ژ ﻮﻣﮑﺎ ﯿﮏ روش ی ﺪدی

(1)

Numerical methods in geomechnics

Hasan Ghasemzadeh

یدﺪ ی شور

ﮏﯿﺎﮑﻣﻮژ

http://wp.kntu.ac.ir/ghasemzadeh

ﯽ ﻮ ﻦﺪ اﺮ ﻪ اﻮ ﯽ هﺎﮕﺸاد

لﻮﺼﻓ و ﻦﻳوﺎﻨﻋ ﺖﺳﺮﻬﻓ

تﻻدﺎﻌﻣ ﻞﺣ رد يدﺪﻋ يﺎﻬﺷور

Numerical Mathematics 2000 Alfio Quarteroni Riccardo Sacco

Fausto Saleri

2 Numerical Methods in

Geomechanics

(2)

يﺮﺒﺟ تﻻدﺎﻌﻣ ﻞﺣ رد يدﺪﻋ يﺎﻬﺷور

لﻮﻬﺠﻣ رد ﻪﻟدﺎﻌﻣ ود باﻮﺟ

3

11 12 1 1

21 22 2 2

a a x b

     

    

 

     

Numerical Methods in Geomechanics

ﺴﻳﺮﺗﺎﻣ دراﺪﻧﺎﺘﺳا ﻞﻜﺷ ﻲ

In matrix form: A x = b

(3)

ﺐﻳاﺮﺿ ﺲﻳﺮﺗﺎﻣ

5

يﺮﻄﻗ ﺲﻳﺮﺗﺎﻣ

11 12 1

21 22 2

1 2

n n

n n nn

a a a

A

a a a

 

 

  

 

 





   



11 22

0 0

_nn

a A a

a

 

 

  

 

 





   



11 12 1

22 2

0

0 0

n n

nn

a a a

a a

A

a

 

 

  

 

 





   



11

21 22

1 2

0 0

0

n n nn

a

a a

A

a a a

 

 

  

 

 





   



6

يﺮﻄﻗ ﻪﺳ ﺲﻳﺮﺗﺎﻣ

11 12 1

21 22 2

1 2

n n

n n nn

a a a

A

a a a

 

 

  

 

 





 



11 12

21 22 23

32 33 34

43

0 0 0

0 0

0 0 0

_nn

a a

a a a

A a a a

a

 

 

  

 

 

 



tridiagonal matrix

(4)

7

ﺪﻨﻛاﺮﭘ ﺲﻳﺮﺗﺎﻣ ه

11 15

21 22

33 34

41

53

0 0 0

0 0

_nn

a a

A a a

a

a a

 

 

  

 

 

 



sparse matrix

11 12 14

21 22 23 2

32 33 34

41 43 44 45

52 54 55

0 0

0

0 0

0

0 0

n

a a a

a a a a

A a a a

a a a a

a a a

 

 

  

 

 

يﺪﻧﺎﺑ ﺲﻳﺮﺗﺎﻣ

banded matrix

1 - ﻢﻴﻘﺘﺴﻣ فﺬﺣ يﺎﻫ شور

2 - يراﺮﻜﺗ يﺎﻫ شور ﻞﺣ رد يدﺪﻋ يﺎﻬﺷور

يﺮﺒﺟ تﻻدﺎﻌﻣ

ﻲﻄﺧ ﺮﻴﻏ تﻻدﺎﻌﻣ

ﻲﻄﺧ تﻻدﺎﻌﻣ

(5)

1 - ﻢﻴﻘﺘﺴﻣ فﺬﺣ يﺎﻫ شور

ﺮﻣاﺮﻛ شور

9

where A

^j

is the n*n matrix obtained by replacing column j in matrix A by the column vector b

computational cost ( n  1)( n  1)!

تﻻدﺎﻌﻣ ﻲﻄﺧ

1 - ﻳاﺮﺿ ﺲﻳﺮﺗﺎﻣ يزﺎﺳ سﻮﻜﻌﻣ ﺐ

2 - سﻮﮔ فﺬﺣ شور

3 - ﺲﻳﺮﺗﺎﻣ ﻪﻳﺰﺠﺗ شور In matrix form: A x = b ﻢﻴﻘﺘﺴﻣ يﺎﻫ شور

Geomechanics

(6)

Find A -1

Compute det(A)

Solve A x = b

Round off error makes the system singular, or numerical instability makes the answer wrong.

1 - ﻳاﺮﺿ ﺲﻳﺮﺗﺎﻣ يزﺎﺳ سﻮﻜﻌﻣ ﺐ

Ax  b

Geomechanics

Gauss Elimination

2 - سﻮﮔ فﺬﺣ شور

Multiplying first eq. by 10e6 subtracting from the second eq.

