[PPT] - The Virtual Element Method for Discrete Fracture Network flow PowerPoint Presentation

SLIDE 1

Intro Model VEM Conforming Mortar Transport References

The Virtual Element Method for Discrete Fracture Network flow simulations

Stefano Berrone Dipartimento di Scienze Matematiche “ Giuseppe Luigi Lagrange ” Politecnico di Torino stefano.berrone@polito.it joint work with Matias Benedetto, Andrea Borio, Sandra Pieraccini, Stefano Scial`

Georgia Tech

Atlanta, October 26th, 2015

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 1

SLIDE 2

Intro Model VEM Conforming Mortar Transport References

Discrete fracture network and flow model:

Figure : Example of DFN

3D network of intersecting fractures in the rock matrix Fractures are represented as planar polygons The flow only/mainly occurs in the fractures, i.e. Rock matrix is considered impervious Flow modeled by Darcy law in the fractures Flux balance and hydraulic head continuity imposed across fracture intersections (traces) No changes on the given DFN geometry stocastically generated (position,

rientation, size, shape) → uncertainty quantification

Many approaches in literature require some geometry modification Complex domain: difficulties in good quality mesh generation

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 2

SLIDE 3

Intro Model VEM Conforming Mortar Transport References Fracture model

Insulated fracture formulation

find H ∈ H1

D(F ) such that:

(K ∇H, ∇v) = (q, v) +GN, v|ΓN

H− 1

2 (ΓN ),H 1 2 (ΓN ), ∀v ∈ V =H1

0,D(F )

H is the hydraulic head on the fracture F ; K is the fracture transmissivity tensor: a symmetric and uniformly positive definite tensor containing the hydrological properties;

∂H ∂ ˆ ν = ˆ

nt K ∇H = GN is the outward co-normal derivative of the hydraulic head and ˆ n the unit outward vector normal to the boundary ΓN.

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 3

SLIDE 4

Intro Model VEM Conforming Mortar Transport References Fracture model

Coupled fracture formulation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

F1 F2 F1 F2 S S S U S

1 =

∂H1

∂ˆ ν1

S

S

U S

2 =

∂H2

∂ˆ ν2

S

S

Ω

For each trace S ∈ S on the fracture Fi i = 1, . . . , N, let use denote by U S

i :=

∂Hi

∂ˆ νi

S

S

U S

i ∈ U S ⊆H− 1

2 (S)

the flux entering in the fracture through the trace S, and Ui ∈ U Si the tuple

f fluxes U S

i ∀S ∈ Si

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 4

SLIDE 5

Intro Model VEM Conforming Mortar Transport References Fracture model

Full fracture formulation

Solving ∀i ∈ I the problem: find Hi ∈ H1

D(Fi) and Ui ∈ Si such that:

(Ki ∇Hi, ∇v) = (qi, v) + Ui, v|Si USi ,USi ′ +GiN, v|ΓiN

H− 1

2 (ΓiN ),H 1 2 (ΓiN ), ∀v ∈ Vi =H1

0,D(Fi)

with additional conditions Hi|S − Hj|S = 0, for i, j ∈ IS, ∀S ∈ S, U S

i + U S j

= 0, for i, j ∈ IS, ∀S ∈ S, provides the hydraulic head H ∈ V = H1

D(Ω).

Hi is the hydraulic head on the fracture Fi, H is the hydraulic head on Ω; Ki is the fracture transmissivity tensor: a symmetric and uniformly positive definite tensor containing the hydrological properties;

∂Hi ∂ ˆ νi = ˆ

nt

i Ki ∇Hi = GiN is the outward co-normal derivative of the

hydraulic head and ˆ ni the unit outward vector normal to the boundary ΓiN.

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 5

SLIDE 6

Intro Model VEM Conforming Mortar Transport References Fracture model

Totally/Partially conforming meshes

Figure : Totally conforming mesh Figure : Partially conforming mesh

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 6

SLIDE 7

Intro Model VEM Conforming Mortar Transport References Fracture model

non-conforming meshes: 120 fractures, 256 traces DFN

Figure : Non-conforming mesh on a 120 fracture DFN

A reformulation as a PDE constrained optimization problem allows non conforming meshes.

