{
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"### Vivaldi antenna\n",
"\n",
"Full parametric Vivaldi antenna mesh using gmsh/OpenCASCADE.\n",
"\n",
"The exponential taper profile follows:\n",
"$$y = \\pm\\frac{w_s}{2} \\cdot e^{C(x - x_0)}$$\n",
"where $C$ is the opening rate (25 here) and $x_0$ is the taper start.\n",
"\n",
"**Coordinate convention** (matches the diagram, x is the long axis):\n",
"- Ground plane centred at origin in XY plane\n",
"- Aperture opens toward +x\n",
"- Cavity is on the −x side\n",
"- Z is vertical (substrate thickness)"
]
},
{
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"source": [
"import gmsh\n",
"import math\n",
"import numpy as np\n",
"from palacetoolkit.mesh import (\n",
" Entity, \n",
" run_meshing_pipeline, \n",
" generate_3d_mesh, \n",
" refine_near_surfaces\n",
")\n",
"from palacetoolkit.viz import run_with_scrollable_output, view_mesh \n"
]
},
{
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"tags": []
},
"source": [
"#### Antenna parameters"
]
},
{
"cell_type": "code",
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"id": "396c6947",
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{
"name": "stdout",
"output_type": "stream",
"text": [
"Ground plane: [-0.1500, 0.1500]\n",
"Taper starts at: x = -0.0930\n",
"Parallel section: x = [-0.1160, -0.0930] length = 23.0 mm (= cavity_to_taper)\n",
"Cavity centre at: x = -0.1280\n",
"Cavity right edge: x = -0.1160 (= x_taper_start - s)\n"
]
}
],
"source": [
"taper_length: float = 0.243 \n",
"aperture_width: float = 0.105 \n",
"opening_rate: float = 25.0 \n",
"slot_width: float = 5e-4 \n",
"cavity_diameter: float = 0.024 \n",
"cavity_to_taper: float = 0.023 \n",
"ground_plane_length: float = 0.300 \n",
"ground_plane_width: float = 0.125 \n",
"h_sub: float = 0.015 \n",
"air_height: float = 0.05 \n",
"air_margin: float = 0.05\n",
"\n",
"freq_ghz = 4.5\n",
"c0 = 3e8\n",
"wavelength = c0 / (freq_ghz * 1e9)\n",
"\n",
"mesh_file = \"vivaldi.msh\"\n",
"\n",
"# Derived coordinates\n",
"Lx = ground_plane_length\n",
"Ly = ground_plane_width\n",
"\n",
"# Left edge of the ground plane.\n",
"x0 = -Lx/2 \n",
"\n",
"# Right edge of the ground plane.\n",
"x1 = Lx/2 \n",
"\n",
"# x_taper_start: where the exponential section begins\n",
"x_taper_start = x1 - taper_length\n",
"\n",
"# x_slot_left: start of parallel section = end of cavity = x_taper_start - s\n",
"x_slot_left = x_taper_start - cavity_to_taper\n",
"\n",
"# cavity centre\n",
"x_cav = x_slot_left - cavity_diameter / 2\n",
"\n",
"print(f\"Ground plane: [{x0:.4f}, {x1:.4f}]\")\n",
"print(f\"Taper starts at: x = {x_taper_start:.4f}\")\n",
"print(f\"Parallel section: x = [{x_slot_left:.4f}, {x_taper_start:.4f}] \"\n",
" f\"length = {cavity_to_taper*1e3:.1f} mm (= cavity_to_taper)\")\n",
"print(f\"Cavity centre at: x = {x_cav:.4f}\")\n",
"print(f\"Cavity right edge: x = {x_slot_left:.4f} (= x_taper_start - s)\")\n"
]
},
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"cell_type": "markdown",
"id": "taper_math",
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"source": [
"#### Exponential taper mathematics\n",
"\n",
"The upper edge of the Vivaldi slot follows:\n",
"$$y_{\\text{upper}}(x) = \\frac{w_s}{2} \\cdot e^{C(x - x_{\\text{ts}})}$$\n",
"scaled so that $y_{\\text{upper}}(x_1) = w_a/2$.\n",
"\n",
"We solve for the normalisation constant $A$:\n",
"$$A = \\frac{w_a/2}{e^{C(x_1 - x_{\\text{ts}})}}$$\n",
"which gives:\n",
"$$y_{\\text{upper}}(x) = A \\cdot e^{C(x - x_{\\text{ts}})}$$\n",
"\n",
"The lower edge is the mirror: $y_{\\text{lower}} = -y_{\\text{upper}}$."
