Electromagnetic scattering - Revision history

Gkosec: /* Anisotropic cylinder */

2019-10-29T09:41:38Z

Anisotropic cylinder

← Older revision		Revision as of 09:41, 29 October 2019
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	<syntaxhighlight lang="c++" line>		<syntaxhighlight lang="c++" line>
	for (int i : inner.interior()) {		for (int i : inner.interior()) {
	//~~Wave eq~~ on the inner circle		//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) +		mxx * op_inner.der2(i, 0, 0) + myy * op_inner.der2(i, 1, 1) +
	(mxy + myx) * op_inner.der2(i, 0, 1)		(mxy + myx) * op_inner.der2(i, 0, 1)

BlazS at 07:08, 13 June 2019

2019-06-13T07:08:23Z

← Older revision		Revision as of 07:08, 13 June 2019
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	</gallery>		</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>.

			=References=
			<references/>

	Go back to [[Medusa#Examples\|Examples]].		Go back to [[Medusa#Examples\|Examples]].

Jureslak at 08:32, 18 May 2019

2019-05-18T08:32:02Z

← Older revision		Revision as of 08:32, 18 May 2019
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			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]].

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	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]].

BlazS at 18:17, 18 April 2019

2019-04-18T18:17:29Z

← Older revision		Revision as of 18:17, 18 April 2019
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	</syntaxhighlight>		</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:InsideField.png
			File:RealInsideField.png
	File:ImagInsideField.png		File:ImagInsideField.png
	~~File:ImagOutsideField.png~~
	~~File:InsideField.png~~
	File:OutsideField.png		File:OutsideField.png
	~~File:RealInsideField.png~~
	File:RealOutsideField.png		File:RealOutsideField.png
			File:ImagOutsideField.png
	</gallery>		</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].

BlazS at 18:05, 18 April 2019

2019-04-18T18:05:29Z

← Older revision		Revision as of 18:05, 18 April 2019
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	\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>

BlazS at 17:28, 18 April 2019

2019-04-18T17:28:41Z

← Older revision		Revision as of 17:28, 18 April 2019
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	the scattered field $u^s$.		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		The fields $u^s$ and $v$ must satisfy the set of PDE

BlazS at 17:23, 18 April 2019

2019-04-18T17:23:42Z

← Older revision		Revision as of 17:23, 18 April 2019
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	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 ==		== Anisotropic cylinder ==
	Let us first quickly derive the problem we are about to solve. Beginning with the electromagnetic wave equation in anisotropic media		Let us first quickly derive the problem we are about to solve. Beginning with the electromagnetic wave equation in anisotropic media

	~~<math>~~		\[
	\label{eq:frekaniwave}		\label{eq:frekaniwave}
	\nabla \times \left( \underline{\varepsilon}^{-1} \nabla \times \boldsymbol{H} \right) = \omega^2 \mu_0 \varepsilon_0 \underline{\mu} \boldsymbol{H},		\nabla \times \left( \underline{\varepsilon}^{-1} \nabla \times \boldsymbol{H} \right) = \omega^2 \mu_0 \varepsilon_0 \underline{\mu} \boldsymbol{H},
	~~</math>~~		\]

	where <math>\underline{\varepsilon}</math> is the relative dielectric tensor, and <math>\underline{\mu}</math> is the magnetic permeability tensor of the anisotropic material		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>~~		\[
	\label{eq:relepsmi}		\label{eq:relepsmi}
	\underline{\varepsilon} =		\underline{\varepsilon} =
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	\mu_{zx} & \mu_{zy} & \mu_{zz}		\mu_{zx} & \mu_{zy} & \mu_{zz}
	\end{pmatrix}.		\end{pmatrix}.
	~~</math>~~		\]

	Let $D \subset \R^2$ be the cross section of an infinitely long anisotropic		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		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 $v \in C^2(\C)$		$\b{TM}^z$ 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		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		outside of the scatterer. Field $u$ can be further decomposed into the incident $u^i$ and
	the scattered field $u^s$.		the scattered field $u^s$.

			[[File:ScatteringDiagram.png\|center\|400px]]

			The fields $u^s$ and $v$ must satisfy the set of PDE
			\begin{align*}
			\nabla \cdot A \nabla v + \epsilon_r k^2 \thinspace v = 0 \qquad &\text{in} \quad D \label{eq:inner} \\
			\nabla^2 u^s + k^2 \thinspace u^s = 0 \qquad &\text{in} \quad \Omega \setminus D \label{eq:outer}
			\end{align*}
			with boundary conditions
			\begin{align*}
			v - u^s =u^i \qquad &\text{on} \quad \partial D \label{eq:BC1} \\
			\frac{\partial v}{\partial n_A\!\!\!} - \frac{\partial u^s}{\partial n} = \frac{\partial u^i}{\partial n} \qquad &\text{on} \quad \partial D \label{eq:BC2} \\
			\frac{\partial u^s}{\partial n} + \left( ik + \frac{1}{2r_2} \right)u^s = 0 \quad &\text{on} \quad \partial \Omega
			\end{align*}
			where $k = \omega \sqrt{\mu_0 \epsilon_0} = \frac{2 \pi}{\lambda}$ is the wave
			number of free space, $\mu_0$ and $\epsilon_0$ are magnetic permeability and
			electric permittivity of free space, while $\epsilon_r$ is the relative electric
			permittivity of the scatterer. Relative magnetic permeability matrix $A$ is of
			the following form
			\begin{equation*}
			A = \frac{1}{\mu_{xx}\mu_{yy} - \mu_{xy}^2}
			\begin{pmatrix}
			\mu_{xx} & \mu_{xy} \\
			\mu_{xy} & \mu_{yy}
			\end{pmatrix}
			\end{equation*}
			and the anisotropic normal derivative is calculated as
			\begin{equation*}
			\frac{\partial v}{\partial n_A\!\!\!} = n \cdot A \nabla {v}.
			\end{equation*}

BlazS at 17:11, 18 April 2019

2019-04-18T17:11:15Z

← Older revision		Revision as of 17:11, 18 April 2019
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	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 ==

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

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

BlazS: /* Anisotropic cylinder */

2019-04-17T17:18:09Z

Anisotropic cylinder

← Older revision		Revision as of 17:18, 17 April 2019
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	</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 $u^s$.