Modélisation et Caractérisation

d’Aspect Xavier Granier <xavier.granier@institutoptique.fr>

Romain Pacanowski <romain.pacanowski@institutoptique.fr>

2

Rappel : luminance réféchie

L(p→o⃗ )=∫ρ( l⃗ →p→o⃗ )⟨ n⃗⋅⃗l ⟩L( l⃗ →p )d l⃗

Lumière incidente - 4DW.m².sr-1

Propriété de réflexions – 6Dsr-1

Facteur géométrique

po⃗

l⃗n⃗

Angle solidesr

3

Propriétés de Réfexion

● Diffus - Lambertien

– Indépendant du point de vue

4

Propriétés de Réfexion

● Diffus - Lambertien

– Indépendant du point de vue

● Miroir – Spéculaire

– Dépendant du point de vue

5

Propriétés de Réfexion

● Diffus - Lambertien

– Indépendant du point de vue

● Miroir – Spéculaire

– Dépendant du point de vue

● Glossy/Brillant

– Dépendant du point de vue

– Lobe

6

BRDF – éléments de radiométrie

pour une longueur d’onde donnée

7

d2

2

dS

d1

1

On considère la réflexion sur une interface entre

deux milieux d’indices différents et un faisceau

incident avec un angle 1

Le flux incident s'écrit (Li est la luminance entrante)

Réfexion - Confguraion

d2 F1=Li (ω1→ s )d2G1

d2 F1=Li (ω1→ s )cos θ1d S dΩ1

Le flux réfléchi s'écrit (dLr est la luminance sortante)

d2 F2=dL r ( s→ω2 )cosθ2d S dΩ2

Supposons que a % soit réfléchi, la luminance sortante est

d Lr (s→ω2 )=α (ω1→s→ω 2 )

cos θ2dΩ2

Li (ω1→s )cosθ1dΩ1

8

Noion de BRDFi

● BRDF = Bidirectional Reflection Distribution Function

● Définition

● Grandeur : sr-1

d Lr ( s→ω2 )=α (ω1→ s→ω2 )cos θ2dΩ2

Li (ω1→ s )cosθ1dΩ1

dLr ( s→ω2)=ρ (ω1→s→ω2 )Li (ω1→s )cosθ1dΩ1

Dimensions BRDFi

● Restriction aux variations directionnelles

– 2D angulaire x 2 = 4D

● Si variations spatiales

– SVBRDF – spatially varying BRDF

ρ (ω1→s→ω2 )⇒ρ (ω1→ω2)

ρ (ω1→ω2 )=ℝ4→ℝ

+

Propriétés Physiques

● Réciprocité (Helmotz)

● Positivité

● Conservation de l'énergie

∀ω1 :∫Ω+ ρ (ω1,ω2 )cosθ2dΩ2≤1

ρ (ω1→ω2 )=ρ (ω2→ω1)=ρ (ω1,ω2 )

