CutFem and Finite Differences for wave equations Gunilla Kreiss - PowerPoint PPT Presentation

CutFem and Finite Differences for wave equations Gunilla Kreiss Uppsala University, Sweden

High order immersed methods Outline Informationsteknologi n Background n Example in 1D: CutFem & SBP-SAT n 4th order and beyond Institutionen för informationsteknologi | www.it.uu.se

Wave equations ! " # !$ " = ! " # !& " + ! " # !( " Informationsteknologi ! " ) = * ! " ) + 2, ! " ) + - ! " # # # # ) !$ " !& " !( " !&!( Institutionen för informationsteknologi | www.it.uu.se

Wave equations ! " # !$ " = ! " # !& " + ! " # !( " Informationsteknologi ! " ) = * ! " ) + 2, ! " ) + - ! " # # # # ) !$ " !& " !( " !&!( Wish List: • High order accuracy • Regular grid and complex geometry: Immersed boundaries/interfaces • Explicit time stepping with k/h ≤ C Institutionen för informationsteknologi | www.it.uu.se

Immersed method? Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se

Immersed method Informationsteknologi • Almost undetermined DoF? • Small time step? Institutionen för informationsteknologi | www.it.uu.se

Immersed methods and waves Finite Difference/Finite Volumes Informationsteknologi Much work by Berger, LeVeque, Shu, Petersson, Lombard, Ditkowski, Tsynkov,…(many more!) FD/FV: no general technique for provably stable high order Institutionen för informationsteknologi | www.it.uu.se

Immersed methods and waves Finite Element methods Informationsteknologi CutFem (Burman,Hansbo,Larson…) , Xfem, etc Mainly developed for elliptic & parabolic Institutionen för informationsteknologi | www.it.uu.se

Immersed methods and waves Finite Element methods Informationsteknologi CutFem (Burman,Hansbo,Larson…) , Xfem, etc Mainly developed for elliptic & parabolic Wave equations? Institutionen för informationsteknologi | www.it.uu.se

Cut FEM for wave equations CutFem for wave equation and elastic wave Informationsteknologi equations Sticko,GK2016, Sticko,GK2017, Sticko,Ludvigsson,GK2018 • Provably stable, immersed, order 2&4, • Explicit, with time step k ~ h independent of cuts Institutionen för informationsteknologi | www.it.uu.se

Cut FEM for wave equations CutFem for wave equation and elastic wave Informationsteknologi equations Sticko,GK2016, Sticko,GK2017, Sticko,Ludvigsson,GK2018 • Provably stable, immersed, order 2&4, • Explicit, with time step k ~ h independent of cuts Compare with FD, SBP-SAT in particular? Order 6,8…? Institutionen för informationsteknologi | www.it.uu.se

FEM and FD, conforming ! "" = ! $$ , 0 ≤ ( ≤ 1, * ≥ 0 ! 0, * = ! 1, * = 0 Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se

FEM, piecewise linear ! "" = ! $$ , 0 ≤ ( ≤ 1, * ≥ 0 ! 0, * = ! 1, * = 0 Informationsteknologi , - (() X i X 1 =0 X i-1 X i+1 X N =1 4 ! 0 (, * = 1 ! - (*), - (() -23 Institutionen för informationsteknologi | www.it.uu.se

FEM with weak boundary condition ! "" = ! $$ , 0 ≤ ( ≤ 1, * ≥ 0 ! 0, * = ! 1, * = 0 Informationsteknologi - , . - ) = − ! $ - + ! $ - . - + ! - . $ - − 2 - , . $ - ! - . - (! "" ⇔ 9 − : 45 "" = −6 + 7 8 + 7 8 ℎ < 5 4 => = ? @ = @ > A( 6 => = ? @ =$ @ >$ A( Institutionen för informationsteknologi | www.it.uu.se

FEM with weak boundary condition ! "" = ! $$ , 0 ≤ ( ≤ 1, * ≥ 0 ! 0, * = ! 1, * = 0 Informationsteknologi - , . - ) = − ! $ - + ! $ - . - + ! - . $ - − 2 - , . $ - ! - . - (! "" ⇔ 9 − : 45 "" = −6 + 7 8 + 7 8 ℎ < 5 0.5 1 4 = ℎ ⋱ 1 0.5 Trapezoidal rule! Institutionen för informationsteknologi | www.it.uu.se

FEM with weak boundary condition , − - !" ## = %", % = −( + * + + * + ℎ / 1 −1 Informationsteknologi −1 2 −1 ( = 1 −1 ⋱ ℎ −1 2 −1 −1 1 1 −1 1 0 0 0 0 3 * + = / = ⋱ ⋱ 4 0 0 0 0 −1 1 1 Q is symmetric and negative definit if g > 1 stability! • FEM with weak BC SBP+SAT • Can extend to multi-D • Institutionen för informationsteknologi | www.it.uu.se

Explicit time stepping: U tt = M -1 Q U Time step restriction? Informationsteknologi Stability region must include [-ic,ic]. ! " # −% &' ( ≤ * Institutionen för informationsteknologi | www.it.uu.se

Explicit time stepping: U tt = M -1 Q U Time step restriction? Informationsteknologi Stability region must include [-ic,ic]. ! " # −% &' ( ≤ * " # −% &' ( ~ℎ &- Institutionen för informationsteknologi | www.it.uu.se

