An h-adaptive unfitted finite element method for interface elliptic - - PowerPoint PPT Presentation

Supported by: Grant-id.: 2017FIB00219

An h-adaptive unfitted finite element method for interface elliptic boundary value problems

Eric Neiva1,3 Santiago Badia2,3 Monash Workshop on Numerical Differential Equations and Applications 2020, MWNDEA 2020, Feb. 2020.

1Universitat Politècnica de Catalunya (UPC), BCN, Spain. 2Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), BCN, Spain. 3Monash University, Clayton, Victoria, Australia.

Towards multiphysics and multiscale applications with h-AgFEM

Unfitted boundaries: ´∆u “ f in Ω u “ uD

• n BΩ

Previous talk Unfitted interfaces:

´∇ ¨ pα∇uq “ f in ΩinYΩout u “ 0

• n BΩ

u “ 0

• n Γ

nout ¨ α∇u “ 0

• n Γ

Building platform POWDER e.g. Powder-bed 3D printing SOLID

In this talk

• E. Neiva ¨ UPC–Monash ¨ 2020

2/16

An overview of unfitted FE methods for interface problems

´∇ ¨ pα∇uq “ f in ΩinYΩout u “ 0 on BΩ u “ 0 on Γ nout ¨ α∇u “ 0 on Γ Interface Poisson problem Structure of approximation Ahpuh, vhq . “ pα∇uh, ∇vhqΩinYΩout ´ @ t tα∇uhu u ¨ n`, vh D

Γ

´ @ uh, t tα∇vhu u ¨ n`D

Γ

` xτuh, vhyΓ SIPM or other Nitsche formulations Approximation method

• Naive SIPM+FEM: τ unbounded

for arbitrarily small cut size [HH02]

• Robustness to cut location via, e.g.,

stabilization (CutFEM) [Bur+15]

• Robustness to material contrast via

harmonic t t¨u u and diag. prec. [Bur+15]

Known results

• E. Neiva ¨ UPC–Monash ¨ 2020

3/16

Is AgFEM suitable for interface elliptic BVPs?

Contents:

• Construction of aggregated FE spaces
• Approximation of the interface problem
• Numerical analysis
• Numerical experiments
• Verification in uniform meshes
• Robustness w.r.t. cut location
• Robustness w.r.t. material contrast
• Robustness and optimality in tree-based meshes
• E. Neiva ¨ UPC–Monash ¨ 2020

4/16

Construction of AgFE spaces in interface problems

Recalling the rationale: improve conditioning by removing problematic DOFs V agg

h

:“ $ & %u P Vh : uˆ “ ÿ

‚Pmasterspˆq

Cˆ‚u‚ @ˆ P P , .

• ‚ well-posed dofs

ˆ problematic dofs (P) touched untouched aggregated

• E. Neiva ¨ UPC–Monash ¨ 2020

5/16

Construction of AgFE spaces in interface problems

Recalling the rationale: improve conditioning by removing problematic DOFs V agg

h

:“ $ & %u P Vh : uˆ “ ÿ

‚Pmasterspˆq

Cˆ‚u‚ @ˆ P P , .

• ‚ well-posed dofs

ˆ problematic dofs (P) touched untouched aggregated

• E. Neiva ¨ UPC–Monash ¨ 2020

5/16

Construction of AgFE spaces in interface problems

Recalling the rationale: improve conditioning by removing problematic DOFs V agg

h

:“ $ & %u P Vh : uˆ “ ÿ

‚Pmasterspˆq

Cˆ‚u‚ @ˆ P P , .

• ‚ well-posed dofs

ˆ problematic dofs (P) touched untouched aggregated

• E. Neiva ¨ UPC–Monash ¨ 2020

5/16

Construction of AgFE spaces in interface problems

Recalling the rationale: improve conditioning by removing problematic DOFs V agg

h

:“ $ & %u P Vh : uˆ “ ÿ

‚Pmasterspˆq

Cˆ‚u‚ @ˆ P P , .

• ‚ well-posed dofs

ˆ problematic dofs (P) touched untouched aggregated

• E. Neiva ¨ UPC–Monash ¨ 2020

5/16

Construction of AgFE spaces in interface problems

Recalling the rationale: improve conditioning by removing problematic DOFs V agg

h

:“ $ & %u P Vh : uˆ “ ÿ

‚Pmasterspˆq

Cˆ‚u‚ @ˆ P P , .

