cretized. The type of discretization used in this model is as shown in Figure 32,
where "E"s represent elements and "n"s represent nodes. Note that the elements
are aligned with the x — y coordinate system.
The elements used in this model belong to the family of elements referred to as
the Lagrange family. The Lagrange family is so called because the basis functions
for the elements in this family can be derived simply by taking the tensor product
of one-dimensional Lagrange polynomials. One-dimensional Lagrange polynomials
y
n43
(Interior node numbers have been omitted for clarity) n44 n45 n46 n47 n48 ----(S)----^---- n49 n36 n29 n22 n15 n8 E31 E25 E19 E32 E26 E20
#
E33 E27 E21 E13 E7 El E14 E8 E15 E9 E2 E3 E34 E28 E22 E16 E10 E4 E35 E29 E23 E17 E1 1 E5 E36 E30 E24 E18 E12 E6 n42 n35 n28 n21 # n14 n1 n3 n4 n5u
yji
(-1.1 —ͨ XFigure 32. FEM Discretization
n1 W---—O n2
(-1.-1) (1.-1)
6.2 Appendix B: The Finite Element Method - 2D ADR Equation
id+i
N,= l[
{x - Xn)n = l \^J ~ ^»)
(12)
where the subscript j is used to specify the shape function for node j in an element
of degree nj.
The simplest two-dimensional element in the Lagrange family is the four-node
rectangular element. Before deriving the basis function for this element, the local
coordinates (^, 77) axe introduced. These are called "isoparametric" coordinates. In
this isoparametric system, the original rectangular subregion reduces to a square
reference element whose corners are located at^ = ±l,J7 = ±l (see Figure 32).
The transformation between the global and isoparametric, or local, coordinates can
be written in the form
i=l
4 (13)
»=i
where Ni{^,r)) is now defined as some as yet unspecified function of coordinates
(^, T]) and is associated with node i of the element.
In order to satisfy the conditions normally imposed on basis functions, Ni must
possess the following properties:Ni= {
1, at node i
0, at the remaining nodes
One possible choice is
E^i = i
1=1
NiU,ri) = ^m(ri) (15)
where V'CO ^^^ ^('?) ^^ ^^® first-degree Lagrange polynomials for node i. Thus,
the resulting basis function for node 1 is
Similarly, the remaining basis functions are
^2(^,^7)= ^(1 + 0(1-'?),
^3(e,^)=j(l + 0(l + '?), (17)
These expressions for iVi, N2, iVs, and N4 can be put in the form
Ni((,r,) = ^il + C(i){l + rjr)i) (18)
Because the four basis functions are formed by taking the tensor product of lin¬
ear Lagrange polynomials, they are often referred to as "bilinear" basis functions
(Huyakorn and Pinder, 1983).For the case of two-dimensional flow, where advection and dispersion occur in
the X and y directions, recall that the ADR equation is given by Equation (3) :
6.2 Appendix B; The Finite Element Method - 2D APR Equation
After applying the method of weighted residuals, Green's formula to the dis¬
persive term, and a Crank-Nicolson finite difference approximation to the time
derivative, the governing equation is reduced to what is referred to as the weak
form. The weak form of Equation (3) is given by the following:
la
la
^y^tWjNjCj +D..^^^-Cj +D^y-^—Cj +
^^Wi^C/+' +VyWi^Cy+' +XWiNjCj'+'yx dy =
2 , dWidNj , dWidNj t
^^ͣ^tWiNjCj -D^.-^^-Cj -D.y^^~Cj - (19)
dy dx ^^ dy dy
v.Wi^Cj^ - VyWi^C/ - XWiNjCj^dx dy
+ / WiDn-^—dS+ / WlDr^^^dS
Jb on Jb dnwhere Wj axe weighting functions of an unspecified form; Nj axe the basis functions
(bilinear in this case, as previously discussed); / and J imply summation over the
number of nodes in the domain V; C is the trial function (9) in terms of C; / + 1
is the unknown time level; and I is the known time level.
Both the C'j'^s and the CjS can be treated as constants and thus removed from
the integrands. For the type of FEM used in this model, known as the Galerkin
FEM, the weighting functions are defined to be equivalent to the basis functions;
{te-o/x
/• /ͣ dNidNj f f dN,dNj. , ^
N,Njdxdy~ (20)
"" / L ^^''^''^ - ^"» / i ^^''^''''-
f [ dNfdNj ffdNidNj.,
+ / WiDn^^dS^ / WiDn^dS
Jb on Jq dnThe basis functions in the above expression are in terms of ^ and 77. Therefore,
the integration has to be carried out in the (^, rj) coordinate system instead of the
(x, y) coordinate system. The required transformation of coordinates can be carried
out by utilizing the Change of Variables Theorem (Bartle, 1976) which states that
/ /= f fog\detDg\ (21)
Jg{A) J A
where, in this case, g : (^, rj) —* (a;, y) represents the function which maps points in
the (^, rj) coordinate system into the (x, y) coordinate system, and Dg is the 2x2 ma¬
trix (also referred to as the Jacobian matrix) representing the derivative of the func¬
tion g. For rectangles aligned with the {x,y) coordinate system, \det Dg\ = -^5-^.
6.2 Appendix B: The Finite Element Method - 2D APR Equation
yg an Jq dn
(22)
The integrands now involve products of known functions — the basis func¬ tions and derivatives of the basis functions. The integration can be carried out using Gauss quadrature techniques. The Gauss quadrature method of numerical integration allows the double integrations to be approximated (exactly for the case of bilinear polynomial basis functions) by a double summation involving sampling, or "Gaussian", points and weighting coefficients. The derivation of these sampling points and weighting coefficients requires the use of orthogonal polynomials. Leg- endre polynomials can be used, which possess the following property
/
1
Thus, the Gauss quadrature method when applied between the limits ±1 is termed
Gauss-Legendre quadrature. Since the basis functions to be integrated are defined
over the square whose corners are located at ^ = ±1, r] = ±1, the integrals can be evaluated numerically using Gauss-Legendre quadrature methods. Gauss-Legendre
quadrature in two spatial dimensions can be expressed as follows
/I -1 mm
-1 -^-1 i=i j=i
(24)
where f{CjiVi) is the value of the function at the "Gaussian" points (j and t^j, m is
the number of Gaussian points, and Wi and Wj {i,j = 1,2,... m) are the weighting
coefficients. Gaussian points are selected in such a way that the weighted sum of
m functional values yields the exact value of a polynomial of degree 2m — 1 or less
(Huyakorn and Finder, 1983).After numerically integrating, we are left with a system of linear equations of
the following form
A fjl+i B C'\ = ib (25)
This system of linear equations can then be solved simultaneously to give values of
the independent variable, C, at each node in the domain.
These nodal values can then be substituted back into the trial function to give
an approximate representation of the unknown function C at any spatial location
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