Control  Volume  Finite  Element  Method  

(element  patch)  

Vornoi  (control  volume)  mesh  is  related  to  triangular  mesh  Centriod  

Control  Volume  Mesh  

Vornoi  grid  is  made  by  connec<ng  The  perpendicular  bisectors  of  the  midpoints  of  a  triangulated  FE  mesh.  

Summing  Fluxes  Over  Support  Area  

€

∂qx∂x

+∂qy∂y

= −R

Control  Volume  Methods  operates  Of  Mass  Conserva<on  based  Equa<ons    

€

∂qx∂x

+∂qy∂y

= Aiqii=1

4

∑

qi  Ai  

Mass  conserva<on:  

Note  that  elemental  fluxes  are  constant  for  each  triangle    

Some  Prac<cal  Issues  Regarding  the  CVFEM:  Compu<ng  Face  Areas,  Δx,  Δy    

€

ΔxF1 =x33−x26−x16

ΔyF1 =y33−y26−y16

ΔxF2 = −x23

+x36

+x16

ΔyF2 = −y23

+y36

+y16

Face  1  

Face  2  

qy  

qx  €

L = Δx 2 + Δy 2

qy  

qx  

Two  faces  per  triangle!  

€

qface1 = qxΔy face1 − qyΔy face1

Node X Y 1 0 0 2 5 0 3 2 2

Δx-Face1: -0.1666 Δy-Face1: 0.6667

Δx-Face2: -1.3333 Δy-Face2: 0.3333

qx -5 q Face1 1.1667 qy -1 Δyqx Face1 6.6667

Δxqy Face1 0.6667

Face1  flux  calcula<ons  

Example  of  Flux  Calcula<on:    

Control  Volume  Finite  Element  Method  Source  Term  Calcula<ons  

€

R⋅ A = AiRii=1

5

∑

R  -­‐  recharge  in  m/s  A  -­‐    the  total  area  of  the  vornoi  element  

Ai  -­‐  area  contribu<on  of  each  triangle  (Ai/3)  in  the  support  area  (nodal  patch)  

Calcula<ng  recharge  for  each  vornoi  node:    

Steady-­‐State  Groundwater  Flow  

€

A∫ ∂

∂xK ∂h

∂x⎡

⎣ ⎢ ⎤

⎦ ⎥ +

∂∂yK ∂h

∂y⎡

⎣ ⎢

⎤

⎦ ⎥ dA =

V∫ RdV

€

A∫ ∂

∂xK ∂h

∂x⎡

⎣ ⎢ ⎤

⎦ ⎥ +

∂∂yK ∂h

∂y⎡

⎣ ⎢

⎤

⎦ ⎥ dA = K∇h ⋅ ndA

A j∫ = 0

j=1

Ni

∑

Consider  Diffusive  Term  First:  

For  each  face  of  an  element:  

€

K∇h⋅ ndAface1∫ + K∇h⋅ ndA

face2∫

K∇h⋅ ndAface1∫ = K ∂ ˆ h

∂xΔy f 1 −K ∂ ˆ h

∂yΔx f 1

= K ∂ψ1

∂xh1 +

∂ψ2

∂xh2 +

∂ψ3

∂xh3

⎡

⎣ ⎢ ⎤

⎦ ⎥ Δy f 1 −K ∂ψ1

∂yh1 +

∂ψ2

∂yh2 +

∂ψ3

∂yh3

⎡

⎣ ⎢

⎤

⎦ ⎥ Δx f 1

K∇h⋅ ndAface2∫ = K ∂ ˆ h

∂xΔy f 22 −K ∂ ˆ h

∂yΔx f 2

= K ∂ψ1

∂xh1 +

∂ψ2

∂xh2 +

∂ψ3

∂xh3

⎡

⎣ ⎢ ⎤

⎦ ⎥ Δy f 2 −K ∂ψ1

∂yh1 +

∂ψ2

∂yh2 +

∂ψ3

∂yh3

⎡

⎣ ⎢

⎤

⎦ ⎥ Δx f 2

€

∂ ˆ h ∂x

=∂ψ i

∂xhi

i=1

3

∑ =βi

2Ae =β1

2Ae h1 +β2

2Ae h2 +β3

2Ae h3

Recall  that  in  the  FE  Method:  

qy  

qx  Two  faces  per  triangle!  