ρ (ω1,ω2 )≥0

11

Modèles empiriques de BRDF

Et autres propriétés

12

Propriétés de Réfexion

● Diffus - Lambertien

– Indépendant du point de vue

13

Difus/Lamberien

n

ρ (ω i ,ωo )= ρd

14

Difus/Lamberien

n

ρ (ω i ,ωo )=αdπ

15

Propriétés de Réfexion

● Diffus - Lambertien

– Indépendant du point de vue

● Miroir – Spéculaire

– Dépendant du point de vue

16

Spéculaire

n

ρ (ω i ,ωo )=δ r , o

ro

17

Propriétés de Réfexion

● Diffus - Lambertien

– Indépendant du point de vue

● Miroir – Spéculaire

– Dépendant du point de vue

● Glossy/Brillant

– Dépendant du point de vue

– Lobe

– Phong

18

Glossy/Brillant

s

n

19

Modèle de Phong – empirique

n⃗ω i

rωo

n

ρ (ω i→ωo )= ρs ⟨ωo⋅r ⟩e

Phong BRDFi

e=1 e=2 e=4

e=16

e=8

n=32 e=64 e=128

e=256 e=512 e=1024

21

Phong – Conservation de l’énergie

n⃗ω i

rωo

n

ρ (ω i ,ωo )=αse+22π

⟨ωo⋅r ⟩e

Généralisaion par combinaison

● Somme de multiples lobes

– Composante lambertienne

– Composante Phong

ρ (ω i ,ωo )=αd1π +αs

e+22π

⟨ωo⋅r ⟩e

BRDFi – Autres caractérisiques

IsotropeAnisotrope

4 3

( , )i o

( , , )i o i o ( , , , )i i o o

Exemples réel d'anisotropies

25

Modèle : théorie des microfacettes

Microfacet Theory

MICROSCOPIC SCALE MESOSCOPIC SCALE MACROSCOPIC SCALE

Microfacet

Microfacet : Idea

Images from Real-Time Rendering. 3rd Editon. A.K.Peters 2008

Microfacet: Roughness impact

Microfacet Theory [Torrance & Sparrow 1967]

• Idea: – surface refecton = collecton of small microfacet

– Surface = height feld

• Statstc descripton of the heightield

microfacet orientaton distributon

• Assumpton: V-groove

Torrance-Sparrow

• Microfacet = perfect mirror

• Half-vector

• Three terms :– Geometric aka Shadowing

– Distributon

– Fresnel

| |i o

i o

h

Distributon Term

• Proporton of surfaces which normal are orientated toward the h vector

• Normalizaton conditon:

( )D h

Distributon Term

• Possible isotropic distributons:

( )D h

22

2

( )

23

2 23

tan /

2 4

( ) : Blinn

exp : Torrance-Sparrow

: Trowbridge-Reitz ( 1)cos 1

1exp : Beckman

cos

e

c

m

h n

c

c

m

a

a

a

a

n

io r

Distributon Term

Other distributons :

• TR [Trowbridge-Reitz 1965]

– average irregularity of curved microsurfaces

• GGX [Walter 2007] (== TR)

Simulaton de la réfracton

• Shifer Gamma Distributon [Bagher 2012] – Distributon plus proche des mesures

• GTR [Disney 2012]– Satsfaire des besoins de contrôlle

Anisotropic Distributon

• e.g., Brushed metals

• Ashikhmin-Shirley [JGT2000]:

2 2cos sin( 2) 2

( ) ( )2

: azimuthal angle of

and control the size of an ellipse

x yx y e e

x y

e eD h h n

h

e e

Anisotropic Distributon

Geometric| Shadowing Term

• = Occlusion between microfacets

• For the light directon:

• For the view directon:

• G = min{1, occ_light,occ_view}

2( )( )occ_light

( )o

o

n h n

h

2( )( )occ_view

( )i

i

n h n

h

G( , )i o

Fresnel Term

• Computes how much a material refect vs. transmit incident light

• Fresnel Equatons:– Dielectric media (non-conductor such as glass)

– Conductors (metals)

• Depends on the polarizaton of the incident light

• Grazing angle efects (view directon):– Increases refecton of the material

Fresnel Term: grazing angle

Classical Phong

Fresnel Term

Fresnel Approximaton [Schlick94]

• Fast Approximaton for un-polarized light– Cheap to compute

• One parameter: – Refecton coefcient at normal incidence

50 0( ) (1 )(1 ( ))r i iF R R h

0R

De nombreux autres modèles

● Ward : basé sur la distribution de Beckman

● He : peut intégrer la polarisation

● Ashikhmin-Shirley : généralisation de Cook-Torrence

Fabricatng D(h)

Weyrich et al. [Sigg. 2009]

