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