cermics.enpc.frcermics.enpc.fr/~cascavik/thesis_slides.pdf · 2018-12-17 · 3/52 the scope of the...

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Hybrid discretization methodsfor Signorini contact and Bingham flow problems

Karol Cascavita1

Alexandre Ern 1 and Xavier Chateau2

1 ENPC (CERMICS) & Inria 2 ENPC (Navier)

December 18th, 2018

K. Cascavita HHO for Signorini and Bingham December 18th, 2018 1 / 52

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Motivations

Variational inequalities involving non smooth solutions

Contact problems

Deformable bodies into contact

Tribology, indentation hardenesstests, bearings, tires

Viscoplastic materials

Non-Newtonian fluids with solid orliquid-like behavior

Civil engineering, materialprocessing, petroleum drilling,food and cosmetics


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The scope of the thesis

Contact problems: Signorini’s unilateral contactContact with a rigid frictionless foundationContact surface not known a prioriNonlinearity on the boundary

u ≤ 0, σn(u) = n · ∇u ≤ 0, uσn(u) = 0

Viscoplasticity: Bingham modelRequire a finite yield stress to flow (solid or fluid-like behavior)Solid/liquid boundary not known a prioriNonlinearity in the domain: constitutive equation σ = 2µ∇su +

√2σ0

∇su

|∇su|`2yielded region (|σ|`2 >

√2σ0)

∇su = 0 unyielded region (|σ|`2 ≤√

2σ0)

σ not uniquely defined in the unyielded region

Finite Elements methods are the usual discretization techniqueUnilateral contact problems [Chouly & Hild 2013]Bingham [Bercovier & Engelman 1980], [Saramito & Roquet 2001-2003]


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From tetrahedral to polyhedral methods

Polyhedral meshes are a very active research topic

Local refinement and coarsening by agglomeration

Low-order schemes (k = 0)

Mimetic Finite Differences [Brezzi, Lipnikov & Shashkov 2005]Hybrid Finite Volumes [Eymard, Gallouet & Herbin 2010]Compatible Discrete Operator (CDO) schemes [Bonelle & Ern 2014]Unified approach [Droniou et al. 2010]

Higher-order schemes (k ≥ 1)

Discontinuous Galerkin methods [Arnold 1982], [Cockburn & Shu 1991]Virtual Element Method [Brezzi & Marini et al 2016]


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Hybrid High-Order methods

Face-based discretization

Main example for this thesis [Di Pietro & Ern 2014]

k = 0: closely related to HFVk ≥ 1: close links to HDG [Cockburn, Gopalakrishnan & Lazarov 2009]

HHO primal formulation 6= HDG mixed formulation

Face-DOFs and Cell-DOFs

Attractive features

Static condensation of cell-DOFsLocal conservationCompact stencil (3D)Dimension independent construction


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Main contributions: Signorini’s contact problem

Contact unilateral conditions

u ≤ 0, σn(u) ≤ 0, uσn(u) = 0

Reformulation [Curnier & Alart 88]:

σn(u) = [σn(u)− γu]R−

Numerical parameter γ > 0 and [x ]R− := min(x , 0).Treated in the consistency term of Nitsche’s method [Chouly & Hild 2013]

Contributions: Nitsche-HHO method

Poisson problemSignorini’s unilateral contact problemNitsche-HHO module in diskpp library[Cascavita, Chouly & Ern 2019]


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Main contributions: Bingham problem

Formulation of HHO within an augmented Lagrangian algorithm

Local mesh adaptation to track yield surface

Bingham module in diskpp library

Validation on several 2D tests cases

(Cascavita, Bleyer, Chateau & Ern, ’18 J Sci Comput)


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1 Nitsche-HHODiffusionSignorini’s unilateral condition

2 HHO for Bingham flowsAntiplanar configurationFull vector setting


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Outline




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Outline for section 1




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Global discrete problem

Model problem: −∆u = f on Ω; u = g on ΓMesh notations

Th Fh = Fbh ∪ F i

h

Ω

Γ

T1

T3

T2

T1

T3

T2

Global space Ukh interface DOFs are single valued

Ukh := Pk(Th)× Pk(Fh)

ukh = (uTh , uFh) ∈ Uk

h

Discrete global bilinear form: ah(uh, vh) =∑T∈Th

aT (uT , vT )

Simple cellwise assembly


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Global discrete problem

Model problem: −∆u = f on Ω; u = g on ΓMesh notations

Th Fh = Fbh ∪ F i

h

Ω

Γ

T1

T3

T2

T1

T3

T2

Global space Ukh interface DOFs are single valued

Ukh := Pk(Th)× Pk(Fh)

UkT := Pk(T )× Pk(F∂T )

ukh = (uTh , uFh) ∈ Uk

h

ukT = (uT , u∂T ) ∈ UkT

Discrete global bilinear form: ah(uh, vh) =∑T∈Th

aT (uT , vT )

