vector(( - R*cos(phi), - R*sin(phi), + R*cos(phiR0/R0), + R*sin(phiR0/R0), Zc, ))) - VTKFile("TokamakCartesianSolution.pvd").write(sol) + VTKFile("TokamakCartesianSolution.pvd").write(uh) -We plot a cross-section of the solution in Cartesian coordinates: +We plot a cross-section "crinkle cut" of the solution in Cartesian coordinates: .. figure:: Example6.png :align: center - :width: 95% + :width: 105% :alt: Finite element solution to 3D periodic Helmholtz problem in Cartesian tokamak geometry. - Finite element solution to 3D periodic Helmholtz problem in Cartesian tokamak geometry with tetrahedral elements. + Cross-section of a finite element solution to a 3D periodic Helmholtz problem in Cartesian tokamak geometry with tetrahedral elements. diff --git a/firedrake/mesh.py b/firedrake/mesh.py index b2034f6634..a3bf059fdc 100644 --- a/firedrake/mesh.py +++ b/firedrake/mesh.py @@ -3527,7 +3527,6 @@ def Mesh(meshfile, **kwargs): "High-order curving of periodic netgen meshes is not supported yet." ) from firedrake.utility_meshes import _postprocess_periodic_mesh - permutation_tol = netgen_flags.get("permutation_tol", None) coordinates = mesh._periodic_coordinates() temp = _postprocess_periodic_mesh(coordinates, mesh.comm, @@ -3540,7 +3539,6 @@ def Mesh(meshfile, **kwargs): temp.netgen_flags = mesh.netgen_flags mesh = temp elif degree != 1: - permutation_tol = netgen_flags.get("permutation_tol", None) cg = netgen_flags.get("cg", None) coordinates = mesh.curve_field( order=degree, diff --git a/tests/firedrake/regression/test_netgen.py b/tests/firedrake/regression/test_netgen.py index 68e9b0f4bf..d0a6e305d5 100644 --- a/tests/firedrake/regression/test_netgen.py +++ b/tests/firedrake/regression/test_netgen.py @@ -449,5 +449,3 @@ def test_netgen_periodic_too_coarse(): ngmesh = OCCGeometry(box).GenerateMesh(maxh=0.4) with pytest.raises(ValueError, match="degenerate"): Mesh(ngmesh) - - From 5410681673cf975d4305fefbc993ee80b49b7b2c Mon Sep 17 00:00:00 2001 From: ThighamOxford Date: Tue, 4 Aug 2026 11:55:46 +0100 Subject: [PATCH 10/13] I have simplified exposition by making examples 3 and 4 the same problem. 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b/demos/periodic_meshes/periodic_meshes.py.rst index 2ca1489bb9..607679fefa 100644 --- a/demos/periodic_meshes/periodic_meshes.py.rst +++ b/demos/periodic_meshes/periodic_meshes.py.rst @@ -112,16 +112,19 @@ We plot the solution below. Finite element solution to 2D periodic Poisson problem. -3D Periodic Poisson Problem ---------------------------- +3D Periodic Helmholtz Problem +----------------------------- + +For the simplest 3D shapes (cube and cuboid) Firedrake is able to generate tetrahedral periodic meshes using the commands ``PeriodicUnitCubeMesh`` and ``PeriodicBoxMesh`` respectively. +For cylindrical objects we have to construct a 2D cross-section and extrude, forming a mesh with prismatic elements. To construct any periodic mesh with tetrahedral elements we need to use Netgen, which is discussed in the next section. -Now we solve the 3D :math:`z`-periodic Poisson problem +For now we solve the 3D :math:`z`-periodic Helmholtz problem .. math:: - - \Delta u = (5 - x^2 - y^2)\sin(z), \quad u(x,y,0) = u(x, y, 2 \pi), \quad u\vert_{\text{walls}} = 0, + (I - \Delta) u = f , \quad u(x,y,0) = u(x, y, 2 \pi), \quad u\vert_{\text{walls}} = 0, -on the domain +on the cylindrical domain .. math:: @@ -132,19 +135,19 @@ on the domain 0\le z\le 2\pi \right\}. -The exact solution is -:math:`u(x,y,z) = (1-x^2-y^2)\sin(z)`. -In the case of cube or cuboid 3D periodic domains, Firedrake is able to make tetrahedral periodic meshes using the commands ``PeriodicUnitCubeMesh`` and ``PeriodicBoxMesh`` respectively. -For this problem we need a periodic cylinder: we first build a 2D mesh for the desired cross section of the cylinder and then we use the command ``ExtrudedMesh`` with flag ``periodic=True`` to generate a mesh of prisms. :: +We can manufacture a right-hand side by choosing the solution +:math:`u_{\text{exact}}(x,y,z) = (1-x^2-y^2)\cos(z)` which gives :math:`f = (I - \Delta) u_{\text{exact}}`. + + +We build a 2D mesh for the desired cross section of the cylinder