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Strip output and adapt eta definition, ref comment from Jack Hale (#269)
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‎chapter2/amr.ipynb‎

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

‎chapter2/amr.py‎

Lines changed: 8 additions & 8 deletions
Original file line numberDiff line numberDiff line change
@@ -192,10 +192,10 @@ def solve(
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# ## Error-indicator
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# In this example, we will use an error-indicator $\eta$ to decide what cells should be refined.
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# Specifically, the estimator is:
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# Specifically, the estimator $\eta$ is defined as:
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#
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# \begin{align*}
198-
# \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)
198+
# \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)
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# + \sum_{E\in\mathcal{F}_i}\frac{h}{2} \vert [\nabla \cdot \mathbf{n}_E ]\vert^2~\mathrm{d}s
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# \end{align*}
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#
@@ -206,7 +206,7 @@ def mark_cells(uh_r: dolfinx.fem.Function, lam: float):
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mesh = uh_r.function_space.mesh
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W = dolfinx.fem.functionspace(mesh, ("DG", 0))
208208
w = ufl.TestFunction(W)
209-
eta = dolfinx.fem.Function(W)
209+
eta_squared = dolfinx.fem.Function(W)
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f = dolfinx.fem.Constant(mesh, 1.0)
211211
h = dolfinx.fem.Function(W)
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h.x.array[:] = mesh.h(mesh.topology.dim, np.arange(len(h.x.array), dtype=np.int32))
@@ -217,14 +217,14 @@ def mark_cells(uh_r: dolfinx.fem.Function, lam: float):
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+ ufl.inner(h("+") / 2 * ufl.jump(ufl.grad(uh_r), n) ** 2, w("+")) * ufl.dS
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+ ufl.inner(h("-") / 2 * ufl.jump(ufl.grad(uh_r), n) ** 2, w("-")) * ufl.dS
219219
)
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dolfinx.fem.petsc.assemble_vector(eta.x.petsc_vec, dolfinx.fem.form(G))
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sqrt_eta = dolfinx.fem.Function(W)
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sqrt_eta.x.array[:] = np.sqrt(eta.x.array[:])
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dolfinx.fem.petsc.assemble_vector(eta_squared.x.petsc_vec, dolfinx.fem.form(G))
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eta = dolfinx.fem.Function(W)
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eta.x.array[:] = np.sqrt(eta_squared.x.array[:])
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sqrt_eta_max = sqrt_eta.x.petsc_vec.max()[1]
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eta_max = eta.x.petsc_vec.max()[1]
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theta = 0.5
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should_refine = ufl.conditional(ufl.gt(sqrt_eta, theta * sqrt_eta_max), 1, 0)
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should_refine = ufl.conditional(ufl.gt(eta, theta * eta_max), 1, 0)
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markers = dolfinx.fem.Function(W)
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ip = W.element.interpolation_points
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if Version(dolfinx.__version__) < Version("0.10.0"):

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