Slides

Compatible Discrete Operators Schemes for the Stokes
Equations
Jérôme Bonelle1,2 and Alexandre Ern2
1
2
EDF R&D
Université Paris Est - CERMICS (ENPC)
WCCM XI - ECCM V - ECFD VI
J. Bonelle and A. Ern (EDF R&D / CERMICS)
CDO for Stokes
July, 24th, 2014 - Barcelona
1 / 23
Context
→ EDF has been developing several in-house Computational Fluid Dynamics
codes for 30 years
Code_Saturne (open-source) single-phase flow solver based on co-located
Finite Volume schemes (since 1998)
Approach close to commercial codes like Star-CD or FLUENT
→ Reopen numerical work to improve numerical methods
Axes: physical fidelity, robustness on complex geometry, efficiency
New developments based on structure-preserving schemes
→ The Compatible Discrete Operator (CDO) framework
Inspired by seminal ideas of Tonti and Bossavit
Elliptic problems (B. & Ern, M2AN, 2014)
Stokes equations (submitted, in revision)
Navier-Stokes (in progress)
J. Bonelle and A. Ern (EDF R&D / CERMICS)
CDO for Stokes
July, 24th, 2014 - Barcelona
2 / 23
Outline
1
Compatible Discrete Operator (CDO) Framework
2
Discretization of the Stokes Equations
3
Analysis of the Vertex-based Schemes
4
Numerical Results
J. Bonelle and A. Ern (EDF R&D / CERMICS)
CDO for Stokes
July, 24th, 2014 - Barcelona
3 / 23
De Rham Complex
→ Degrees of freedom (DoFs) are defined by De Rham maps
→ Definition of DoFs in agreement with the physical nature of fields
Potential
(scalar)
at a point
Flux
(vector)
curl
along a line
RE (u)|e =
RV (p)|v = p(v )
V
Circulation
(vector)
grad
GRAD
e
div
across a surface
RF (φ)|f =
u · τe
CURL
E
f
inside a volume
RC (s)|c =
φ · νf
F
Density
(scalar)
DIV
c
s
C
Discrete differential operators
1
Metric-free operators: algebraically defined by incidence matrices
2
Commuting property with De Rham’s maps
3
GRAD · RV = RE · grad,
CURL · RE = RF · curl,
DIV · RF = RC · div
Cochain complex: CURL · GRAD ≡ 0F and DIV · CURL ≡ 0C
J. Bonelle and A. Ern (EDF R&D / CERMICS)
CDO for Stokes
July, 24th, 2014 - Barcelona
5 / 23
Two Meshes: a Primal and a Dual Mesh
Primal mesh : M = {V, E, F, C}
Dual mesh : M = {V, E, F, C}
→ Carry the information on geometry,
material properties and BCs
→ Only for defining the scheme
→ Several definitions: barycentric,
voronoï. . .
→ Only mesh seen by the end-user
→ One-to-one pairing: v ↔ c˜(v ), e ↔ f˜(e), f ↔ ˜e (f ) and c ↔ v˜ (c)
→ Transfer of orientation: τ e → ν f˜(e) and ν f → τ ˜e (f )
v•
◦
◦
◦
◦
v
˜(c1 )
•
◦
τe
◦
• τ e˜(f)
◦
˜e(f)
v
˜(c2 )
◦
J. Bonelle and A. Ern (EDF R&D / CERMICS)
νf
v
˜(c3 )
•
ν ˜f(e)
˜f(e)
f
˜c(v)
e
τ e˜(f)
•
v
˜(c4 )
CDO for Stokes
July, 24th, 2014 - Barcelona
6 / 23
Discrete Setting
DoFs on the dual mesh V, E, F, C are also defined by de Rham maps
V
GRAD
E
CURL
F
DIV
C
C
DIV
F
CURL
E
GRAD
V
Duality Products
Adjunction properties
˜ ∈ {V˜C, E F˜, F˜E , C˜V }
XY
a, b
˜ :=
XY
− GRAD(p), φ
ax by˜ (x) = a b
t
CURL(u), v
x∈X
where a ∈ X , b ∈ Y, X ∈ {V, E, F, C}
J. Bonelle and A. Ern (EDF R&D / CERMICS)
− DIV(φ), p
CDO for Stokes
˜ = p, DIV(φ)
EF
˜
VC
˜ = u, CURL(v)
FE
˜ = φ, GRAD(p)
CV
July, 24th, 2014 - Barcelona
˜
EF
˜
FE
7 / 23
Discrete Hodge Operators HXα˜Y
V
E
F
C
HVα˜C
HEαF˜
HFα˜E
HCα˜V
C
F
E
V
• Link DoF spaces in duality X Y˜ ∈ {V˜C, E F˜, F˜E , C˜V }
• Depend on a metric induced by a material property α
• HX Y˜ is built up from a cellwise assembly process
α
• Definition hinges on two local design properties
1
2
Stability Upper/lower eigenvalues are uniformly bounded
P0 -consistency Exactly represents piecewise constant field on each c ∈ C
• Multiple definitions → multiple schemes
J. Bonelle and A. Ern (EDF R&D / CERMICS)
CDO for Stokes
July, 24th, 2014 - Barcelona
8 / 23
Synthesis of the Discrete Setting
V
GRAD
HVα˜C
C
E
CURL
HEαF˜
DIV
F
F
DIV
HFα˜E
CURL
E
C
HCα˜V
GRAD
V
Discrete Differential Operators
Discrete Hodge Operators
Topological laws
Constitutive relations
Error-free
Approximation
Unique definition
Multiple definitions
J. Bonelle and A. Ern (EDF R&D / CERMICS)
CDO for Stokes
July, 24th, 2014 - Barcelona
9 / 23
CDO and Some Related Approaches
HO/LO
Setting
Element
Meshes
Key
Op.
CDO
HFV
MFV
DDFV
MFD
VEM
HHO
FEEC
MSE
LO
NC
Poly.
P+D
disc.
Hodge
LO
NC
Poly.
P
grad
reco.
LO
NC
Poly.
P+D+
grad/div
reco.
LO/HO
C
Poly.
P
inner
prod.
HO
C
Poly.
P
inner
prod.
HO
NC
Poly.
P
grad
reco.
HO
C
Spe.
P
cochain
proj.
HO/LO = Higher-order/Lower-order
NC/C = nonconforming/conforming reconstruction operator
Poly. / Spe. = Polyhedral / Specific (i.e. tetrahedral, hexahedral meshes)
P/D/ = primal / dual / diamond meshes are explicitly considered
• HFV/MFV Hybrid/Mixed Finite Volume: Droniou, Eymard, Herbin, Gallouët
• DDFV Discrete Duality Finite Volume: Andreianov, Hubert, Krell et al.
• MFD Mimetic Finite Differences & VEM Virtual Element Method:
Beirão Da Veiga, Brezzi, Lipnikov, Shashkov, Manzini et al.
• HHO Hybrid High-Order : Di Pietro & Ern
• FEEC Finite Element Exterior Calculus: Arnold, Falk, Whinteret al.
• MSE Mimetic Spectral Element: Kreeft, Gerritsma, Pahla et al.
J. Bonelle and A. Ern (EDF R&D / CERMICS)
CDO for Stokes
July, 24th, 2014 - Barcelona
10 / 23
Stokes Equations for Incompressible Flows
Three-field Curl
Two-field Curl
Classical
Let u be the velocity, p the pressure, ω the vorticity and f the external load
(u, p)
(u, p)
(u, ω, p)
− ∆(u) + grad(p)
=f
− div(u)
=0
curl curl(u) + grad(p)
=f
− div(u)
=0