Simple cellwise assemblyK. Cascavita HHO for Signorini and Bingham December 18th, 2018 11 / 52

12/52

Local operators

T

F∂T = F1,F2,F3,F4

F4F2

F1

F3

Rk+1T : computed from local Neumann problems (mean value condition)

(∇Rk+1T (uT ),∇z)T = −(uT ,∆z)T + (u∂T ,nT · ∇z)∂T ∀z ∈ Pk+1(T )

The local contributions

aT (uT , vT ) := (∇Rk+1T (uT ),∇Rk+1

T (vT ))T

Stability norm: |vT |2UkT

:= ‖∇vT‖2T + ‖h− 1

2 (v∂T − vT )‖2∂T


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Local operators

T

F∂T

vT = (vT , v∂T )




aT (uT , vT ) := (∇Rk+1T (uT ),∇Rk+1

T (vT ))T


:= ‖∇vT‖2T + ‖h− 1

2 (v∂T − vT )‖2∂T


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Local operators

T

F∂T

vT = (vT , v∂T )

T

vF4 vF2

vF1

vF3

vT

∈ UkT = Pk(T )× Pk(F∂T ) (ex. k = 0)




aT (uT , vT ) := (∇Rk+1T (uT ),∇Rk+1

T (vT ))T


:= ‖∇vT‖2T + ‖h− 1

2 (v∂T − vT )‖2∂T


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Local operators

T

F∂T

vT = (vT , v∂T )

T

vF4 vF2

vF1

vF3

vT

∈ UkT = Pk(T )× Pk(F∂T ) (ex. k = 0) Pk+1(T )

Local reconstructionoperator

Rk+1T

T




aT (uT , vT ) := (∇Rk+1T (uT ),∇Rk+1

T (vT ))T


:= ‖∇vT‖2T + ‖h− 1

2 (v∂T − vT )‖2∂T


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Local operators

T

F∂T

vT = (vT , v∂T )

T

vF4 vF2

vF1

vF3

vT



Rk+1T

T




aT (uT , vT ) := (∇Rk+1T (uT ),∇Rk+1

T (vT ))T

but ∇Rk+1T (uT ) = 0 ; uT = uF = cst


:= ‖∇vT‖2T + ‖h− 1

2 (v∂T − vT )‖2∂T


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Local operators

T

F∂T

vT = (vT , v∂T )

T

vF4 vF2

vF1

vF3

vT



Rk+1T

T




aT (uT , vT ) := (∇Rk+1T (uT ),∇Rk+1

T (vT ))T + (h−1T Sk

∂T (uT ),Sk∂T (vT ))∂T

Sk∂T (uT ) := (u∂T − uT )∂T︸︷︷︸

HDG−like term

− (πk∂TR

k+1T (uT )− πk

TRk+1T (uT ))∂T︸︷︷︸

HHO−higher−order term


:= ‖∇vT‖2T + ‖h− 1

2 (v∂T − vT )‖2∂T


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Imposing weakly Dirichlet boundary conditions

Nitsche-FEM:

aγ,h(uh, vh) =∑T∈Th

(∇uh,∇vh)T︸︷︷︸Galerkin

−∑F∈Fb

h

(σn(uh), vh︸︷︷︸consistency

)F

+(uh, γvh − σn(vh)︸︷︷︸

penalty & symmetry

)F

Nitsche-HHO


aT (uT , vT )︸︷︷︸Galerkin

−∑F∈Fb

h

(σHHO

n (uT ),︸︷︷︸consistency

)F

+(, γ − σHHO

n (vT )︸︷︷︸penalty & symmetry

)F

Numerical parameter γ = γ0h−1 and γ0 > 0

Normal stress σHHOn


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Nitsche-FEM:



−∑F∈Fb

h


)F


penalty & symmetry

)F

Nitsche-HHO



−∑F∈Fb

h

(σHHO

n (uT ), v?︸︷︷︸consistency

)F

+(u?, γv? − σHHO


)F


Normal stress σHHOn


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Nitsche-FEM:



−∑F∈Fb

h


)F


penalty & symmetry

)F

Nitsche-HHO with Face-based trace



−∑F∈Fb

h

(σHHO

n (uT ), vF︸︷︷︸consistency

)F

+(uF , γvF − σHHO


)F


Normal stress σHHOn (vT ) = σn(Rk+1

T (uT ))


13/52


Nitsche-FEM:



