Error estimation Adaptive remeshing Remapping. Lionel Fourment Mines ParisTech CEMEF Sophia Antipolis, France

Transcription

1 Error stimation Adaptiv rmsing Rmapping Lionl Fourmnt Mins ParisTc CEMEF Sopia Antipolis, Franc

2 Error stimation Wic masur? L 2 H Enrgy 2 = u 2 u dω Ω Error stimation / Adaptiv rmsing / Rmapping 2 i i ( u u ) = u u + dω i x Ω i= = ( ) : ( ε ε ) dω E Ω = ( ) : ( ) dω E K Ω 2 2 2

3 Error stimation A priori rror stimation (linar problms) p C C p E Wit a singularity C 3 ( λ ) MIN p, Error stimation / Adaptiv rmsing / Rmapping

4 Error stimation / Adaptiv rmsing / Rmapping Error stimation A priori rror stimation (NON linar problms) Norm W,m+ E m 2 4 m p m C + ( ) 3 m m i i m i m i u u u u d x Ω ω = = + ( ) 3 3 m K ε = ɺ

5 Error stimation Error stimator / indicator θ E ( ) α, α, α θ α θ 2 E 2 ξ = θ E limξ = 0 Error stimation / Adaptiv rmsing / Rmapping

6 Error stimation Babusa s rror stimator div = T =.n u = u0 f on Ω on Γ on Γ 0 ( ) u, : ε u dω + f.u dω T.u ds = * * * * Ω Ω Γ 0 0 BUT r = div f 0 is not continuous Error stimation / Adaptiv rmsing / Rmapping

7 Error stimation Babusa s rror stimator ( ) r = div f = div bab 2 2 ( θ ) = α ( ) ( ) ω + α2 ( f ) 2 2 r d J ds f Ω Ω Ω J γ r r' n γ ( ) ' f n on Ω f J f = T n on Ω Γ 0 on Ω Γ 0 n γ γ - ' ' n γ Error stimation / Adaptiv rmsing / Rmapping

8 Error stimation Ziniwicz & Zu rror stimator = 2 = K ( ) : ( ) dω ( θ ) ( θ ) 2 = Rcovrd strss tnsor ɶ Ω 2 ( ) = ( ) ( ) θ K ɶ : ɶ dω ɶ Ω C p+ ɶ 0 α C α p Error stimation / Adaptiv rmsing / Rmapping <

9 Error stimation Ziniwicz & Zu rror stimator limξ = 0 ɶ = + ɶ ɶ ɶ + ɶ ɶ θ + ɶ ɶ ɶ ξ + ɶ ɶ C p+ α α = = 0 p C 0 Error stimation / Adaptiv rmsing / Rmapping

10 Error stimation Rcovrd tnsor? ɶ p+ C α Continuous tnsor ɶ ( ) ( ) u x = u N x ɶ? ( x) ɶ N ( x) = Error stimation / Adaptiv rmsing / Rmapping

11 Error stimation Rcovrd tnsor? Supr convrgnt valus ɶ p+ C α Intgration point: supr convrgnt valus of int Finit lmnt patc Ω Continuous local intrpolation ɶ 3 0 i P a a a xi = = + i= Error stimation / Adaptiv rmsing / Rmapping

12 Error stimation Rcovrd tnsor? ɶ p+ C α Minimization π SPR ( a ) = ( x ) 2 ɶ int Ω ( a ) 0 ( X ) a = = ( ) 2 int int π i, = 0 linar systm i a SPR i ( a ) i=,4 Ω Error stimation ( x) N ( x) 2 ɶ ɶ ( ) = ( ) ( ) = θ K ɶ : ɶ dω Error stimation / Adaptiv rmsing / Rmapping Ω

13 Adaptiv rmsing Adaptiv msing -rfinmnt Adaptd Optimal θ θ imp opt Nblt ( ) opt t ( ) ( θ ) = C MIN Nblt, θ θ imp imp θ = θ Nblt opt p θ opt θ opt = imp ( ) ( opt ) p 2 ( θ ) ( θ ) θ = θ = Nbltθ = 2 opt Error stimation / Adaptiv rmsing / Rmapping

14 Adaptiv Rmsing I - Optimal ms for prscribd accuracy θ imp opt Nblt Nblt 2 Nblt 2d 2 p opt imp ( ) 2 p d p ( ) ( ) 2 p d = + θ θ + θ = Adaptiv rmsing topological ms gnrator (MTC) Uncontrolld numbr of lmnts Unfasibl calculations Error stimation / Adaptiv rmsing / Rmapping

15 Adaptiv Rmsing II - Optimal ms for prscribd numbr of lmnts Nblt imp θ min 2 p+ d p Nblt 2d 2d imp d 2 p+ d ( Nblt ) (θ ) = = Ms siz map for t ms rgnrator 2 Nblt 2d 2 p opt imp ( ) 2 p d p ( ) ( ) 2 p d = + θ θ + θ = Unncssary accurat III - Hybrid Combin bot approacs if possibl, impos θ imp ls, impos Nblt imp Error stimation / Adaptiv rmsing / Rmapping