=0

 − ω + curl(u)
curl(ω) + grad(p) = f


− div(u)
=0
J. Bonelle and A. Ern (EDF R&D / CERMICS)
CDO for Stokes
(MFV) Droniou & Eymard
(MFD) Beirão da Veiga et al.
(DDFV) Krell & Manzini
(HFV) Di Pietro & Lemaire
(FE) Bramble & Lee
(FE) Abboud et al.
(FV) Eymard et al.
(FE) Nédélec; Dubois
(Spectral) Bernardi & Chorfi
(MSEM) Kreeft & Gerritsma
(DDFV) Delcourte & Omnes
July, 24th, 2014 - Barcelona
12 / 23
Overview of CDO Schemes for Stokes
Vertex-based Pressure
Cell-based Pressure
Two-field curl formulation
• Pressure DoFs ∈ V
• Velocity DoFs ∈ E
(circulation)
System size: #V + #E
Three-field curl formulation
• Pressure DoFs ∈ V (1:1 with C)
• Velocity DoFs ∈ F (flux)
• Vorticity DoFs ∈ E
System size: #C + #F + #E
Main Features
→ Exact mass and momentum balances on polyhedral meshes
→ 1st order CV rate for smooth enough solutions
on velocity, vorticity and the pressure gradient
→ Robust treatment of the external load with large divergence-free or curl-free
part
J. Bonelle and A. Ern (EDF R&D / CERMICS)
CDO for Stokes
July, 24th, 2014 - Barcelona
13 / 23
Two-field Curl
Vertex-based Pressure Schemes
u, p ∗ := pρ , (ρ ≡ 1 and µ ≡ 1)
curl(µ curl(u)) + ρ grad(p ∗ ) = ρf
− div(ρu) = 0
p∗ ∈ V
u∈E
GRAD
• Pressure DoFs: p∗ at primal vertices
• Velocity DoFs: u at primal edges
• HEρF˜ and HFµ˜E
CURL
HEρF˜
HV˜C
Exact mass balance
C
HFµ˜E
Svb ∈ F
DIV
C
DIV
F
CURL
E
HC˜V
GRAD
V
Exact momentum balance

