cermics.enpc.frcermics.enpc.fr/~cascavik/THESIS_Slides.pdf · 2018-12-17 · 3/52 The scope of the...
Transcript of 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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 2 / 52
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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]
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 4 / 52
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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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 5 / 52
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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]
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 6 / 52
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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)
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 7 / 52
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1 Nitsche-HHODiffusionSignorini’s unilateral condition
2 HHO for Bingham flowsAntiplanar configurationFull vector setting
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Outline
1 Nitsche-HHODiffusionSignorini’s unilateral condition
2 HHO for Bingham flowsAntiplanar configurationFull vector setting
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 9 / 52
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Outline for section 1
1 Nitsche-HHODiffusionSignorini’s unilateral condition
2 HHO for Bingham flowsAntiplanar configurationFull vector setting
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 10 / 52
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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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 11 / 52
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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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 11 / 52
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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
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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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 12 / 52
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Local operators
T
F∂T
vT = (vT , v∂T )
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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 12 / 52
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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)
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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 12 / 52
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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
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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 12 / 52
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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
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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 12 / 52
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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
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
but ∇Rk+1T (uT ) = 0 ; uT = uF = cst
Stability norm: |vT |2UkT
:= ‖∇vT‖2T + ‖h− 1
2 (v∂T − vT )‖2∂T
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 12 / 52
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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
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 + (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
Stability norm: |vT |2UkT
:= ‖∇vT‖2T + ‖h− 1
2 (v∂T − vT )‖2∂T
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 12 / 52
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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
aγ,h(uh, vh) =∑T∈Th
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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 13 / 52
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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
aγ,h(uh, vh) =∑T∈Th
aT (uT , vT )︸ ︷︷ ︸Galerkin
−∑F∈Fb
h
(σHHO
n (uT ), v?︸ ︷︷ ︸consistency
)F
+(u?, γv? − σHHO
n (vT )︸ ︷︷ ︸penalty & symmetry
)F
Numerical parameter γ = γ0h−1 and γ0 > 0
Normal stress σHHOn
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 13 / 52
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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 with Face-based trace
aγ,h(uh, vh) =∑T∈Th
aT (uT , vT )︸ ︷︷ ︸Galerkin
−∑F∈Fb
h
(σHHO
n (uT ), vF︸ ︷︷ ︸consistency
)F
+(uF , γvF − σHHO
n (vT )︸ ︷︷ ︸penalty & symmetry
)F
Numerical parameter γ = γ0h−1 and γ0 > 0
Normal stress σHHOn (vT ) = σn(Rk+1
T (uT ))
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 13 / 52
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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 with Cell-based trace
aγ,h(uh, vh) =∑T∈Th
aT (uT , vT )︸ ︷︷ ︸Galerkin
−∑F∈Fb
h
(σHHO
n (uT ), vT︸ ︷︷ ︸consistency
)F
+(uT , γvT − σHHO
n (vT )︸ ︷︷ ︸penalty & symmetry
)F
Numerical parameter γ = γ0h−1 and γ0 > 0
Normal stress σHHOn needs to be modified
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 13 / 52
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14/52
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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 14 / 52
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15/52
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)
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 15 / 52
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16/52
Bound on consistency error
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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 16 / 52
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16/52
Bound on consistency error
ComputationEh(vh) = ˆ
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
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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 16 / 52
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16/52
Bound on consistency error
ComputationEh(vh) = ˆ
γ,h(vh)︸ ︷︷ ︸rhs
−∑
T∈Th
(∇Pk+1T (u),∇Rk+1
T (vT ))T
︸ ︷︷ ︸Galerkin
+∑
F∈Fbh
(σn(Pk+1
T (u)), vF)F︸ ︷︷ ︸
consistency
+(πkF (u), σHHO
n (vT )− γvF)F︸ ︷︷ ︸
penalty & symmetry
− stab
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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 16 / 52
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16/52
Bound on consistency error
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
︸ ︷︷ ︸Galerkin
+∑
F∈Fbh
(σn(Pk+1
T (u)), vF)F︸ ︷︷ ︸
consistency
+∑
F∈Fbh
(πkF (u), σHHO
n (vT )− γvF)F︸ ︷︷ ︸
penalty & symmetry
− stab
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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 16 / 52
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16/52
Bound on consistency error
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
︸ ︷︷ ︸Galerkin
+∑
F∈Fbh
(σn(Pk+1
T (u)), vF)F︸ ︷︷ ︸
consistency
+∑
F∈Fbh
(πkF (u), σHHO
n (vT )− γvF)F︸ ︷︷ ︸
penalty & symmetry
− stab
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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 16 / 52
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16/52
Bound on consistency error
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
︸ ︷︷ ︸Galerkin
+∑
F∈Fbh
(σn(Pk+1
T (u)), vF)F︸ ︷︷ ︸
consistency
− stab
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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 16 / 52
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16/52
Bound on consistency error
ComputationEh(vh) =
∑T∈T
h
(u − πk
F (u), σHHOn (vT )− γvF
)F︸ ︷︷ ︸
penalty & symmetry
+∑
T∈Th
(∇u,∇vT )T − (σn(u), vT )∂T
−∑
T∈Th
(∇Pk+1T (u),∇Rk+1
T (vT ))T
︸ ︷︷ ︸Galerkin
+∑
F∈Fbh
(σn(Pk+1
T (u)), vF)F︸ ︷︷ ︸
consistency
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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 16 / 52
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16/52
Bound on consistency error
ComputationEh(vh) =
∑T∈T
h
(u − πk
F (u), σHHOn (vT )− γvF
)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
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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 16 / 52
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16/52
Bound on consistency error
ComputationEh(vh) =
∑T∈T
h
(u − πk
F (u), σHHOn (vT )− γvF
)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
︸ ︷︷ ︸Galerkin: def ∇Rk+1
T (vT )
+∑
F∈Fbh
(σn(Pk+1
T (u)), vF)F︸ ︷︷ ︸
consistency
