Hivel2D users manual
Finite Element Model
This system of partial differential equations is solved using the finite element method.
The finite element approach taken is a Petrov-Galerkin formulation that incorporates a
combination of the Galerkin test function and a non Galerkin component to control
oscillations due to convection. An integration by parts procedure is used to develop the
weak form of the equations. The weak form which facilitates the specification of
boundary conditions is:
Q φi
φi
Q
∑ t x
∫ ψ i
F y+ ϕ i A
-
F x-
y
x
e Ω e
(8)
(F n
)
Q
∫φ
+ ψ i H d Ωe +
+ F yny
dl = 0
B
x
x
i
y
Γe
where the variables are understood to be discrete values and
e = subscript indicating a particular element
Ω = domain
ψ i = φ i I + ϕ i = test function
φ i = Galerkin part of the test function
I = identity matrix
ϕi = non-Galerkin part of the test function
(nx, ny) = ^ = unit vector outward normal to the boundary Γe
n
and
Fx
A=
Q
(9)
Fy
B=
Q
5
Chapter 2 HIVEL 2D overview
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