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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
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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