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "taper_func",
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"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"y at taper start : +0.00012 (expected ≈ +0.00025)\n",
"y at aperture : +0.05250 (expected ≈ +0.05250)\n"
]
}
],
"source": [
"def taper_y(x: float, sign: float = 1.0) -> float:\n",
" \n",
" # Normalisation: amplitude A chosen so y(x1) = aperture_width/2\n",
" A = (aperture_width / 2) / math.exp(opening_rate * (x1 - x_taper_start))\n",
" \n",
" return sign * A * math.exp(opening_rate * (x - x_taper_start))\n",
"\n",
"# Quick sanity checks\n",
"print(f\"y at taper start : {taper_y(x_taper_start):+.5f} (expected ≈ {slot_width/2:+.5f})\")\n",
"print(f\"y at aperture : {taper_y(x1):+.5f} (expected ≈ {aperture_width/2:+.5f})\")"
]
},
{
"cell_type": "markdown",
"id": "gmsh_init_md",
"metadata": {
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},
"tags": []
},
"source": [
"#### Initialise gmsh"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "7054c50f",
"metadata": {
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},
"outputs": [],
"source": [
"gmsh.initialize()\n",
"gmsh.model.add(\"vivaldi_antenna\")\n",
"kernel = gmsh.model.occ "
]
},
{
"cell_type": "markdown",
"id": "volumes_md",
"metadata": {
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},
"source": [
"#### Build the 3-D volumes (substrate + air sphere)"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "c3ee11e8",
"metadata": {
"execution": {
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"status": "completed"
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"tags": []
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Substrate tag: 1\n",
"Air sphere tag: 2\n"
]
}
],
"source": [
"# Bounding box extents \n",
"total_xmin = x0 - air_margin\n",
"total_xmax = x1 + air_margin\n",
"total_ymin = -Ly/2 - air_margin\n",
"total_ymax = Ly/2 + air_margin\n",
"total_zmax = h_sub + air_height\n",
"\n",
"# Substrate \n",
"substrate = kernel.addBox(\n",
" x0, -Ly/2, 0,\n",
" Lx, Ly, h_sub\n",
")\n",
"\n",
"# Air sphere (replace air box, consistent with patch_antenna workflow)\n",
"airsphere_radius = max(abs(total_xmin), abs(total_xmax), abs(total_ymin), abs(total_ymax), total_zmax)\n",
"air_sphere = kernel.addSphere(0.0, 0.0, 0.0, airsphere_radius)\n",
"\n",
"print(\"Substrate tag:\", substrate)\n",
"print(\"Air sphere tag:\", air_sphere)"
]
},
{
"cell_type": "markdown",
"id": "copper_md",
"metadata": {
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},
"source": [
"#### Build the copper patch (ground plane + taper slot + cavity)\n",
"\n",
"Strategy:\n",
"1. Start with a full rectangular ground-plane surface.\n",
"2. Subtract the exponential slot (built from a spline boundary).\n",
"3. Subtract the circular cavity.\n",
"4. The result is the physical copper surface at z = h_sub."
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "copper_patch",
"metadata": {
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"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Parallel section: x=[-0.1160, -0.0930] y=±0.00025\n",
"Exponential section: x=[-0.0930, 0.1500]\n",
"Aperture width at x1: 105.00 mm (target 105.00 mm)\n",
"Top rectangle tag: 8\n",
"Slot surface tag: 9\n",
"Cavity surface tag: 10\n"
]
}
],
"source": [
"# Top rectangle. We´ll cut the slot and cavity out of this.\n",
"top_rect = kernel.addRectangle(x0, -Ly/2, h_sub, Lx, Ly)\n",
"\n",
"# Parallel section geometry.\n",
"p_ul = kernel.addPoint(x_slot_left, +slot_width/2, h_sub)\n",
"p_ur = kernel.addPoint(x_taper_start, +slot_width/2, h_sub)\n",
"p_lr = kernel.addPoint(x_taper_start, -slot_width/2, h_sub)\n",
"p_ll = kernel.addPoint(x_slot_left, -slot_width/2, h_sub)\n",
"\n",
"# Straight lines\n",
"line_top_par = kernel.addLine(p_ul, p_ur) # upper parallel edge\n",
"line_bot_par = kernel.addLine(p_lr, p_ll) # lower parallel edge (reversed for CCW)\n",
"line_left = kernel.addLine(p_ll, p_ul) # left closing line (at x_slot_left)\n",
"\n",
"# Interpolate points along the exponential taper curve from x_taper_start to x1.\n",
"N_pts = 100\n",
"xs = np.linspace(x_taper_start, x1, N_pts)\n",
"\n",
"# Upper spline\n",
"upper_inner = [kernel.addPoint(float(x), taper_y(float(x), +1.0), h_sub)\n",
" for x in xs[1:]]\n",
"upper_spline = kernel.addSpline([p_ur] + upper_inner)\n",
"\n",
"# Lower spline: starts at p_lr, reversed direction for CCW loop.\n",
