1 Lecture 4 FEM in 1D: FEM and solution of algebraic systems (Lecture notes taken by Peter Gerakios and Steven Burgess) • Bilinear and linear description of FEM: a(u,φ) = l(φ). General description of weak equation is as follows: ሺ ,ݑሻ ൌ ሺሻ, ݓݎሺ ,ݑሻ א, ݏݕݏ ,ݎݎݐ ሺ, ሻ 0 ሺሻ א ݏݎ. An energy equation J(u) = ½ a(u,u) – l(u) can be shown to be admissible. Example 1. In the string equation: ሺ ,ݑሻ ൌ න ݑ௫ ௫ ሺሻ ൌ න ܬሺݑሻൌ 2 න ݑ௫ ଶ െ න ݑܬሺݑሻൌ 1 2 ሺݑ ,ݑሻ െ ሺݑሻ Example 2. Another example is the fin problem: െ ݑܭ௫௫ ݑܥൌ ݑܭ௫ ሺ0ሻ ൌ ൫ ݑ௦ െ ݑሺܮሻ൯ ሺ ,ݑሻ ൌ ܭන ݑ௫ ௫ ܥන ݑݑሺܮሻሺܮሻ ሺሻ ൌ න ݑ௦ ሺܮሻ It can be shown that the energy equation J(u) = ½ a(u,u) – l(u) is also admissible.
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Lecture 4
FEM in 1D: FEM and solution of algebraic systems
(Lecture notes taken by Peter Gerakios and Steven Burgess)
• Bilinear and linear description of FEM: a(u,φ) = l(φ).
General description of weak equation is as follows:
, ,
, , , , 0 .
An energy equation J(u) = ½ a(u,u) – l(u) can be shown to be admissible.
Example 1. In the string equation:
,
2
12 ,
Example 2. Another example is the fin problem:
0
,
It can be shown that the energy equation J(u) = ½ a(u,u) – l(u) is also admissible.
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Example 3. Another example can be found in two dimensions:
u u f |
We can use Green’s theorem to get: Let | 0.
,
Note: J(u) is admissible if
12 ,
where both a(.,.) and l(.) are real-valued functions. Use the linear and symmetric properties
12 , ,
12 ,
0 , , 0 0 0 ,
, 0
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• Element matrices from a(u,φ) = l(φ).
When we restrict u to some specific element, .
The functions (Nie) are the linear shape functions
,
The summation can be pulled out due to linearity
,
This can be written as
, , , ,
And this gives the element matrix k:
, ,, , and
, ,
OR, alternatively using the variational approach:
where the k coefficients are identical to the a(Ni,Nj)’s.
Nth
W
Note: In transhe plane vers
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ical line of h
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height one ex
, the elemendes to descri
, ,
xtends from
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ire three nodsegment. Fo
,
er 1, and
des to describr example:
,
4
be
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• General Outline of FEM.
Goal is to begin with a differential equation and rewrite in the weak or energy form. There are six major components/steps to this procedure.
I. Input Data a. i = nod(e , ie), where i is the system node #, ie is the element node #, and e is the
element matrix # b. the components of the points in the domain c. other parameters in the model
II. Find element matrices a. , , b. The dimension of matrix k depends on the dimensions of the model as well as
the shape functions.
III. Assemble matrix Ann x nn and right hand side dnn x 1 where nn is the total number of nodes a. This can be done via the assembly by elements or nodes
IV. Adjust for given boundary conditions a. ,
# .
V. “Solve” Au = d
VI. Analyze Output -- Graphically
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• Gauss elimination.
Solve
First step is to find U and then use backward substitution to solve
However, the operation count is of the order
Example:
The first step would be to obtain a sequence of row operations to obtain U.
∏ ∏
Consider the following augmented system:
3 1 01 3 1
0 1 3|123
3 1 0
083 1
0 1 3
|1733
3 1 0
083 1
0 0218
|
173
318
Now utilize backward substitution to solve for the three unknowns.
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• Tridiagonal systems.
0
0
0 00
0
000 0
And this gives the equation L(Ux) = d. Let Ux = y and solve Ly = d by forward substitution. Now solve Ux = y by backward substitution. This is possible if the alpha’s and gamma’s are known.
Tridiagonal algorithm:
, ,
2,
1, 1
Note: For a point-wise tridiagonal system, the operation count is of order 5n.
• SOR iterative method.
Let Au = d be written in component form
.
The SOR method should be used for sparse matrices (for each i most aij are zero) and for diagonally dominate matrices (for each i aii ≥∑ ). Even in these cases the method may not converge to the solution!
Let uim+1/2 be the solution of
∑ / ∑ . (1)
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Next, let ω be greater than or equal to 1.0 and less than 2.0. Good choices of this parameter can accelerate convergence (it takes less steps to reach a convergence test).
1 / . (2)
SOR Algorithm.
Choose an initial vector u0
For m = 1, maxm
For i = 1, n
Solve (1) for uim+1/2
Solve (2) for uim+1
End
Test for convergence
End.
Example. Consider the algebraic system n = 3 unknowns
3 1 01 3 1
0 1 3
123
.
Use the initial choice u0 = [ 1 1 1]T and do one iteration.
3u11 = 1 + 1 u2
0 = 1 + 1, which implies u11 = 2/3
3u21 = 2 + 1 u1
1 + 1 u30 = 2 + 2/3 +1, which implies u2
1=11/9
3u31 = 3 + 1 u2
1 = 3 + 11/9, which implies u31=38/27.
Next repeat this until difference in successive iterations is “suitably” small.