Variante Théorie Microfacete

• Oren-Nayar: Microfacet = perfect Lambertan

rougher

Reflectionoff a cylinder

Lambertian

( , ) ( ) ( ) with : roughness parameterdi o

kA B

Oren-Nayar

Oren

10o 40o 20o 0o

44

BRDF - paramétrisation

Paramétrisaton de la BRDF

La défniton de la BRDF n'impose rien sur le repérage des directons de l'hémisphère

• Paramétrisatons possibles:– 4D: classique, Rusinkiewicz

– 3D Barycentrique : Arvo,.. Diference Vector

– 2D: Romeiro

• Intérêt : – Mathématque : Séparabilité, Compression

– Physique: pilotage intelligent de banc d'acquisiton

Paramétrisaton classique

• Les directons et dans un repère local à la surface en coordonnées :– Sphériques

– Cartésiennes

Paramétrisaton de Rusinkiewicz

• Deux nouveaux vecteurs h et d :

Infuence de la paramétrisaton

49

BRDF : Acquisition

•Les détecteurs optques mesurent

– des flux luum i neuux

•Mesure relatve par rapport à une référence

étalon de BRDF.

Nov 2008

Principe de la mesure

m

m

faisceauincident normale

faisceaudiffusé

i

mmiiinc

mmmmmii d

BRDF

).cos(,,

,,,,,,''

Étalons/Étalonnage

Infra-rouge: infragold de LabSphere

Nov 2008 CEA/CESTA/LTO

Visible:Spectralon

BRDF Acquisiton

Two approaches:

• Goniorefectometer

• Digital Camera CCD

Goniorefectometer[LFTW05]

BRDF Acquisiton

54

Isotropic BRDF Database [Matusik]

Approche CCD

Isotropic BRDF Database [Matusik]

• Base MERL-MIT

• 100 matériaux isotropes mesurés

• Paramétrisaton Rusinkiewicz

BRDFs mesurées

• Objectf: représentaton efcace compacte

utle pour le rendu et/ou l’éditon

Mesures

Approximaton par

modèles analytques

Projectionavec

bases de fonction

CalculCoût mémoire !

Motvaton

BRDF = 30 à 100 Mo

BRDFs mesurées

Motiatone

Mesures

Approximatonpar

modèles analytques

Projection avec

bases de fonction

Rendu

Représentatons existantes

• Harmoniques sphériques [Cabral87,Westn92]

• Polynôlmes de Zernike [Koenderink96]

• sRBF [Zickler05]

• Ondeletes sphériques [Schröeder95]

Basusldulfonectones

Nombre de coefcients augmente quadratquement avec la spécularité [MTR2008]

Harmoniques sphériques

BRDFs mesurées

Motiatone

Mesures

Approximatonpar

modèles analytques

Projection avec

bases de fonction

Rendu

Approximaton (Fi ttineg)

• Approximaton linéaire – Polynôlmes,…

• Approximaton non-linéaire– Modèles Phong, Ward, A&S

• Paramètre du modèle est un exposant == non-linéaire

– Outls mathématques• Levenberg Marquardt , SQP Convergence locale

Approximaton des mesures

• Schéma au tableau– Problématque

– Convergence locale vs. globale

Analyse de la base MERL-MIT [Ngan2005]

• Fitng de 5 modèles• He, Ward, Ashikhmin, Lafortune, Blinn-Phong

Analyse de la base MERL-MIT [Ngan2005]

Conclusion :– He et Cook-Torrance semble le plus apte

– Aucun modèle n'est bon pour certains matériaux

– Problèmes de reproductbilité des résultats

Analyse de la base Merl-MIT [Romeiro2009]

• Beaucoup de matériaux sont reperésentables par 2 angles sur les 4 de Rusinkiewicz :

Analyse de la base Merl-MIT [Romeiro2009]

