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Finite element methods

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.

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Finite element methods

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  1. Finite element methods László Szirmay-Kalos

  2. 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

  3. 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)

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

  5. 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*

  6. Adjoint base • 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’

  7. 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’>

  8. Projected rendering equation = linear equation for Lj Rij = <tbj ,bi’> L = Le+ RL 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)

  9. Galerkin’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)

  10. Example: Diffuse caseGalerkin+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

  11. 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

  12. Example: Diffuse casePoint 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

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