"lower_inner = [kernel.addPoint(float(x), taper_y(float(x), -1.0), h_sub)\n",
" for x in xs[1:]]\n",
"\n",
"# In the loop we traverse lower in reverse (aperture → taper_start),\n",
"# so we list points aperture-end first.\n",
"lower_spline = kernel.addSpline(lower_inner[::-1] + [p_lr])\n",
"\n",
"# Aperture closing line (right edge, x = x1)\n",
"p_apt = upper_inner[-1] # top-right aperture point\n",
"p_apb = lower_inner[-1] # bottom-right aperture point\n",
"line_aperture = kernel.addLine(p_apt, p_apb)\n",
"\n",
"slot_loop = kernel.addCurveLoop([\n",
" line_left, # up the left edge (x_slot_left, bot→top)\n",
" line_top_par, # rightward along upper parallel\n",
" upper_spline, # upper exponential curve to aperture\n",
" line_aperture, # down the aperture edge\n",
" lower_spline, # lower exponential back to x_taper_start (reversed)\n",
" line_bot_par, # leftward along lower parallel back to start\n",
"])\n",
"\n",
"slot_surf = kernel.addPlaneSurface([slot_loop])\n",
"\n",
"# Circular cavity\n",
"cav_r = cavity_diameter / 2\n",
"cav_cx = x_cav\n",
"cav_cy = 0.0\n",
"\n",
"cav_circle = kernel.addCircle(cav_cx, cav_cy, h_sub, cav_r)\n",
"cav_loop = kernel.addCurveLoop([cav_circle])\n",
"cav_surf = kernel.addPlaneSurface([cav_loop])\n",
"\n",
"print(f\"Parallel section: x=[{x_slot_left:.4f}, {x_taper_start:.4f}] y=±{slot_width/2:.5f}\")\n",
"print(f\"Exponential section: x=[{x_taper_start:.4f}, {x1:.4f}]\")\n",
"print(f\"Aperture width at x1: {2*taper_y(x1,1)*1e3:.2f} mm (target {aperture_width*1e3:.2f} mm)\")\n",
"print(\"Top rectangle tag: \", top_rect)\n",
"print(\"Slot surface tag: \", slot_surf)\n",
"print(\"Cavity surface tag: \", cav_surf)"
]
},
{
"cell_type": "markdown",
"id": "feed_md",
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"source": [
"#### Feed port\n",
"\n",
"From the inset diagram, the feed port is a **square face** in the YZ plane\n",
"at the left edge of the ground plane (`x = x0`). It is centred on the slot\n",
"(`y = 0`) and spans the substrate thickness in z.\n",
"\n",
"- `feed_offset` is the **x-axis** offset that positions where along the slot\n",
" the port is referenced — here it locates the port at `x = x0 + feed_offset`\n",
" inside the ground plane, but the excitation face itself sits flush at\n",
" `x = x0` (the left wall of the computational domain).\n",
"- The port is square: width = height = `slot_width` in the YZ cross-section.\n",
"- It is centred at `y = 0, z = h_sub / 2` (mid-height of the substrate)."
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "feed_strip",
"metadata": {
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},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Feed port surface tag: 11\n",
"Port centre: x=-0.0932 y=0 z=0.01500 (top of substrate)\n",
"Port x range: [-0.0935, -0.0930]\n",
"Port y range: [-0.00025, 0.00025]\n",
"Port size: 0.50 mm × 0.50 mm (square)\n",
"Sanity: port_x_ctr (-0.0932) should be between cavity right edge (-0.1160) and taper start (-0.0930)\n"
]
}
],
"source": [
"port_size = slot_width # square side length [m]\n",
"port_x_ctr = x_taper_start - port_size / 2 # port centre along x\n",
"port_x0 = port_x_ctr - port_size / 2 # left x of port\n",
"port_y0 = -port_size / 2 # bottom y (centred on slot)\n",
"\n",
"# Flat XY-plane rectangle at z = h_sub — no rotation needed.\n",
"feed_port_surf = kernel.addRectangle(\n",
" port_x0, # x start\n",
" port_y0, # y start (centred: -w_s/2 .. +w_s/2)\n",
" h_sub, # z = top of substrate\n",
" port_size, # dx = slot_width (square in x)\n",
" port_size, # dy = slot_width (square in y, touches both copper edges)\n",
")\n",
"\n",
"print(f\"Feed port surface tag: {feed_port_surf}\")\n",
"print(f\"Port centre: x={port_x_ctr:.4f} y=0 z={h_sub:.5f} (top of substrate)\")\n",
"print(f\"Port x range: [{port_x0:.4f}, {port_x0+port_size:.4f}]\")\n",
"print(f\"Port y range: [{port_y0:.5f}, {-port_y0:.5f}]\")\n",
"print(f\"Port size: {port_size*1e3:.2f} mm × {port_size*1e3:.2f} mm (square)\")\n",
"print(f\"Sanity: port_x_ctr ({port_x_ctr:.4f}) should be between \"\n",
" f\"cavity right edge ({x_cav + cavity_diameter/2:.4f}) \"\n",
" f\"and taper start ({x_taper_start:.4f})\")"
]
},
{
"cell_type": "markdown",
"id": "boolean_md",
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"tags": []
},
"source": [
"#### Boolean operations — assemble the copper patch\n",
"\n",
"Cut the slot and cavity out of the ground-plane rectangle.\n",
"The feed strip is kept separate (it is a distinct conductor patch)."