• ‚ well-posed dofs

ˆ problematic dofs (P) touched untouched aggregated

• E. Neiva ¨ UPC–Monash ¨ 2020

5/16

Construction of AgFE spaces in interface problems

Recalling the rationale: improve conditioning by removing problematic DOFs V agg

h

:“ $ & %u P Vh : uˆ “ ÿ

‚Pmasterspˆq

Cˆ‚u‚ @ˆ P P , .

• ‚ well-posed dofs

ˆ problematic dofs (P) touched untouched aggregated

• E. Neiva ¨ UPC–Monash ¨ 2020

5/16

Construction of AgFE spaces in interface problems

In our context: constrain by cell aggregation at both subregions

Ñ V in,agg

h

Ñ V out,agg

h

V agg

h

“ V in,agg

h

ˆ V out,agg

h

Analogously, given n different materials, V agg

h

“ V 1,agg

h

ˆ . . . ˆ V n,agg

h

.

• E. Neiva ¨ UPC–Monash ¨ 2020

6/16

Approximation of interface problem with Nitsche’s method

Local FE operators: For any K P Th, we define AKpuh, vhq . “ abulk

K

puh, vhq ` aΓ

Kpuh, vhq,

lKpvhq . “ lbulk

K

pvhq, with abulk

K

puh, vhq “ ´ αin∇uin

h , ∇vin h

¯

ΩinXK `

` αout∇uout

h , ∇vout h

˘

ΩoutXK,

lbulk

K

pvhq “ A f in, vin

h

E

ΩinXK `

@ f out, vout

h

D

ΩoutXK,

and aΓ

Kpuh, vhq “ ´

@ nout ¨ t tα∇uhu u, vh D

ΓXK ´

@ uh, nout ¨ t tα∇vhu u D

ΓXK ` xτKuh, vhyΓXK,

where τK “ βKh´1

K , with βK ą 0 large enough, is a stabilization parameter and weighted average

t t¨u u given by αout αin ` αout p¨qin ` αin αin ` αout p¨qout.

• E. Neiva ¨ UPC–Monash ¨ 2020

7/16

Approximation of interface problem with Nitsche’s method

Local FE operators: For any K P Th, we define AKpuh, vhq . “ abulk

K

puh, vhq ` aΓ

Kpuh, vhq,

lKpvhq . “ lbulk

K

pvhq, with abulk

K

puh, vhq “ ´ αin∇uin

h , ∇vin h

¯

ΩinXK `

` αout∇uout

h , ∇vout h

˘

ΩoutXK,

lbulk

K

pvhq “ A f in, vin

h

E

ΩinXK `

@ f out, vout

h

D

ΩoutXK,

and aΓ

Kpuh, vhq “ ´

@ nout ¨ t tα∇uhu u, vh D

ΓXK ´

@ uh, nout ¨ t tα∇vhu u D

ΓXK ` xτKuh, vhyΓXK,

where τK “ βKh´1

K , with βK ą 0 large enough, is a stabilization parameter and weighted average

t t¨u u given by αout αin ` αout p¨qin ` αin αin ` αout p¨qout.

• E. Neiva ¨ UPC–Monash ¨ 2020

7/16

interface-AgFEM inherits key boundary-AgFEM inverse inequalities

Trace inverse inequality for unfitted boundary [BVM18] For any vh P V agg

h

and K P T Γ

h ,

}n ¨ ∇vh}2

0,ΓDXK ď CBh´1 K }∇vh}2 0,ΩK,

where ΩK is the domain of the aggregate where K belongs and CB independent of mesh size and cut location. Trace inverse inequality for unfitted interface For any vh P V agg

h

and K P T Γ

h ,

› ›nout ¨ t tα∇vhu u › ›2

0,ΓXK ď CBh´1 K

αinαout αin ` αout ´ }∇vh}2

0,ΩKin ` }∇vh}2 0,ΩKout

¯ , where Ωin{out

K

is the aggregate domain at Ωin{out and CB independent of mesh size and cut location.

