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Aug 18, 2025
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Strip output and adapt eta definition, ref comment from Jack Hale #269

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72 changes: 36 additions & 36 deletions chapter2/amr.ipynb
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Original file line number Diff line number Diff line change
Expand Up @@ -2,7 +2,7 @@
"cells": [
{
"cell_type": "markdown",
"id": "4535be91",
"id": "0",
"metadata": {},
"source": [
"# Adaptive mesh refinement with NetGen and DOLFINx\n",
Expand All @@ -18,7 +18,7 @@
},
{
"cell_type": "markdown",
"id": "b0282fe2",
"id": "1",
"metadata": {},
"source": [
"In this tutorial, we will consider an adaptive mesh refinement method, applied to\n",
Expand All @@ -31,7 +31,7 @@
{
"cell_type": "code",
"execution_count": null,
"id": "32fe6a58",
"id": "2",
"metadata": {},
"outputs": [],
"source": [
Expand All @@ -50,7 +50,7 @@
},
{
"cell_type": "markdown",
"id": "4782ba3b",
"id": "3",
"metadata": {},
"source": [
"## Generating a higher-order mesh with NetGen\n",
Expand All @@ -60,7 +60,7 @@
{
"cell_type": "code",
"execution_count": null,
"id": "d6046434",
"id": "4",
"metadata": {},
"outputs": [],
"source": [
Expand All @@ -80,7 +80,7 @@
},
{
"cell_type": "markdown",
"id": "e041abef",
"id": "5",
"metadata": {},
"source": [
"## Loading a mesh into DOLFINx\n",
Expand All @@ -92,7 +92,7 @@
{
"cell_type": "code",
"execution_count": null,
"id": "fc64bf64",
"id": "6",
"metadata": {},
"outputs": [],
"source": [
Expand All @@ -101,7 +101,7 @@
},
{
"cell_type": "markdown",
"id": "67cded8b",
"id": "7",
"metadata": {},
"source": [
"Next, we generate the mesh with the function :py:func:`ngsPETSc.utils.fenicsx.GeometricModel.model_to_mesh`.\n",
Expand All @@ -112,7 +112,7 @@
{
"cell_type": "code",
"execution_count": null,
"id": "dbae564a",
"id": "8",
"metadata": {},
"outputs": [],
"source": [
Expand All @@ -121,7 +121,7 @@
},
{
"cell_type": "markdown",
"id": "f90ca932",
"id": "9",
"metadata": {},
"source": [
"We use pyvista to visualize the mesh."
Expand All @@ -130,7 +130,7 @@
{
"cell_type": "code",
"execution_count": null,
"id": "34407248",
"id": "10",
"metadata": {
"tags": [
"hide-input"
Expand All @@ -154,7 +154,7 @@
},
{
"cell_type": "markdown",
"id": "6a18b03f",
"id": "11",
"metadata": {},
"source": [
"We have read in any cell and facet markers that have been defined in the NetGen model,\n",
Expand All @@ -166,7 +166,7 @@
{
"cell_type": "code",
"execution_count": null,
"id": "adc50da0",
"id": "12",
"metadata": {},
"outputs": [],
"source": [
Expand All @@ -176,7 +176,7 @@
},
{
"cell_type": "markdown",
"id": "0a3449b0",
"id": "13",
"metadata": {},
"source": [
"Again, we visualize the curved mesh with pyvista."