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "boolean_ops",
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"tags": []
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Copper patch surfaces after boolean cut: \n",
" dim=2, tag=8\n"
]
}
],
"source": [
"kernel.synchronize()\n",
"\n",
"# Cut slot + cavity from ground-plane rectangle\n",
"# BooleanCut returns (result_dimtags, map)\n",
"copper_patch, _ = kernel.cut(\n",
" [(2, top_rect)], # object: full rectangle\n",
" [(2, slot_surf), (2, cav_surf)], # tools: slot + cavity\n",
" removeObject=True, removeTool=True\n",
")\n",
"\n",
"kernel.synchronize()\n",
"print(\"Copper patch surfaces after boolean cut:\")\n",
"for dim, tag in copper_patch:\n",
" print(f\" dim={dim}, tag={tag}\")"
]
},
{
"cell_type": "markdown",
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"tags": []
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"source": [
"#### Entity definition. \n",
"\n",
"In order to get a good mesh we need to fragment it and restore it, run_meshing_pipeline does this and also defines the physical groups."
]
},
{
"cell_type": "code",
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"id": "7691895e",
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"status": "completed"
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"tags": []
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" Physical group 'substrate' (dim=3): pg=1, tags=[1] \n",
" Physical group 'air_sphere' (dim=3): pg=2, tags=[2]\n",
" Physical group 'copper_patch' (dim=2): pg=3, tags=[8]\n",
" Physical group 'feed_port' (dim=2): pg=4, tags=[11]\n",
" Physical group 'air_sphere__substrate' (dim=2): pg=5, tags=[12, 13, 14, 15, 5, 16, 17, 18]\n",
" Physical group 'air_sphere__None' (dim=2): pg=6, tags=[19]\n",
" ppw_near=300 ppw_far=5\n",
" SizeMax=0.0133 transition=0.0167\n",
" global: 16 curves, SizeMin=0.0002\n",
"Info : Meshing 1D...\n",
"Info : [ 0%] Meshing curve 4 (Line)\n",
"Info : [ 10%] Meshing curve 8 (Line)\n",
"Info : [ 10%] Meshing curve 9 (Line)\n",
"Info : [ 20%] Meshing curve 11 (Line)\n",
"Info : [ 20%] Meshing curve 12 (Line)\n",
"Info : [ 20%] Meshing curve 13 (Line)\n",
"Info : [ 30%] Meshing curve 14 (Line)\n",
"Info : [ 30%] Meshing curve 15 (Line)\n",
"Info : [ 30%] Meshing curve 16 (Line)\n",
"Info : [ 40%] Meshing curve 17 (Line)\n",
"Info : [ 40%] Meshing curve 18 (Line)\n",
"Info : [ 40%] Meshing curve 19 (Line)\n",
"Info : [ 50%] Meshing curve 20 (Line)\n",
"Info : [ 50%] Meshing curve 21 (Line)\n",
"Info : [ 60%] Meshing curve 22 (Circle)\n",
"Info : [ 60%] Meshing curve 23 (BSpline)\n",
"Info : [ 60%] Meshing curve 24 (Line)\n",
"Info : [ 70%] Meshing curve 25 (BSpline)\n",
"Info : [ 70%] Meshing curve 26 (Line)\n",
"Info : [ 70%] Meshing curve 27 (Line)\n",
"Info : [ 80%] Meshing curve 28 (Line)\n",
"Info : [ 80%] Meshing curve 29 (Line)\n",
"Info : [ 80%] Meshing curve 30 (Line)\n",
"Info : [ 90%] Meshing curve 32 (Circle)\n",
"Info : [100%] Meshing curve 34 (Line)\n",
"Info : [100%] Meshing curve 35 (Line)\n",
"Info : Done meshing 1D (Wall 1.88283s, CPU 1.81569s)\n",
"Info : Meshing 2D...\n",
"Info : [ 0%] Meshing surface 5 (Plane, MeshAdapt)\n",
"Info : [ 10%] Meshing surface 8 (Plane, MeshAdapt)\n",
"Info : [ 20%] Meshing surface 11 (Plane, MeshAdapt)\n",
"Info : [ 30%] Meshing surface 12 (Plane, MeshAdapt)\n",
"Info : [ 40%] Meshing surface 13 (Plane, MeshAdapt)\n",
"Info : [ 50%] Meshing surface 14 (Plane, MeshAdapt)\n",
"Info : [ 60%] Meshing surface 15 (Plane, MeshAdapt)\n",
"Info : [ 70%] Meshing surface 16 (Plane, MeshAdapt)\n",
"Info : [ 80%] Meshing surface 17 (Plane, MeshAdapt)\n",
"Info : [ 90%] Meshing surface 18 (Plane, MeshAdapt)\n",
"Info : [100%] Meshing surface 19 (Sphere, MeshAdapt)\n",
"Info : Done meshing 2D (Wall 5.92237s, CPU 5.90935s)\n",
"Info : Meshing 3D...\n",
"Info : 3D Meshing 2 volumes with 1 connected component\n",