−∑F∈Fb

h


)F


penalty & symmetry

)F

Nitsche-HHO with Cell-based trace



−∑F∈Fb

h

(σHHO

n (uT ), vT︸︷︷︸consistency

)F

+(uT , γvT − σHHO


)F


Normal stress σHHOn needs to be modified


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Error estimate

RHSˆγ,h(vh) :=

∑T∈Th

(f , vT )T −∑F∈Fb

h

(g , σHHO

n (vT )− γv)F

Global problem Find uh ∈ Uk

h such that

aγ,h(uh, vh) = ˆγ,h(vh) ∀vh ∈ Uk

h

Consistency error: For any test function vh ∈ Ukh

Eh(vh) := ˆγ,h(vh)− aγ,h(I kh (u), vh)

with local projection operator I kT (u) = (πkT (u), πk

∂T (u|∂T )) ∈ UkT


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Error estimate

Key property: Pk+1T := Rk+1

T I kT is the local elliptic projector

Approximation property: letting δ := u − Pk+1T (u), we have

‖δ‖T + h12

T‖δ‖∂T + hT‖∇δ‖T + h32

T‖σn(δ)‖∂T . hk+2T |u|Hk+2(T )

Bound on consistency error

|Eh(vh)| .(∑T∈Th

‖∇δ‖2T + hT‖σn(δ)‖2

∂T

) 12 ‖vh‖Uk

h

=: ‖δ‖†‖vh‖Ukh

This leads to the optimal energy-error estimate(∑T∈Th

‖∇(u − Rk+1T (uT ))‖2

T

) 12

. O(hk+1)


16/52


ComputationEh(vh) = ˆ

γ,h(vh)− aγ,h(I kh (u), vh)

Re-arranging yields Eh(vh) = T1 + T2 − stab

IBP on (−∆u, vT )T + def. of ∇Rk+1T (vT ) =⇒ |T1| . ‖δ‖†‖vh‖Uk

h

Crucially T2 = 0: this is the property lost in the nonlinear casestab . ‖δ‖†‖vh‖Uk

has in the usual HHO analysis


16/52



γ,h(vh)︸︷︷︸rhs

−∑

T∈Th

(∇Rk+1T I kT (u),∇Rk+1

T (vT ))T

︸︷︷︸Galerkin

+∑

F∈Fbh

(σn(Rk+1

T I kT (u)), vF)F︸︷︷︸

consistency

+(πkF (u), σHHO

n (vT )− γvF)F︸︷︷︸

penalty & symmetry

− stab



h




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γ,h(vh)︸︷︷︸rhs

−∑

T∈Th

(∇Pk+1T (u),∇Rk+1

T (vT ))T


+∑

F∈Fbh

(σn(Pk+1

T (u)), vF)F︸︷︷︸

consistency

+(πkF (u), σHHO

n (vT )− γvF)F︸︷︷︸

penalty & symmetry

− stab



h




16/52


ComputationEh(vh) =

∑T∈T

h

(−∆u, vT )T −∑

F∈Fbh

(u, σHHO

n (vT )− γvF)F︸︷︷︸

rhs

−∑

T∈Th

(∇Pk+1T (u),∇Rk+1

T (vT ))T


+∑

F∈Fbh

(σn(Pk+1

T (u)), vF)F︸︷︷︸

consistency

+∑

F∈Fbh

(πkF (u), σHHO

n (vT )− γvF)F︸︷︷︸

penalty & symmetry

− stab



h




16/52


ComputationEh(vh) =

∑T∈T

h

(u − πk

F (u), σHHOn (vT )− γvF

)F︸︷︷︸

penalty & symmetry

+∑

T∈Th

(−∆u, vT )T

−∑

T∈Th

(∇Pk+1T (u),∇Rk+1

T (vT ))T


+∑

F∈Fbh

(σn(Pk+1

T (u)), vF)F︸︷︷︸

consistency

− stab



h




16/52


ComputationEh(vh) =

∑T∈T

h

(u − πk


)F︸︷︷︸

penalty & symmetry

+∑

T∈Th

(∇u,∇vT )T − (σn(u), vT )∂T

−∑

T∈Th

(∇Pk+1T (u),∇Rk+1

T (vT ))T


+∑

F∈Fbh

(σn(Pk+1

T (u)), vF)F︸︷︷︸

consistency



h




16/52


ComputationEh(vh) =

∑T∈T

h

(u − πk


)F︸︷︷︸

penalty & symmetry

+∑

T∈Th

(∇u,∇vT )T − (σn(u), vT )∂T

−∑

T∈Th

(∇Pk+1T (u),∇vT )T + (σn(Pk+1

T (u)), v∂T − vT )∂T

︸︷︷︸Galerkin: def ∇Rk+1

T (vT )

+∑

F∈Fbh

(σn(Pk+1

T (u)), vF)F︸︷︷︸

consistency



h




16/52


ComputationEh(vh) =

∑T∈T

h

(u − πk


)F︸︷︷︸

penalty & symmetry

+∑

T∈Th

(∇u,∇vT )T − (σn(u), vT − v∂T )∂T −∑

F∈Fbh

(σn(u), vF )F

−∑

T∈Th

(∇Pk+1T (u),∇vT )T + (σn(Pk+1

T (u)), v∂T − vT )∂T


T (vT )

+∑

F∈Fbh

(σn(Pk+1

T (u)), vF)F︸︷︷︸

consistency



h




16/52


ComputationEh(vh) =

∑T∈T

h

(u − πk


)F︸︷︷︸

penalty & symmetry

+∑

T∈Th

(∇δ,∇vT )T − (σn(δ), vT − v∂T )∂T −∑

F∈Fbh

(σn(δT ), vF )F



h




17/52

Problems with face-based traces

Unfortunaly we cannot prove optimal estimates for Signorini’s problem

We can do so with cell traces!