Ratonal BRDF

• BRDF représentées avec foncton ratonnelles

• Fitng avec convergence globale garante

• Modèles orientés mesures

pas de paramètres de contrôlle

Intérêt Fonctons Ratonnelles

• Approximaton avec 7 coefcients

Ratonal Functons

Polynomial Functons

DataErreur Max

• Polynôlmes : 0.0689

• Ratonnelles : 0.0017

Ratonal BRDF : RésultatsTRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. XXXX, NO. XXXXXX, XXX 20XX 2

Original datasize: BRDF =99MB, TabulatedCDF+PDFw30MB Our approach: BRDF =1.67KB, InverseCDF =0.600KB

Fig. 1. Monte-Carlo rendering with 2048 samples/pixel for a scene with three measured BRDFs from the MERL-MITdatabase (bl ue-metal l i c on the dragon, bei ge- f abri c on the floor, ni ckel on the sphere). Our approximation of theBRDFs andthe inverseCDFs, basedonRational Functions, provides efficientimportancesamplingwithanegligiblememoryfootprint: with less than 1 KB of storage, our IS technique (right) ofers equivalent quality (mean Lab diference is 0.77 andmax7.03 on low-dynamic range images) compared to the reference solution (left) obtained fromtabulated data ofw30MB.These tabulated CDF+PDF data have been generated by resampling the BRDF in (✓v,✓l,φl) at 90⇥90⇥180. Furthermore,the rendering time of our approach is 10% faster.

As detailed in Section 4.1, performing IS requires the in-verseof theBRDF’sCumulativeDistributionFunction (CDF).There are basically two approaches to compute the inverse oftheCDF. Thefirst is tofit themeasureddatawithananalyticalBRDF model [7]–[10], [24], [28] thato↵ersareadily invertibleCDF. In addition to the previously-mentioned weaknessesof non-linear fitting, all these approaches (except for [28])ignore the cosine factor that scales the BRDF according tothe incident light direction, reducing the efficiency of IS forgrazing angles of light. The second approach consists oftabulating the CDF into a sorted data structure (e.g., binarysearch tree) and computing the inverse function on-the-flyin this structure [21], [29], [30]. A major benefit of thisapproach is that the cosine scaling factor can be triviallyincluded, greatly improvingtheefficiencyof IS. Unfortunately,thestoragecost isseveral ordersof magnitudehigher thanwiththe first approach, and the iterative data retrieval process hasa non-constant computation cost.

In this paper we introduce the following contributions:

• ageneral frameworkbasedonRational Functions(RFs),which efficiently represents BRDFs and CDFs withouthaving to separate di↵use and specular components.

• anassociated fitting techniquethat scaleswith thedesiredaccuracy and memory footprint. The involved optimiza-tion is that of a strictly convex function of which theglobal minimum is guaranteed to be reached, providedthat a feasible solution exists.

• a new Monte-Carlo estimator for importance samplingrendering, which does not require to store thePDF whencombined with our representation.

2 Rational Functions Framework

In approximation theory Rational Functions are recognizedfor their greater expressivity compared to polynomials. They

are preferred in several numerical approximation problems inscientific computing [31]. A Rational Function (RF) of afinitedimensional vector xxx of real variables xi is:

rn,m(xxx)=pn,m(xxx)qn,m(xxx)

=

nX

j=0

pjbj(xxx)

mX

k=0

qkbk(xxx)

(1)

wherethen+1(resp.m+1) coefficientsof thenumerator (resp.denominator) arerepresentedby thereal numbers pj (resp. qk),and where bj(xxx) and bk(xxx) are multivariate basis functions.We use the multinomials in this paper, because they can beevaluatedefficiently.Weorder themby increasing total degree,for example in the bivariate case: b0=1,b1=x1,b2=x2,b3=x21,b4=x

22,b5=x1x2,b6=x

31,... Therefore, for agiven degree

wefavor adding first smoother basis functions (e.g., x21) ratherthan more oscillating ones (e.g., x1x2). Furthermore, bothpn,m(xxx)/qn,m(xxx) and ↵pn,m(xxx)/↵qn,m(xxx) take the same functionvalues for finitenonzero↵, and thecoefficients pj andqk needonly be determined up to a multiplicative constant that canbe used to normalize the representation of rn,m(xxx). Thereforern,m(xxx) has no more than n+m+1 free coefficients.

