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

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

Sep 07, 2023 238 likes •629 views

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 fr

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CutFem and Finite Differences for wave equations

Gunilla Kreiss

Uppsala University, Sweden

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

High order immersed methods

Outline

n Background n Example in 1D: CutFem & SBP-SAT n 4th order and beyond

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Wave equations

!"# !$" = !"# !&" + !"# !(" !" !$" # ) = * !" !&" # ) + 2, !" !&!( # ) + - !" !(" # )

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Wave equations

!"# !$" = !"# !&" + !"# !(" !" !$" # ) = * !" !&" # ) + 2, !" !&!( # ) + - !" !(" # ) Wish List:

  • High order accuracy
  • Regular grid and complex geometry: Immersed

boundaries/interfaces

  • Explicit time stepping with k/h ≤ C
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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Immersed method?

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

  • Almost undetermined DoF?
  • Small time step?

Immersed method

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Immersed methods and waves

Finite Difference/Finite Volumes Much work by Berger, LeVeque, Shu, Petersson, Lombard, Ditkowski, Tsynkov,…(many more!) FD/FV: no general technique for provably stable high order

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Immersed methods and waves

Finite Element methods CutFem (Burman,Hansbo,Larson…), Xfem, etc Mainly developed for elliptic & parabolic

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Immersed methods and waves

Finite Element methods CutFem (Burman,Hansbo,Larson…), Xfem, etc Mainly developed for elliptic & parabolic Wave equations?

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Cut FEM for wave equations

CutFem for wave equation and elastic wave equations

Sticko,GK2016, Sticko,GK2017, Sticko,Ludvigsson,GK2018

  • Provably stable, immersed, order 2&4,
  • Explicit, with time step k ~ h independent of

cuts

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Cut FEM for wave equations

CutFem for wave equation and elastic wave 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…?

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

FEM and FD, conforming

!"" = !$$, 0 ≤ ( ≤ 1, * ≥ 0 ! 0, * = ! 1, * = 0

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

FEM, piecewise linear

!"" = !$$, 0 ≤ ( ≤ 1, * ≥ 0 ! 0, * = ! 1, * = 0

X1=0 Xi-1 Xi+1 Xi

,-(()

XN=1

!0 (, * = 1

  • 23

4

!-(*),-(()

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

FEM with weak boundary condition

!"" = !$$, 0 ≤ ( ≤ 1, * ≥ 0 ! 0, * = ! 1, * = 0 (!""

  • , .-) = − !$
  • , .$
  • + !$
  • .- + !-.$
  • − 2
  • !-.-

⇔ 45"" = −6 + 78 + 78

9 − :

ℎ < 5 4=> = ? @=@>A( 6=> = ? @=$@>$A(

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

FEM with weak boundary condition

!"" = !$$, 0 ≤ ( ≤ 1, * ≥ 0 ! 0, * = ! 1, * = 0 (!""

  • , .-) = − !$
  • , .$
  • + !$
  • .- + !-.$
  • − 2
  • !-.-

⇔ 45"" = −6 + 78 + 78

9 − :

ℎ < 5

4 = ℎ 0.5 1 ⋱ 1 0.5

Trapezoidal rule!

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

FEM with weak boundary condition

!"## = %", % = −( + *+ + *+

, − -

ℎ /

( = 1 ℎ 1 −1 −1 2 −1 −1 ⋱ −1 2 −1 −1 1 *+=

3 4

1 −1 ⋱ −1 1 / = 1 ⋱ 1

  • Q is symmetric and negative definit if g > 1 stability!
  • FEM with weak BC SBP+SAT
  • Can extend to multi-D
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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Explicit time stepping: Utt = M-1Q U Time step restriction?

Stability region must include [-ic,ic]. ! "# −%&'( ≤ *

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Explicit time stepping: Utt = M-1Q U Time step restriction?

Stability region must include [-ic,ic]. ! "# −%&'( ≤ * "# −%&'( ~ℎ&-

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Explicit time stepping: Utt = M-1Q U Time step restriction?

Stability region must include [-ic,ic]. ! "# −%&'( ≤ * "# −%&'( ~ℎ&- ! ℎ ≤ .