Cell-based unknowns now in Pk+1(T )

Still locally eliminated by static condensation

Similar ideas in [Burman & Ern, 2018] for CutHHO


18/52

Analysis for Cell-based traces

New reconstruction

to bound T1 as above, we need to modify the reconstructionwe modify the contribution of boundary faces

(∇Rk+1T (uT ),∇z)T = −(uT ,∆z)T + (u∂T , nT · ∇z)∂T

Pk+1T\Γ := Rk+1

T\Γ IkT\Γ is no longer the local elliptic projection ... but optimal

approximation properties are preserved!

Approximation property: letting δ∗ := u − Pk+1T\Γ(u), we still have

‖δ∗‖T + h12

T‖δ∗‖∂T + hT‖∇δ∗‖T + h32

T‖σn(δ∗)‖∂T . hk+2T |u|Hk+2(T )


‖∇(u − Rk+1T\Γ(uT ))‖2

T

) 12

. O(hk+1)


18/52

Analysis for Cell-based traces

New reconstruction

to bound T1 as above, we need to modify the reconstructionwe modify the contribution of boundary faces

(∇Rk+1T\Γ(uT ),∇z)T = −(uT ,∆z)T + (u∂T , nT · ∇z)∂T\Γ

Pk+1T\Γ := Rk+1

T\Γ IkT\Γ is no longer the local elliptic projection ... but optimal

approximation properties are preserved!

Approximation property: letting δ∗ := u − Pk+1T\Γ(u), we still have

‖δ∗‖T + h12

T‖δ∗‖∂T + hT‖∇δ∗‖T + h32

T‖σn(δ∗)‖∂T . hk+2T |u|Hk+2(T )


‖∇(u − Rk+1T\Γ(uT ))‖2

T

) 12

. O(hk+1)


19/52

Consistency error

. . . still defined as Eh(vh) := ˆγ,h(vh)− aγ,h(I kh (u), vh)

Face-based trace method

Eh(vh) =∑

T∈Th

(u − πk


)F

+∑

T∈Th

(∇u,∇vT )T − (σn(u), vT − v∂T )∂T −∑

F∈Fbh

(σn(u), vF )F

−∑

T∈Th

(∇Pk+1T (u),∇vT )T + (σn(Pk+1

T (u)), v∂T − vT )∂T


T (vT )

+∑

F∈Fbh

(σn(Pk+1

T (u)), vF)F︸︷︷︸

consistency

Cell-based trace method

Eh(vh) =∑

T∈Th

(u − πk+1

T (u), σHHOn (vT )− γvT

)F

+∑

T∈Th

(∇u,∇vT )T − (σn(u), vT )∂T

−∑

T∈Th

(∇Pk+1T (u),∇vT )T + (σn(Pk+1

T (u)), v∂T − vT )∂T


T (vT )

+∑

F∈Fbh

(σn(Pk+1

T\Γ(u)), vT)F︸︷︷︸

consistency


19/52

Consistency error



Eh(vh) =∑

T∈Th

(u − πk


)F

+∑

T∈Th

(∇u,∇vT )T − (σn(u), vT − v∂T )∂T −∑

F∈Fbh

(σn(u), vF )F

−∑

T∈Th

(∇Pk+1T (u),∇vT )T + (σn(Pk+1

T (u)), v∂T − vT )∂T


T (vT )

+∑

F∈Fbh

(σn(Pk+1

T (u)), vF)F︸︷︷︸

consistency


Eh(vh) =∑

T∈Th

(u − πk+1


)F

+∑

T∈Th

(∇u,∇vT )T − (σn(u), vT )∂T\Γ −∑

F∈Fbh

(σn(u), vT )F

−∑

T∈Th

(∇Pk+1T (u),∇vT )T + (σn(Pk+1

T (u)), v∂T − vT )∂T


T (vT )

+∑

F∈Fbh

(σn(Pk+1

T\Γ(u)), vT)F︸︷︷︸

consistency


19/52

Consistency error



Eh(vh) =∑

T∈Th

(u − πk


)F

+∑

T∈Th

(∇u,∇vT )T − (σn(u), vT − v∂T )∂T −∑

F∈Fbh

(σn(u), vF )F

−∑

T∈Th

(∇Pk+1T (u),∇vT )T + (σn(Pk+1

T (u)), v∂T − vT )∂T


T (vT )

