Electromagnetic scattering - Revision history

Gkosec: /* Anisotropic cylinder */

2019-10-29T09:41:38Z

‎Anisotropic cylinder

BlazS at 07:08, 13 June 2019

2019-06-13T07:08:23Z

Jureslak at 08:32, 18 May 2019

2019-05-18T08:32:02Z

BlazS at 18:17, 18 April 2019

2019-04-18T18:17:29Z

BlazS at 18:05, 18 April 2019

2019-04-18T18:05:29Z

BlazS at 17:28, 18 April 2019

2019-04-18T17:28:41Z

BlazS at 17:23, 18 April 2019

2019-04-18T17:23:42Z

BlazS at 17:11, 18 April 2019

2019-04-18T17:11:15Z

Jureslak: Jureslak moved page Coupled domains - electromagnetic scattering to Electromagnetic scattering without leaving a redirect

2019-04-18T10:28:48Z

Jureslak moved page Coupled domains - electromagnetic scattering to Electromagnetic scattering without leaving a redirect

BlazS: /* Anisotropic cylinder */

2019-04-17T17:18:09Z

‎Anisotropic cylinder

@@ Line 71: / Line 71: @@
 <syntaxhighlight lang="c++" line>
      for (int i : inner.interior()) {
-         //Wave eq on the inner circle
+         //Eq on the inner circle
          mxx * op_inner.der2(i, 0, 0) + myy * op_inner.der2(i, 1, 1) +
          (mxy + myx) * op_inner.der2(i, 0, 1)

@@ Line 121: / Line 121: @@
 </gallery>
-The example can be found here [https://gitlab.com/e62Lab/medusa/blob/dev/examples/electromagnetic_scattering/anisotropic_cylinder.cpp anisotropic_cylinder.cpp], and the matlab plot script can be found here [https://gitlab.com/e62Lab/medusa/blob/dev/examples/electromagnetic_scattering/anisotropic_cylinder.m anisotropic_cylinder.m].
+The example can be found here [https://gitlab.com/e62Lab/medusa/blob/dev/examples/electromagnetic_scattering/anisotropic_cylinder.cpp anisotropic_cylinder.cpp], and the matlab plot script can be found here [https://gitlab.com/e62Lab/medusa/blob/dev/examples/electromagnetic_scattering/anisotropic_cylinder.m anisotropic_cylinder.m]. A more comprehensive description of thre problem can be found in the paper <ref name="StojanovicRef">Stojanovič, B., & Slak, J., & Kosec, G. (2019). RBF-FD Solution of Electromagnetic Scattering Problem. MIPRO 2019 : 42nd International Convention, May 20 -24, 2019, Opatija, Croatia : proceedings.</ref>.
 Go back to [[Medusa#Examples|Examples]].

← Older revision		Revision as of 08:32, 18 May 2019
Line 1:		Line 1:
		+	Go back to [[Medusa#Examples\|Examples]].
		+
	In this example we will show how to solve an electromagnetic scattering problem in Medusa. The example uses both complex numbers as well as domain coupling, so we recommend firstly reading tutorials on [[Coupled domains]] and [[Complex-valued problems]].		In this example we will show how to solve an electromagnetic scattering problem in Medusa. The example uses both complex numbers as well as domain coupling, so we recommend firstly reading tutorials on [[Coupled domains]] and [[Complex-valued problems]].

Line 120:		Line 122:

	The example can be found here [https://gitlab.com/e62Lab/medusa/blob/dev/examples/electromagnetic_scattering/anisotropic_cylinder.cpp anisotropic_cylinder.cpp], and the matlab plot script can be found here [https://gitlab.com/e62Lab/medusa/blob/dev/examples/electromagnetic_scattering/anisotropic_cylinder.m anisotropic_cylinder.m].		The example can be found here [https://gitlab.com/e62Lab/medusa/blob/dev/examples/electromagnetic_scattering/anisotropic_cylinder.cpp anisotropic_cylinder.cpp], and the matlab plot script can be found here [https://gitlab.com/e62Lab/medusa/blob/dev/examples/electromagnetic_scattering/anisotropic_cylinder.m anisotropic_cylinder.m].
		+
		+	Go back to [[Medusa#Examples\|Examples]].