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

CutFEM,

X1=0 XN=1+h XN-1=1 X1

Same weak form: integrate only part of [xN-1,xN] !"" = !$$, 0 ≤ ( ≤ 1 + +ℎ, - ≥ 0 ! 0, - = ! 1 + +ℎ, - = 0

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

CutFEM,

X1=0 XN=1+h XN-1=1 X1

Same weak form: integrate only part of [xN-1,xN]

dh

!"" = !$$, 0 ≤ ( ≤ 1 + +ℎ, - ≥ 0 ! 0, - = ! 1 + +ℎ, - = 0

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Same weak form: integrate only part of [xN-1,xN] ! " #

CutFEM

X1=0 XN=1+h XN-1=1 X1

dh

$%% = $'', 0 ≤ + ≤ 1 + .ℎ, 0 ≥ 0 $ 0, 0 = $ 1 + .ℎ, 0 = 0

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

CutFem = SBP?

! " = 1 + & −& −& & ( )* = −1 + & 1 − & −& & + , = (1 − &)/ &(1 − &) &(1 − &) &/ Interpret as SBP finite difference method!

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

CutFem = SBP?

! " = 1 + & −& −& & ( )* = 1 − & −1 + & & −& + , = (1 − &)/ &(1 − &) &(1 − &) &/ Problem when & ≪ 1: 2 almost singular 3 ≪ ℎ

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

CutFem = SBP?

! " = 1 + & −& −& & ( )* = 1 − & −1 + & & −& + , = (1 − &)/ &(1 − &) &(1 − &) &/ Problem when & ≪ 1: 2 almost singular 3 ≪ ℎ

Stabilize!

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Jump stabilization: add to weak form

CutFEM

X1=0 XN=1+h XN-1=1 X1

!"" = !$$, 0 ≤ ( ≤ 1 + +ℎ, - ≥ 0 ! 0, - = ! 1 + +ℎ, - = 0

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

CutFEM

X1=0 XN=1+h XN-1=1 X1

Jump stabilization: add to weak form Penalize jumps in normal derivative !"" = !$$, 0 ≤ ( ≤ 1 + +ℎ, - ≥ 0 ! 0, - = ! 1 + +ℎ, - = 0

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

CutFEM

X1=0 XN=1+h XN-1=1 X1

Jump stabilization: add to lower right of M and Q !"" = !$$, 0 ≤ ( ≤ 1 + +ℎ, - ≥ 0 ! 0, - = ! 1 + +ℎ, - = 0

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

CutFEM

X1=0 XN=1+h XN-1=1 X1

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!

!"" = !$$, 0 ≤ ( ≤ 1 + +ℎ, - ≥ 0 ! 0, - = ! 1 + +ℎ, - = 0

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

1) Standard FEM FD with alternating stencils

  • Equivalent to a strange SBP-SAT
  • Time step k/h ≤ cp decreses with p
  • Condition number k(M) grows with p

Higher order

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Higher order

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Cut Hermite FEM

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Cut Hermite FEM

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Cut Hermite FEM

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Cut Hermite FEM: Mass matrix condition number

  • Mass matrix conditioning is independent of h and d.
  • Robust with respect to parameter

0.01 0.02 0.03 0.04 0.05

gridsize

4.62026 4.62028 4.6203 4.62032 4.62034 4.62036 4.62038

cond #

104 10-10 10-5 100

size of cut (fraction of h)

1 2 3 4 5

cond #

104 10-10 10-5 100

regularization parameter

104 106 108 1010 1012

cond #

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Cut Hermite FEM: time step

Time-step restriction is independent of d and h!

d=1 h=2-7

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Summary

§ The second order CutFem method is an

immersed SBP-SAT Finite Difference method

§ The Cut Hermite Finite Element Method is

stable, explicit time steps independent of cut,

§ Is Cut Hermite Finite Element Method an

immersed 4th order SBP-SAT Finite Difference method of compact type?

§ Time step restriction when order increases?

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Informationsteknologi

Institutionen för informationsteknologi | www.it.uu.se

Summary

§ The second order CutFem method is an

immersed SBP-SAT Finite Difference method

§ The Cut Hermite Finite Element Method is

stable, explicit time steps independent of cut,

§ Is Cut Hermite Finite Element Method an

immersed 4th order SBP-SAT Finite Difference method of compact type?

§ Time step restriction when order increases?

Thank you!