"Info : Tetrahedrizing 18829 nodes...\n",
"Info : Done tetrahedrizing 18837 nodes (Wall 0.295651s, CPU 0.290184s)\n",
"Info : Reconstructing mesh...\n",
"Info : - Creating surface mesh\n",
"Info : - Identifying boundary edges\n",
"Info : - Recovering boundary\n",
"Info : Done reconstructing mesh (Wall 0.580503s, CPU 0.575125s)\n",
"Info : Found volume 2\n",
"Info : Found volume 1\n",
"Info : It. 0 - 0 nodes created - worst tet radius 34.952 (nodes removed 0 0)\n",
"Info : It. 500 - 496 nodes created - worst tet radius 3.19998 (nodes removed 0 4)\n",
"Info : It. 1000 - 992 nodes created - worst tet radius 3.29385 (nodes removed 0 8)\n",
"Info : It. 1500 - 1488 nodes created - worst tet radius 2.98787 (nodes removed 0 12)\n",
"Info : It. 2000 - 1980 nodes created - worst tet radius 2.21376 (nodes removed 0 20)\n",
"Info : It. 2500 - 2480 nodes created - worst tet radius 2.03058 (nodes removed 0 20)\n",
"Info : It. 3000 - 2980 nodes created - worst tet radius 1.97895 (nodes removed 0 20)\n",
"Info : It. 3500 - 3476 nodes created - worst tet radius 1.8197 (nodes removed 0 24)\n",
"Info : It. 4000 - 3972 nodes created - worst tet radius 1.74223 (nodes removed 0 28)\n",
"Info : It. 4500 - 4469 nodes created - worst tet radius 1.67595 (nodes removed 0 31)\n",
"Info : It. 5000 - 4968 nodes created - worst tet radius 1.61457 (nodes removed 0 32)\n",
"Info : It. 5500 - 5467 nodes created - worst tet radius 1.86937 (nodes removed 0 33)\n",
"Info : It. 6000 - 5967 nodes created - worst tet radius 1.51066 (nodes removed 0 33)\n",
"Info : It. 6500 - 6467 nodes created - worst tet radius 1.46905 (nodes removed 0 33)\n",
"Info : It. 7000 - 6967 nodes created - worst tet radius 1.43197 (nodes removed 0 33)\n",
"Info : It. 7500 - 7461 nodes created - worst tet radius 1.39799 (nodes removed 0 39)\n",
"Info : It. 8000 - 7961 nodes created - worst tet radius 1.36684 (nodes removed 0 39)\n",
"Info : It. 8500 - 8455 nodes created - worst tet radius 1.33936 (nodes removed 0 45)\n",
"Info : It. 9000 - 8955 nodes created - worst tet radius 1.31112 (nodes removed 0 45)\n",
"Info : It. 9500 - 9452 nodes created - worst tet radius 1.28486 (nodes removed 0 48)\n",
"Info : It. 10000 - 9952 nodes created - worst tet radius 1.26075 (nodes removed 0 48)\n",
"Info : It. 10500 - 10452 nodes created - worst tet radius 1.2391 (nodes removed 0 48)\n",
"Info : It. 11000 - 10950 nodes created - worst tet radius 1.21802 (nodes removed 0 50)\n",
"Info : It. 11500 - 11450 nodes created - worst tet radius 1.19886 (nodes removed 0 50)\n",
"Info : It. 12000 - 11950 nodes created - worst tet radius 1.1814 (nodes removed 0 50)\n",
"Info : It. 12500 - 12450 nodes created - worst tet radius 1.16408 (nodes removed 0 50)\n",
"Info : It. 13000 - 12950 nodes created - worst tet radius 1.14617 (nodes removed 0 50)\n",
"Info : It. 13500 - 13450 nodes created - worst tet radius 1.13022 (nodes removed 0 50)\n",
"Info : It. 14000 - 13950 nodes created - worst tet radius 1.11679 (nodes removed 0 50)\n",
"Info : It. 14500 - 14450 nodes created - worst tet radius 1.10362 (nodes removed 0 50)\n",
"Info : It. 15000 - 14950 nodes created - worst tet radius 1.1308 (nodes removed 0 50)\n",
"Info : It. 15500 - 15448 nodes created - worst tet radius 1.07942 (nodes removed 0 52)\n",
"Info : It. 16000 - 15948 nodes created - worst tet radius 1.06716 (nodes removed 0 52)\n",
"Info : It. 16500 - 16448 nodes created - worst tet radius 1.0563 (nodes removed 0 52)\n",
"Info : It. 17000 - 16948 nodes created - worst tet radius 1.04573 (nodes removed 0 52)\n",
"Info : It. 17500 - 17448 nodes created - worst tet radius 1.03526 (nodes removed 0 52)\n",
"Info : It. 18000 - 17948 nodes created - worst tet radius 1.02558 (nodes removed 0 52)\n",
"Info : It. 18500 - 18448 nodes created - worst tet radius 1.01597 (nodes removed 0 52)\n",
"Info : It. 19000 - 18948 nodes created - worst tet radius 1.00774 (nodes removed 0 52)\n",