2

1.000001 1

x  

1

0

x 

Back substituting wrong

Pivoting and stability

(7)

LU Decomposition

LU = A

Thus A x = (LU) x = L (U x) = b

Let y = U x, then solve y in L y = b by forward substitution

Solve x in U x = y by backward substitution

3 - ﺲﻳﺮﺗﺎﻣ ﻪﻳﺰﺠﺗ شور تﻻدﺎﻌﻣ

ﻲﻄﺧ

11 11 1 11 1

1 1

0

n n

n nn nn n nn

l u u a a

l l u a a

    

     

    

    

  

        

  

Ax  b

Geomechanics

1- Doolittle factorization 2-Crout factorization 3-LDM T Factorization

ﺲﻳﺮﺗﺎﻣ ﻪﻳﺰﺠﺗ تﻻدﺎﻌﻣ

ﻲﻄﺧ

A LU factorization of A =

solving the following nonlinear system of n

²

equations

Geomechanics

(8)

Doolittle factorization

 supposing that the first k − 1 columns of L and U are available and setting

 equations can be solved in a sequential way with respect to the boxed variables

 we thus obtain first the k-th row of U and then the k-th column of L, as follows:

for k = 1, . . . , n

 computational cost

kk 1 l 

n

2

Geomechanics

Crout’s Algorithm

 Set for all i

 computing first the k-th column of L and then the k-th row of U:

for k = 1, . . . , n

ii 1 l 

تﻻدﺎﻌﻣ

ﻲﻄﺧ

(9)

LDM T Factorization

 when A is symmetric M=L

 computational cost

11 1 11 11 1

1 1 1

0 0 0

0

T

n

n nn n n nn n nn

l d m a a

l l d m m a a

 

     

 

        

     

       

       

  

         

  

LDM

T

 A

3

3 n

Geomechanics

Pivoting

Pivoting is essential for stability

Interchange rows to get largest u ii

Implicit pivoting (when comparing for the biggest elements in a column, use the

normalized one so that the largest coefficient in an equation is 1)

18 Numerical Methods in Geomechanics

(10)

Compute A -1

Let B = A -1

then AB = I (I is identity matrix) or A [b 1 , b 2 , …,b N ] = [e 1 , e 2 , …,e N ] or A b j = e j for j = 1, 2, …, N

where b j is the j-th column of B.

I.e., to compute A -1 , we solve a linear system N times, each with unit vector e j .

Compute det(A)

Definition of determinant

Properties of determinant

Since det(LU) = det(L)det(U), thus det(A) = det(U) =

1

N

jj j

u





1 2

1, 2, ,

det( ) ( 1)

n

P

i i n i

P

A    a a  a تﻻدﺎﻌﻣ

ﻲﻄﺧ

(11)

Solution of linear systems, Ax = b

Least-square problem, min ||Ax-b|| 2

Eigenvalue problems, Ax =x

Singular value decomposition, A = UsV T .

Lapack is a free, high quality linear algebra solver package (downloadable at www.netlib.org/lapack/).

ﺲﻳﺮﺗﺎﻣ ﻚﻳ ندﺮﻛ يﺮﻄﻗ ياﺮﺑ بﺮﺿ ﺲﭘ و بﺮﺿ ﺶﻴﭘ يﺎﻫ ﺲﻳﺮﺗﺎﻣ

Geomechanics

1 - ﻪﻨﻳﺰﻫ :

رد ﺮﻫ ﻪﻠﺣﺮﻣ زﺎﻴﻧ ﻪﺑ ﻪﺒﺳﺎﺤﻣ هﺪﻧﺎﻤﻴﻗﺎﺑ هﺎﮕﺘﺳد

ﻲﻣ ﺷﺎﺑ ﺪ . رد درﻮﻣ

ﻚﻳ ﺲﻳﺮﺗﺎﻣ ﻞﻣﺎﻛ

ﻪﻨﻳﺰﻫ ﻪﺒﺳﺎﺤﻣ نآ ياﺮﺑ ﺮﻫ راﺮﻜﺗ زا ﻪﺒﺗﺮﻣ n 2

ﺖﺳا

ﻪﻛ ﺎﺑ شور ﻢﻴﻘﺘﺴﻣ ﻪﻛ رد نآ ﻪﻨﻳﺰﻫ

3 ﻞﻛ n 3 / ﺪﺷﺎﺒﻴﻣ 2 ﻞﺑﺎﻗ ﻪﺴﻳﺎﻘﻣ

ﺖﺳا . ﻦﻳاﺮﺑﺎﻨﺑ ﺮﮔا

داﺪﻌﺗ راﺮﻜﺗ درﻮﻣ زﺎﻴﻧ ﻲﻳاﺮﮕﻤﻫ ﻪﺘﺴﺑاو

ﺑ n ﻪ ﻲﻟو

ﺮﺘﻤﻛ زا نآ ﺪﺷﺎﺑ شور يﺎﻫ يراﺮﻜﺗ ﻲﻣ

ﺪﻨﻧاﻮﺗ ﺎﺑ يﺎﻬﺷور ﻘﺘﺴﻣ

ﻢﻴ ﺖﺑﺎﻗر

ﺪﻨﻨﻛ .

