Added symmetry to KochCurve preset

examples/Supplementary code for Proc R Soc paper 'Integral equation methods for acoustic scattering by fractals'.ipynb

@@ -6,9 +6,9 @@
    "source": [
     "# Supplementary code for  *Integral equation methods for acoustic scattering by fractals*\n",
     "\n",
-    "This notebook contains sample code which can be used to reproduce the field plots in Section 5(a) of the Proc. R. Soc. paper **Integral equation methods for acoustic scattering by fractals**, by *A. M. Caetano, S. N. Chandler-Wilde, X. Claeys, A. Gibbs, D. P. Hewett and A. Moiola*.\n",
+    "This notebook contains sample code which can be used to reproduce the 2D field plots in Section 5(a) of the Proc. R. Soc. paper **Integral equation methods for acoustic scattering by fractals**, by *A. M. Caetano, S. N. Chandler-Wilde, X. Claeys, A. Gibbs, D. P. Hewett and A. Moiola*.\n",
     "\n",
-    "All results in the above paper were produced using the open-source Julia package [IFSintegrals](https://github.com/AndrewGibbs/IFSintegrals). A technical understanding of this package is **not** required to understand the key ideas in this notebook. Throughout this notebook, we will make frequent references to equation numbers of the associated paper."
+    "All results in the above paper were produced using the open-source Julia package [IFSintegrals](https://github.com/AndrewGibbs/IFSintegrals). A technical understanding of this package is **not** required to understand the key ideas in this notebook. Throughout this notebook, we will make frequent references to equation and other numbers of the associated paper."
    ]
   },
   {
@@ -49,12 +49,13 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The following values of ```eg_index``` correspond to different scattering configurations, each in $\\mathbb{R}^2$:\n",
-    "1. The Koch curve (see Example 2.3)\n",
-    "2. Cantor set  with contraction factor $\\rho=1/3$\n",
-    "3. Cantor dust with contraction factor $\\rho=1/3$\n",
-    "4. A Koch snowflake, constructed as the union of three Koch curves (see Fig. 8(b))\n",
-    "5. The union of a Cantor set with contraction factor $\\rho=1/3$ and a Koch curve.\n",
+    "The following values of ```eg_index``` correspond to different scattering configurations, each in $\\mathbb{R}^2$. Plots for 1,3,4 and 5 are included in the paper:\n",
+    "1. The Koch curve (see Example 2.3 and Figure 5(a))\n",
+    "2. Cantor set $C$ with contraction factor $\\rho=1/3$ (see Example 2.2)\n",
+    "3. Cantor dust (i.e. $C\\times C$) with contraction factor $\\rho=1/3$\n",
+    "4. A Koch snowflake boundary, constructed as the union of three Koch curves (see Fig. 8(b))\n",
+    "5. A solid Koch snowflake (see Example 2.4 and Fig 8(a))\n",
+    "6. The union of a Cantor set with contraction factor $\\rho=1/3$ and a Koch curve.\n",
     "\n",
     "Now we define our scatterer $\\Gamma$ and plot it."
    ]
@@ -67,18 +68,20 @@
    "source": [
     "if eg_index == 1 # Koch curve\n",
     "    Γ = KochCurve()\n",
-    "elseif eg_index == 2 # Cantor dust\n",
+    "elseif eg_index == 2 # Cantor set\n",
     "    Γ = CantorSet() + [0,0]\n",
-    "elseif eg_index == 3 # Cantor set\n",
+    "elseif eg_index == 3 # Cantor dust\n",
     "    Γ = CantorDust()\n",
-    "elseif eg_index == 4 # Koch snowflake, as a union of Koch curves\n",
+    "elseif eg_index == 4 # Koch snowflake boundary, as a union of Koch curves\n",
     "    Koch_height = 0.288675134594813\n",
     "    γ = KochCurve()+[-1/2,Koch_height]\n",
