Computational electromagnetics - Revision history

BlazS: /* Streched Coordinate Perfectly Matched Layer */

2020-09-27T15:21:08Z

‎Streched Coordinate Perfectly Matched Layer

BlazS: /* Streched Coordinate Perfectly Matched Layer */

2020-09-27T08:11:27Z

‎Streched Coordinate Perfectly Matched Layer

BlazS: /* Streched Coordinate Perfectly Matched Layer */

2020-09-27T08:10:14Z

‎Streched Coordinate Perfectly Matched Layer

BlazS: /* Streched Coordinate Perfectly Matched Layer */

2020-09-27T08:07:27Z

‎Streched Coordinate Perfectly Matched Layer

BlazS: /* Case studies */

2020-08-11T17:34:56Z

‎Case studies

BlazS: /* References */

2020-08-11T14:44:05Z

‎References

BlazS: /* Modeling in the frequency domain */

2020-08-11T14:39:00Z

‎Modeling in the frequency domain

BlazS at 13:52, 11 August 2020

2020-08-11T13:52:17Z

BlazS: /* Absorbing boundary conditions */

2020-08-11T09:19:03Z

‎Absorbing boundary conditions

BlazS: /* Transmittance and Reflectivity */

2020-08-11T09:06:57Z

‎Transmittance and Reflectivity

@@ Line 238: / Line 238: @@
 	\frac{1}{s_x} \frac{\partial}{\partial x} \left(\frac{1}{s_x}\dpar{E_x}{x}\right) +
 	\frac{1}{s_y s_x}\frac{\partial E_y}{\partial x \partial y} +
-	\frac{1}{s_z s_x}\frac{\partial E_z}{\partial z \partial z} \\
+	\frac{1}{s_z s_x}\frac{\partial E_z}{\partial x \partial z} \\
 	\frac{1}{s_x s_y}\frac{\partial E_x}{\partial y \partial x} +
 	\frac{1}{s_y} \frac{\partial}{\partial y} \left(\frac{1}{s_y}\dpar{E_y}{y}\right) +

@@ Line 246: / Line 246: @@
 	\frac{1}{s_z} \frac{\partial}{\partial z} \left(\frac{1}{s_z}\dpar{E_z}{z}\right)
 	\end{bmatrix} -
-	\\ -\omega^2 \eps \mu \b E
+	\omega^2 \eps \mu \b E
 	\begin{bmatrix}
 	E_x \\

← Older revision		Revision as of 08:10, 27 September 2020
Line 264:		Line 264:

	The above graphs display a simple use case of SC-PML with radial geometry. Instead of being surrounded by a box, the computational domain is surrounded by an annulus of perfectly matched layer, that contains both $s_x$ and $s_y$ components and therefore dampens the wave in both $x$ and $y$ directions throughout the PML. Such approaches are not usually taken in conventional FDTD codes because of difficult implementation, this is not the case when using the Medusa library.		The above graphs display a simple use case of SC-PML with radial geometry. Instead of being surrounded by a box, the computational domain is surrounded by an annulus of perfectly matched layer, that contains both $s_x$ and $s_y$ components and therefore dampens the wave in both $x$ and $y$ directions throughout the PML. Such approaches are not usually taken in conventional FDTD codes because of difficult implementation, this is not the case when using the Medusa library.
		+
		+	<syntaxhighlight lang="c++" line>
		+	// set equation with source
		+	double l = std::pow(omega/c0, 2.0);
		+	op.lap(source) + l*op.value(source) = -1.0;
		+
		+	for(int i: inter){ // does not contain source node!
		+	op.lap(i) + l*op.value(i) = 0.0;
		+	}
		+	for(int i: PML){ //under assumption sx = sy!
		+	1.0/(sx[i]sx[i])op.lap(i) + l*op.value(i) +
		+	((-1.0)/(sx[i]sx[i]sx[i])exop.d1(sx, 0, i)op.der1(i, 0)
		+	+(-1.0)/(sx[i]sx[i]sx[i])exop.d1(sx, 1, i)op.der1(i, 1)) = 0.0;
		+	}
		+	for (int i: disc.boundary()){
		+	op.value(i) = 0.0;
		+	}
		+	</syntaxhighlight>

	===== Benchmarking =====		===== Benchmarking =====

@@ Line 259: / Line 259: @@
 	\end{bmatrix}
 \end{multline}
 The expression further simplifies in cases where some of $s_w$'s are constant (cartesian coordinates).
 ===== Benchmarking =====
 Parameters of the PML are usually determined empirically, separately for each simulation, however, there are some rules of thumb. PMLs by Berenger's definition are exact for a continuous equation. When the equation is discretized reflections appear at the boundary of the PML. A lot of care needs to be taken so that the reflections are minimized. This is usually achieved by giving  $\sigma_{w}(l)$ a polynomial relationship with the distance $l$ from the PML boundary

@@ Line 3: / Line 3: @@
 The following pages describe the basics of computational electromagnetics, starting with relevant derivations and the basics of classical electromagnetism. The subpages include case studies with analytical solutions if they exist and numerical solutions.
 * [[Triple dielectric step in 1D]]
 * [[Scattering from an infinite cylinder]]
 * [[Point source near an anisotropic lens]]

← Older revision		Revision as of 14:44, 11 August 2020
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	==References==		==References==
		+	# Taflove, A. and Hagness, S.C., 2005. Computational electrodynamics: the finite-difference time-domain method. Artech house.
		+	# Shin, W. and Fan, S., 2012. Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell’s equations solvers. Journal of Computational Physics, 231(8), pp.3406-3431.
		+	# Podgornik, Rudolf, and Andrej Vilfan. Elektromagnetno polje. DMFA-založništvo, 2012.
		+	# John, S. G. J., et al. "Photonic crystals: molding the flow of light." Princeton University of press. 2008.

@@ Line 164: / Line 164: @@
 \end{equation}
 in practice, so that it is not needed to store the signals on both edges which may take up considerable computer memory, the sum can be calculated after each step in the simulation to save space.
 ==Modeling in the frequency domain==
-=== Frequency response calculations ===
+In the frequency domain, two types of calculations are most common. Within the field of photonic crystals, bandstructures of periodic materials are sought by solving a generalized eigenvalue problem $Ax = \omega^2Bx$. In engineering applications, a response of a structure to a periodical source $\b J(x) e^{-i\omega t}$ is more often calculated. We will focus on the second type of calculation.
 ==== Formulation ====
-==== Modeling sources ====
+We need to solve the equation that can be derived by combining Maxwell's equations and constitutive relations
 ==== Absorbing boundary conditions ====
 ==References==

← Older revision		Revision as of 09:19, 11 August 2020
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	==== Absorbing boundary conditions ====		==== Absorbing boundary conditions ====
		+	One of the crucial developments in computational electromagnetics is the ability to simulate ''open domains'', which approximate infinite problems.

	==Modeling in the frequency domain==		==Modeling in the frequency domain==