Explicit time stepping: U tt = M -1 Q U Time step restriction? Informationsteknologi Stability region must include [-ic,ic]. ! " # −% &' ( ≤ * " # −% &' ( ~ℎ &- ! ℎ ≤ . Institutionen för informationsteknologi | www.it.uu.se

CutFEM, ! "" = ! $$ , 0 ≤ ( ≤ 1 + +ℎ, - ≥ 0 ! 0, - = ! 1 + +ℎ, - = 0 Informationsteknologi X N-1 =1 X N =1+h X 1 =0 X 1 Same weak form: integrate only part of [x N-1 ,x N ] Institutionen för informationsteknologi | www.it.uu.se

CutFEM, ! "" = ! $$ , 0 ≤ ( ≤ 1 + +ℎ, - ≥ 0 ! 0, - = ! 1 + +ℎ, - = 0 Informationsteknologi d h X N-1 =1 X N =1+h X 1 =0 X 1 Same weak form: integrate only part of [x N-1 ,x N ] Institutionen för informationsteknologi | www.it.uu.se

CutFEM $ %% = $ '' , 0 ≤ + ≤ 1 + .ℎ, 0 ≥ 0 $ 0, 0 = $ 1 + .ℎ, 0 = 0 Informationsteknologi d h X N-1 =1 X N =1+h X 1 =0 X 1 Same weak form: integrate only part of [x N-1 ,x N ] ! " # Institutionen för informationsteknologi | www.it.uu.se

CutFem = SBP? = 1 + & −& ! Informationsteknologi " −& & ) * = −1 + & 1 − & ( −& & (1 − &) / &(1 − &) + , = & / &(1 − &) Interpret as SBP finite difference method! Institutionen för informationsteknologi | www.it.uu.se

CutFem = SBP? = 1 + & −& ! Informationsteknologi " −& & ) * = 1 − & −1 + & ( & −& (1 − &) / &(1 − &) + , = & / &(1 − &) Problem when & ≪ 1: 2 almost singular 3 ≪ ℎ Institutionen för informationsteknologi | www.it.uu.se

CutFem = SBP? = 1 + & −& ! Informationsteknologi " −& & ) * = 1 − & −1 + & ( & −& (1 − &) / &(1 − &) + , = & / &(1 − &) Problem when & ≪ 1: 2 almost singular 3 ≪ ℎ Stabilize! Institutionen för informationsteknologi | www.it.uu.se

CutFEM ! "" = ! $$ , 0 ≤ ( ≤ 1 + +ℎ, - ≥ 0 ! 0, - = ! 1 + +ℎ, - = 0 Informationsteknologi X N-1 =1 X N =1+h X 1 =0 X 1 Jump stabilization: add to weak form Institutionen för informationsteknologi | www.it.uu.se

CutFEM ! "" = ! $$ , 0 ≤ ( ≤ 1 + +ℎ, - ≥ 0 ! 0, - = ! 1 + +ℎ, - = 0 Informationsteknologi X N-1 =1 X N =1+h X 1 =0 X 1 Jump stabilization: add to weak form Penalize jumps in normal derivative Institutionen för informationsteknologi | www.it.uu.se

CutFEM ! "" = ! $$ , 0 ≤ ( ≤ 1 + +ℎ, - ≥ 0 ! 0, - = ! 1 + +ℎ, - = 0 Informationsteknologi X N-1 =1 X N =1+h X 1 =0 X 1 Jump stabilization: add to lower right of M and Q Institutionen för informationsteknologi | www.it.uu.se

CutFEM ! "" = ! $$ , 0 ≤ ( ≤ 1 + +ℎ, - ≥ 0 ! 0, - = ! 1 + +ℎ, - = 0 Informationsteknologi X N-1 =1 X N =1+h X 1 =0 X 1 Jump stabilization: add to lower right of M and Q Stability by ”fem machinery”, k independent of d , multi-D. Second order immersed SBP-SAT method! Institutionen för informationsteknologi | www.it.uu.se

Higher order 1) Standard FEM FD with alternating stencils Informationsteknologi • Equivalent to a strange SBP-SAT • Time step k/h ≤ c p decreses with p • Condition number k ( M ) grows with p Institutionen för informationsteknologi | www.it.uu.se

Higher order Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se

Cut Hermite FEM Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se

Cut Hermite FEM Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se

Cut Hermite FEM Informationsteknologi Institutionen för informationsteknologi | www.it.uu.se

Cut Hermite FEM: Mass matrix condition number 10 4 10 4 4.62038 5 10 12 Informationsteknologi 4.62036 4 10 10 4.62034 3 cond # cond # cond # 4.62032 10 8 2 4.6203 10 6 1 4.62028 10 4 0 4.62026 10 -10 10 -5 10 0 10 -10 10 -5 10 0 0.01 0.02 0.03 0.04 0.05 gridsize size of cut (fraction of h) regularization parameter Mass matrix conditioning is independent of h and d . • Robust with respect to parameter • Institutionen för informationsteknologi | www.it.uu.se

Cut Hermite FEM: time step d =1 h=2 -7 Informationsteknologi Time-step restriction is independent of d and h ! Institutionen för informationsteknologi | www.it.uu.se