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CubicSplineSmoothingKernel Class Reference

#include <CubicSplineSmoothingKernel.hpp>

Inheritance diagram for CubicSplineSmoothingKernel:
Inheritance graph
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Public Member Functions

double columnDensity (double q) const override
 
double density (double u) const override
 
double generateRadius () const override
 
virtual double columnDensity (double q) const =0
 
virtual double density (double u) const =0
 
virtual double generateRadius () const =0
 
- 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 ItemgetItemProperty (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)
 
Itemoperator= (const Item &)=delete
 
Itemparent () 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

 CubicSplineSmoothingKernel ()
 
void setupSelfBefore () override
 
- Protected Member Functions inherited from SmoothingKernel
 SmoothingKernel ()
 
Randomrandom () const
 
void setupSelfBefore () override
 
- Protected Member Functions inherited from SimulationItem
 SimulationItem ()
 
virtual bool offersInterface (const std::type_info &interfaceTypeInfo) const
 
virtual void setupSelfAfter ()
 
virtual void setupSelfBefore ()
 
- Protected Member Functions inherited from Item
 Item ()
 

Private Types

using BaseType = SmoothingKernel
 
using ItemType = CubicSplineSmoothingKernel
 

Private Attributes

Array _Xv
 

Friends

class ItemRegistry
 

Detailed Description

The CubicSpineSmoothingKernel class is a subclass of the abstract SmoothingKernel class, and describes smoothing kernels defined by the standard cubic spline density,

\[ W(u) = \begin{cases}\; \dfrac{8\,(1-6\,u^2+6\,u^3)}{\pi} & \quad\text{if }0\leq u \leq \tfrac12, \\ \; \dfrac{16\,(1-u)^3}{\pi} & \quad \text{if }\tfrac12 \leq u \leq 1, \\ \; 0 & \quad \text{else}. \end{cases} \]

It can be checked that this function is continuous at \(u=\tfrac12\) and that it satisfies the required normalization

\[ 4\pi \int_0^\infty W(u)\, u^2\, {\text{d}}u = 1. \]

Constructor & Destructor Documentation

◆ CubicSplineSmoothingKernel()

CubicSplineSmoothingKernel::CubicSplineSmoothingKernel ( )
inlineprotected

Default constructor for concrete Item subclass CubicSplineSmoothingKernel : "a cubic spline smoothing kernel" .

Member Function Documentation

◆ columnDensity()

double CubicSplineSmoothingKernel::columnDensity ( double  q) const
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 cubic spline smoothing kernel, we obtain

\[\Sigma(q) = \begin{cases} \; \dfrac{2}{\pi}\left[(4+26\,q^2)\sqrt{1-q^2} - (1+26\,q^2)\sqrt{1-4\,q^2} \right. \\ \left. \qquad\quad-18\,q^4\ln \left(\dfrac{2\,q} {1+\sqrt{1-4\,q^2}}\right) -6\,q^2(4+q^2) \ln\left(\dfrac{2+2\sqrt{1-q^2}} {1+\sqrt{1-4\,q^2}}\right) \right] & \quad{\text{if }}0\leq q\leq \tfrac12, \\[1em] \; \dfrac{4}{\pi}\left[(2+13\,q^2)\sqrt{1-q^2} + 3\,q^2(4+q^2)\ln\left(\dfrac{q} {1+\sqrt{1-q^2}}\right)\right] & \quad{\text{if }}\tfrac12\leq q\leq 1, \\ \; 0 & \quad{\text{else}}. \end{cases} \]

Implements SmoothingKernel.

◆ density()

double CubicSplineSmoothingKernel::density ( double  u) const
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.

◆ generateRadius()

double CubicSplineSmoothingKernel::generateRadius ( ) const
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

\[ {\cal{X}} = 4\pi\,W(u)\,u^2\, {\text{d}}u \]

for \(u\). For the cubic spline smoothing kernel, we use a precomputed grid with values on which we interpolate to solve this equation.

Implements SmoothingKernel.

◆ setupSelfBefore()

void CubicSplineSmoothingKernel::setupSelfBefore ( )
overrideprotectedvirtual

This function sets up a grid that will be used to sample random radii from the smoothing kernel.

Reimplemented from SimulationItem.


The documentation for this class was generated from the following file: