VEM 3D: set up, implementation and applications · VEM 3D: set up, implementation and applications...

VEM 3D: set up, implementation and applications

F. Dassi1 and S. Scialo2

1 Dipartimento di Matematica e ApplicazioniUniversita Milano - Bicocca

[email protected] Dipartimento di Scienze Matematiche “ Giuseppe Luigi Lagrange ”

Politecnico di [email protected]

[email protected]

joint work withL. Beirao da Veiga, S. Berrone, A. Borio, F. Brezzi, A. D’Auria, L. Mascotto, L. D. Marini, S. Pieraccini,

A. Russo, G. Vacca and F. Vicini

GNCS 2018 Montecatini - Italy

14th February 2018

F. Dassi and S. Scialo (MIBI-DISMA PoliTo) February 14, 2018 1 / 43

Progetto GNCS 2017

Titolo:“Tecniche numeriche avanzate basate su discretizzazioni con elementi poligonali/poliedrici percontesti applicativi caratterizzati da una elevata complessita geometrica”.Responsabile: Stefano BerroneUnita del progetto:

Dipartimento di Scienza Metematiche - Politecnico di Torino.Partecipanti strutturati: Stefano Berrone, Claudio Canuto, Sandra Pieraccini, Stefano Scialo.Partecipanti non strutturati: Alessandro D’Auria, Fabio Vicini.

Dipartimento di Matematica e Applicazioni - Universita di Milano - Bicocca.Partecipanti strutturati: Lourenco Beirao da Veiga, Alessandro Russo.Partecipanti non strutturati: Franco Dassi, Giuseppe Vacca.

Istituto di Matematica Applicata e Tecnologie Informatiche - CNR.Partecipanti strutturati: Silvia Bertoluzza, Paola Pietra.Partecipanti non strutturati: Yumeng Zhang.

Dipartimento di Matematica - Universita di MilanoPartecipanti strutturati: Carlo Lovadina.Partecipanti non strutturati: Lorenzo Mascotto.

MOX - Dipartimento di Matematica - Politecnico di MilanoPartecipanti strutturati: Marco Verani.


What is the Virtual Element Method (VEM)?

Novel way to solve Partial Differential Equations in 2d and 3d


Why VEM?

general polyhedral meshes


Why VEM?

no need to avoid hanging nodes/edges


Why VEM?

better isotropy


Why VEM?

generality of the discretization space


Why VEM?

naturally more robust with respect to mesh distortion ordegeneration than FEM


Why VEM?

similar to FEM for coding


Why VEM?

compatibility with FEM


Why VEM?

Novel way to solve PDEs with these advantages

general polyhedral meshes

no need to avoid hanging nodes/edges

better isotropy

generality of the discretization space

naturally more robust with respect to mesh distortion ordegeneration than FEM

similar to FEM for coding

compatibility with FEM


VEM in the world

more than 40 publications in the last 4 years!


Talk outline

1 Introduction

2 Virtual elements 3dDefinition of V k

h on a polyhedron PDefinition of Π∇k,P for a polyhedron

3 Application of 3D VEM


Introduction

Introduction

Basic ingredients of VEM (in 3d)

VEM space, V kh

projection operators


Introduction


VEM space, V kh



Introduction


VEM space, V kh



Introduction


VEM space, V kh



Introduction


VEM space, V kh



Virtual elements

Virtual elements 3d

Notation for polygons and polyhedrons

polyhedron P polygon f

|P | = volume |f | = area


Virtual elements 3d

Notation for 3d monomials

Let P ⊂ R3, k ∈ N\{0} and α = (α1, α2, α3) be a multi-index, wedefine the scaled monomials

mα :=

(x− xPhP

)α1(y − yPhP

)α2(z − zPhP

)α3

,

and the spaces

Pk (P ) = span{mα , 0 ≤ |α| ≤ k} .

where|α| = α1 + α2 + α3 .

The case f ⊂ R2 is analogous but with 2d coordinates system!


Virtual elements 3d



mα :=

(x− xPhP

)α1(y − yPhP

)α2(z − zPhP

)α3

,

and the spaces

Pk (P ) = span{mα , 0 ≤ |α| ≤ k} .

where|α| = α1 + α2 + α3 .



Virtual elements 3d



mα :=

(x− xPhP

)α1(y − yPhP

)α2(z − zPhP

)α3

,

and the spaces

Pk (P ) = span{mα , 0 ≤ |α| ≤ k} .

where|α| = α1 + α2 + α3 .



Virtual elements 3d Definition of V kh on a polyhedron P

Basic ingredients of VEM

VEM space, Vkh




Virtual element space for a generic polyhedronWe define the space

V kh (P ) =

{v ∈ H1(P ) : v|f ∈ V k

h (f) ∀f ∈ ∂P ,

∆v ∈ Pk−2(P )},

where

V kh (f) =

{w ∈ H1(f) ∩ C0(f) : w|e ∈ Pk(e), ∀e ∈ ∂f,

∆w ∈ Pk(f) ,∫f

(Π∇k,fw

)pk =

∫f

w pk, ∀pk ∈ Pk(f)\Pk−2(f)

}.



Dofs for v ∈ V kh (P )




- the values at v(ν) ∀ν vertex of P ,

- the values at k − 1 nodes on eachedge of P , v(νie),






- face moments∫f

vpk−2 ∀pk−2 ∈ Pk−2(f),∀f face of P






- face moments∫f


- internal moments∫P

v pk−2 ∀pk−2 ∈ Pk−2(P )






- face moments∫f


- internal moments∫P

v pk−2 ∀pk−2 ∈ Pk−2(P )



One more consideration on the space V kh (f )

V kh (f) =

{w ∈ H1(f) ∩ C0(f) : w|e ∈ Pk(e), ∀e ∈ ∂f,

∆w ∈ Pk(f) ,∫f

(Π∇k,fw

)pk =

∫f


}.



One more consideration on the space V kh (f )

V kh (f) =

{w ∈ H1(f) ∩ C0(f) : w|e ∈ Pk(e), ∀e ∈ ∂f,

∆w ∈ Pk(f) ,∫f

(Π∇k,fw

)pk =

∫f


}.



Enhancing idea



Enhancing idea



Enhancing idea



Comparison between VEM face spaces

V kh (f) V k

h (f)

w ∈ H1(f) ∩ C0(f) w ∈ H1(f) ∩ C0(f)

w|e ∈ Pk(e) ∀e ∈ ∂f w|e ∈ Pk(e) ∀e ∈ ∂f∆w ∈ Pk(f)

∆w ∈ Pk−2(f)∫f

(Π∇k,fw

)pk =

∫f

w pk

same dofs

both define Π∇k,f via dofs

but only via V kh (f) we can get the 3d projection operator!




V kh (f) V k

h (f)

w ∈ H1(f) ∩ C0(f) w ∈ H1(f) ∩ C0(f)


∆w ∈ Pk−2(f)∫f

(Π∇k,fw

)pk =

∫f

w pk

same dofs






V kh (f) V k

h (f)

w ∈ H1(f) ∩ C0(f) w ∈ H1(f) ∩ C0(f)


∆w ∈ Pk−2(f)∫f

(Π∇k,fw

)pk =

∫f

w pk

same dofs






V kh (f) V k

h (f)

w ∈ H1(f) ∩ C0(f) w ∈ H1(f) ∩ C0(f)


∆w ∈ Pk−2(f)∫f

(Π∇k,fw

)pk =

∫f

w pk

same dofs




Virtual elements 3d Definition of Π∇k,P for a polyhedron


VEM space, V kh




H1-projector operator for polyhedrons

We can define a projection Π∇k,P : V kh (P )→ Pk(P ) by

∫P

∇(v − Π∇k,Pv) · ∇pk = 0 ∀pk ∈ Pk(P ) ,

for k = 1:∑ν vertex

ofP

(v(ν)− Π∇k,Pv(ν)

)= 0,

for k ≥ 2:∫P

(v − Π∇k,Pv) = 0.



Yes, we can find the projection!

Consider the monomial basis {mα}|α|≤k

Π∇k,Pv =∑|α|≤k

cvαmα .

To find cvα we consider the (k + 1)(k + 2)(k + 3)/6 conditions

∫P

∇(v − Π∇k,Pv) · ∇mβ = 0 ∀|β| ≤ k ,


ofP


)= 0,

for k ≥ 2:∫p

(v − Π∇k,Pv) = 0.




Consider the monomial basis {mα}|α|≤k

Π∇k,Pv =∑|α|≤k

cvαmα .

To find cvα we consider the (k + 1)(k + 2)(k + 3)/6 conditions

∫P

∇(v − Π∇k,Pv) · ∇mβ = 0 ∀|β| ≤ k ,


ofP


)= 0,

for k ≥ 2:∫p

(v − Π∇k,Pv) = 0.




∫P

∇(v − Π∇k,Pv) · ∇mβ = 0

∫P

∇

v − ∑|α|≤k

cvαmα

· ∇mβ = 0

∑|α|≤k

cvα =

∑|α|≤k

cvα




∫P

∇(v − Π∇k,Pv) · ∇mβ = 0

∫P

∇

v − ∑|α|≤k

cvαmα

· ∇mβ = 0

∑|α|≤k

cvα =

∑|α|≤k

cvα




∫P

∇(v − Π∇k,Pv) · ∇mβ = 0

∫P

∇

v − ∑|α|≤k

cvαmα

· ∇mβ = 0

∑|α|≤k

cvα

∫P

∇mα · ∇mβ =

∫P

∇v · ∇mβ

∑|α|≤k

cvα

∫P

∇mα · ∇mβ = −∫P

∆mβ v︸︷︷︸dofs

+

∫∂P

(∇mβ · n) v




∫P

∇(v − Π∇k,Pv) · ∇mβ = 0

∫P

∇

v − ∑|α|≤k

cvαmα

· ∇mβ = 0

∑|α|≤k

cvα

∫P

∇mα · ∇mβ︸︷︷︸exactly computable

=

∫P

∇v · ∇mβ

∑|α|≤k

cvα




∫P

∇(v − Π∇k,Pv) · ∇mβ = 0

∫P

∇

v − ∑|α|≤k

cvαmα

· ∇mβ = 0

∑|α|≤k

cvα

∫P

∇mα · ∇mβ =

∫P

∇v · ∇mβ

∑|α|≤k

cvα

∫P

∇mα · ∇mβ = −∫P

∆mβ v︸︷︷︸dofs

+

∫∂P

(∇mβ · n) v




∫∂P

(∇mβ · n) v =∑f∈∂P

∫f

(∇mβ · nf ) v

but

∇mβ · nf ∈ Pk−2(f)

∇mβ · nf ∈ Pk−1(f)\Pk−2(f)

so ∑f∈∂P

∫f

(∇mβ · nf ) v =∑f∈∂P

∫f

(∇mβ · nf ) Π∇k,fv




∫∂P

(∇mβ · n) v =∑f∈∂P

∫f

(∇mβ · nf ) v

but∇mβ · nf ∈ Pk−2(f)

∇mβ · nf ∈ Pk−1(f)\Pk−2(f)

so ∑f∈∂P

∫f

(∇mβ · nf ) v =∑f∈∂P

∫f





∫∂P

(∇mβ · n) v =∑f∈∂P

∫f

(∇mβ · nf ) v

but∇mβ · nf ∈ Pk−2(f) ⇒ face moments

∇mβ · nf ∈ Pk−1(f)\Pk−2(f)

so ∑f∈∂P

∫f

(∇mβ · nf ) v =∑f∈∂P

∫f





∫∂P

(∇mβ · n) v =∑f∈∂P

∫f

(∇mβ · nf ) v

but∇mβ · nf ∈ Pk−2(f) ⇒ face moments

∇mβ · nf ∈ Pk−1(f)\Pk−2(f) ⇒ property of V kh (f)

so ∑f∈∂P

∫f

(∇mβ · nf ) v =∑f∈∂P

∫f





To fix the constant, case k = 1, we use∑ν vertex

ofP

Π∇k,Pv(ν) =∑ν vertex

ofP

v(ν)

∑ν vertex

ofP

∑|α|≤k

cvα =∑ν vertex

ofP





ofP


ofP

v(ν)

∑ν vertex

ofP

∑|α|≤k

cvαmα(ν) =∑ν vertex

ofP

v(ν)





ofP


ofP

v(ν)

∑ν vertex

ofP

∑|α|≤k

cvα mα(ν)︸︷︷︸exactly

computable

=∑ν vertex

ofP

v(ν)︸︷︷︸dofs




To fix the constant, case k ≥ 2, we use∫P

Π∇k,Pv =

∫P

v

∑|α|≤k

cvα =





Π∇k,Pv =

∫P

v

∑|α|≤k

cvα

∫P

mα =

∫P

v





Π∇k,Pv =

∫P

v

∑|α|≤k

cvα

∫P

mα︸︷︷︸exactly

computable

=

∫P

v︸︷︷︸exactly

computablefrom dofs




VEM space, V kh



Application of 3D VEM


Problem description

F

D1

D2

O

D

Let us consider the following problem in D = F ∪(⋃

i=1,2Di)

−∇ · (Ki∇Hi) = fi in Di, i = 1, 2−∇π · (KF∇πHF ) = fF − (Q1 +Q2) in FH1 = H2 = HF on FQi = (Ki∇Hi) · ni i = 1, 2H = 0 on ∂D

being fi, i = 1, 2 and fF known source terms and ni the unit normal vector to F pointingoutward from Di.



Functional setting

Let us introduce the following functional spaces, for i = 1, . . . , 2:

Vi ={v ∈ H1

0(Di) : v|F ∈ H10(F)

}VF = H1

0(F) and W = H−1(F)

than it is possible to rewrite the problem as:find (H1, H2, HF ) ∈ V1 ×V2 ×VF and Q1, Q2 ∈W, such that

(Ki∇Hi,∇v)Di

= (fi, v)Di+⟨Qi, v|F

⟩F , ∀v ∈ Vi, i = 1, 2

(KF∇πHF ,∇πv)F = (fF , v)F −∑2i=1 〈Qi, v〉F , ∀v ∈ VF⟨

Hi|F −HF , λ⟩F = 0 ∀λ ∈W, i = 1, 2



Functional settingProblem:find (H1, H2, HF ) ∈ V1 ×V2 ×VF and Q1, Q2 ∈W, such that

(Ki∇Hi,∇v)Di

= (fi, v)Di+⟨Qi, v|F

⟩F , ∀v ∈ Vi, i = 1, 2


Hi|F −HF , λ⟩F = 0 ∀λ ∈W, i = 1, 2

is well posed, as it can be shown observing that:

V ={v : vi = v|Di

∈ Vi, vF ∈ VF , vi = vF = v|F , i = 1, 2}⊂ H1

0(D)

is an Hilbert space with the norm

a(v, v) = (K∇v,∇v)D +(KF∇πv|F ,∇πv|F ,

)F , ∀v ∈ V,

and the problem is equivalent to the unique minimum in V of the quadratic functional

E(w) =1

2

∫D

(∣∣∣√K∇w∣∣∣2 − 2fw

)+

1

2

∫F

(∣∣∣√KF∇πw|F∣∣∣2 − 2fFw

).



Typical application domains

F2



Typical application domains



Problem discretization

Two different discretization techniques:

VEM based technique coupling VEM-3D to VEM-2D - VEM-VEM approachEfficient generation of a conforming mesh of complex domainsClassical approach (DD)

Optimization-based approach coupling BEM-3D to FEM-2D - OPT approach

(but also FEM-3D to FEM-2D, FEM-3D to VEM-2D, ...)Generation of a mesh non conforming to the interfaces (trivial process!)Flexible and scalable approach



VEM mesh

B1

B2

F



VEM mesh

B1

B2

F



VEM-VEM approach



VEM-VEM approachProblem find (H1, H2, HF ) ∈ V1 ×V2 ×VF and Q1, Q2 ∈W, such that

(Ki∇Hi,∇v)Di= (fi, v)Di

+⟨Qi, v|F

⟩F , ∀v ∈ Vi, i = 1, 2


Hi|F −HF , λ⟩F = 0 ∀λ ∈W, i = 1, 2

after discretization with VEM-3D on the blocks and VEM-2D on the interfaces, is re-written as the followingsaddle-point problem: (

Ab LTAfL O

)(HbHfQ

)=

(qbqf0

)



Numerical results - Problem 1 (6 Blocks, 3 Fractures, 2 Traces)

VEM Order 1

VEM Order 2




VEM Order 1

VEM Order 2



OPT-approachA cost functional is introduced to enforce matching conditions at the interfaces:

J(H,HF , U) =∑`∈L?

∑k∈KΓ`

∥∥∥H`k −HF%(`)

∥∥∥2

H`Γ

+M∑m=1

(∥∥∥HFi|Sm−HFj |Sm

∥∥∥2

HmS

+∥∥Umi + Umj

∥∥2

Um

)

minU,Q

J(H,HF , U) such that:

- for all k = 1, . . . , nD and for all vk ∈ Vk

(K∇Hk,∇vk)Dk= (fk, vk)Dk

+∑`∈Lk

(Q`k, v

`k

)Γ`

,

- and for all i = 1, . . . , I and for all vFi∈ VFi

:

(KFi∇HFi

,∇vFi

)Fi

=∑`∈LFi

(−Q`, vFi

)Fi

+M∑m=1

(Umi , vFi

)Sm

,

being Vk , VFisuitable function spaces.






References

M. F. Benedetto, S. Berrone, A. Borio, S. Pieraccini, and S. Scialo, A hybrid mortar Virtual Element Methodfor discrete fracture network simulations, J. Comput. Phys., 306 (2016), pp. 148-166.

M. F. Benedetto, S. Berrone, A. Borio, S. Pieraccini, and S. Scialo, Order preserving SUPG stabilization forthe virtual element formulation of advection-diffusion problems, Comput. Methods Appl. Mech.Engrg., 311 (2016), pp. 18-40.

S. Berrone, A. Borio, and S. Scialo, A posteriori error estimate for a PDE constrained optimizationformulation for the flow in DFNs, SIAM Journal Numer. Anal., 54 (2016), pp. 242-261.

S. Berrone, S. Pieraccini, and S. Scialo, Towards effective flow simulations in realistic discrete fracturenetworks, J. Comput. Phys., 310 (2016), pp. 181-201.

S. Berrone, S. Pieraccini, and S. Scialo, Non-stationary transport phenomena in networks of fractures:Effective simulations and stochastic analysis, Comput. Methods Appl. Mech. Engrg., 315 (2017), pp.1098-1112.

S. Berrone, S. Pieraccini, and S. Scialo, Flow simulations in porous media with immersed intersectingfractures, J. Comput. Phys., 345 (2017), pp. 768-791.

M. F. Benedetto, A. Borio, S. Scialo, Mixed Virtual Elements for discrete fracture network simulations FiniteElem. Anal. Des., 134 (2017), pp. 55-67.


Date post:	18-Feb-2019
Category:	Documents
Upload:	dodiep
View:	220 times
Download:	0 times

VEM 3D: set up, implementation and applications · VEM 3D: set up, implementation and applications...

Documents