Finish blog post

2023-10-02.html

@@ -113,29 +113,29 @@ <h1>Strongly singular integrals over curved elements</h1>
 
                     <h2>Background on integral equations</h2>
 
-                    <p>The Helmholtz equation, which has the form
+                    <p>The <a href="https://en.wikipedia.org/wiki/Helmholtz_equation">Helmholtz equation</a>, which has the form
                     $$
                     \Delta u + k^2u = 0,
                     $$
-                    appears when one looks for time-harmonic solutions to the wave equation&mdash;if \(v(\boldsymbol{x},t)=u(\boldsymbol{x})e^{-i\omega t}\) is a solution to \(v_{tt}=c^2\Delta v\), then \(u\) satisfies the Helmholtz equation with wavenumber \(k=\omega /c\). It is of fundamental importance in science and engineering, with applications as diverse as noise scattering, radar and sonar technology, and seismology. For instance, given an incident acoustic wave \(u^i\) that is a solution to Helmholtz equation in \(\mathbb{R}^3\), the outgoing scattered field \(u^s\) generated by a bounded obstacle \(D\) is also a solution to Helmholtz equation, in \(\mathbb{R}^3\setminus\overline{D}\), with \(u^i+u^s=0\) on the boundary \(\partial D\) (for sound-soft scattering).</p>
+                    appears when one looks for time-harmonic solutions to the <a href="https://en.wikipedia.org/wiki/Wave_equation">wave equation</a>&mdash;if \(v(\boldsymbol{x},t)=u(\boldsymbol{x})e^{-i\omega t}\) is a solution to \(v_{tt}=c^2\Delta v\), then \(u\) satisfies the Helmholtz equation with <a href="https://en.wikipedia.org/wiki/Wavenumber">wavenumber</a> \(k=\omega /c\). It is of fundamental importance in science and engineering, with applications as diverse as noise scattering, <a href="https://en.wikipedia.org/wiki/Radar">radar</a> and <a href="https://en.wikipedia.org/wiki/Sonar">sonar</a> technology, and seismology. For instance, given an incident acoustic wave \(u^i\) that is a solution to the Helmholtz equation in \(\mathbb{R}^3\), the outgoing scattered field \(u^s\) generated by a bounded obstacle \(D\) is also a solution to Helmholtz equation, in \(\mathbb{R}^3\setminus\overline{D}\), with \(u^i+u^s=0\) on the boundary \(\partial D\) (for sound-soft scattering).</p>
 
-                    <p>For obstacles \(D\) whose complements are connected, a popular technique for calculating scattered fields is based on integral equations. As an example, one can show that the solution \(u^s\) to the sound-soft scattering problem of the previous paragraph may be obtained by solving
+                    <p>For obstacles \(D\) whose complements are <a href="https://en.wikipedia.org/wiki/Connected_space">connected</a>, a popular technique for calculating scattered fields is based on <a href="https://en.wikipedia.org/wiki/Integral_equation">integral equations</a>. As an example, one can show that the solution \(u^s\) to the sound-soft scattering problem of the previous paragraph may be obtained by solving
                     $$
-                    \int_{\partial D}\left\{\frac{\partial G(\boldsymbol{x},\boldsymbol{y})}{\partial\boldsymbol{n}(\boldsymbol{y})} - i\eta G(\boldsymbol{x},\boldsymbol{y})\right\}\varphi^s(\boldsymbol{y})dS(\boldsymbol{y}) + \frac{\varphi^s(\boldsymbol{x})}{2} = -u^i(\boldsymbol{x}), \quad \boldsymbol{x}\in\partial D,
+                    \int_{\partial D}\left\{\frac{\partial G(\boldsymbol{x},\boldsymbol{y})}{\partial\boldsymbol{n}(\boldsymbol{y})} - i\eta G(\boldsymbol{x},\boldsymbol{y})\right\}\varphi^s(\boldsymbol{y})dS(\boldsymbol{y}) + \frac{\varphi^s(\boldsymbol{x})}{2} = -u^i(\boldsymbol{x}), \;\; \boldsymbol{x}\in\partial D,
                     $$
                     for some arbitrary real number \(\eta\neq0\) such that \(\eta\mathrm{Re}(k)\geq0\). Once the equation is solved for \(\varphi^s\), the scattered field is given by 
                     $$
-                    u^s(\boldsymbol{x}) =\int_{\partial D}\left\{\frac{\partial G(\boldsymbol{x},\boldsymbol{y})}{\partial\boldsymbol{n}(\boldsymbol{y})} - i\eta G(\boldsymbol{x},\boldsymbol{y})\right\}\varphi^s(\boldsymbol{y})dS(\boldsymbol{y}), \quad \boldsymbol{x}\in\mathbb{R}^3\setminus{\overline{D}}.
+                    u^s(\boldsymbol{x}) =\int_{\partial D}\left\{\frac{\partial G(\boldsymbol{x},\boldsymbol{y})}{\partial\boldsymbol{n}(\boldsymbol{y})} - i\eta G(\boldsymbol{x},\boldsymbol{y})\right\}\varphi^s(\boldsymbol{y})dS(\boldsymbol{y}), \;\; \boldsymbol{x}\in\mathbb{R}^3\setminus{\overline{D}}.
                     $$
-                    The function \(G\) is the Green's function of the Helmholtz equation in 3D,
+                    The function \(G\) is the <a href="https://en.wikipedia.org/wiki/Green%27s_function">Green's function</a> of the Helmholtz equation in 3D,
                     $$
                     G(\boldsymbol{x},\boldsymbol{y}) = \frac{1}{4\pi}\frac{e^{ik\vert\boldsymbol{x}-\boldsymbol{y}\vert}}{\vert\boldsymbol{x}-\boldsymbol{y}\vert}.
                     $$
-                    One of the challenges one faces when solving integral equations is the computation of singular integrals&mdash;when \(\boldsymbol{x}\) is close or equal to \(\boldsymbol{y}\), the Green's function and its derivatives become (numerically) unbounded.
+                    </p>
 