and then we use the command ``ExtrudedMesh`` with flag ``periodic=True`` to generate a mesh of prisms. :: - R = 2.0 refinements = 2 base = UnitDiskMesh(refinements) mesh = ExtrudedMesh( base, layers=32, - layer_height=2*pi*R/32, + layer_height=2*pi/32, periodic=True, ) @@ -158,13 +161,16 @@ The two important arguments are ``layers`` and ``layer_height``, which tell Fire v = TestFunction(V) uh = Function(V, name="Numerical") - u_exact = (1 - x*x - y*y) * sin(z) + u_exact = (1 - x*x - y*y) * cos(z) - f = (5 - x*x - y*y) * sin(z) + f = u_exact - div(grad(u_exact)) - a = inner(grad(u), grad(v))*dx + u = TrialFunction(V) + v = TestFunction(V) + a = (inner(u, v) + inner(grad(u), grad(v))) * dx L = inner(f, v) * dx + bc = DirichletBC(V, 0.0, "on_boundary") solve( @@ -174,25 +180,26 @@ The two important arguments are ``layers`` and ``layer_height``, which tell Fire ) print(f"L2 error = {errornorm(u_exact, uh, norm_type='L2'):.3e}") - VTKFile("output/cylinder_poisson.pvd").write(uh) - + VTKFile("output/cylinder_fd_Helmholtz.pvd").write(uh) We plot the solution below. .. figure:: Example3.png :align: center - :alt: Cross-section of the finite element solution to a 3D periodic Poisson problem. + :alt: Cross-section of the finite element solution to a 3D periodic Helmholtz problem. - Cross-section of the finite element solution to a 3D periodic Poisson problem using prismatic elements. + Cross-section of the finite element solution to a 3D periodic Helmholtz problem using prismatic elements. Periodic Meshes From Netgen --------------------------- +Netgen can help us generate a periodic mesh of tetrahedral elements. + Netgen can identify pairs of vertices lying on opposite boundaries of a geometry as being *the same* point. When such a mesh is imported into Firedrake, the identified vertices are merged in the mesh topology, so that a continuous (CG) function space automatically shares its degrees of freedom across the seam: the mesh is -genuinely **periodic**. This is exactly the representation Firedrake uses for its built-in ``PeriodicBoxMesh``, and it is now available for any Netgen geometry carrying +genuinely **periodic**. This is exactly the representation Firedrake uses for its built-in ``PeriodicBoxMesh``, and it is also available for any Netgen geometry carrying periodic identifications. Identifications are declared on the geometry, before meshing, with the OCC @@ -211,28 +218,21 @@ where ``transformation`` is the rigid motion (typically a translation) that maps Netgen then meshes the two boundaries compatibly and records the vertex pairs; Firedrake consumes them automatically -- no extra flag on the ``Mesh`` constructor is required. -As a physically motivated example we build the *periodic cylinder*, the classic reduced ("screw pinch") model -of a tokamak plasma column. A tokamak is a torus, so the plasma is periodic in the toroidal direction; in the -large-aspect-ratio limit one straightens a toroidal section into a cylinder and identifies its two circular -ends, recovering periodicity along the axis. - -We take the axial (toroidal) coordinate to run over :math:`[0, 2\pi)` -and identify the two end caps by a translation of :math:`2\pi R_0` along ``Z``. -:math:`R_0` is called the major radius and ensures that the torus that the cylinder defines is the correct length. :: +To construct a periodic cylinder of length :math:`2\pi` we identify the two end caps of the cylinder by a translation of :math:`2\pi` along ``z``. :: from netgen.occ import Cylinder, OCCGeometry, Pnt, Z, gp_Trsf, gp_Vec from netgen.meshing import IdentificationType - R0 = 1 - height = 2 * pi * R0 + height = 2 * pi cyl = Cylinder(Pnt(0, 0, 0), Z, r=1.0, h=height) + # Label the lateral wall, then the two end caps that we will identify. for face in cyl.faces: face.name = "wall" cyl.faces.Min(Z).name = "bottom" cyl.faces.Max(Z).name = "top" + # Identify the bottom cap with the top cap: a translation of 2*pi along Z - # maps one onto the other, making the axial direction periodic. cyl.faces.Min(Z).Identify(cyl.faces.Max(Z), "toroidal", IdentificationType.PERIODIC, gp_Trsf.Translation(gp_Vec(0, 0, height))) @@ -249,27 +249,16 @@ and identify