"Info : 3D refinement terminated (38264 nodes total):\n",
"Info : - 26 Delaunay cavities modified for star shapeness\n",
"Info : - 52 nodes could not be inserted\n",
"Info : - 216865 tetrahedra created in 2.73033 sec. (79428 tets/s)\n",
"Info : 4 node relocations\n",
"Info : Done meshing 3D (Wall 4.25849s, CPU 4.26071s)\n",
"Info : Optimizing mesh...\n",
"Info : Optimizing volume 1\n",
"Info : Optimization starts (volume = 0.0005625) with worst = 0.000289261 / average = 0.592864:\n",
"Info : 0.00 < quality < 0.10 : 297 elements\n",
"Info : 0.10 < quality < 0.20 : 801 elements\n",
"Info : 0.20 < quality < 0.30 : 1681 elements\n",
"Info : 0.30 < quality < 0.40 : 5167 elements\n",
"Info : 0.40 < quality < 0.50 : 8958 elements\n",
"Info : 0.50 < quality < 0.60 : 11495 elements\n",
"Info : 0.60 < quality < 0.70 : 11119 elements\n",
"Info : 0.70 < quality < 0.80 : 9732 elements\n",
"Info : 0.80 < quality < 0.90 : 5468 elements\n",
"Info : 0.90 < quality < 1.00 : 1491 elements\n",
"Info : 2287 edge swaps, 100 node relocations (volume = 0.0005625): worst = 0.000309648 / average = 0.608497 (Wall 0.0431121s, CPU 0.043229s)\n",
"Info : 2423 edge swaps, 115 node relocations (volume = 0.0005625): worst = 0.000329345 / average = 0.6088 (Wall 0.0576463s, CPU 0.057853s)\n",
"Info : 2441 edge swaps, 120 node relocations (volume = 0.0005625): worst = 0.000329345 / average = 0.608884 (Wall 0.0702314s, CPU 0.070517s)\n",
"Info : 0.00 < quality < 0.10 : 6 elements\n",
"Info : 0.10 < quality < 0.20 : 1 elements\n",
"Info : 0.20 < quality < 0.30 : 375 elements\n",
"Info : 0.30 < quality < 0.40 : 5510 elements\n",
"Info : 0.40 < quality < 0.50 : 9027 elements\n",
"Info : 0.50 < quality < 0.60 : 11469 elements\n",
"Info : 0.60 < quality < 0.70 : 11237 elements\n",
"Info : 0.70 < quality < 0.80 : 9789 elements\n",
"Info : 0.80 < quality < 0.90 : 5513 elements\n",
"Info : 0.90 < quality < 1.00 : 1489 elements\n",
"Info : Optimizing volume 2\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"Warning : 2 ill-shaped tets are still in the mesh\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Info : Optimization starts (volume = 0.0328953) with worst = 0.000224913 / average = 0.678141:\n",
"Info : 0.00 < quality < 0.10 : 589 elements\n",
"Info : 0.10 < quality < 0.20 : 1579 elements\n",
"Info : 0.20 < quality < 0.30 : 3071 elements\n",
"Info : 0.30 < quality < 0.40 : 7814 elements\n",
"Info : 0.40 < quality < 0.50 : 15648 elements\n",
"Info : 0.50 < quality < 0.60 : 23015 elements\n",
"Info : 0.60 < quality < 0.70 : 27328 elements\n",
"Info : 0.70 < quality < 0.80 : 31555 elements\n",
"Info : 0.80 < quality < 0.90 : 34975 elements\n",
"Info : 0.90 < quality < 1.00 : 15082 elements\n",
"Info : 4755 edge swaps, 225 node relocations (volume = 0.0328953): worst = 0.000316063 / average = 0.692736 (Wall 0.126227s, CPU 0.125273s)\n",
"Info : 4882 edge swaps, 254 node relocations (volume = 0.0328953): worst = 0.000316063 / average = 0.692877 (Wall 0.162867s, CPU 0.161823s)\n",
"Info : 0.00 < quality < 0.10 : 8 elements\n",
"Info : 0.10 < quality < 0.20 : 0 elements\n",
"Info : 0.20 < quality < 0.30 : 240 elements\n",
"Info : 0.30 < quality < 0.40 : 7963 elements\n",
"Info : 0.40 < quality < 0.50 : 15629 elements\n",
"Info : 0.50 < quality < 0.60 : 23060 elements\n",
"Info : 0.60 < quality < 0.70 : 27511 elements\n",
"Info : 0.70 < quality < 0.80 : 31853 elements\n",
"Info : 0.80 < quality < 0.90 : 35235 elements\n",
"Info : 0.90 < quality < 1.00 : 14954 elements\n",
"Info : Done optimizing mesh (Wall 0.5531s, CPU 0.529983s)\n",
"Info : 38460 nodes 253619 elements\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"Warning : 2 ill-shaped tets are still in the mesh\n",
"Warning : ------------------------------\n",
"Warning : Mesh generation error summary\n",
"Warning : 2 warnings\n",
"Warning : 0 errors\n",
"Warning : Check the full log for details\n",
