Finite element methods László Szirmay-Kalos

Representation of functions by finite data Finite function series: L(p)  Lj bj (p) 1 box 1 tent b1 b1 b2 b2 b3 b3 Piece-wise constant Piece-wise linear

Representation 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 function space L*(p) = Lj bj (p)  L L L +Le b2 b1 L* Original rendering equation Finite element approximation

Projected rendering equation L* L*(p) = Lj bj (p) Basis functions b2 +Le b1 L* b2’ F L* b1’ Adjoint base +Le* L* = Le* +F L*

L* Adjoint base <bi , bj’> = 1 if i=j and 0 otherwise b2 L* b2’ Equality is required in a subspace of adjoint basis functions: b1’, b2’ ,..., bn’ orthogonality: <bi , bj’> = 1 if i=j and 0 otherwise b2 L* +Le L* b2’ b1 projection b1’

Derivation of the projected rendering equation FEM: Projecting to an adjoint base: < •, bi’> L(p)  Lj bj (p) p=(x,w) Lj bj (p)  Lje bj (p) + t Lj bj (p) Li = Lie +  Lj <tbj ,bi’>

Projected rendering equation = linear equation for Lj Rij = <tbj ,bi’> L = Le + R L FEM: 1. define basis functions and adjoint basis function tesselation, function shape 2. Evaluate Rij 3. Solve the linear equation for L1, L2 ,…, Ln 4. For any p: L(p)  Lj bj (p)

Galerkin’s method <bi ,bi’>=1  bi’ = bi /<bi ,bi> 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 case Galerkin+constant basis <u,v>=Su(x)v(x)dx  <bi,bi> = Ai Aj bi is 1 on patch i w’ h(x,-w’) ’ Ai x <tbj,bi’>= 1/Ai Ai bj (h(x,-w’)) fr(x) cos’ dw’dx

Solid angle  Area integral Aj  h(x,-w’) = y w’ ’ Ai dw’= dy cos / |x - y|2 x <tbj,bi>=1/AiAiAjv(x,y) fr(x) dydx = ai Fij cos’ cos  |x - y|2 Patch-patch form factor: Albedo: cos’ cos  ai = fri  Fij=1/Ai AiAj v(x,y) dydx  |x - y|2

Example: Diffuse case Point collocation+linear basis bi bi’= (x - xi) Aj w’ h(x,-w’) ’ Ai xi <tbj,bi’>=  bj (h(xi,-w’)) fr(xi) cos’ dw’ cos’ cos  = Aiv(xi,y) bj (y) fr(xi) dy = ai Fij point-patch |xi - y|2