the two end caps by a translation of :math:`2\pi R_0` along ``Z``. Because the two end caps have been identified, no boundary markers survive on them: the seam has become an *interior* set of facets, and the only labelled boundary that remains is the lateral wall. This is what makes -a continuous field wrap around continuously in the axial direction. We can verify the geometry survived the -merge intact -- the volume of the cylinder is :math:`\pi r^2 h = 2\pi^2 R0` -- while the ends carry no exterior -facets:: +a continuous field wrap around continuously in the ``z``-direction. We can verify the geometry survived the +merge intact --- the volume of the cylinder is :math:`\pi r^2 h = 2\pi^2`: :: volume = assemble(Constant(1.0) * dx(domain=msh)) print( f"cylinder volume: {volume:.4f} " - f"(exact 2*pi**2*R0 = {2*pi**2*R0:.4f})" + f"(exact 2*pi**2 = {2*pi**2:.4f})" ) -To show that the periodicity is doing real work, we solve a Helmholtz problem whose exact solution is periodic -in the axial coordinate and vanishes on the lateral wall, - -.. math:: - - u_{\text{ex}}(x, y, z) = \cos(z/R_0) \bigl(1 - x^2 - y^2\bigr), - -so that we can impose a homogeneous Dirichlet condition on the wall while relying on the identified ends for -continuity along the axis. We look up the id of the ``"wall"`` boundary with ``GetRegionNames`` (as in the -Poisson example above) and manufacture the right-hand side :math:`f = u_{\text{ex}} - \Delta u_{\text{ex}}` for -:math:`(I - \Delta)u = f`:: +We solve again the same Helmholtz problem as in the previous section. :: V = FunctionSpace(msh, "CG", 2) x, y, z = SpatialCoordinate(msh) @@ -280,6 +269,8 @@ Poisson example above) and manufacture the right-hand side :math:`f = u_{\text{e v = TestFunction(V) a = (inner(u, v) + inner(grad(u), grad(v))) * dx L = inner(f, v) * dx + +We look up the id of the ``"wall"`` boundary with ``GetRegionNames`` :: labels = [i + 1 for i, name in enumerate(ngmsh.GetRegionNames(codim=1)) if name == "wall"] bc = DirichletBC(V, 0, labels) @@ -303,11 +294,17 @@ We plot the solution below with a "crinkle cut" cross-section to inspect the sol Netgen Tokamak Example --------------------------- +Helmholtz problems on periodic cylinders are used in simplified models of magnetic confinement fusion devices, where the straight cylinder acts as a large-aspect-ratio approximation of a torus. +To obtain a more realistic geometry, we construct the domain in cylindrical coordinates and choose a cross-section representative of a tokamak plasma. After solving we transform the solution to Cartesian coordiantes. + +We describe the tokamak in cylindrical coordinates :math:`(R,\phi,Z)` where :math:`\phi` is the toroidal angle. +The cross-section is constructed in the :math:`(R,Z)` plane. Rather than extruding by :math:`\phi` we extrude by the corresponding toroidal arc-length + +.. math:: + + s = R_0 \phi, -We can also solve a Helmholtz problem on a realistic tokamak geometry, although we no longer have an analytical solution to test against. -We now work with the physical cylindrical coordinates :math:`(R,\phi,Z)`. -The cross-section is constructed in the :math:`(R,Z)` plane and extruded -through one period in :math:`\phi`. +where :math:`R_0` is called the major radius. This makes one complete revolution of the tokamak have length :math:`2 \pi R_0`. We parametrize the tokamak cross-section using the formula .. math:: @@ -382,23 +379,23 @@ where :math:`\delta` is the triangularity, :math:`\kappa` is the elongation, :ma The physical cylindrical coordinate convention is :math:`(R,\phi,Z)`. However, the Netgen mesh stores its coordinate components in the order -:math:`(R,Z,\phi)`: the first two components are the coordinates of the -cross-section and the third is the extrusion coordinate. We therefore unpack -``SpatialCoordinate`` as ``R, Zc, phi``. +:math:`(R,Z,s)`: the first two components are the coordinates of the +cross-section and the third is the toroidal arc-length. We therefore unpack +``SpatialCoordinate`` as ``R, Zc, s``. We solve the Helmholtz problem .. math:: - (I-\Delta)u=f + (I-\Delta)u=f, \quad u(r, 0, z) = u(r, 2 \pi R_0, z), \quad u|_{\text{walls}} = 0, with