"Warning : ------------------------------\n"
]
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Info : Optimizing mesh (Netgen)...\n",
"Info : Optimizing volume 1\n",
"Info : CalcLocalH: 17912 Points 54439 Elements 30154 Surface Elements \n",
"Info : Remove Illegal Elements \n",
"Info : 5520 illegal tets \n",
"Info : SplitImprove \n",
"Info : badmax = 313633 \n",
"Info : 751 splits performed \n",
"Info : SwapImprove \n",
"Info : 456 swaps performed \n",
"Info : SwapImprove2 \n",
"Info : 7 swaps performed \n",
"Info : 4151 illegal tets \n",
"Info : SplitImprove \n",
"Info : badmax = 73860.9 \n",
"Info : 772 splits performed \n",
"Info : SwapImprove \n",
"Info : 154 swaps performed \n",
"Info : SwapImprove2 \n",
"Info : 141 swaps performed \n",
"Info : 2307 illegal tets \n",
"Info : SplitImprove \n",
"Info : badmax = 9194.41 \n",
"Info : 566 splits performed \n",
"Info : SwapImprove \n",
"Info : 82 swaps performed \n",
"Info : SwapImprove2 \n",
"Info : 66 swaps performed \n",
"Info : 680 illegal tets \n",
"Info : SplitImprove \n",
"Info : badmax = 9194.41 \n",
"Info : 199 splits performed \n",
"Info : SwapImprove \n",
"Info : 15 swaps performed \n",
"Info : SwapImprove2 \n",
"Info : 12 swaps performed \n",
"Info : 52 illegal tets \n",
"Info : SplitImprove \n",
"Info : badmax = 9194.41 \n",
"Info : 16 splits performed \n",
"Info : SwapImprove \n",
"Info : 2 swaps performed \n",
"Info : SwapImprove2 \n",
"Info : 1 swaps performed \n",
"Info : 3 illegal tets \n",
"Info : SplitImprove \n",
"Info : badmax = 9194.41 \n",
"Info : 1 splits performed \n",
"Info : SwapImprove \n",
"Info : 0 swaps performed \n",
"Info : SwapImprove2 \n",
"Info : 0 swaps performed \n",
"Info : 0 illegal tets \n",
"Info : Volume Optimization \n",
"Info : CombineImprove \n",
"Info : 737 elements combined \n",
"Info : ImproveMesh \n",
"Info : Total badness = 144887 \n",
"Info : Total badness = 132150 \n",
"Info : SplitImprove \n",
"Info : badmax = 8364.09 \n",
"Info : 1 splits performed \n",
"Info : ImproveMesh \n",
"Info : Total badness = 132174 \n",
"Info : Total badness = 131064 \n",
"Info : SwapImprove \n",
"Info : 3859 swaps performed \n",
"Info : SwapImprove2 \n",
"Info : 3 swaps performed \n",
"Info : ImproveMesh \n",
"Info : Total badness = 122719 \n",
"Info : Total badness = 117831 \n",
"Info : CombineImprove \n",
"Info : 129 elements combined \n",
"Info : ImproveMesh \n",
"Info : Total badness = 116163 \n",
"Info : Total badness = 115773 \n",
"Info : SplitImprove \n",
"Info : badmax = 7026.46 \n",
"Info : 0 splits performed \n",
"Info : ImproveMesh \n",
"Info : Total badness = 115773 \n",
"Info : Total badness = 115745 \n",
"Info : SwapImprove \n",
"Info : 1511 swaps performed \n",
"Info : SwapImprove2 \n",
"Info : 0 swaps performed \n",
"Info : ImproveMesh \n",
"Info : Total badness = 113432 \n",
"Info : Total badness = 111438 \n",
"Info : CombineImprove \n",
"Info : 33 elements combined \n",
"Info : ImproveMesh \n",
"Info : Total badness = 111069 \n",
"Info : Total badness = 110950 \n",
"Info : SplitImprove \n",
"Info : badmax = 7006.38 \n",
"Info : 0 splits performed \n",
"Info : ImproveMesh \n",
"Info : Total badness = 110950 \n",
"Info : Total badness = 110944 \n",
"Info : SwapImprove \n",
"Info : 732 swaps performed \n",
"Info : SwapImprove2 \n",
"Info : 0 swaps performed \n",
"Info : ImproveMesh \n",
"Info : Total badness = 110077 \n",
"Info : Total badness = 109160 \n",
"Info : Optimizing volume 2\n",
"Info : CalcLocalH: 35431 Points 156676 Elements 37650 Surface Elements \n",
"Info : Remove Illegal Elements \n",
"Info : 0 illegal tets \n",
"Info : Volume Optimization \n",
"Info : CombineImprove \n",
"Info : 644 elements combined \n",
"Info : ImproveMesh \n",
"Info : Total badness = 263887 \n",
"Info : Total badness = 237532 \n",
"Info : SplitImprove \n",
"Info : badmax = 20417.9 \n",
"Info : 1 splits performed \n",