+∑

F∈Fbh

(σn(Pk+1

T (u)), vF)F︸︷︷︸

consistency


Eh(vh) =∑

T∈Th

(u − πk+1


)F

+∑

T∈Th

(∇u,∇vT )T − (σn(u), vT − v∂T )∂T\Γ −∑

F∈Fbh

(σn(u), vT )F

−∑

T∈Th

(∇Pk+1T (u),∇vT )T + (σn(Pk+1

T (u)), v∂T − vT )∂T


T (vT )

+∑

F∈Fbh

(σn(Pk+1

T\Γ(u)), vT)F︸︷︷︸

consistency


19/52

Consistency error



Eh(vh) =∑

T∈Th

(u − πk


)F

+∑

T∈Th

(∇u,∇vT )T − (σn(u), vT − v∂T )∂T −∑

F∈Fbh

(σn(u), vF )F

−∑

T∈Th

(∇Pk+1T (u),∇vT )T + (σn(Pk+1

T (u)), v∂T − vT )∂T


T (vT )

+∑

F∈Fbh

(σn(Pk+1

T (u)), vF)F︸︷︷︸

consistency


Eh(vh) =∑

T∈Th

(u − πk+1


)F

+∑

T∈Th

(∇u,∇vT )T − (σn(u), vT − v∂T )∂T\Γ −∑

F∈Fbh

(σn(u), vT )F

−∑

T∈Th

(∇Pk+1T\Γ(u),∇vT )T + (σn(Pk+1

T\Γ(u)), v∂T − vT )∂T\Γ


T\Γ(vT )

+∑

F∈Fbh

(σn(Pk+1

T\Γ(u)), vT)F︸︷︷︸

consistency


20/52

Diffusion: test case

Analytical solution u in the unit square

u = cos(πx) cos(πy)

Energy error vs h:0.1 0.2 0.4 0.6

10-6

10-4

10-2

100

Err

or

Hexagons

k = 0 k = 1 k = 2 k = 3

0.785

2.087

3.235

4.198

0.025 0.05 0.1 0.2

h

10-8

10-6

10-4

10-2

100

Err

or

Face-based traces

1.122

2.003

2.911

3.873

0.025 0.05 0.1 0.2

h

10-8

10-6

10-4

10-2

100

Err

or

Cell-based traces

2.889

3.876

1.984

1.101


21/52





22/52

Signorini’s unilateral contact problem

Model problem

−∆u = f in Ω

u = g on ΓD

Contact conditions

u ≤0 on ΓS

σn(u) ≤0 on ΓS

uσn(u) =0 on ΓS

Reformulation of Curnier & Alart:

σn(u) = [σn(u)− γu]R− on ΓS

ΩΓD

ΓS

ΓD

ΓD

u=0σn≤0

u<0σn=0

No contactContact

Γ = ΓD ∪ ΓS


23/52

Nitsche for Signorini’s problem

Nitsche-FEM [Chouly & Hild 2013]

aγ,h(uh; vh) :=∑T∈Th

(∇uh,∇vh)T −∑

F∈Fb,Sh

1

γ

(σn(uh), σn(vh)

)︸︷︷︸

symmetry

+∑

F∈Fb,Sh

1

γ

([σn(uh)− γuh)]R− , σn(uh)︸︷︷︸

penalty

− γvh)F︸︷︷︸

consistency

Nitsche-HHO [Cascavita et al. 2019]

aγ,h(uh; vh) :=∑T∈Th


−∑

F∈Fb,Sh

1

γ

(σHHO

n (uT ), σHHO

n (vT )︸︷︷︸symmetry

)F

+∑

F∈Fb,Sh

1

γ

([σHHO

n (uT )− γuT (F )

]R− , σ

HHO

n (vT )︸︷︷︸coercivity

− γvT︸︷︷︸consistency

)F

Optimal energy-error estimate: u ∈ Hk+1(Ω)∑T∈Th

‖∇(u − Rk+1T\

(uT ))‖2T .

∑T∈Th

h2(k+1)T |u|2Hk+2(T )


24/52

Signorini: Manufactured solution test case

Analytical solution u

u(r , θ) = −r 112 sin

(11

2θ

), with θ ∈ [−π, 0]

-1

-0.5-2

-1

0

1

1

2

3

0.5

4

0 0-0.5 -1

-1

-0.5

0

0.5

1

1.5

2

2.5

3

-

= 0

0

u

n(u)

Contact:

u

n(u) =

0

0

No contact:

-1-2

y

1 -0.5

0u

2

x

0

4

0-1

-1

0

1

2

3


25/52

Signorini: Results

Meshes:

-1 -0.5 0 0.5 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Energy error vs h: 0.1 0.2 0.4 0.610

-6

10-4

10-2

100

Err

or

Hexagons

k = 0 k = 1 k = 2 k = 3

0.785

2.087

3.235

4.198

0.05 0.1 0.2

h

10-6

10-4

10-2

100

Err

or

Triangles

2.960

1.921

3.945

0.908

0.025 0.05 0.1 0.2

h

10-8

10-6

10-4

10-2

100

Err

or

Square

1.922

2.957

3.969

0.939

0.1 0.2 0.4 0.6

h

10-6

10-4

10-2

100

Err

or

Hexagons

k = 0

k = 1

k = 2

k = 3

0.785

2.087

3.235

4.198


26/52

Outline




27/52

Model problem

Conservation laws: Total stress tensor σtot and velocity field u

div σtot + f = 0 in Ω

div u = 0 in Ω

Constitutive equation (liquid / solid) σ = 2µ∇su +√

2σ0∇su

|∇su|`2

yielded (|σ|`2 >√

2σ0)

∇su = 0 unyielded (|σ|`2 ≤√

2σ0)

Viscosity µ > 0, yield stress limit σ0 > 0, σ = σtot − (1/3)tr(σtot)Id .