Expand All @@ -185,7 +185,7 @@
{
"cell_type": "code",
"execution_count": null,
"id": "1eb97f0f",
"id": "14",
"metadata": {
"tags": [
"hide-input"
Expand All @@ -206,7 +206,7 @@
},
{
"cell_type": "markdown",
"id": "7613cc7f",
"id": "15",
"metadata": {},
"source": [
"## Solving the eigenvalue problem\n",
Expand All @@ -224,7 +224,7 @@
},
{
"cell_type": "markdown",
"id": "717f48a2",
"id": "16",
"metadata": {
"lines_to_next_cell": 2
},
Expand All @@ -236,7 +236,7 @@
{
"cell_type": "code",
"execution_count": null,
"id": "d59bc925",
"id": "17",
"metadata": {
"lines_to_next_cell": 2
},
Expand Down Expand Up @@ -310,17 +310,17 @@
},
{
"cell_type": "markdown",
"id": "1c251b79",
"id": "18",
"metadata": {
"lines_to_next_cell": 2
},
"source": [
"## Error-indicator\n",
"In this example, we will use an error-indicator $\\eta$ to decide what cells should be refined.\n",
"Specifically, the estimator is:\n",
"Specifically, the estimator $\\eta$ is defined as:\n",
"\n",
"\\begin{align*}\n",
" \\eta = \\sum_{K\\in \\mathcal{T}_h(\\Omega)}\\left(h^2\\int_K \\vert \\lambda u_h + \\Delta u_h\\vert^2~\\mathrm{d}x\\right)\n",
" \\eta^2 = \\sum_{K\\in \\mathcal{T}_h(\\Omega)}\\left(h^2\\int_K \\vert \\lambda u_h + \\Delta u_h\\vert^2~\\mathrm{d}x\\right)\n",
"+ \\sum_{E\\in\\mathcal{F}_i}\\frac{h}{2} \\vert [\\nabla \\cdot \\mathbf{n}_E ]\\vert^2~\\mathrm{d}s\n",
"\\end{align*}\n",
"\n",
Expand All @@ -330,15 +330,15 @@
{
"cell_type": "code",
"execution_count": null,
"id": "95f75bd8",
"id": "19",
"metadata": {},
"outputs": [],
"source": [
"def mark_cells(uh_r: dolfinx.fem.Function, lam: float):\n",
" mesh = uh_r.function_space.mesh\n",
" W = dolfinx.fem.functionspace(mesh, (\"DG\", 0))\n",
" w = ufl.TestFunction(W)\n",
" eta = dolfinx.fem.Function(W)\n",
" eta_squared = dolfinx.fem.Function(W)\n",
" f = dolfinx.fem.Constant(mesh, 1.0)\n",
" h = dolfinx.fem.Function(W)\n",
" h.x.array[:] = mesh.h(mesh.topology.dim, np.arange(len(h.x.array), dtype=np.int32))\n",
Expand All @@ -349,14 +349,14 @@
" + ufl.inner(h(\"+\") / 2 * ufl.jump(ufl.grad(uh_r), n) ** 2, w(\"+\")) * ufl.dS\n",
" + ufl.inner(h(\"-\") / 2 * ufl.jump(ufl.grad(uh_r), n) ** 2, w(\"-\")) * ufl.dS\n",
" )\n",
" dolfinx.fem.petsc.assemble_vector(eta.x.petsc_vec, dolfinx.fem.form(G))\n",
" sqrt_eta = dolfinx.fem.Function(W)\n",
" sqrt_eta.x.array[:] = np.sqrt(eta.x.array[:])\n",
" dolfinx.fem.petsc.assemble_vector(eta_squared.x.petsc_vec, dolfinx.fem.form(G))\n",
" eta = dolfinx.fem.Function(W)\n",
" eta.x.array[:] = np.sqrt(eta_squared.x.array[:])\n",
"\n",
" sqrt_eta_max = sqrt_eta.x.petsc_vec.max()[1]\n",
" eta_max = eta.x.petsc_vec.max()[1]\n",
"\n",
" theta = 0.5\n",
" should_refine = ufl.conditional(ufl.gt(sqrt_eta, theta * sqrt_eta_max), 1, 0)\n",
" should_refine = ufl.conditional(ufl.gt(eta, theta * eta_max), 1, 0)\n",
" markers = dolfinx.fem.Function(W)\n",
" ip = W.element.interpolation_points\n",
" if Version(dolfinx.__version__) < Version(\"0.10.0\"):\n",
Expand All @@ -367,7 +367,7 @@
},
{
"cell_type": "markdown",
"id": "d9243490",
"id": "20",
"metadata": {},
"source": [
"## Running the adaptive refinement algorithm\n",
Expand All @@ -376,7 +376,7 @@
},
{
"cell_type": "markdown",
"id": "ab8f6701",
"id": "21",
"metadata": {},
"source": [
"We will track the progress of the adaptive mesh refinement as a GIF."
