Upload
others
View
19
Download
0
Embed Size (px)
Citation preview
Modélisation et Caractérisation
d’Aspect Xavier Granier <[email protected]>
Romain Pacanowski <[email protected]>
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