• E. Neiva ¨ UPC–Monash ¨ 2020

8/16

interface-AgFEM inherits key boundary-AgFEM inverse inequalities

Trace inverse inequality for unfitted boundary [BVM18] For any vh P V agg

h

and K P T Γ

h ,

}n ¨ ∇vh}2

0,ΓDXK ď CBh´1 K }∇vh}2 0,ΩK,

where ΩK is the domain of the aggregate where K belongs and CB independent of mesh size and cut location. Trace inverse inequality for unfitted interface For any vh P V agg

h

and K P T Γ

h ,

› ›nout ¨ t tα∇vhu u › ›2

0,ΓXK ď CBh´1 K

αinαout αin ` αout ´ }∇vh}2

0,ΩKin ` }∇vh}2 0,ΩKout

¯ , where Ωin{out

K

is the aggregate domain at Ωin{out and CB independent of mesh size and cut location.

• E. Neiva ¨ UPC–Monash ¨ 2020

8/16

Summary of numerical analysis

Well-posedness and a priori error estimates Let V phq . “ V agg

h

` H1

0pΩq X H2pΩin Y Ωoutq and define for any v P V agg h

the norms |v|2

˚

. “ ÿ

KPT Γ

βKh´1

K }v}2 0,ΓXK,

}v}2

V phq .

“ |v|2

1,ΩinYΩout ` |v|2 ˚,

and |||v|||2

V phq

. “ }v}2

V phq `

ÿ

KPTcut

h2

K

ˆˇ ˇ ˇvinˇ ˇ ˇ

2 2,Ω`XK `

ˇ ˇvoutˇ ˇ2

2,Ω´XK

˙ . The following results hold: Apuh, uhq Á }uh}2

V phq

for all uh P V agg

h

(stability if βK Á 2αinαout

αin`αout )

Apu, vq À |||u|||2

V phq|||v|||2 V phq

for all u, v P V phq (continuity) }u ´ uh}1,ΩinYΩout À hp for all uh P V agg

h

and u P V phq (optimal convergence in H1) }u ´ uh}0,ΩinYΩout À hp`1 for all u P V phq and uh P V agg

h

(optimal convergence in L2) where the constants are independent of cut location.

• E. Neiva ¨ UPC–Monash ¨ 2020

9/16

Summary of numerical analysis

Well-posedness and a priori error estimates Let V phq . “ V agg

h

` H1

0pΩq X H2pΩin Y Ωoutq and define for any v P V agg h

the norms |v|2

˚

. “ ÿ

KPT Γ

βKh´1

K }v}2 0,ΓXK,

}v}2

V phq .

“ |v|2

1,ΩinYΩout ` |v|2 ˚,

and |||v|||2

V phq

. “ }v}2

V phq `

ÿ

KPTcut

h2

K

ˆˇ ˇ ˇvinˇ ˇ ˇ

2 2,Ω`XK `

ˇ ˇvoutˇ ˇ2

2,Ω´XK

˙ . The following results hold: Apuh, uhq Á }uh}2

V phq

for all uh P V agg

h

(stability if βK Á 2αinαout

αin`αout )

Apu, vq À |||u|||2

V phq|||v|||2 V phq

for all u, v P V phq (continuity) }u ´ uh}1,ΩinYΩout À hp for all uh P V agg

h

and u P V phq (optimal convergence in H1) }u ´ uh}0,ΩinYΩout À hp`1 for all u P V phq and uh P V agg

h

(optimal convergence in L2) where the constants are independent of cut location.

• E. Neiva ¨ UPC–Monash ¨ 2020

9/16

Numerical verification experiments

10´16 10´14 10´12 10´10 10´8 10´6 10´4 10´2 100 100 101 102 103 104 |u ´ uh|H1{|u|H1 DOFs1{d u P V agg

h

, Q1 and uniform refinements tol “ 10´9 tol “ 10´12 α`{α´ “ 10´6 α`{α´ “ 1 α`{α´ “ 106 10´16 10´14 10´12 10´10 10´8 10´6 10´4 10´2 100 100 101 102 103 |u ´ uh|H1{|u|H1 DOFs1{d u P V agg

h

, Q2 and uniform refinements tol “ 10´9 tol “ 10´12 α`{α´ “ 10´6 α`{α´ “ 1 α`{α´ “ 106 10´8 10´7 10´6 10´5 10´4 10´3 10´2 10´1 100 100 101 102 103 104 |u ´ uh|H1{|u|H1 DOFs1{d u R V agg