Expand All @@ -385,7 +385,7 @@
{
"cell_type": "code",
"execution_count": null,
"id": "293b959a",
"id": "22",
"metadata": {},
"outputs": [],
"source": [
Expand All @@ -395,7 +395,7 @@
},
{
"cell_type": "markdown",
"id": "5fe0ac8b",
"id": "23",
"metadata": {
"lines_to_next_cell": 2
},
Expand All @@ -407,7 +407,7 @@
{
"cell_type": "code",
"execution_count": null,
"id": "969aaaa5",
"id": "24",
"metadata": {
"tags": [
"hide-input"
Expand Down Expand Up @@ -444,7 +444,7 @@
},
{
"cell_type": "markdown",
"id": "d798464c",
"id": "25",
"metadata": {},
"source": [
"We set some parameters for checking convergence of the algorithm, and provide the exact eigenvalue\n",
Expand All @@ -460,7 +460,7 @@
{
"cell_type": "code",
"execution_count": null,
"id": "ebd3fef7",
"id": "26",
"metadata": {
"tags": [
"scroll-output"
Expand Down Expand Up @@ -492,7 +492,7 @@
},
{
"cell_type": "markdown",
"id": "5bac3843",
"id": "27",
"metadata": {},
"source": [
"<img src=\"./amr.gif\" alt=\"gif\" class=\"bg-primary mb-1\" width=\"800px\">"
Expand Down
16 changes: 8 additions & 8 deletions chapter2/amr.py
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Original file line number Diff line number Diff line change
Expand Up @@ -192,10 +192,10 @@ def solve(

# ## Error-indicator
# In this example, we will use an error-indicator $\eta$ to decide what cells should be refined.
# Specifically, the estimator is:
# Specifically, the estimator $\eta$ is defined as:
#
# \begin{align*}
# \eta = \sum_{K\in \mathcal{T}_h(\Omega)}\left(h^2\int_K \vert \lambda u_h + \Delta u_h\vert^2~\mathrm{d}x\right)
# \eta^2 = \sum_{K\in \mathcal{T}_h(\Omega)}\left(h^2\int_K \vert \lambda u_h + \Delta u_h\vert^2~\mathrm{d}x\right)
# + \sum_{E\in\mathcal{F}_i}\frac{h}{2} \vert [\nabla \cdot \mathbf{n}_E ]\vert^2~\mathrm{d}s
# \end{align*}
#
Expand All @@ -206,7 +206,7 @@ def mark_cells(uh_r: dolfinx.fem.Function, lam: float):
mesh = uh_r.function_space.mesh
W = dolfinx.fem.functionspace(mesh, ("DG", 0))
w = ufl.TestFunction(W)
eta = dolfinx.fem.Function(W)
eta_squared = dolfinx.fem.Function(W)
f = dolfinx.fem.Constant(mesh, 1.0)
h = dolfinx.fem.Function(W)
h.x.array[:] = mesh.h(mesh.topology.dim, np.arange(len(h.x.array), dtype=np.int32))
Expand All @@ -217,14 +217,14 @@ def mark_cells(uh_r: dolfinx.fem.Function, lam: float):
+ ufl.inner(h("+") / 2 * ufl.jump(ufl.grad(uh_r), n) ** 2, w("+")) * ufl.dS
+ ufl.inner(h("-") / 2 * ufl.jump(ufl.grad(uh_r), n) ** 2, w("-")) * ufl.dS
)
dolfinx.fem.petsc.assemble_vector(eta.x.petsc_vec, dolfinx.fem.form(G))
sqrt_eta = dolfinx.fem.Function(W)
sqrt_eta.x.array[:] = np.sqrt(eta.x.array[:])
dolfinx.fem.petsc.assemble_vector(eta_squared.x.petsc_vec, dolfinx.fem.form(G))
eta = dolfinx.fem.Function(W)
eta.x.array[:] = np.sqrt(eta_squared.x.array[:])

sqrt_eta_max = sqrt_eta.x.petsc_vec.max()[1]
eta_max = eta.x.petsc_vec.max()[1]

theta = 0.5
should_refine = ufl.conditional(ufl.gt(sqrt_eta, theta * sqrt_eta_max), 1, 0)
should_refine = ufl.conditional(ufl.gt(eta, theta * eta_max), 1, 0)
markers = dolfinx.fem.Function(W)
ip = W.element.interpolation_points
if Version(dolfinx.__version__) < Version("0.10.0"):
Expand Down