Require a finite yield stress to flow (liquid or solid-like behavior)

Liquid/Solid boundary not known a priori


27/52

Model problem

Conservation laws: Total stress tensor σtot and velocity field u

div σtot + f = 0 in Ω

div u = 0 in Ω

Constitutive equation (liquid / solid)σ = 2µ∇su︸︷︷︸

viscous term

+√

2σ0∇su

|∇su|`2︸︷︷︸plastic term

yielded (|σ|`2 >√

2σ0)

∇su = 0 unyielded (|σ|`2 ≤√

2σ0)

Viscosity µ > 0, yield stress limit σ0 > 0, σ = σtot − (1/3)tr(σtot)Id .

Require a finite yield stress to flow (liquid or solid-like behavior)

Liquid/Solid boundary not known a priori


28/52

State of the art

Mathematical formulation:

Regularization of constitutive law:

Artificial parameter [Bercovier & Engelman 1980, Papanastasiou 1987]The yield surface is not well captured

Augmented Lagrangian algorithm (ALG):

[Hestenes 1969, Powell 1969], for Bingham [Fortin & Glowinski 1983]Favorite method for viscoplastic flows [Saramito & Wachs 2017]

Recent improvements:

Adaptive augmentation [Bartels & Milicevic 2017],Accelerated version [Treskatis, Moyers-Gonzalez & Price 2016]Lagrangian & conic programming [Bleyer, Maillard, de Buhan & Coussot 2015]Damped Newton algorithm [Saramito 2016]


29/52

Augmented Lagrangian formulation

Continuous velocity field

u = arg minv∈V0(Ω)

∫Ω

H(∇sv)dΩ− (f, v)Ω

with the subspace of velocities with zero divergence

V0(Ω) := v ∈ H10(Ω) | ∇ · v = 0

and dissipation energy density

H(g) = µ|g|2`2 +√

2σ0|g|`2

Augmented Lagrangian: L : V0(Ω)× L2(Ω;Rd×d

s )× L2(Ω;Rd×d

s )→ R

L(u,γ,σ) :=

∫Ω

H(γ)dΩ− (f,u)Ω + (σ,∇su− γ)Ω + α‖∇su− γ‖2Ω

Auxiliary variable γ interpreted as strain rateLagrange multiplier σ turns out to be the stressAugmentation parameter α > 0


30/52

Continuous ALG

Compute γn+1 ∈ L2(Ω;Rd×ds ) such that:

γn+1 :=

1

2(α + µ)

(|θn|`2 −

√2σ0

) θn

|θn|`2

unyielded region

0 yielded region

where θn := σn + 2α∇sun

Seek (un+1, pn+1) ∈ H10(Ω)× L2

0(Ω) s.t.

2α(∇sun+1,∇sv)Ω − (pn+1,∇·v)Ω = (f , v)Ω − (σn − 2αγn+1,∇sv)Ω

(q,∇·un+1)Ω = 0

Update the Lagrange multiplier σn+1 ∈ L2(Ω;Rd×ds ):

σn+1 := σn + 2α(∇sun+1 − γn+1)


31/52





32/52

Antiplanar flows

Constitutive equation: σ = µ∇u + σ0∇u|∇u|`2

yielded

∇u = 0 unyielded


u = arg minv∈H1

0 (Ω)

∫Ω

H(∇v)dΩ− (f , v)Ω

f

ez

ey

ex

u(x) = (0, 0, u(x))T

Augmented Lagrangian L : H10 (Ω)× L

2(Ω)× L2(Ω)→ R

L(u,γ,σ) :=

∫Ω

H(γ)dΩ− (f , u)Ω + (σ,∇u − γ)Ω +α

2‖∇u − γ‖2

Ω


33/52

Lowest-order HHO

Cell velocities (scalar) are locally eliminated

Global linear system for face velocities

γ and σ vector locally evaluated

Velocity u γ − σ(ALG)

Local gradient reconstruction operator (HFV) GT : UT → P0(T ;Rd)

GT (vT ) =∑

F∈F∂T

|F |d−1

|T |d(vF − vT )nT ,F

Discrete bilinear form

aT (uT , vT ) = (GT (uT ),GT (vT ))T + (h−1T S∂T (uT ),S∂T (vT ))∂T


34/52

Lowest-order HHO with ALG


u = arg minv∈H1

0 (Ω)