source term -:math:`f(R,\phi,Z)=RZ\cos(\phi)`. :: +:math:`f(R,s,Z)=RZ\cos(s)`. :: V = FunctionSpace(msh, "CG", 2) - R, Zc, phi = SpatialCoordinate(msh) + R, Zc, s = SpatialCoordinate(msh) - f = R * Zc * cos(phi) + f = R * Zc * cos(s) u = TrialFunction(V) v = TestFunction(V) @@ -416,7 +413,7 @@ with source term solve(a == L, uh, bcs=bc) VTKFile("output/tokamak_helmholtz.pvd").write(uh) -We plot the output in :math:`(R, \phi, Z)`, a cylindrical representation: +We plot the output in :math:`(R, s, Z)`, a cylindrical representation: .. figure:: Example5.png @@ -431,16 +428,16 @@ Cartesian visualisation by interpolating the cylindrical-to-Cartesian map .. math:: - (R,\phi,Z) \longmapsto - \bigl( R\cos(\phi / R_0), R\sin(\phi / R_0), Z \bigr) + (R,s,Z) \longmapsto + \bigl( R\cos(s / R_0), R\sin(s/ R_0), Z \bigr) into the mesh coordinate field. The mesh components must again be unpacked in -their stored order, :math:`(R,Z,\phi)`:: +their stored order, :math:`(R,Z,s)`:: - R, Zc, phiR0 = SpatialCoordinate(msh) + R, Zc, s = SpatialCoordinate(msh) msh.coordinates.interpolate(as_vector(( - R*cos(phiR0/R0), - R*sin(phiR0/R0), + R*cos(s/R0), + R*sin(s/R0), Zc, ))) VTKFile("TokamakCartesianSolution.pvd").write(uh) From a077ec3709e67bee76f172f57ec1266ca3473883 Mon Sep 17 00:00:00 2001 From: ThighamOxford Date: Tue, 4 Aug 2026 11:58:45 +0100 Subject: [PATCH 11/13] Fixed a code typo --- demos/periodic_meshes/periodic_meshes.py.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demos/periodic_meshes/periodic_meshes.py.rst b/demos/periodic_meshes/periodic_meshes.py.rst index 607679fefa..96bdeaef6b 100644 --- a/demos/periodic_meshes/periodic_meshes.py.rst +++ b/demos/periodic_meshes/periodic_meshes.py.rst @@ -262,7 +262,7 @@ We solve again the same Helmholtz problem as in the previous section. :: V = FunctionSpace(msh, "CG", 2) x, y, z = SpatialCoordinate(msh) - u_exact = cos(z/R0) * (1 - x**2 - y**2) + u_exact = cos(z) * (1 - x**2 - y**2) f = u_exact - div(grad(u_exact)) u = TrialFunction(V) From da596ef6013c76d8c107c9d03e4ef8f7324cbc7e Mon Sep 17 00:00:00 2001 From: Thomas Higham Date: Thu, 6 Aug 2026 16:02:34 +0100 Subject: [PATCH 12/13] Update demos/periodic_meshes/periodic_meshes.py.rst Co-authored-by: Umberto Zerbinati --- demos/periodic_meshes/periodic_meshes.py.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demos/periodic_meshes/periodic_meshes.py.rst b/demos/periodic_meshes/periodic_meshes.py.rst index 96bdeaef6b..9ffdd3ff42 100644 --- a/demos/periodic_meshes/periodic_meshes.py.rst +++ b/demos/periodic_meshes/periodic_meshes.py.rst @@ -215,7 +215,7 @@ Identifications are declared on the geometry, before meshing, with the OCC ) where ``transformation`` is the rigid motion (typically a translation) that maps ``shape_a`` onto ``shape_b``. -Netgen then meshes the two boundaries compatibly and records the vertex pairs; Firedrake consumes them +Netgen then meshes the two boundaries compatibly and records the vertex pairs. Firedrake consumes them automatically -- no extra flag on the ``Mesh`` constructor is required. To construct a periodic cylinder of length :math:`2\pi` we identify the two end caps of the cylinder by a translation of :math:`2\pi` along ``z``. :: From e565df29b4c8872d162dea3299d9b13aea39e883 Mon Sep 17 00:00:00 2001 From: Thomas Higham Date: Thu, 6 Aug 2026 16:02:52 +0100 Subject: [PATCH 13/13] Update demos/periodic_meshes/periodic_meshes.py.rst Co-authored-by: Pablo Brubeck --- demos/periodic_meshes/periodic_meshes.py.rst | 6 +----- 1 file changed, 1 insertion(+), 5 deletions(-) diff --git a/demos/periodic_meshes/periodic_meshes.py.rst b/demos/periodic_meshes/periodic_meshes.py.rst index 9ffdd3ff42..7aa9e2a350 100644 --- a/demos/periodic_meshes/periodic_meshes.py.rst +++ b/demos/periodic_meshes/periodic_meshes.py.rst @@ -173,11 +173,7 @@ The two important arguments are ``layers`` and ``layer_height``, which tell Fire bc = DirichletBC(V, 0.0, "on_boundary") - solve( - a == L, - uh, - bcs=bc, - ) + solve(a == L, uh, bcs=bc) print(f"L2 error = {errornorm(u_exact, uh, norm_type='L2'):.3e}") VTKFile("output/cylinder_fd_Helmholtz.pvd").write(uh)

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