"Info : ImproveMesh \n",
"Info : Total badness = 237611 \n",
"Info : Total badness = 234276 \n",
"Info : SwapImprove \n",
"Info : 8283 swaps performed \n",
"Info : SwapImprove2 \n",
"Info : 0 swaps performed \n",
"Info : ImproveMesh \n",
"Info : Total badness = 219376 \n",
"Info : Total badness = 214404 \n",
"Info : CombineImprove \n",
"Info : 135 elements combined \n",
"Info : ImproveMesh \n",
"Info : Total badness = 213139 \n",
"Info : Total badness = 212597 \n",
"Info : SplitImprove \n",
"Info : badmax = 7390.53 \n",
"Info : 0 splits performed \n",
"Info : ImproveMesh \n",
"Info : Total badness = 212597 \n",
"Info : Total badness = 212495 \n",
"Info : SwapImprove \n",
"Info : 1586 swaps performed \n",
"Info : SwapImprove2 \n",
"Info : 0 swaps performed \n",
"Info : ImproveMesh \n",
"Info : Total badness = 211012 \n",
"Info : Total badness = 209983 \n",
"Info : CombineImprove \n",
"Info : 32 elements combined \n",
"Info : ImproveMesh \n",
"Info : Total badness = 209706 \n",
"Info : Total badness = 209607 \n",
"Info : SplitImprove \n",
"Info : badmax = 7234.06 \n",
"Info : 0 splits performed \n",
"Info : ImproveMesh \n",
"Info : Total badness = 209607 \n",
"Info : Total badness = 209588 \n",
"Info : SwapImprove \n",
"Info : 574 swaps performed \n",
"Info : SwapImprove2 \n",
"Info : 0 swaps performed \n",
"Info : ImproveMesh \n",
"Info : Total badness = 209152 \n",
"Info : Total badness = 208774 \n",
"Info : Done optimizing mesh (Wall 12.149s, CPU 12.1483s)\n",
"Info : Writing 'vivaldi.msh'...\n",
"Mesh saved to vivaldi.mshInfo : Done writing 'vivaldi.msh'\n",
"\n",
" Nodes: 39057\n",
" Elements: 250451\n",
"Info : Writing 'vivaldi.msh'...\n",
"Info : Done writing 'vivaldi.msh'\n"
]
}
],
"source": [
"entities = [\n",
" Entity(\"copper_patch\", dim=2, btype=\"pec\", mesh_order=1, tags=[copper_patch[0][1]]),\n",
" Entity(\"substrate\", dim=3, btype=\"dielectric\", mesh_order=1, tags=[substrate], loss_tan=0.0009, eps_r=2.2, mu_r=1.0),\n",
" Entity(\"air_sphere\", dim=3, btype=\"dielectric\", mesh_order=2, tags=[air_sphere], loss_tan=0.0, eps_r=1.0, mu_r=1.0),\n",
" Entity(\"feed_port\", dim=2, btype=\"waveport\", mesh_order=0, tags=[feed_port_surf]),\n",
"]\n",
"\n",
"pg_map = run_meshing_pipeline(entities)\n",
"refine_near_surfaces(entities[0].dimtags + entities[-1].dimtags, \n",
" wavelength, \n",
" ppw_near=300, \n",
" ppw_far=5, \n",
" transition_distance=wavelength/4,\n",
" set_as_background=True)\n",
"\n",
"generate_3d_mesh(entities, mesh_file, optimize=True)\n",
"gmsh.option.setNumber(\"Mesh.MshFileVersion\", 2.2)\n",
"gmsh.write(mesh_file)\n",
"gmsh.finalize()"
]
},
{
"cell_type": "markdown",
"id": "mesh_md",
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"duration": 0.003282,
"end_time": "2026-05-27T20:02:30.343738+00:00",
"exception": false,
"start_time": "2026-05-27T20:02:30.340456+00:00",
"status": "completed"
},
"tags": []
},
"source": [
"#### Mesh generation"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "93234454",
"metadata": {
"execution": {
"iopub.execute_input": "2026-05-27T20:02:30.351224Z",
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"start_time": "2026-05-27T20:02:30.346970+00:00",
"status": "completed"
},
"tags": []
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Loading mesh file: vivaldi.msh\n",
"Groups to render transparent: air_sphere__None\n",
"\n",
"Mesh loaded successfully with 2 cell blocks\n",
"Found 37650 triangles total\n",
"Physical group tags in mesh: {3: 'copper_patch', 4: 'feed_port', 5: 'air_sphere__substrate', 6: 'air_sphere__None'}\n"
]
},
{
"data": {
"text/html": [
""
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"source": [
"view_mesh(mesh_file, transparent_groups=\"air_sphere__None\", transparent_alpha=0)"
]
}
],
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"display_name": ".venv",
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