Observa<on:  For  linear,  triangular  finite  element  grids,  q  is  constant  across  the  element:  

€

qx = −Kx∂ ˆ h ∂x

=∂ψ i

∂xhi

i=1

3

∑ = −Kxβi

2Ae =β1

2Ae h1 +β2

2Ae h2 +β3

2Ae h3

⎡

⎣ ⎢ ⎤

⎦ ⎥

€

qy = −Ky∂ ˆ h ∂y

=∂ψ i

∂yhi

i=1

3

∑ = −Kyγ i

2Ae =γ1

2Ae h1 +γ 2

2Ae h2 +γ 3

2Ae h3

⎡

⎣ ⎢ ⎤

⎦ ⎥

€

qface1 = qxΔy face1 − qyΔy face1

€

K∇h⋅ ndAA∫ = −a1h1 + a2h2 + a3h3

a1 = K −∂ψ1∂x

Δy f1 +∂ψ1∂y

Δx f 1 −∂ψ1∂x

Δy f 2 +∂ψ1∂y

Δx f 2⎡

⎣ ⎢

⎤

⎦ ⎥

a2 = K ∂ψ2

∂xΔy f1 −

∂ψ2

∂yΔx f 1 +

∂ψ2

∂xΔy f 2 −

∂ψ2

∂yΔx f 2

⎡

⎣ ⎢

⎤

⎦ ⎥

a3 = K ∂ψ3

∂xΔy f1 −

∂ψ3

∂yΔx f 1 +

∂ψ3

∂xΔy f 2 −

∂ψ3

∂yΔx f 2

⎡

⎣ ⎢

⎤

⎦ ⎥

Groundwater  Flow  

diagonal    node  

support  nodes…this  is  a  local  node  numbering  scheme  

Steady  State  Advec<on-­‐Dispersion  

€

A∫ ∂

∂xD ∂c∂x

⎡

⎣ ⎢ ⎤

⎦ ⎥ +

∂∂yD ∂c∂y

⎡

⎣ ⎢

⎤

⎦ ⎥ dA = qx

∂c∂x

+V∫ qy

∂c∂ydV

€

D∇c ⋅ ndAA∫ − q∇c ⋅ ndA

A∫ = −a1c1 + a2c2 + a3c3 − qf1c f 1 − qf 2c f 2

a1 = D −∂ψ1∂x

Δy f 1 +∂ψ1∂y

Δx f 1 −∂ψ1∂x

Δy f 2 +∂ψ1∂y

Δx f 2⎡

⎣ ⎢

⎤

⎦ ⎥

a2 = D ∂ψ2

∂xΔy f 1 −

∂ψ2

∂yΔx f 1 +

∂ψ2

∂xΔy f 2 −

∂ψ2

∂yΔx f 2

⎡

⎣ ⎢

⎤

⎦ ⎥

a3 = D ∂ψ3

∂xΔy f 1 −

∂ψ3

∂yΔx f 1 +

∂ψ3

∂xΔy f 2 −

∂ψ3

∂yΔx f 2

⎡

⎣ ⎢

⎤

⎦ ⎥

Groundwater  Flow  

diagonal    node  

support  nodes…this  is  a  local  node  numbering  scheme  

€

c f1 = c1ψ1 + c2ψ2 + c3ψ3 = c1512

+ c2512

+ c3212

c f 2 = c1ψ1 + c2ψ2 + c3ψ3 = c1512

+ c2212

+ c3512

€

qface1 = qxΔy face1 − qyΔy face1qface2 = qxΔy face2 − qyΔy face2