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 7

SLIDE 8

Intro Model VEM Conforming Mortar Transport References

DFN flow simulations Virtual Element Method

Virtual Element Methods allows meshes with Polygonal elements with a different number of edges, elements with aligned edges.

s s s s s s s s s s ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣

Figure : General P1 VEM element E

So we can easily obtain polygonal partially o totally conforming meshes

n DFNs that can be easily obtained starting from independent

triangular meshes on the fractures. We need many informations on the traces → conforming meshes are recommended.

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 8

SLIDE 9

Intro Model VEM Conforming Mortar Transport References

The space discretization: VEM

1

Start from a given triangular mesh, built without taking into account trace positions or conformity requirements.

2

Whenever a trace intersects one element edge, a new node is

created. New nodes are also

created at trace tips. If the trace tip falls in the interior of an element, the segment-trace is prolonged up to the opposite mesh edge.

3

Elements cut by prolonged segment-traces are then split into new “sub-elements”, which become elements in their own

right. Convex polygons are
btained.

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 9

SLIDE 10

Intro Model VEM Conforming Mortar Transport References

The space discretization: VEM

1

Start from a given triangular mesh, built without taking into account trace positions or conformity requirements.

2

Whenever a trace intersects one element edge, a new node is

created. New nodes are also

created at trace tips. If the trace tip falls in the interior of an element, the segment-trace is prolonged up to the opposite mesh edge.

3

Elements cut by prolonged segment-traces are then split into new “sub-elements”, which become elements in their own

right. Convex polygons are
btained.

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 10

SLIDE 11

Intro Model VEM Conforming Mortar Transport References

The space discretization: VEM

1

Start from a given triangular mesh, built without taking into account trace positions or conformity requirements.

2

Whenever a trace intersects one element edge, a new node is

created. New nodes are also

created at trace tips. If the trace tip falls in the interior of an element, the segment-trace is prolonged up to the opposite mesh edge.

3

Elements cut by prolonged segment-traces are then split into new “sub-elements”, which become elements in their own

right. Convex polygons are
btained.

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 11

SLIDE 12

Intro Model VEM Conforming Mortar Transport References

Mesh smoothing

Independent smoothing for each fracture to improve the quality of the mesh.

Figure : Mesh modifications.

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 12

SLIDE 13

Intro Model VEM Conforming Mortar Transport References Conforming Mesh Equations Convergence Robustness

DFNs with Conforming Virtual Element Method of several orders

Once we have obtained a partially conforming mesh a simple step forward allows us to obtain a globally conforming mesh simply adding to the elements of each fracture with an edge on the trace the nodes on the trace of the twin fracture: Partially conforming mesh → Totally conforming mesh.

r r r r r r r r r

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . .

Fi Fj

r r r r r r r r r r r r r r

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . .

Fi Fj

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 13

SLIDE 14

Intro Model VEM Conforming Mortar Transport References Conforming Mesh Equations Convergence Robustness

Figure : Final globally conforming VEM mesh

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 14

SLIDE 15

Intro Model VEM Conforming Mortar Transport References Conforming Mesh Equations Convergence Robustness

Let us define the space of “continuous” test functions V =

v : v|Fi∈H1

0,D(Fi), ∀i=1, .., N,

γS(v|Fi)=γS(v|Fj), ∀S ∈Si, i, j =IS

,

find Hi ∈ H1

D(Fi), ∀i = 1, ..., N, such that ∀v ∈ V : N

i=1
Fi

Ki∇Hi∇v|FidFi =

N

i=1
Fi

qiv|FidFi +Gi,N, v|ΓNi

H− 1

2 (ΓNi ),H 1 2 (ΓNi )

,

γT (Hi) = γT (Hj), ∀S ∈ S, {i, j} = I(S).

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 15

SLIDE 16

Intro Model VEM Conforming Mortar Transport References Conforming Mesh Equations Convergence Robustness

K

LT L h λ

=

f

.

K=       K1 · · · K2 · · · . . . . . . . . . ... . . . · · · · · · KN       , f=       f1 . . . . . . fN       , h=       h1 . . . . . . hN       , L=       L1 . . . . . . Lndoft       . Lt =

dofi

dofj · · · 1 · · · −1 · · ·

q

q q q q q q q q q q q q q

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .

Fi Fj

dofi dofj

The multiplier λt penalizes the jump hdofi − hdofi.

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 16

SLIDE 17

Intro Model VEM Conforming Mortar Transport References Conforming Mesh Equations Convergence Robustness

Figure : Spatial distribution of fractures for benchmark problem Non redundant Lagrange multipliers at the cross-point

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 17

SLIDE 18

Intro Model VEM Conforming Mortar Transport References Conforming Mesh Equations Convergence Robustness

(a) Order 1 (b) Order 2 (c) Order 3 (d) Order 4 Figure : Convergence curves for benchmark problem - Fracture 1

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 18

SLIDE 19

Intro Model VEM Conforming Mortar Transport References Conforming Mesh Equations Convergence Robustness

Figure : Order 1

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 19

SLIDE 20

Intro Model VEM Conforming Mortar Transport References Conforming Mesh Equations Convergence Robustness

Figure : Order 2

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 20

SLIDE 21

Intro Model VEM Conforming Mortar Transport References Conforming Mesh Equations Convergence Robustness

Figure : Order 3

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 21

SLIDE 22

Intro Model VEM Conforming Mortar Transport References Conforming Mesh Equations Convergence Robustness

Figure : Order 4

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 22

SLIDE 23

Intro Model VEM Conforming Mortar Transport References Conforming Mesh Equations Convergence Robustness

27 Fractures

Figure : Spatial distribution of fractures for a DFN with 27 fractures

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 23

SLIDE 24

Intro Model VEM Conforming Mortar Transport References Conforming Mesh Equations Convergence Robustness

(a) Mesh (b) Detail Figure : DFN 27: Trace very close to an edge. Smoothing can prevent these situations

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 24

SLIDE 25

Intro Model VEM Conforming Mortar Transport References Conforming Mesh Equations Convergence Robustness

(a) Order 1 (b) Order 3 (c) Order 4 (d) Order 5 Figure : DFN 27: Comparison of results for critical situations

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 25

SLIDE 26

Intro Model VEM Conforming Mortar Transport References Conforming Mesh Equations Convergence Robustness

120 Fractures

Figure : Spatial distribution of fractures for a DFN with 120 fractures

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 26

SLIDE 27

Intro Model VEM Conforming Mortar Transport References Conforming Mesh Equations Convergence Robustness

Troublesome situations

(a) Mesh (b) Detail Figure : DFN 120: Detail of two very close and almost parallel traces (“almost rectangular element” with aspect ratio 177, 8 edges; if k = 4 we have 38 DOFs)

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 27

SLIDE 28

Intro Model VEM Conforming Mortar Transport References Conforming Mesh Equations Convergence Robustness

(a) Order 1 (b) Order 2 (c) Order 3 (d) Order 4 Figure : DFN 120: Comparison of results for critical situations

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 28

SLIDE 29

Intro Model VEM Conforming Mortar Transport References Conforming Mesh Equations Convergence Robustness

134 Fractures

Figure : Spatial distribution of fractures for a DFN with 134 fractures

up to 24 traces in a fracture, minimum angle=0.41◦, T raceLenghtMax/T raceLenghtmin ≈ 4.5E3.

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 29

SLIDE 30

Intro Model VEM Conforming Mortar Transport References Conforming Mesh Equations Convergence Robustness

(a) Mesh (b) Detail Figure : DFN 134: Detail of two traces meeting at a very small angle

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 30

SLIDE 31

Intro Model VEM Conforming Mortar Transport References Conforming Mesh Equations Convergence Robustness

(a) Order 3 (b) Order 4 Figure : DFN 134: Comparison of results for problematic situations

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 31

SLIDE 32

Intro Model VEM Conforming Mortar Transport References Equations Convergence

DFNs with Mortar Virtual Element Method of several orders

A partially conforming mesh allows us to apply the Mortar method without looking for a totally conforming mesh.

ai (hδ, vδ) +

S∈Si bS

vδ|S, λδ,S
= (f, vδ)Fi

i = 1, . . . , N, bS

[

[hδ] ]S, µδ,S

= 0

∀S ∈ S .

Figure : Spatial distribution of fractures for benchmark problem

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 32

SLIDE 33

Intro Model VEM Conforming Mortar Transport References Equations Convergence

ErrΛ

L2

2 =

S∈S
e⊂S

Λ − λ2

e ,

ErrΛ

H− 1

2

2 =

S∈S
e⊂S

|e| Λ−λ2

e .

h λ on S1 VEM order Mortar basis L2 Norm H1 Norm L2 Norm H− 1

2 Norm

1 M0 1.00 (1) 0.50 (0.5) 1.19 1.79 1 M1 1.00 (1) 0.50 (0.5) 1.26 1.87 2 M0 1.38 (1.5) 0.91 (1) 0.98 1.54 2 M1 1.50 (1.5) 1.01 (1) 1.54 2.05 2 M2 1.51 (1.5) 1.01 (1) 2.45 3.02

Table : Benchmark problem: convergence rates with respect to #hdofs and #λdofs for several VEM orders and Mortar bases. The numbers in parentheses indicate the expected rates.

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 33

SLIDE 34

Intro Model VEM Conforming Mortar Transport References Equations Convergence

Figure : Benchmark problem: computed and exact fluxes

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 34

SLIDE 35

Intro Model VEM Conforming Mortar Transport References Simulation Convection Diffusion SUPG DFN Transport Equation

Transport of a passive scalar

Figure : DFN Figure : Hydraulic head Figure : Darcy velocity field β = − K ∇h

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 35

SLIDE 36

Intro Model VEM Conforming Mortar Transport References Simulation Convection Diffusion SUPG DFN Transport Equation

Convection Diffusion with VEM (preliminary results)

We start discretizing the steady convection diffusion equation in the convection dominated case: −∇ · ν ∇c + β · ∇c = f. Homogeneous Dirichlet boundary conditions, β = [1, 1], ν = 10−4 I, P e ≈ 104. SUPG-like stabilization needed.

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 36

SLIDE 37

Intro Model VEM Conforming Mortar Transport References Simulation Convection Diffusion SUPG DFN Transport Equation

Figure : “Mole Antonelliana” and night skyline, Torino Figure : “Mole mesh” on a rectangular domain

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 37

SLIDE 38

Intro Model VEM Conforming Mortar Transport References Simulation Convection Diffusion SUPG DFN Transport Equation

Convection-Diffusion no stabilization versus SUPG stabilization

Figure : Solution without stabilization Figure : Solution with SUPG-like stabilization

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 38

SLIDE 39

Intro Model VEM Conforming Mortar Transport References Simulation Convection Diffusion SUPG DFN Transport Equation

SUPG stabilization, rate of convergence

Figure : Theoretical and numerical rates

f convergence for VEM of order 1

Figure : Theoretical and numerical rates

f convergence for VEM of order 2

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 39

SLIDE 40

Intro Model VEM Conforming Mortar Transport References Simulation Convection Diffusion SUPG DFN Transport Equation

DFN Transport Equation

∂c ∂t − ∇ · ν ∇c + β · ∇c = f.

Figure : Unsteady, VEM, No stabilization, P e = 104 Figure : Unsteady, VEM, SUPG-like stabilization, P e = 104

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 40

SLIDE 41

Intro Model VEM Conforming Mortar Transport References

References Thank you!

References

Benedetto M.F., Berrone S., Pieraccini S., Scial`

S., The Virtual Element

Method for discrete fracture network simulations. Computer Methods in Applied Mechanics and Engineering., 2014, vol. 280 n. 1, pp. 135-156. Benedetto M.F., Berrone S., Borio A., Scial`

S., A Globally Conforming Method

for Solving Flow in Discrete Fracture Network Using the Virtual element Method. Accepted in Finite Elements in Analysis & Design. Benedetto M.F., Berrone S., Borio A., Pieraccini S., Scial`

S., A Hybrid Virtual

Element Method for Discrete Fracture Network Simulations. Submitted, August 2015. Benedetto M.F., Berrone S., Borio A., Pieraccini S., Scial`

S., The Virtual

Element Method for Transport Problems in Fractured Media. In preparation.

Stefano Berrone Polytopal Element Methods in Mathematics and Engineering 41