16 Rmsing Larg matrial dformations larg ms distortions Rmsing triggr rmsing: lmnt distortion pntration into obstacl (ms rfinmnt) rror stimation prscribd frquncy Gnration of nw ms surfac & volum Error stimation / Adaptiv rmsing / Rmapping

17 Rmsing topology Initial stag Topologic ms gnrator itrations Final ms Error stimation / Adaptiv rmsing / Rmapping ms

18 Rmsing drfinmnt Error stimation / Adaptiv rmsing / Rmapping

19 Adaptiv rmsing Boundary ms Volum ms Error stimation / Adaptiv rmsing / Rmapping

20 Rmapping Transfr of data 2 typs of variabls Continuous (P, for instanc) Discontinuous (P0, for instanc) Error stimation / Adaptiv rmsing / Rmapping

21 Rmsing Larg matrial dformations larg ms distortions Rmsing triggr rmsing: lmnt distortion pntration into obstacl (ms rfinmnt) rror stimation prscribd frquncy Gnration of nw ms surfac & volum Error stimation / Adaptiv rmsing / Rmapping

22 Rmapping Continuous variabls (P) Invrs intrpolation ( numrical diffusion) Nbno (, ) N (, ) ϕ ξ η = ϕ ξ η ϕ = ϕ l ( X l ) = ( ξ, η ) suc tat: X X N ( ξ, η ) Nbno ( X ) N (, ) ϕl = ϕ l = ϕ ξl ηl = Nbno = l l l l l = Nbno (, ) l N l (, ) ϕ ξ η = ϕ ξ η l= X Error stimation / Adaptiv rmsing / Rmapping

23 Rmapping Continuous variabls (P) Invrs intrpolation ( numrical diffusion) Global last squar ( spcial tratmnt of boundary) ϕ = ϕ l ( ) ϕ suc tat: MIN Φ ϕ l ( X l ) (( ) ) ( ) 2 = = Φ ϕ ϕ ϕ ω l l,nbno d Ω ( ϕ ) l l =, Nbno Nbno (, ) l N l (, ) ϕ ξ η = ϕ ξ η l= (( ) ) l l =,Nbno l =,Nbno Error stimation / Adaptiv rmsing / Rmapping

24 Rmapping Continuous variabls (P) Invrs intrpolation ( numrical diffusion) Global last squar ( spcial tratmnt of boundary) Supr convrgnt Patc Rcovry Tcniqus. P fild locally P2 fild (igr ordr) 2. P2 fild projction on t P fild Error stimation / Adaptiv rmsing / Rmapping

25 Rmapping Discontinuous variabls (P0) Invrs intrpolation discontinuous strong diffusion P0 transfr g = 0 0 g Nw ms fast and asy to implmnt Prvious ms consistant (balanc quation) igly diffusiv Error stimation / Adaptiv rmsing / Rmapping

26 Rmapping Discontinuous variabls (P0) Invrs intrpolation discontinuous strong diffusion P0 P and tn, invrs intrpolation diffusion Nbno (, ) ɶ N (, ) ɶ ϕ ξ η = ϕ ξ η = (( ɶ ) ) ( ɶ ) 2 Φ ϕ = ϕ ϕ dω =,Nbno MIN ( ɶ ϕ ) =,Nbno Φ Ω (( ɶ ϕ ) ) =,Nbno = Error stimation / Adaptiv rmsing / Rmapping Nbno ( ξ, η ) suc tat: X X N ( ξ, η ) Nbno = = ( X ) ɶ N (, ) ɶ ϕ = ϕ ξ η

27 Rmapping Discontinuous variabls (P0) Invrs intrpolation discontinuous strong diffusion P0 P (SPR), invrs intrpolation muc bttr Nbno (, ) ɶ N (, ) ɶ ϕ ξ η = ϕ ξ η = Ω Suprconvrgnt Patc Rcovry ϕ 3 = + 0 i i i ( x) a a ( x x ) i= 0 i MIN π ( a, ) ( ) 0 i a = ϕ x ϕ ( a, a ) 2 π Ω ( 0 i, ) ( 0 i a a π a, a ) = = 0 ( ) 0 i a a 0 ɶ ϕ Error stimation / Adaptiv rmsing / Rmapping = a 2

28 Rmapping Discontinuous variabls (P0) Invrs intrpolation discontinuous strong diffusion P0 P (SPR), invrs intrpolation muc bttr Nbno (, ) ɶ N (, ) ɶ ϕ ξ η = ϕ ξ η = Ω = Error stimation / Adaptiv rmsing / Rmapping Nbno ( ξ, η ) suc tat: X X N ( ξ, η ) Nbno = = ( X ) ɶ N (, ) ɶ ϕ = ϕ ξ η