                     <h2>Challenges</h2>
 
-                    <p>One of the challenges one faces when solving integral equations is the computation of singular integrals&mdash;when \(\boldsymbol{x}\) is close or equal to \(\boldsymbol{y}\), the Green's function and its derivatives become (numerically) unbounded.here are many specialized methods to compute such integrals, including singularity subtraction, singularity cancellation, and the continuation approach.  We described, in a previous <a href="https://hadrien-montanelli.github.io/2021-11-25.html">blog post</a>, algorithms to compute weakly singular integrals using singularity subtraction and the continuation approach. The term <i>weakly singular</i> refers to integrals with singularities of the same type as the Green's function. We describe, here, our method to <i>strongly singular</i> integrals, which have singularities of the same nature as the derivatives of the Green's function.</p>
+                    <p>One of the challenges one faces when solving integral equations is the computation of <a href="https://en.wikipedia.org/wiki/Singular_integral">singular integrals</a>&mdash;when \(\boldsymbol{x}\) is equal to \(\boldsymbol{y}\), the Green's function and its derivatives are unbounded. There are many specialized methods to compute such integrals, including singularity subtraction, singularity cancellation, and the continuation approach. We described, in a previous <a href="https://hadrien-montanelli.github.io/2021-11-25.html">blog post</a>, algorithms to compute weakly singular integrals using singularity subtraction and the continuation approach. The term <i>weakly singular</i> refers to integrals with singularities of the same type as the Green's function. We describe, here, our method to <i>strongly singular</i> integrals, which have singularities of the same nature as the derivatives of the Green's function.</p>
 
                     <h2>Computing strongly singular integrals</h2>
 
@@ -158,7 +158,7 @@ <h2>Computing strongly singular integrals</h2>
                     for some scalar \(h\), which may be negative, obtained via
                     $$
                     h = (\boldsymbol{x}_0 - F(\boldsymbol{\widehat{x}}_0)) \cdot \frac{\boldsymbol{\widehat{n}}_0}{\vert\boldsymbol{\widehat{n}}_0\vert}.
-                    $$.
+                    $$
                     </p>
 