∫Ω

H(∇v)dΩ− (f , v)Ω

Discrete velocity field

uh = arg minvh∈Uh,0

∑T∈Th

∫T

H(GT (vT ))− (f , vT )T +α

2‖h−

12

T S∂T (vT )‖2∂T

Augmented Lagrangian: L : H1

0 (Ω)× L2(Ω)× L

2(Ω)→ R

L(v ,γ,σ) :=

∫Ω

H(γ)dΩ− (f , v)Ω + (σ,∇v − γ)Ω +α

2‖∇v − γ‖2

Ω

Discrete Augmented Lagrangian: Lh : Uh,0 × R2|Th| × R2|Th| → R

Lh(vh,γTh ,σTh) :=∑T∈Th

∫T

H(γT )− (f , vT )T +α

2‖h−

12

T S∂T (vT )‖2∂T

+ (σT ,GT (vT )− γT )T +α

2‖GT (vT )− γT‖2

T


35/52

Discrete iterative algorithm

1 Find γn+1Th ∈ R2|Th| with θn

T := σnT + αGT (unT ), locally

γn+1T :=

1

(α + µ)(|θn

T |`2 − σ0)θnT

|θnT |`2

yielded

0 unyielded

2 Solve global problem: Seek un+1h ∈ Uh,0 solving

ah(uh, vh) =∑T∈Th

(f , vT )T − (σn

T − αγn+1T ,GT (vT ))T

3 Update Lagrange multiplier σn+1

Th ∈ R2|Th| locally

σn+1T = σn

T + α(GT (un+1T )− γn+1

T )


36/52

Mesh adaptation

General idea: SOLVE → FLAG + MARK → REFINE

Flagging cells criterion: Use |θnT |`2 ≶ σ0 for solid/liquid status

TFLAG MARK

X

REFINE

T

Yield surface

Yielded region

(liquid)

Unyielded region

(solid)

T

T

T

T

Control on local refinement variations (limit number of hanging nodes)


37/52

Numerical Results

Test cases

Test case 1: Circular cross section, known solution

Test case 2: Eccentric annulus cross section, unknown solution

Bi = 2σ0/fR (R characteristic length, f flow driving force)

test case 1 test case 2(one-eigth of domain) (one-half of domain)

(Bi = 0.3, 0.9) (Bi = 0.2)


38/52

Refinement

Stress norm |σ|`2 : colormap for [σ0, σ0 + δ], δ = 0.005

Known analytical solution: yield surface

Bi Initial mesh 5th Adaptive level Zoom

0.3

0.9


39/52

Eccentric annulus cross section

Bi = 0.2 and σ0 = 0.1

Stress norm |σ|`2 : colormap [σ0,σ0 + δ], δ = 0.005


40/52

Capture of yield surfaces

Top: uniform triangulation, h = 7.5e-3 and 93 KDOF

Bottom: adaptive mesh T6, hmin = 7.8e-4 and 113 KDOF

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1


41/52





42/52

Extension to vector flows

Difficulties:

Incompressibility condition ∇ · u = 0

HHO method with pressure [Di Pietro, Ern, Linke & Schieweck 2016]

Symmetric gradient: Korn’s inequality

Reconstruction with k = 0:

Korn’s inequality not granted a prioriExperiments show squares still work (not triangles)ALG variables treated in the mesh cells using affine cell unknowns

Reconstruction with k ≥ 1:

Korn’s inequality granted [Di Pietro & Ern]ALG variables evaluated at Gauss points


43/52

HHO methods with ALG

Cell velocities (vector) are locally eliminated

Cell pressures (scalar)

Global linear system for face velocities & constant pressures

γ and σ tensors handled by ALG

Example in the lowest-order case

Velocity u Pressure p γ − σ

(ALG)

Uh := P1h(Th;Rd)× P0(Fh;Rd) Ph := P0

h(Th;R) Σh = P(Th;Rd×ds )


44/52

HHO methods with ALG

Extension to higher-order

Local symmetric gradient operator E kT (vT ) : Uk

T → Pk(T ;Rd×ds )

(E kT (vT ), z)T = −(vT ,∇ · z)T + (v∂T , znT )∂T ∀z ∈ Pk(T ;Rd×d

s )

Global contributions:

Viscous term

ah(uh, vh) =∑T∈Th

2µ(E kT (uT ),E k

T (vT ))T + stab term

Divergence term (incompressibility)

bh(vh, qh) =∑T∈Th

(tr(E kT (vT )), qT )T


45/52

Augmented Lagrangian formulation

Explicitly enforce homogeneous Dirichlet BCs leads to Ukh,0

Discrete velocity field

uh = arg minvh∈U

kh,0

∑T∈Th

QT ((Ek

T (vT ))2)− (f , vT )T + β‖h−12

T Sk∂T (vT )‖2

∂T

,

β = µ improves convergence

Augmented Lagrangian: Lh : Ukh,0 × Σk

h × Σkh → R

Lh(uh, γh, σh) :=∑T∈Th

QT (γT ) + β‖h−

12

T Sk∂T (uT )‖∂T − (f,uT )T

+ QT (σT : EkT (uT )− γT ) + αQT ((Ek

T (uT )− γT )2)

.