h

, Q2 and uniform refinements Oph2q α`{α´ “ 10´6 α`{α´ “ 1 α`{α´ “ 106 10´8 10´7 10´6 10´5 10´4 10´3 10´2 10´1 100 100 101 102 103 |u ´ uh|H1{|u|H1 DOFs1{d u R V agg

h

, Q1 and uniform refinements Oph1q α`{α´ “ 10´6 α`{α´ “ 1 α`{α´ “ 106

• E. Neiva ¨ UPC–Monash ¨ 2020

10/16

Robustness w.r.t. to cut location in popcorn Q2 with u R V agg

h

Condition numbers w/ standard FEM: condestpAstdq ´6 ´4 ´2 2 4 6 Material constrast: log10 αout{αin ´1 ´0.5 0.5 1 Bounding box scaling: a Ñ 1 ` ah 105 1010 1015 1020 1025 1030 1035 1040 1045 Condition numbers w/ AgFEM: condestpAaggq ´6 ´4 ´2 2 4 6 Material constrast: log10 αout{αin ´1 ´0.5 0.5 1 Bounding box scaling: a Ñ 1 ` ah 105 1010 1015 1020 1025 1030 1035 1040 1045

• E. Neiva ¨ UPC–Monash ¨ 2020

11/16

Robustness w.r.t. to material contrast in popcorn Q2 with u R V agg

h

Condition numbers w/ AgFEM: condestpAaggq ´6 ´4 ´2 2 4 6 Material constrast: log10 αout{αin ´1 ´0.5 0.5 1 Bounding box scaling: a Ñ 1 ` ah 107 108 109 1010 1011 1012 1013 1014 Condition numbers w/ AgFEM: condestpD´1Aaggq ´6 ´4 ´2 2 4 6 Material constrast: log10 αout{αin ´1 ´0.5 0.5 1 Bounding box scaling: a Ñ 1 ` ah 103 104 105

• E. Neiva ¨ UPC–Monash ¨ 2020

12/16

Robustness and optimality in h-adaptivity: Fichera-corner

10´7 10´6 10´5 10´4 10´3 10´2 10´1 100 101 101 102 103 104 |αpu ´ uhq|H1{|αu|H1 DOFs1{d Q2 and uniform refinements Oph

2 3q

Oph2q α`{α´ “ 1 α`{α´ “ 103 α`{α´ “ 106

• E. Neiva ¨ UPC–Monash ¨ 2020

13/16

Robustness and optimality in h-adaptivity: Fichera-corner

10´7 10´6 10´5 10´4 10´3 10´2 10´1 100 101 101 102 103 104 |αpu ´ uhq|H1{|αu|H1 DOFs1{d Q2 and adaptive refinements Oph

2 3q

Oph2q α`{α´ “ 1 α`{α´ “ 103 α`{α´ “ 106

• E. Neiva ¨ UPC–Monash ¨ 2020

13/16

Conclusions and further work

AgFEM shows good properties, in the context of interface elliptic BVPs, but further study is required:

• 1. more experimentation on tree-based meshes
• 2. accuracy of the quantities at the interface (traces, fluxes...)
• 3. comparison against other well-behaved methods
• 4. multimaterial, elasticity, moving interfaces, scalability...
• E. Neiva ¨ UPC–Monash ¨ 2020

14/16

References

Erik Burman et al. “CutFEM: discretizing geometry and partial differential equations”. In: International Journal for Numerical Methods in Engineering 104.7 (2015),

• pp. 472–501.

Santiago Badia, Francesc Verdugo, and Alberto F Martín. “The aggregated unfitted finite element method for elliptic problems”. In: Computer Methods in Applied Mechanics and Engineering 336 (2018), pp. 533–553. Anita Hansbo and Peter Hansbo. “An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems”. In: Computer methods in applied mechanics and engineering 191.47-48 (2002), pp. 5537–5552.

• E. Neiva ¨ UPC–Monash ¨ 2020

15/16

Acknowledgements

• EN gratefully acknowledges the support received from the Catalan Government through a FI fellowship (2019 FI-B2-00090;

2018 FI-B1-00095; 2017 FI-B-00219).

• The authors thankfully acknowledge the computer resources at Marenostrum IV and the technical support provided by

BSC under the RES (Spanish Supercomputing Network). RES-ActivityID: IM-2019-3-0008. Computed with www.fempar.org

Thank you!

• E. Neiva ¨ UPC–Monash ¨ 2020

16/16