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The Finite Element Method
Matrix algebra
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Matrix Algebra
A matrix is an m x n array of numbers
arranged in mrows and ncolumns. m = n A square matrix.
m = 1 A row matrix. n = 1 A column matrix.
aij Element of matrix a row i, column j
Matrix algebraComputational Mechanics, AAU, EsbjergThe Finite Element Method
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Matrix Operations
Multiplication of a matrix by a scalar.
[a] = k [c] aij = kcij
Addition of matrices.
Matrices must be of same order (m x n) Add them term by term
[c] = [a] +[b] cij
= aij
+ bij
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Matrix Operation
Multiplication of two matrices
If [a] is m x n then [b] must have nrows [c] = [a] [b]
n
ij ie ej
e 1
c a b
==
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Matrix Operations
Transpose of a matrix:
Interchange of rows and columns
If [a] is m x nthen [a]T is n x m
If [a] = [a]T then [a] is symmetric.
[a] must be a square matrix
T
ij jia a
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Matrix Operations
The identity matrix (or unit matrix) is
denoted by the symbol [I]: [a][I] = [I][a] = [a]
[ ] 1 0 0I 0 1 00 0 1
=
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Matrix Operations
The inverse of a matrix is such that:
[ ][ ] 1a a [I ]=Matrix algebra
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Matrix Operations
Differentiating a matrix:
[ ] ijdad adx dx
= Matrix algebra
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Matrix Operations
Differentiating a matrix:
11 12
21 22
11 12
21 22
a a x1U [x y]
a a2 y
U
a a xx
U a a y
y
=
= Matrix algebra
Computational Mechanics, AAU, EsbjergThe Finite Element Method
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Matrix Operations
Integrating a matrix:
ij[a]dx a dx Matrix algebra
Computational Mechanics, AAU, EsbjergThe Finite Element Method
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The Inverse of a Matrix
Need to find the determinant
Need to find the co-factors of [a]
determinant of matrixa [a]
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Cofactors
Cofactors of [aij] are given by:
Then :
[ ]ijwhere matrix d is the first minor
of a and is matrix a
with row i and column j deleted.
i j
ijC ( 1) d=
[ ]T1
ijC[a ]a
=Matrix algebra
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Cramers Rule
[ ]o r i n i n d e x n o t a t i o n :
L e t m a t r i x b e m a t r i x
w i t h c o l u m n i r e p a c e d b y .
T h e n :
n
i j j i
j 1
( i )
( i )
i
a { x } { c }
a x c
[ d ] [ a ]
{ c }
dx
a
=
=
=
=
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Example:
Consider the following equations:
1 2 3
1 2 3
2 3
x 3x 2x 2
2x 4x 2x 14x x 3
+ = + =+ =
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Example:
=
3
1
2
x
x
x
140
242
231
3
2
1
:ormmatrixIn
Matrix algebraComputational Mechanics, AAU, EsbjergThe Finite Element Method
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Solving:
1410
41
140
242
231
143
241
232
a
d
x
1
1 .
)(
=
=
=
Matrix algebraComputational Mechanics, AAU, EsbjergThe Finite Element Method
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Solving:
1.1
140
242
231
130212
221
a
dx
)2(
2 =
=
Matrix algebraComputational Mechanics, AAU, EsbjergThe Finite Element Method
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Solving:
4.1
140
242
231
340
142
231
a
d
x
)3(
3
=
Matrix algebraComputational Mechanics, AAU, EsbjergThe Finite Element Method
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Inversion
[ ]{ } { }[ ] [ ]{ } [ ] { }[ ]{ } [ ] { }{ } [ ] { }
1 1
1
1
a x c
a a x a c
I x a c
x a c
==
==
Matrix algebraComputational Mechanics, AAU, EsbjergThe Finite Element Method
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Example
=
3
1
2
x
x
x
140
242
231
3
2
1
Matrix algebraComputational Mechanics, AAU, EsbjergThe Finite Element Method
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Example
=
=
41
11
14
3
1
2
204080
201020
201121
x
x
x
3
2
1
.
.
.
...
...
...
Matrix algebraComputational Mechanics, AAU, EsbjergThe Finite Element Method
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Solving:
41
140
242
231
340
142
231
a
d
x
3
3 .
