Stability analysis of ALE-Methods for Advection-Diffusion problems

Transcription

1 Bielefeld Universiy Deparmen of Mahemaics Presened a he COMSOL Conference 2008 Hannover Sabiliy analysis of ALE-Mehods for Advecion-Diffusion problems D2 PHYSICS A. Weddemann*, V. Thümmler *weddeman@physik.uni-bielefeld.de Bielefeld Universiy

2 Ouline Moivaion 2. Inroducion o ALE-mehods 1. Basic ideas 2. Weak form of advecion-diffusion problems on moving domains 3. Numerical calculaions wih COMSOL 1. Weak form modeling 2. Predefined ALE-mode 4. Conclusions and Oulook

3 Moivaion 3 Domain moion: - Level se -mehod - ALE-mehods Typical sysems: - Buildings in wind - Haemodynamics While for ALE-mehods based on finie volume schemes cerain condiions on he numerical schemes (geomerical conversaion laws) have been shown [Far], similar resuls for FEM-schemes are sill missing. [Far] Farha e al., Compu. Meh. Appl. Mech. Engrg., 190

4 Inroducion 4 ALE-mehods in FEM-frameworks We consider a general parabolic advecion-diffusion problem ha can be wrien as: u ( x, )+ [ u ]( x, )= f ( x, ) L Ω n x bounded I=[ 0, E ] 2 n where f L(Ω I, ) n L[ u]: I n ellipic differenial operaor Expec he domain o evolve in respec o ime, where he domain displacemen is given by an ALE-funcion a: a :Ω I n ˆ 0 α :Ω Ω0 ( ξ, ) a a( ξ, ) ( x, ) a α( x, ) a( α(,),)= ξ ξ Ω ˆ ={( x, ), x Ω, I} Ω τ = a(,) τ

5 Basic ideas of ALE-mehods 5 ALE-mehods in FEM-frameworks The basic idea of ALE-mehods is o use differen coordinae sysems, a reference and a spaial sysem. Calculaion ransformed Example: o reference sysem! ξ 2 Ω x 2 0 a( ξ,) a(ξ,) coninuous, inverible α(ξ,) coninuous u (, )+ [ u](, )=0 x L x Ω α( x,) Ω ξ 1 x 1 Ω u ψ( x,) ( x,) dx+ ψ( x,) L[ u]( x,) dx=0 u a ψ( a( ξ, )) ( a( ξ, ), ) de ( ξ, dξ ) ξ + ((,)) [ ]((,),)de a ψ a ξ L u a ξ (,) ξ dξ ξ =0 weak formulaion domain ransformaion

6 Limiaions of ALE-mehods 6 Limiaions of ALE-mehods: - Topological changes: ALE-mehods in FEM-frameworks paricles moving owards each oher - Very srong displacemens: paricle moving oo far in one direcion

7 Weak form of ALE-equaions 7 The original equaion leads o he weak equaion ALE-mehods in FEM-frameworks u ψ( x,) ( x,) dx + ψ( x,) L[ u]( x,) dx = ψ( x,) f( x,) dx Ω Ω Ω Recasing he equaion in he reference sysem esfuncions can be chosen independen of [Nob]: uˆ a ψξ ˆ() ( ξ,)de ( ξ,) dξ ξ a + ψξ ˆ( ) L[ u]( ξ,) de ( ξ,) dξ ξ ˆ ˆ a = ψξ () f( ξ,)de ( ξ,) dξ ξ [Nob] F. Nobile, PhD Thesis where vˆ ( ξ,) = v( a( ξ, ),)

8 Model definiion 8 Assume he following model sysem uni square Ω =[0,1] [0,1] u D x u=f x Ω, ˆ I ux, ( )=0 x Ω u( ξ, 0) =16 ξ (1- ξ ) ξ (1- ξ ) Ω ξ 0 Advecion-diffusion problem L[ u]( x,)=- Δ u( x,) x on he Analyic soluion is given by ˆ 1 u(,) ξ = 161+ sin(5 π) 2 ξ (1- ξ ) ξ (1- ξ ) xi α i( x,)= 2-cos(10 π ) a ( ξ,)= ξ (2-cos(10 π)) fˆ ( ξ,) = 40πcos(5 π) ξ (1- ξ ) ξ (1- ξ ) i D (1+ 0.5sin(5 π)) + ( ξ 2 1(1- ξ1)+ ξ2(1- ξ2)) (2-cos(10 π)) 160 π (1+ 0.5sin(5 π)) sin(10 π) - 2-cos(10 π) ξξ 1 2 (2-3ξ 1-3 ξ ξξ 1 2 ) i

9 Weak formulaion 9 The weak form in ALE-formulaion is given by: (2-cos(10 )) ˆ ˆ ( ) u π ψξ ( ξdξ, ) D ψˆ uˆ - ( ξ) ( ξ, dξ ) 2-cos(10 π) i ξi ξi uˆ -10π sin(10 π) ψξ ˆ( ) ξi ( ξdξ, ) ξ =(2-cos(10 π)) ψξ ˆ( ) fˆ ( ξ, dξ ) The model was solved using implici Euler scheme for D = 0.01 and D = 1 for {Δ = 1/(20k), k = 1, 2,, 15} i i Advecion-diffusion problem Max. elem. size a boundaries: elemens 8397 degrees of freedom

10 Numerical resuls 10 Advecion-diffusion problem Global error analysis D = 1 D = 0.01 Srong deviaions from he analyic soluion can be found even for very fine imeseps!!

11 Numerical resuls 11 Global error analysis Advecion-diffusion problem - peaks occur periodically - even finer imeseps or higher BDF-order makes hem disappear

12 Numerical resuls 12 Local error analysis D = 0.01 Advecion-diffusion problem Δ =1/300 Δ =1/500 - errors resul from srong poinwise deviaions close o he moving boundaries

13 Numerical resuls 13 Advecion-diffusion problem Global error analysis min. elemen size a upper and righ boundary Number of degrees of elemens freedom even for very fine room discreizaion small deviaions can be found

14 Predefined ALE-mode 14 Solving he equaion wih he predefined ALE-mode in COMSOL: Ω where u ψ( x, ) ( xdx, ) ψˆ u -D ( x, ) ( xdx, ) x x = ψ( x, ) fxdx (, ) Ω i Ω i i = α (, ) j x x x ξ i j i j Advecion-diffusion problem Of course, equivalen on he coninuous level, bu no for he discree schemes!

15 Conclusion & Oulook D2 PHYSICS 15 Bielefeld Universiy Conclusion Large convecion leads o insabiliies in ALE-schemes Resuls depend srongly on discreizaion scheme: choose proper formulaion in COMSOL Muliphysics. An increase of he BDF-order can lead o a worse resul. Oulook Find condiions for a numerical scheme wih improved numerical sabiliy Incorporae differen sabilizaion mehods used on fixed domains, e.g. Perov- Galerkin-discreizaion