@@ Line 108: / Line 108: @@
 </syntaxhighlight>
-Solving the system gives the complex valued solution to both fields.
+Solving the system gives the complex valued solution to both fields. The solution is plotted below.
-<gallery>
+<gallery class="centered" mode=packed heights=500px>
 File:ImagInsideField.png
 File:OutsideField.png
 File:RealOutsideField.png
 </gallery>

← Older revision		Revision as of 18:05, 18 April 2019
Line 64:		Line 64:
	\frac{\partial v}{\partial n_A\!\!\!} = n \cdot A \nabla {v}.		\frac{\partial v}{\partial n_A\!\!\!} = n \cdot A \nabla {v}.
	\end{equation*}		\end{equation*}
		+
		+	The system is implemented below
		+
		+	<syntaxhighlight lang="c++" line>
		+	for (int i : inner.interior()) {
		+	//Wave eq on the inner circle
		+	//(mxxv_xx + myyv_yy + mxyv_xy + myxv_yx) + k0^2erv = 0
		+	mxx * op_inner.der2(i, 0, 0) + myy * op_inner.der2(i, 1, 1) +
		+	(mxy + myx) * op_inner.der2(i, 0, 1)
		+	+ k0 * k0 * er * op_inner.value(i) = 0;
		+	}
		+	for (int i : inner.boundary()) {
		+	//dirichlet, u_inside - u_scattered = u_incident <- incoming wave from t0 angle
		+	double x = inner.pos(i, 0);
		+	double y = inner.pos(i, 1);
		+	double theta = atan2(y, x); // get phase
		+
		+	// in the upper side of the matrix
		+	std::complex<double> incident = std::exp(1.0i * k0 * (x * std::cos(t0) + y * std::sin(t0)));
		+	op_inner.value(i) + (-1) * op_outer.value(inner_bnd[i], N_outer + i) = incident;
		+
		+	// Neumann boundary condition on the inside circle
		+	auto norm = inner.normal(i);
		+	double n0 = norm[0];
		+	double n1 = norm[1];
		+
		+	// Anisotropic normal derivative
		+	// (n0mxx+n1myx)v_x + (n0mxy+n1myy)v_y + du/dn = d(incident)/dn
		+	op_outer.neumann(inner_bnd[i], outer.normal(inner_bnd[i]))
		+	+ (n0 * mxx + n1 * myx) * op_inner.der1(i, 0, -N_outer + inner_bnd[i])
		+	+ (n0 * mxy + n1 * myy) * op_inner.der1(i, 1, -N_outer + inner_bnd[i])
		+	= 1.0i * k0 * std::cos(theta - t0) *
		+	std::exp(1.0i * k0 * (x * std::cos(t0) + y * std::sin(t0)));
		+	}
		+	for (int i : outer.interior()) {
		+	// wave equation for the outer region
		+	op_outer.lap(i) + k0 * k0 * op_outer.value(i) = 0;
		+	}
		+	for (int i : outer_bnd) {
		+	// Sommerfeld boundary condition
		+	op_outer.neumann(i, outer.normal(i)) + (k0 * 1.0i + 1 / (2 * r2)) * op_outer.value(i) = 0.0;
		+	}
		+	</syntaxhighlight>
		+
		+	Solving the system gives the complex valued solution to both fields.
		+
		+	<gallery>
		+	File:ImagInsideField.png
		+	File:ImagOutsideField.png
		+	File:InsideField.png
		+	File:OutsideField.png
		+	File:RealInsideField.png
		+	File:RealOutsideField.png
		+	</gallery>

@@ Line 35: / Line 35: @@
 the scattered field $u^s$.
-[[File:ScatteringDiagram.png|center|400px]]
+[[File:ScatteringDiagram.png|center|300px]]
 The fields $u^s$ and $v$ must satisfy the set of PDE

