Fem Matrix Algebra

8/3/2019 Fem Matrix Algebra

1/41

The Finite Element Method

Matrix algebra

Matrix algebraComputational Mechanics, AAU, EsbjergThe Finite Element Method


2/41

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



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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

Computational Mechanics, AAU, EsbjergThe Finite Element Method


8/41

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



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Matrix Operations

Integrating a matrix:

ij[a]dx a dx Matrix algebra



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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



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Solving:

1410

41

140

242

231

143

241

232

a

d

x

1

1 .

)(

=

=

=



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Solving:

1.1

140

242

231

130212

221

a

dx

)2(

2 =

=



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Solving:

4.1

140

242

231

340

142

231

a

d

x

)3(

3

=



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Inversion

[ ]{ } { }[ ] [ ]{ } [ ] { }[ ]{ } [ ] { }{ } [ ] { }

1 1

1

1

a x c

a a x a c

I x a c

x a c

==

==



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Example

=

3

1

2

x

x

x

140

242

231

3

2

1



21/41

Example

=

=

41

11

14

3

1

2

204080

201020

201121

x

x

x

3

2

1

.

.

.

...

...

...



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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



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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



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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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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



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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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Repeat the process for the remaining

rows until we have a triangularized systemof equation.


=

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:


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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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

==

=+=

==



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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

====


E l


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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

=

==


E l


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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


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