46/52

Lid driven cavity

Benchmark for the vector case [Syrakos et al 2013]

Velocity field for Bi = 2 (left).

Velocity profile along the axis x = 1/2 (right).

Excellent agreement with reference solutions

0 0.2 0.4 0.6 0.8 1

x

0

0.5

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

y

-0.5 0 0.5 1

ux/V

Bi = 50

Bi = 2

Syrakos et al. (2013)


47/52

Capture of yield surfaces

Stress norm |σ|`2 : colormap for [σ0, σ0 + δ], δ = 0.005Quadrangular meshes (similar results in hexagonal meshes)

Coarse mesh Fine mesh h = 1/256

Bi = 2

Bi = 50


48/52

Vane-in-cup geometry problem

No-slip condition u = 0

R Re i

u = x

Physical parameters:

Angular velocity ω = 1,The radius Ri = 4 for the innerboundary,The radius Re = 6 outerboundary.

Adimensional setting:

Reference length R = Re

Reference velocity V = ωRi

Bingham numberBi = σ0R

µV= σ0(Re−Ri )

µωRi= σ0

2


49/52

Vane-in-cup geometry problem

Unknown analytical solution

Comparison with [Ovarlez et al. 2011] (Herschel-Bulkley material)

Velocity field for Bi = 1 (center) and for Bi = 10 (right)

that of Yan and James (1997) is that the blade thickness is zero in this last study.It is particularly striking and counterintuitive that Rc is largest at the angular position

(θ = 30˚) where shear at Ri is smallest (similar observation was made by Potanin (2010)).As in Sec. IIIA3, this points out the importance of the extensional flow in this geometry, withθ-dependent normal stress differences which have to be taken into account in the yield criterion,and which thus impact the yield surface. It thus seems that the link between the yield stressτy and the torque T measured at yield with a vane-in-cup geometry is still an open question,although the classical formula probably provides a sufficiently accurate determination of τy inpractice.

Figure 9: Two-dimensional plot of the limit between rigid motion and shear (circles) and betweenshear and rest (triangles) for a yield stress fluid (concentrated emulsion) sheared in the six-bladedvane-in-cup geometry at 0.1 rpm (left) and 1 rpm (right). The grey rectangles correspond tothe blades.

The same 2D map as above is plotted for Ω = 1 rpm in Fig. 9; the same phenomena areobserved, with enhanced departure from cylindrical symmetry, consistent with the observationthat Rl decreases when Ω increases. This result was also unexpected, as simulations find uniformflows for shear-thinning material of index n ≤ 0.5 [Barnes and Carnali (1990); Savarmand et

al. (2007)]; we would have expected the same phenomenology in a Herschel-Bulkley material ofindex n = 0.5 (and thus Rl to tend to Ri when increasing Ω). This observation also shows thata Couette analogy can hardly be defined for studying the flow properties of such materials ina vane-in-cup geometry because the equivalent Couette geometry radius Ri,eq would probablydepend also on Ω (as recently shown by Zhu et al. (2010)).

Let us finally note that this departure from cylindrical symmetry has important impact onthe migration of particles in a yield stress fluid (see below).

C Concentrated suspension

In this section, we investigate the behavior of a concentrated suspension of noncolloidal particlesin a yield stress fluid (at a 40% volume fraction).

A detailed study of their velocity profiles would a priori present here limited interest: suchmaterials present the same nonlinear macroscopic behavior as the interstitial yield stress fluid,and their rheological properties (yield stress, consistency) depend moderately on the particlevolume fraction [Mahaut et al. (2008a); Chateau et al. (2008)].

On the other hand, noncolloidal particles in suspensions are known to be prone to shear-induced migration, which leads to volume fraction heterogeneities. This phenomenon is well

16


50/52

Wrap up: Signorini’s contact problem

Conclusions

Optimal error estimates for Nitsche-HHO

Trace type Diffusion SignoriniFace-based X ×Cell-based X X

Optimal numerical convergence rates.

Perspectives

Extension to vector deformations

Tresca and Coulomb friction laws.


51/52

Wrap up: Bingham problem

Conclusions

Polyhedral meshes for local refinement to track yield surface

Agreement with analytical solutions, literature and experiments

Perspectives

Development

A posteriori error estimate to further drive mesh adaptationExploit hp-refinement within HHO methods

Applications

Extension to 3D vector flowsHerschel-Bulkley or visco-elasto-plasticity models.Simulation of an air bubble immersed in a non-Newtonian fluid, surface-tension


52/52

Thank you for your attention


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