RFsareideal for approximatingdatathatexhibit steepchangeswhich are characteristic for specular lobes. An illustration ofapproximation of lobe-like functions using RFs is given inFigure 2, where it can be observed that a low degree RFcan easily represent abrupt variations followed by regionsof almost constant values, whereas a polynomial with thesame number of coefficients cannot. Such combinations ofsteep changes with flat regions are quite common in mea-sured BRDF data and their corresponding CDF. However, incomputer graphics, RFs have seldom been employed (exceptfor the ad hoc BRDF model proposed by Schlick [8]).

Algorithm 1 presents an overview of our fitting procedurebasedonthework of Salazar Celis et al. [32]. A preprocessing

TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. XXXX, NO. XXXXXX, XXX 20XX 2

Original datasize: BRDF =99MB, TabulatedCDF+PDF w30MB Our approach: BRDF =1.67KB, InverseCDF =0.600KB

Fig. 1. Monte-Carlo rendering with 2048 samples/pixel for a scene with three measured BRDFs from the MERL-MITdatabase (bl ue-metal l i c on the dragon, bei ge- f abri c on the floor, ni ckel on the sphere). Our approximation of theBRDFs andthe inverseCDFs, basedonRational Functions, provides efficient importance samplingwithanegligiblememoryfootprint: with less than 1 KB of storage, our IS technique (right) ofers equivalent quality (mean Lab diference is 0.77 andmax7.03 on low-dynamic range images) compared to the reference solution (left) obtained fromtabulated data ofw30MB.These tabulated CDF+PDF data have been generated by resampling the BRDF in (✓v,✓l,φl) at 90⇥90⇥180. Furthermore,the rendering time of our approach is 10% faster.

As detailed in Section 4.1, performing IS requires the in-verseof theBRDF’sCumulativeDistributionFunction (CDF).There arebasically two approaches to compute the inverse oftheCDF. Thefirst is tofit themeasureddatawithananalyticalBRDF model [7]–[10], [24], [28] thato↵ersareadily invertibleCDF. In addition to the previously-mentioned weaknessesof non-linear fitting, all these approaches (except for [28])ignore the cosine factor that scales the BRDF according tothe incident light direction, reducing the efficiency of IS forgrazing angles of light. The second approach consists oftabulating the CDF into a sorted data structure (e.g., binarysearch tree) and computing the inverse function on-the-flyin this structure [21], [29], [30]. A major benefit of thisapproach is that the cosine scaling factor can be triviallyincluded, greatly improvingtheefficiencyof IS. Unfortunately,thestoragecost isseveral ordersof magnitudehigher thanwiththe first approach, and the iterative data retrieval process hasa non-constant computation cost.

In this paper we introduce the following contributions:

• ageneral frameworkbasedonRational Functions (RFs),which efficiently represents BRDFs and CDFs withouthaving to separate di↵use and specular components.

• anassociated fitting techniquethat scaleswith thedesiredaccuracy and memory footprint. The involved optimiza-tion is that of a strictly convex function of which theglobal minimum is guaranteed to be reached, providedthat a feasible solution exists.

• a new Monte-Carlo estimator for importance samplingrendering, which does not require to store thePDF whencombined with our representation.