)(
=
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Gaussian Elimination
General System of n equations with n unknowns:
=
n
2
1
n
2
1
nn2n1n
n22221
n11211
c
c
c
x
x
x
aaa
aaa
aaa
MM
K
MMM
K
K
Matrix algebraComputational Mechanics, AAU, EsbjergThe Finite Element Method
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Steps in Gaussian Elimination
Eliminate the coefficient of x1 in every
equation except the first one. Select a11 asthe pivot element. Add the multiple -a21/a11 of the first row to
the second row. Add the multiple -a31/a11 of the first row to
the third row.
Continue this procedure through the nth row
Matrix algebraComputational Mechanics, AAU, EsbjergThe Finite Element Method
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After this Step:
=
n
2
1
n
2
1
nn2n
n222
n11211
c
c
c
x
x
x
aa0
aa0
aaa
MM
K
MMM
K
K
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Steps in Gaussian Elimination
Eliminate the coefficient of x2 in every
equation below the second one. Select a22
as the pivot element.
Add the multiple -a 32/
a 22
of the second rowto the third row.
Add the multiple -a 42/a 22 of the second row
to the fourth row. Continue this procedure through the nth row
Matrix algebraComputational Mechanics, AAU, EsbjergThe Finite Element Method
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After This Step:
=
n
2
1
n
2
1
nn3n
n333
n22322
n1131211
c
c
c
x
x
x
aa00
aa00
aaa0
aaaa
MM
K
MMMM
K
L
K
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Steps in Gaussian Elimination
Repeat the process for the remaining
rows until we have a triangularized systemof equation.
Matrix algebraComputational Mechanics, AAU, EsbjergThe Finite Element Method
=
1nn
4
3
2
1
n
4
3
2
1
1nnn
n444
n33433
n2242322
n114131211
c
c
c
cc
x
x
x
xx
a0000
aa000
aaa00
aaaa0aaaaa
MM
L
MMMMM
L
L
LL
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Solve Using Back-substitution
=
+
n
1irrir1n,1
ii
i
1nnn
1nn
n
xaaa
1x
a
cx
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Example
=
6
4
9
x
x
x
111
012
122
3
2
1
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Solving:
Eliminate the coefficient of x1 in every
equation except the first one. Select a11 =2 as the pivot element.
Add the multiple -a21/a11 = -2/2 = -1 of the firstrow to the second row.
Add the multiple -a31/a11 = -1/2=-0.5 of the
first row to the third row.
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Step 1
5.1
5
9
x
x
x
5.000
110
122
3
2
1
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Steps in Gaussian Elimination
Eliminate the coefficient of x2 in every
equation below the second one. Select a22
as the pivot element. (Already done in thisexample.)
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Step 2
5.1
5
9
x
x
x
5.000
110
122
3
2
1
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Solve Using Back-substitution( )( )
( )
( )1
2
3)2(29x
21
35x
3
21
23
a
cx
2
2
33
33
==
=+=
==
Matrix algebraComputational Mechanics, AAU, EsbjergThe Finite Element Method
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Gauss-Seidel Iteration
( )( )
( )1n1n.n22n11nnnn
n
nn2323121222
2
nn1313212111
1
xaxaxaca
1x
xaxaxaca
1
x
xaxaxaca
1
x
:forminequationsWrite
L
M
L
L
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Gauss-Seidel Iteration
Assume a set of initial values for unknowns.Substitute into RHS of first equation. Solve fornew value of x1
Use new value of x1and assumed values of
other xs to solve for x2 in second equation. Continue till new values of all variables are
obtained.
Iterate until convergence.
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Example
1x1x1x2
1x
6x2x6xx4x
5xx4x
2xx4
4321
43
432
321
21
====
Matrix algebraComputational Mechanics, AAU, EsbjergThe Finite Element Method
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Example
( ) ( )( ) ( )( ) ( ))( ) ( ) 16.067.1221x221x
672.1168.164
1xx64
1x
68.114
354
1xx54
1x
43
1241
x241
x
34
423
312
21
=
==
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Example
( ) ( )( ) ( )( ) ( ))( ) ( ) 28.0.0944.12
21x2
21x
944.116.0899.1641xx641x
899.1672.1922.054
1xx54
1x
922.068.124
1x24
1x
34
423
312
21
=
==
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Iteration x1 x2 x3 x4
0 0.5 1.0 1.0 -1.01 0.75 1.68 1.672 -0.16
2 0.922 1.899 1.944 -0.028
3 0.975 1.979 1.988 -0.0064 0.988 1.9945 1.9983 -0.0008
Exact 1.0 2.0 2.0 0.04
Matrix algebraComputational Mechanics, AAU, EsbjergThe Finite Element Method