 CURL · HF˜E · CURL(u) + HE F˜ · GRAD(p∗ )
µ
 − DIV · HE F˜ (u)
ρ
J. Bonelle and A. Ern (EDF R&D / CERMICS)
ρ
CDO for Stokes
= Svb (f )
= 0C
July, 24th, 2014 - Barcelona
14 / 23
Cell-based Pressure Schemes
Three-field Curl
φ := ρu, ω ∗ := µω, p ∗ :=
p
ρ
•
•
•
•

−1 ∗
−1

 − µ ω + curl(ρ φ) = 0


ρ−1 curl(ω ∗ ) + grad(p ∗ ) = f
− div(φ) = 0
GRAD
V
Exact mass balance
ω∗ ∈ E
CURL
DIV
F
DIV
φ∈F
˜
HµE F
−1
HV˜C
C
Pressure DoFs: p∗ at dual vertices
Velocity DoFs: φ at primal faces
Vorticity DoFs: ω ∗ at primal edges
˜E
HEµF˜−1 and HFρ−1
˜E
HF
ρ−1
Scb ∈ E
CURL
Vorticity definition
C
HCV˜
GRAD
p∗ ∈ V
Exact momentum balance

EF
˜
∗
F˜
E

−
H
= 0F
−1 (ω ) + CURL · Hρ−1 (φ)

