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The SKIRT project
advanced radiative transfer for astrophysics
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The SKIRT project
advanced radiative transfer for astrophysics
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#include <ScaledGaussianSmoothingKernel.hpp>
Public Member Functions | |
| double | columnDensity (double q) const override |
| double | density (double u) const override |
| double | generateRadius () const override |
| Public Member Functions inherited from SimulationItem | |
| template<class T> | |
| T * | find (bool setup=true) const |
| template<class T> | |
| T * | interface (int levels=-999999, bool setup=true) const |
| virtual string | itemName () const |
| void | setup () |
| string | typeAndName () const |
| Public Member Functions inherited from Item | |
| Item (const Item &)=delete | |
| virtual | ~Item () |
| void | addChild (Item *child) |
| const vector< Item * > & | children () const |
| virtual void | clearItemListProperty (const PropertyDef *property) |
| void | destroyChild (Item *child) |
| virtual bool | getBoolProperty (const PropertyDef *property) const |
| virtual vector< double > | getDoubleListProperty (const PropertyDef *property) const |
| virtual double | getDoubleProperty (const PropertyDef *property) const |
| virtual string | getEnumProperty (const PropertyDef *property) const |
| virtual int | getIntProperty (const PropertyDef *property) const |
| virtual vector< Item * > | getItemListProperty (const PropertyDef *property) const |
| virtual Item * | getItemProperty (const PropertyDef *property) const |
| virtual string | getStringProperty (const PropertyDef *property) const |
| int | getUtilityProperty (string name) const |
| virtual void | insertIntoItemListProperty (const PropertyDef *property, int index, Item *item) |
| Item & | operator= (const Item &)=delete |
| Item * | parent () const |
| virtual void | removeFromItemListProperty (const PropertyDef *property, int index) |
| virtual void | setBoolProperty (const PropertyDef *property, bool value) |
| virtual void | setDoubleListProperty (const PropertyDef *property, vector< double > value) |
| virtual void | setDoubleProperty (const PropertyDef *property, double value) |
| virtual void | setEnumProperty (const PropertyDef *property, string value) |
| virtual void | setIntProperty (const PropertyDef *property, int value) |
| virtual void | setItemProperty (const PropertyDef *property, Item *item) |
| virtual void | setStringProperty (const PropertyDef *property, string value) |
| void | setUtilityProperty (string name, int value) |
| virtual string | type () const |
Protected Member Functions | |
| ScaledGaussianSmoothingKernel () | |
| void | setupSelfBefore () override |
| Protected Member Functions inherited from SmoothingKernel | |
| SmoothingKernel () | |
| Random * | random () const |
| void | setupSelfBefore () override |
| Protected Member Functions inherited from SimulationItem | |
| SimulationItem () | |
| virtual bool | offersInterface (const std::type_info &interfaceTypeInfo) const |
| virtual void | setupSelfAfter () |
| Protected Member Functions inherited from Item | |
| Item () | |
Private Types | |
| using | BaseType |
| using | ItemType |
Private Attributes | |
| Array | _Xv |
Friends | |
| class | ItemRegistry |
An instance of the ScaledGaussianSmoothingKernel describes a scaled Gaussian smoothing kernel with finite support, as presented by Altay and Theuns 2013 (MNRAS 434,748). For the normalized radius \(u=r/h\), the kernel profile is given by:
\[ W(u) = \begin{cases}\; \mathcal{N}\,\exp(-\frac{u^2}{2\sigma^2}) & \quad\text{if }0\leq u \leq 1, \\ \; 0 & \quad \text{else}. \end{cases} \]
where
\[ \begin{aligned} \sigma &= \frac{1}{2\sqrt{2}\,\pi^{1/6}} \\ \mathcal{N} &= \frac{8}{\pi}\,\left[\mathrm{erf}(t)-\frac{2t\exp(-t^2)}{\sqrt{\pi}} \right]^{-1} \quad \text{with} \quad t=2\pi^{1/6}. \end{aligned} \]
It can be verified that this function satisfies the required normalization
\[ 4\pi \int_0^\infty W(u)\, u^2\, {\text{d}}u = 1. \]
With this scaling and cutoff, the Gaussian profile approximates the standard cubic spline kernel profile to within about three percent for all radii. The reason for using a Gaussion kernel instead of the standard cubic spline kernel in some applications is that a spherical Gaussian profile (assuming infinite support, i.e. not cut off at the smoothing length) can be separated into Gaussian component profiles along each of the coordinate axes, It thus becomes easy to calculate the mass inside a cuboidal box (such as a grid cell) or to determine the surface density for the projection on a rectangle (such as a pixel).
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inlineprotected |
Default constructor for concrete Item subclass ScaledGaussianSmoothingKernel: "a scaled Gaussian smoothing kernel".
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overridevirtual |
This function returns the column density \(\Sigma(q) = 2 \int_{q}^1 \frac{W(u)\,u \,{\text{d}}u} {\sqrt{u^2-q^2}}\) of the smoothing kernel as a function of the normalized impact radius \(q=r_\text{i}/h\). For the scaled Gaussian smoothing kernel, we obtain
\[\Sigma(q) = \begin{cases} \; {\cal{N}}\,\sqrt{2\pi}\,\sigma \exp\left(-\frac{q^2}{2\sigma^2}\right) \mathrm{erf}\left(\frac{\sqrt{1-q^2}} {\sqrt{2}\sigma}\right) & \quad{\text{if }} 0\leq q\leq 1, \\ \; 0 & \quad{\text{else}}. \end{cases} \]
Implements SmoothingKernel.
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overridevirtual |
This function returns the density \(W(u)\) of the smoothing kernel as a function of the normalized radius \(u\). It just implements the analytical formula given in the class header.
Implements SmoothingKernel.
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overridevirtual |
This function generates a random normalized radius \(u\) from the smoothing kernel, by drawing a number from the one-dimensional probability density \(p(u)\,{\text{d}}u = 4\pi\,W(u)\,u^2\, {\text{d}}u\). This is accomplished by generating a uniform deviate \({\cal{X}}\), and solving the equation
\[ {\cal{X}} = \int_0^u 4\pi\,W(u')\,u'^2\, {\text{d}}u' \]
for \(u\). For the scaled gaussian smoothing kernel, we use a precomputed grid with values on which we interpolate to solve this equation.
Implements SmoothingKernel.
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overrideprotectedvirtual |
This function sets up a grid that will be used to sample random radii from the smoothing kernel.
Reimplemented from SimulationItem.