2 Rational Functions Framework

In approximation theory Rational Functions are recognizedfor their greater expressivity compared to polynomials. They

are preferred in several numerical approximation problems inscientific computing [31]. A Rational Function (RF) of afinitedimensional vector xxx of real variables xi is:

rn,m(xxx)=pn,m(xxx)qn,m(xxx)

=

nX

j=0

pjbj(xxx)

mX

k=0

qkbk(xxx)

(1)

wherethen+1(resp.m+1) coefficientsof thenumerator (resp.denominator) arerepresentedby thereal numbers pj (resp.qk),and where bj(xxx) and bk(xxx) are multivariate basis functions.We use the multinomials in this paper, because they can beevaluatedefficiently.Weorder themby increasing total degree,for example in the bivariate case: b0=1,b1=x1,b2=x2,b3=x21,b4=x

22,b5=x1x2,b6=x

31,... Therefore, for a given degree

wefavor adding first smoother basis functions (e.g., x21) ratherthan more oscillating ones (e.g., x1x2). Furthermore, bothpn,m(xxx)/qn,m(xxx) and ↵pn,m(xxx)/↵qn,m(xxx) take the same functionvalues for finitenonzero↵, and thecoefficients pj andqk needonly be determined up to a multiplicative constant that canbe used to normalize the representation of rn,m(xxx). Thereforern,m(xxx) has no more than n+m+1 free coefficients.

RFsareideal for approximatingdatathatexhibit steepchangeswhich are characteristic for specular lobes. An illustration ofapproximation of lobe-like functions using RFs is given inFigure 2, where it can be observed that a low degree RFcan easily represent abrupt variations followed by regionsof almost constant values, whereas a polynomial with thesame number of coefficients cannot. Such combinations ofsteep changes with flat regions are quite common in mea-sured BRDF data and their corresponding CDF. However, incomputer graphics, RFs have seldom been employed (exceptfor the ad hoc BRDF model proposed by Schlick [8]).

Algorithm 1 presents an overview of our fitting procedurebasedonthework of Salazar Celis et al. [32]. A preprocessing

Mesures99 MB

Approximatons RF1,7 KB

69

Extension à la réfraction

70

d2

2

dS

d1

1

n2

n1

On considère une interface entre deux milieux

d’indice n1 et n

2 et un faisceau incident avec un

angle 1

Le flux incident s'écrit (Li est la luminance entrante)

Extension - Réfracion

d2 F 1=Li (ω1→s )d2G1

d2 F1=Li (ω1→ s )cosθ1d S dΩ1Le flux transmis s'écrit (dL

t )

d2 F2=d Lt ( s→ω2 )cos θ2d S dΩ2

Supposons que t % soit transmis, la luminance sortante est

d Lt ( s→ω2 )=τ (ω1→ s→ω2 )cos θ2dΩ2

Li (ω1→ s )cosθ1dΩ1

71

Noion de BTDFi

● Bidirectional Transmission Distribution Function

– Grandeur : sr-1

● Propriétés

– Réciprocité (Loi de Kirchoff)

– Conservation de l'énergie

d Lt (s→ω2 )=τ (ω1→ s→ω2 )cosθ2dΩ2

Li (ω1→s )cos θ1dΩ1

d Lt (s→ω2 )=ρ (ω1→ s→ω2 )Li (ω1→s )cosθ1dΩ1

∀ω1:∫Ω- ρ (ω1→ s→ω2 )cosθ2dΩ2≤1

ρ (ω1→s→ω2 )/n22=ρ (ω 2→s→ω1 )/n1

2=ρ* ( s ,ω1,ω 2)

72

Généralisaion : noion de BSDFi

● Bidirectional Scattering Distribution Function

– Grandeur : sr-1

● Propriétés

– Conservation de l'énergie

d Ls ( s→ω2 )=ρ (ω1→ s→ω2 )Li (ω1→ s )cosθ1dΩ1

∀ω1:∫Ω ρ (ω1→ s→ω2 )cosθ2dΩ2≤1

73

Vers l'éclairement global

74

Éclairement

75

Éclairement global/indirect

Éclairement direct

Éclairement indirect

Éclairement globalÉclairement direct

A

76

Éclairement global

L(p→ o⃗ )