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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 16 / 52
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16/52
Bound on consistency error
ComputationEh(vh) =
∑T∈T
h
(u − πk
F (u), σHHOn (vT )− γvF
)F︸ ︷︷ ︸
penalty & symmetry
+∑
T∈Th
(∇δ,∇vT )T − (σn(δ), vT − v∂T )∂T −∑
F∈Fbh
(σn(δT ), vF )F
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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 16 / 52
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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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 17 / 52
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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 )
This leads to the optimal energy-error estimate(∑T∈Th
‖∇(u − Rk+1T\Γ(uT ))‖2
T
) 12
. O(hk+1)
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 18 / 52
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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 )
This leads to the optimal energy-error estimate(∑T∈Th
‖∇(u − Rk+1T\Γ(uT ))‖2
T
) 12
. O(hk+1)
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 18 / 52
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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 (u), σHHOn (vT )− γvF
)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
︸ ︷︷ ︸Galerkin: def ∇Rk+1
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
︸ ︷︷ ︸Galerkin: def ∇Rk+1
T (vT )
+∑
F∈Fbh
(σn(Pk+1
T\Γ(u)), vT)F︸ ︷︷ ︸
consistency
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 19 / 52
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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 (u), σHHOn (vT )− γvF
)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
︸ ︷︷ ︸Galerkin: def ∇Rk+1
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\Γ −∑
F∈Fbh
(σn(u), vT )F
−∑
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\Γ(u)), vT)F︸ ︷︷ ︸
consistency
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 19 / 52
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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 (u), σHHOn (vT )− γvF
)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
︸ ︷︷ ︸Galerkin: def ∇Rk+1
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 − 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
︸ ︷︷ ︸Galerkin: def ∇Rk+1
T (vT )
+∑
F∈Fbh
(σn(Pk+1
T\Γ(u)), vT)F︸ ︷︷ ︸
consistency
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 19 / 52
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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 (u), σHHOn (vT )− γvF
)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
︸ ︷︷ ︸Galerkin: def ∇Rk+1
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 − 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
︸ ︷︷ ︸Galerkin: def ∇Rk+1
T\Γ(vT )
+∑
F∈Fbh
(σn(Pk+1
T\Γ(u)), vT)F︸ ︷︷ ︸
consistency
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 19 / 52
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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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 20 / 52
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Outline for section 2
1 Nitsche-HHODiffusionSignorini’s unilateral condition
2 HHO for Bingham flowsAntiplanar configurationFull vector setting
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 21 / 52
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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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 22 / 52
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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
aT (uT , vT )︸ ︷︷ ︸Galerkin
−∑
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 )
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 23 / 52
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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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 24 / 52
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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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 25 / 52
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Outline
1 Nitsche-HHODiffusionSignorini’s unilateral condition
2 HHO for Bingham flowsAntiplanar configurationFull vector setting
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 26 / 52
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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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 27 / 52
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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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 27 / 52
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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]
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 28 / 52
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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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 29 / 52
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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)
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 30 / 52
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Outline for section 1
1 Nitsche-HHODiffusionSignorini’s unilateral condition
2 HHO for Bingham flowsAntiplanar configurationFull vector setting
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 31 / 52
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Antiplanar flows
Constitutive equation: σ = µ∇u + σ0∇u|∇u|`2
yielded
∇u = 0 unyielded
Continuous velocity field
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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 32 / 52
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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
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Lowest-order HHO with ALG
Continuous velocity field
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
K. Cascavita HHO for Signorini and Bingham December 18th, 2018 34 / 52
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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 )
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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)
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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)
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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
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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
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Eccentric annulus cross section
Bi = 0.2 and σ0 = 0.1
Stress norm |σ|`2 : colormap [σ0,σ0 + δ], δ = 0.005
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Eccentric annulus cross section
Bi = 0.2 and σ0 = 0.1
Stress norm |σ|`2 : colormap [σ0,σ0 + δ], δ = 0.005
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Eccentric annulus cross section
Bi = 0.2 and σ0 = 0.1
Stress norm |σ|`2 : colormap [σ0,σ0 + δ], δ = 0.005
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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
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Outline for section 2
1 Nitsche-HHODiffusionSignorini’s unilateral condition
2 HHO for Bingham flowsAntiplanar configurationFull vector setting
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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
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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 )
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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
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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)
.
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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)
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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
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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
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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
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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.
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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
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Thank you for your attention
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