µ

F˜
E
∗
∗
H
−1 · CURL(ω ) + GRAD(p )
ρ



− DIV(φ)
J. Bonelle and A. Ern (EDF R&D / CERMICS)
CDO for Stokes
= Scb (f )
= 0C
July, 24th, 2014 - Barcelona
15 / 23
Discrete Poincaré Inequalities
Stability & well-posedness hinge on these inequalities + Hodge stability
→ Mesh regularity: Assume there exists a shape-regular simplicial submesh
ax 2
→ • |||a|||22,Xc = x∈Xc hc3 ( |x|
) where a ∈ X ∈ {V, E, F} and X ∈ {V, E, F}
• |||a|||22,X collects local contributions
1
Discrete Poincaré-Wirtinger inequality
There exists CP > 0 s.t. ∀p ∈ V verifying
p, HV˜C (1) V˜C = 0.
(0)
2
(0)
|||p|||2,V ≤ CP |||GRAD(p)|||2,E
Discrete Poincaré inequality for CURL
There exists CP > 0 s.t. ∀u ∈ E verifying
˜ (v)
u, HEαF
EF
˜ = 0 where v ∈ Ker CURL
Sketch
(1)
(1)
|||u|||2,E ≤ CP |||CURL(u)|||2,F
→ Consider conforming reconstructions on polyhedral meshes (Christiansen)
• Use the compatibility property with the differential operators
→ Use the continuous Poincaré inequalities
J. Bonelle and A. Ern (EDF R&D / CERMICS)
CDO for Stokes
July, 24th, 2014 - Barcelona
17 / 23
Error Estimates
α, X Y˜ (A) = 0, ∀A ∈ [P0 (C)]3
XY
˜ · R (A)
α, X Y˜ (A) := RY (α A) − Hα
X
P0 -consistency
of HXα Y˜
1st order CV on
smooth solutions
Let (u, p) be the exact solution and ω = curl(u), g = grad(p)
Let (u, p) solve the discrete system and ω = CURL(u), g = GRAD(p)
Pressure Gradient
Vorticity
Velocity
where |||•|||α :=
|||RE (g) − g)|||ρ
E
|||RF (ω) − ω|||µ
E + ||| µ, F ˜E (ω)|||(µ)−1
|||RE (u) − u|||ρ
Y
˜
•, HX
α (•)
˜
XY
˜ · R (f )
(PL) HEρ F
E
E + ||| µ, F ˜E (ω)|||(µ)−1 + ||| ρ, E F
˜ (u)|||(ρ)−1
, |||•|||(α)−1 :=
Y
˜ −1 (•), •
(HX
α )
Y
˜
EF
˜ F˜
E
for HX
α ∈ {Hρ , Hµ }
E := ||| ρ, E F
˜ (curl ω)|||(ρ)−1
Better when
curl(ω)
grad(p)
⇒ Large curl-free part
E := ||| ρ, E F
˜ (grad p)|||(ρ)−1
Better when
grad(p)
curl(ω)
⇒ Large divergence-free part
Svb (ρ, f )
(DL) RF (ρf )
˜
XY
J. Bonelle and A. Ern (EDF R&D / CERMICS)
CDO for Stokes
July, 24th, 2014 - Barcelona
18 / 23
3D Taylor-Green Testcase
Analytical solutions for p, u and ω are a product of sin and cos
Vertex-based Pressure Scheme
•
•
•
•
BCs: Fully Natural u · ν and ω ∗ × ν
HEρF˜ and HFµ˜E built with the Discrete Geometric Approach (Codecasa et al.)
Algorithm: Uzawa - Augmented Lagrangian (Preconditioned CG as inner solver)
System size: nsys = #E + #V
→ Influence of the Primal external load (PL) and Dual external load (DL)
→ Mesh sequences from the FVCA benchmark
Prismatic meshes
J. Bonelle and A. Ern (EDF R&D / CERMICS)
Polyhedral meshes
CDO for Stokes
Polyhedral meshes with hanging nodes
July, 24th, 2014 - Barcelona
20 / 23
Convergence Rates
J. Bonelle and A. Ern (EDF R&D / CERMICS)
External Load
Load
Er(p)
Er(u)
Er(ω)
(PL)
1.8
2.0
1.9
(DL)
2.0
2.0
1.9
(PL)
1.7
1.8
1.7
(DL)
2.1
1.8
1.7
(PL)
1.8
1.1
1.0
(DL)
2.2
1.1
1.0
CDO for Stokes
July, 24th, 2014 - Barcelona
21 / 23
f = χu curl(Ψu ) + χp grad(θp )
Eﬀectofofthe
theDiscretization
Discretization
theExternal
ExternalLoad
Load
Eﬀect
ofofthe
curl-free
divergence-free
Prism
Meshes
with
Polygonal
Basis
Prism
Meshes
with
Polygonal
Basis
The Hodge-Helmholtz
decomposition
f is not
in the
discretization
curl
curl(u)
ρ grad(p)
+p χgrad(θ
curl
·µ ·µ
curl(u)
++
ρ grad(p)
==
χof
curl(Ψ
+)used
χ
u curl(Ψ
p grad(θ
uχ
p) p)
u) u
� �
χu
χp large divergence-free
(DL) RF
suited
χp(f
1isand
variable
χu χu
χ=
1better
and
variable
p) =
��
p
104104
103103
� �
χu large curl-free
F
˜ R (f
χu χ=u 1=
1
p and
(PL) variable
HEvariable
is
better
suited
E χp)χand
χρf ρf
101 101
102102
Er(u)
Er(u)
Er(p)
Er(p)
100 100
101101
100100
10−1
10−1
10−1 −1
10
10−2
10−2
10−2
10−2
104104
105 105
#V#V
(DL)
χ=
(DL)
χ{
={
105 105
#E #E
2
4
4 {1,
: 10: 210
, 2 , χ:p10∈
} 4 } 10 , 10 } and χu = 1
: 10
: CDO
1,: 1,for Stokes
: 10: 210
, 2 , : 10: 410
} 4 } July, 24th, 2014 - Barcelona
22 / 23
2
4
χp = 1 and χ(PL)
{1,
χ=
u ∈
(PL)
χ10{
= ,{10 :}1,: 1,
J. Bonelle and A. Ern (EDF R&D / CERMICS)
104 104
Thank you for your attention
Contact: bonellej “at” cermics.enpc.fr
Homepage: http://cermics.enpc.fr/~bonellej/home.html
1
Analysis of Compatible Discrete Operator Schemes for Elliptic Problems on
Polyhedral Meshes, B. & Ern, M2AN, 2014
2
Analysis of Compatible Discrete Operator Schemes for the Stokes Equations on
Polyhedral Meshes, B. & Ern, 2014 (in revision, IMA JNA), HAL preprint
J. Bonelle and A. Ern (EDF R&D / CERMICS)
CDO for Stokes
July, 24th, 2014 - Barcelona
23 / 23

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