[Kajya 1996]

77

Éclairement global

L(p→ o⃗ )=Le(p→ o⃗ )

[Kajya 1996]

78

Éclairement global

L(p→ o⃗ )=Le(p→ o⃗ )+∫Ω ρ(ω⃗→ p→ o⃗ )⟨n⃗⋅⃗ω⟩L( ω⃗→ p )d ω⃗

Lumière incidente4D

Propriété de réflexions6D

Facteur géométrique

[Kajya 1996]

79

L'équaion du rendu [Kajiya 1986]

● Hypothèses

– Équilibre lumineux

– Une longueur d'onde

● Luminance émise (W.m-2.sr-1)

– Luminance propre

– Luminance réfléchie● Toutes les contributions

L ( s→o )=L p ( s→o )+∫Ω ρ ( i← s→o ) ⟨ i⋅n⟩L (−i→ s )d i

d i

s

n io

80

Rappel : équation du rendu

● Basé sur les valeurs de radiance

● Terme géométrique

d i d s'

s

n

n' g ( s , s ' )=⟨ i⋅n⟩ ⟨− i⋅n' ⟩

‖s−s '‖2 V ( s , s ' )

L ( s→ o )=Lp ( s→o )+∫S ρ ( i← s→o )g ( s , s ' )L ( s '→−i )d s '

L ( s→o )=L p ( s→o )+∫Ω ρ ( i← s→o ) ⟨ i⋅n⟩L (−i→ s )d i

81

Hypothèse diffuse [Goral84]

● Indépendance à la direction

● Notion de radiosité (exitance : W.m-2)

● Albédo = % énergie réfléchie

B ( s )=∫Ω

L ( s→ω ) ⟨n⋅ω ⟩dω=πLd ( s )

ρ ( i← s→o )=ρd ( s )

L ( s→ω )=Ld ( s )

αd (s )=∫Ω ρd (s ) ⟨n⋅ω ⟩dω=πρd (s )

82

Hypothèse diffuse [Goral84]

● Nouvelle équation (1)

● Notion d'irradiance (éclairement)

– W.m-2

● Nouvelle équation (2)

B ( s )=B p ( s )+αd ( s ) I ( s )

I ( s )=1π∫S g ( s , s ' )B ( s ' )d s '

B (s )=B p ( s )+αd ( s )1π∫S g ( s , s ' )B ( s ' )d s '

83

Discréisaion

● Hypothèse : une valeur constante par élément

– B(s) = Bi sur S

i

– Bp(s) = E

i sur S

i

● Calcul de l'irradiance

I ( s )=1π∫S g ( s , s ' )B ( s ' ) d s '

⇒ I ( s )=1π∑i

Bi∫S i g ( s , si )d si

Discréisaion (suite)

● Une valeur moyenne par élément

– Exitances

– Albédo

– Éclairement

Bi=1Si∫Si

B∂ Si

E i=1Si∫Si

I ∂ Si

⇒ E i=1Si

1π∑ j

B j∫Si∫S j

cos θ . cosθ '

d2∂ S i∂ S j

⇒ E i=∑ jF ij B j

Bp , i=1S i∫S i

B p∂ Si

ρi=1S i∫S i

ρ∂ S i

Équaion matricielle [Goral1984]

● Forme matricielle

● Facteur de forme

– % d'énergie transférée

– Relation avec l'étendue géométrie

● Étendue géométrique entre Si et S

j normalisée par

l'étendue géométrique portée la surface Si dans toutes

les direction : Si

– Propriétés

Bi=B p ,i+ρi∑ jF ij B j

F ij=1πSi∫S i∫S j

cos θi cos θj

d ij2 ds j dsi

πS i Fij=Gij

∑jFij≤1

JOHN R. HOWELLA catalog of Radiation Heat Transfer Configuration Factors

http://www.engr.uky.edu/rtl/Catalog/

Si F ij=S j F ji