@@ Line 1: / Line 1: @@
-In this example we will show how to solve an electromagnetic scattering problem in Medusa. The example uses both complex numbers as well as domain coupling, so we recommend firstly reading [[Coupled domains]] and [[Complex Numbers]]
+In this example we will show how to solve an electromagnetic scattering problem in Medusa. The example uses both complex numbers as well as domain coupling, so we recommend firstly reading tutorials on [[Coupled domains]] and [[Complex-valued problems]].
 == Anisotropic cylinder ==
 Let us first quickly derive the problem we are about to solve. Beginning with the electromagnetic wave equation in anisotropic media
-<math>
+\[
 \label{eq:frekaniwave}
 \nabla \times \left( \underline{\varepsilon}^{-1} \nabla \times \boldsymbol{H} \right) = \omega^2 \mu_0 \varepsilon_0 \underline{\mu} \boldsymbol{H},
-</math>
+\]
 where <math>\underline{\varepsilon}</math> is the relative dielectric tensor, and <math>\underline{\mu}</math> is the magnetic permeability tensor of the anisotropic material
-<math>
+\[
 \label{eq:relepsmi}
 \underline{\varepsilon} =
@@ Line 26: / Line 25: @@
 \mu_{zx} & \mu_{zy} & \mu_{zz}
 \end{pmatrix}.
-</math>
+\]
 Let $D \subset \R^2$ be the cross section of an infinitely long anisotropic
 dielectric cylindrical scatterer with its axis alligned with the $z$-axis, surrounded by a free space, with an outward
-normal $n$ on boundary $\partial D$. The scatterer is excited by an $e^{i \omega t}$ time-harmonic plane wave with
+normal $n$ on boundary $\partial D$. The cylinder is isotropic along its axis, and is excited by an $e^{i \omega t}$ time-harmonic plane wave with
 $\b{TM}^z$ polarization, with $\omega$ standing for its angular frequency. Let $v \in C^2(\C)$
 denote the complex valued field inside the scatterer and $u \in C^2(\C)$ the field
 outside of the scatterer. Field $u$ can be further decomposed into the incident $u^i$ and
 the scattered field $u^s$.

← Older revision		Revision as of 17:11, 18 April 2019
Line 1:		Line 1:
−	In this example we will ~~showcase the ability of the medusa library~~ to solve ~~coupled domain problems. We will solve the problem of~~ electromagnetic scattering ~~on an anisotropic cylinder~~.	+	In this example we will show how to solve an electromagnetic scattering problem in Medusa. The example uses both complex numbers as well as domain coupling, so we recommend firstly reading [[Coupled domains]] and [[Complex Numbers]]

	== Anisotropic cylinder ==		== Anisotropic cylinder ==

← Older revision	Revision as of 10:28, 18 April 2019
(No difference)

← Older revision		Revision as of 17:18, 17 April 2019
Line 9:		Line 9:
	</math>		</math>

−	where <math>\underline{\varepsilon}</math> is the relative dielectric tensor, and <math>\underline{\mu}</math> is the magnetic permeability tensor of the anisotropic material.	+	where <math>\underline{\varepsilon}</math> is the relative dielectric tensor, and <math>\underline{\mu}</math> is the magnetic permeability tensor of the anisotropic material
		+
		+	<math>
		+	\label{eq:relepsmi}
		+	\underline{\varepsilon} =
		+	\begin{pmatrix}
		+	\varepsilon_{xx} & \varepsilon_{xy} & \varepsilon_{xz} \\
		+	\varepsilon_{yx} & \varepsilon_{yy} & \varepsilon_{yz} \\
		+	\varepsilon_{zx} & \varepsilon_{zy} & \varepsilon_{zz}
		+	\end{pmatrix}
		+	, \qquad
		+	\underline{\mu} =
		+	\begin{pmatrix}
		+	\mu_{xx} & \mu_{xy} & \mu_{xz} \\
		+	\mu_{yx} & \mu_{yy} & \mu_{yz} \\
		+	\mu_{zx} & \mu_{zy} & \mu_{zz}
		+	\end{pmatrix}.
		+	</math>
		+
		+	Let $D \subset \R^2$ be the cross section of an infinitely long anisotropic
		+	dielectric cylindrical scatterer with its axis alligned with the $z$-axis, surrounded by a free space, with an outward
		+	normal $n$ on boundary $\partial D$. The scatterer is excited by an $e^{i \omega t}$ time-harmonic plane wave with
		+	$\b{TM}^z$ polarization, with $\omega$ standing for its angular frequency. Let $v \in C^2(\C)$
		+	denote the complex valued field inside the scatterer and $u \in C^2(\C)$ the field
		+	outside of the scatterer. Field $u$ can be further decomposed into the incident $u^i$ and
		+	the scattered field $u^s$.