finite element methods
DESCRIPTION
Finite element methods. L ászló Szirmay-Kalos. Representation of functions by finite data. Finite function series: L ( p ) L j b j ( p ). 1. box. 1. tent. b 1. b 1. b 2. b 2. b 3. b 3. Piece-wise constant. Piece-wise linear. Representation of the radiance. - PowerPoint PPT PresentationTRANSCRIPT
Finite element methodsFinite element methods
László Szirmay-Kalos
Representation of functions by Representation of functions by finite datafinite data
Finite function series: L(p) Lj bj (p)
1b1
b2
b3
Piece-wise constant
1 b1
b2
b3
Piece-wise linear
box tent
Representation of the radianceRepresentation of the radiance
Finite elements: L(p) Lj bj (p)
– bj: total function system
box, tent, harmonic, Chebishev, etc.
diffuse radiosity: piece-wise constant non-diffuse case:
– partitioned hemisphere (piece-wise constant),– directional distributions (spherical harmonics)– illumination networks (links)
Rendering equation in Rendering equation in function spacefunction space
L*(p) = Lj bj (p)
L L
+Le
Original renderingequation
L
Finite element approximation
b1
b2
L*
Projected rendering equationProjected rendering equation
L* = Le* +F L*
L*(p) = Lj bj (p)
b2
L*+Le
L*
b1
b1’b2’
Adjointbase
F L*+Le*
Basis functions
Adjoint baseAdjoint base
Equality is required in a subspace of adjoint basis functions: b1’, b2’ ,..., bn’
orthogonality: <bi , bj’> = 1 if i=j and 0 otherwise
+Le
L*L*
projectionb1
b2
b1’
b2’
Derivation of the projected Derivation of the projected rendering equationrendering equation
FEM:
Projecting to an adjoint base: < •, bi’>Lj bj (p) Lj
e bj (p) + Lj bj (p)
L(p) Lj bj (p)
Li = Lie + Lj <bj ,bi’>
p=(x,)
Projected rendering equation Projected rendering equation
= linear equation for = linear equation for LLjj
L = Le + RL Rij = <bj ,bi’>
FEM: 1. define basis functions and adjoint basis function
tesselation, function shape2. Evaluate Rij
3. Solve the linear equation for L1, L2 ,…, Ln
4. For any p: L(p) Lj bj (p)
Galerkin’s methodGalerkin’s method
The base and the adjoint base are the same except for a normalization constant:
–<bi ,bi’>=1 bi’ = bi /<bi ,bi>– Error is orthogonal to the original base
Point collocation method– equality is required at finite dot points pi
– bi’ (p) = (p - pi)
Example: Diffuse caseExample: Diffuse caseGalerkin+constant basisGalerkin+constant basis
<u,v>=Su(x)v(x)dx <bi,bi> = Ai
<bj,bi’>= 1/Ai Aibj (h(x,-’) fr(x) cos’ d’dx
bi is 1 on patch iAj
Ai
x
’ h(x,-’)’
Solid angle Solid angle Area integral Area integral
cos’ cos
Aj
Ai
x
’h(x,-’) = y
’ d’= dy cos / |x - y|2
<bj,bi>=1/AiAiAjv(x,y) fr(x) dydx = ai Fij
|x - y|2
cos’ cos ai = fri Fij=1/Ai AiAj v(x,y) dydx
|x - y|2
Albedo: Patch-patch form factor:
Example: Diffuse caseExample: Diffuse casePoint collocation+linear basisPoint collocation+linear basis
Aj
<bj,bi’>= bj (h(xi,-’) fr(xi) cos’ d’
Ai
xi
’ h(x,-’)’
cos’ cos = Aiv(xi,y) bj (y) fr(xi) dy = ai Fij
point-patch
|xi - y|2
bi’= (x - xi) bi