     "    γ₁ = γ\n",
     "    γ₂ = rotation2(2π/3)*γ\n",
     "    γ₃ = rotation2(4π/3)*γ\n",
     "    Γ = UnionInvariantMeasure([γ₁,γ₂,γ₃])\n",
-    "elseif eg_index == 5 # Cantor set union Koch curve\n",
+    "elseif eg_index == 5\n",
+    "    Γ = KochFlake()\n",
+    "elseif eg_index == 6 # Cantor set union Koch curve\n",
     "    γ₁ = CantorSet() + [0,-0.2] # embed Cantor set in ℝ² and translate\n",
     "    γ₂ = KochCurve()\n",
     "    Γ = UnionInvariantMeasure([γ₁, γ₂])\n",
@@ -92,7 +95,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "We note that for cases ```eg_index```$=4,5$, we are translating the fractals by adding a vector, and ```eg_index```$=4$ also has a rotation. By default, the Cantor set is defined as a subset of $\\mathbb{R}$. By adding $[0,0]$ in line 4 (similarly in line 15), the Cantor set is embedded in $\\mathbb{R}^2$."
+    "We note that for cases ```eg_index```$=4,6$, we are translating the fractals by adding a vector, and ```eg_index```$=4$ also has a rotation. By default, the Cantor set is defined as a subset of $\\mathbb{R}$. By adding $[0,0]$ in line 4 (similarly in line 17), the Cantor set is embedded in $\\mathbb{R}^2$."
    ]
   },
   {
@@ -143,15 +146,15 @@
    "metadata": {},
    "outputs": [],
    "source": [
-    "h = 0.05 # max mesh width of BEM elements\n",
-    "hQ = 0.01; # max mesh width for quadrature in BEM"
+    "h = 0.05 # max element diameter of the IEM elements\n",
+    "hQ = 0.01; # max element diameter for IEM quadrature in "
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Define the boundary integral operator $A$ of (3.15), and its discretised stucture $A_h$, which contains essential information about the mesh, Galerkin matrix (4.16), etc. Then solve the system to obtain the discrete solution $\\phi_h$."
+    "Define the boundary integral operator $A$ of (3.15), and its discretised stucture $A_h$, which contains essential information about the mesh, Galerkin matrix (4.16), etc. Then solve the system to obtain the discrete solution $\\phi_N$."
    ]
   },
   {
@@ -169,7 +172,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The final line above represents the Galerkin system (4.15). Now we can construct an approximation to the total field using (3.14) and (3.10):"
+    "The final line above solves the Galerkin system (4.15). Now we can construct an approximation to the total field using (3.14) and (3.10):"
    ]
   },
   {
@@ -178,7 +181,7 @@
    "metadata": {},
    "outputs": [],
    "source": [
-    "𝘈ϕ_N = SingleLayerPotentialHelmholtz(ϕ_N,k; h_quad = hQ)# returns function\n",
+    "𝘈ϕ_N = SingleLayerPotentialHelmholtz(ϕ_N,k; h_quad = hQ)# returns potential as function\n",
     "u_N(x) = uⁱ(x) + 𝘈ϕ_N(x) # total field"
    ]
   },
@@ -227,7 +230,7 @@
    "outputs": [],
    "source": [
     "heatmap(x₁,x₂,transpose(real(u_N.(X))), aspect_ratio = 1, \n",
-    "    title=\"Total field\", label=false, c = :jet)\n",
+    "    title=\"Rela part of total field\", label=false, c = :jet)\n",
     "plot!(Γ,color = \"black\", markersize=0.5, label=false)"
    ]
   }

src/fractal_presets.jl

@@ -309,7 +309,10 @@ function KochCurve()
                             false true true true;
                             false false true true]
 
-    return InvariantMeasure(IFS, connectedness = touching_singularities);
+    G = get_group_operations2D(GroupGenerator2D(1,[π/2],[0.5,0]))
+    return InvariantMeasure(IFS, 
+                            connectedness = touching_singularities,
+                            symmetry_group = G);
 end