                     <p><i>Step 3.</i> We compute the strongly singular term,
@@ -168,7 +168,7 @@ <h2>Computing strongly singular integrals</h2>
                     where \(J_0\) is the Jacobian matrix at \(\boldsymbol{\widehat{x}}_0\) and \(\varphi_0=\varphi(\boldsymbol{\widehat{x}}_0)\), as well as the weakly singular term \(T_{-1}\). We subtract them from/add them to the integrand,
                     $$
                     I(\boldsymbol{x}_0) = \, \int_{\widehat{T}}\left[\frac{(F(\boldsymbol{\widehat{x}})-\boldsymbol{x}_0)\cdot\boldsymbol{\widehat{n}}(\boldsymbol{\widehat{x}})}{\vert F(\boldsymbol{\widehat{x}})-\boldsymbol{x}_0\vert^3}\varphi(\boldsymbol{\widehat{x}}) - T_{-2}(\boldsymbol{\widehat{x}},h) - T_{-1}(\boldsymbol{\widehat{x}},h)\right]dS(\boldsymbol{\widehat{x}}) \\
-                    + \int_{\widehat{T}}T_{-2}(\boldsymbol{\widehat{x}},h)dS(\boldsymbol{\widehat{x}}) + \int_{\widehat{T}}T_{-1}(\boldsymbol{\widehat{x}},h)dS(\boldsymbol{\widehat{x}}). \nonumber
+                    + \int_{\widehat{T}}T_{-2}(\boldsymbol{\widehat{x}},h)dS(\boldsymbol{\widehat{x}}) + \int_{\widehat{T}}T_{-1}(\boldsymbol{\widehat{x}},h)dS(\boldsymbol{\widehat{x}}).
                     $$
                     The first integral is regularized&mdash;it may be computed with Gauss quadrature on triangles. The last two integrals are singular or near-singular and will be computed in steps 4&ndash;5. 
                     </p>
@@ -192,17 +192,17 @@ <h2>Computing strongly singular integrals</h2>
 
                     <h2>Numerical experiments</h2>
 
-                    <p>We consider the sound-soft scattering of a plane wave \(u^i(r,\theta)=e^{ikr\cos\theta}\) by the unit sphere. We utilize the combined boundary integral equation with \(\eta=k/2\) and discretize it with a boundary element method with quadratic basis functions (\(p=2\)) and quadratic triangles (\(q=2\)). We take \(k=2\pi\), solve for \(\varphi^s\), and evaluate the far-field pattern
+                    <p>We consider the sound-soft scattering of a plane wave \(u^i(r,\theta)=e^{ikr\cos\theta}\) by the unit sphere. We choose \(\eta=k/2\) and discretize the boundary integral equation with a boundary element method with linear basis functions/planar triangles (\(p=q=1\)) and quadratic basis functions/triangles (\(p=q=2\)). We take \(k=2\pi\), solve for \(\varphi^s\), and evaluate the far-field pattern
                     $$
-                    u_\infty(\boldsymbol{\theta}) = \frac{1}{4\pi}\int_{\partial D} e^{-ik\boldsymbol{\theta}\cdot\boldsymbol{y}}\varphi^s(\boldsymbol{y})dS(\boldsymbol{y}), \quad \boldsymbol{\theta}\in\mathbb{S}^2,
+                    u_\infty(\boldsymbol{\theta}) = \frac{1}{4\pi}\int_{\partial D} e^{-ik\boldsymbol{\theta}\cdot\boldsymbol{y}}\varphi^s(\boldsymbol{y})dS(\boldsymbol{y}), \;\; \boldsymbol{\theta}\in\mathbb{S}^2,
                     $$
-                    for an increasing number of triangles. We plot the relative error in the far-field pattern below. We observe quartic superconvergence as the mesh size \(h\to0\).</p>
+                    for an increasing number of triangles. We observe quartic superconvergence for quadratic elements as the mesh size \(h\to0\).</p>
 
                     <center>
                     <img src="/blog/test_cv_sphere_2pi.jpg" class="img-responsive">
                     </center>
 
-                    <p>We now take \(k\in\{2\pi,4\pi,8\pi,16\pi\}\), solve for \(\varphi^s\) and seek the number of degrees of freedom needed to reach a relative error on the far-field pattern around \(10^{-3}\) for each \(k\). We report the results below for linear basis functions/planar triangles (\(p=q=1\)) and quadratic basis functions/triangles (\(p=q=2\)). We observe that using quadratic basis functions and triangles reduces the number of degrees of freedom by a factor of about four, while also decreasing computer time by a similar factor at high frequency.</p>
+                    <p>We now take \(k\in\{2\pi,4\pi,8\pi,16\pi\}\), solve for \(\varphi^s\) and seek the number of degrees of freedom needed to reach a relative error on the far-field pattern around \(10^{-3}\) for each \(k\). We observe that using quadratic elements reduces the number of degrees of freedom by a factor of about four, while also decreasing computer time by a similar factor at high frequency.</p>
 
                     <center>
                     <img src="/blog/test_cv_sphere_DoFs.jpg" class="img-responsive">