2 - هﺪﻨﻛاﺮﭘ يﺎﻬﺴﻳﺮﺗﺎﻣ :

ﺒﺳﺎﻨﻣ ﻲﻳارﺎﻛ ﻢﻴﻘﺘﺴﻣ يﺎﻬﺷور ﺪﻧراﺪﻧ ﻲ

تﻻدﺎﻌﻣ ﻞﺣ رد راﺮﻜﺗ شور ﻲﻄﺧ

Geomechanics

(12)

ﻲﻣ ﻪﺘﺧﺎﺳ باﻮﺟ رادﺮﺑ ﻪﺘﺷر ﻚﻳ يراﺮﻜﺗ يﺎﻬﺷور رد دﻮﺷ

ﺪﻳآ ﻲﻣ ﺖﺳﺪﺑ باﻮﺟ ﺎﻬﻧآ ﻲﻳاﺮﮕﻤﻫ زا ﻪﻛ

ﺖﺳﺪﺑ هﺪﻧﺎﻤﻴﻗﺎﺑ لﺮﺘﻨﻛ ﺎﺑ ﻲﻳاﺮﮕﻤﻫ ﺪﻳآ ﻲﻣ

A x = b

( k )

x l i m x k

 

( k )

x

x n  x  

Geomechanics

و راﺮﻜﺗ ﺲﻳﺮﺗﺎﻣ B زا ﻪﻛ ﺖﺳا يرادﺮﺑ f

ﺪﻳآ ﻲﻣ ﺖﺳﺪﺑ b .

A x = b

( ) ( k ) ( k )

x

⁰

g i v e n, x

^¹

 B x  f , k  0

ﺮﮔا دﻮﺑ ﺪﻫاﻮﺧ رﺎﮔزﺎﺳ ﻪﻟدﺎﻌﻣ هﺎﮕﺘﺳد ﺎﺑ قﻮﻓ راﺮﻜﺗ شور و B

ﻪﻛ ﺪﻨﺷﺎﺑ ﻲﺗرﻮﺻ ﻪﺑ f

f  ( I  B ) A

^¹

b . x  B x  f ﺎﻳ

ﺖﺴﻴﻧ ﻲﻳاﺮﮕﻤﻫ ﻲﻓﺎﻛ طﺮﺷ يرﺎﮔزﺎﺳ

( k ) ( k )

e  x  x ما k ﻪﻠﺣﺮﻣ يﺎﻄﺧ

( k )

l i m( e

k

)



 0 x ( )

⁰

ﻪﻴﻟوا سﺪﺣ ﺮﻫ ياﺮﺑ ﻲﻳاﺮﮕﻤﻫ

تﻻدﺎﻌﻣ ﻞﺣ رد راﺮﻜﺗ شور

ﻲﻄﺧ

(13)

لﺎﺜﻣ

ﺷاد ﺪﻫاﻮﺨﻧ ﻲﻳاﺮﮕﻤﻫ ﻪﻴﻟوا سﺪﺣ ﺮﻫ ياﺮﺑ شور ﻦﻳا ﻲﻟو ﺖﺳا رﺎﮔزﺎﺳ ﺖ

x  b 2

( k ) ( k )

x

^¹

  x  b راﺮﻜﺗ شور

x ( ) 0  0 x

^{( k )}²

 0 ,x

^{( k}² ^¹⁾

 b, k  01 , ,. . . . x

( )⁰

 1 b

2 اﺮﮕﻤﻫ

Geomechanics

ﻲﻳاﺮﮕﻤﻫ طﺮﺷ

ﻒﻳﺮﻌﺗ : ﺲﻳﺮﺗﺎﻣ ﻲﻔﻴﻃ عﺎﻌﺷ A x = b

راﺮﻜﺗ ﻪﻟدﺎﻌﻣ

i m i

( B ) m a x ( B )

 



  1

( ) ( k ) ( k )

x

⁰

g i v e n, x

^¹

 B x  f , k  0

ﻪﻴﻀﻗ : ﺑ ﺎﻬﺑاﻮﺟ رادﺮﺑ ﺪﺷﺎﺑ رﺎﮔزﺎﺳ راﺮﻜﺗ ﻪﻟدﺎﻌﻣ ﺮﮔا مﺎﻤﺗ يازﺎ

ﺮﮔا دﻮﺑ ﺪﻫاﻮﺧ اﺮﮕﻤﻫ ﻪﻴﻟوا يﺎﻬﺳﺪﺣ

( k )

x

( B )

  1

Geomechanics

(14)

ﻲﻄﺧ يراﺮﻜﺗ يﺎﻬﺷور

ﺐﺳﺎﻨﻣ ﺲﻳﺮﺗﺎﻣ ود ﻪﺑ ﻪﻳﺰﺠﺗ N , P

P ﻪﻛ دﺮﻔﻨﻣ ﺮﻴﻏ ) ﺮﻳﺬﭘ سﻮﻜﻌﻣ (

ﺖﺳا

A x = b

( k ) ( k )

P x

^¹

 N x  b A   P N

k  0 B  P

^¹

N f  P

^¹

b

( k ) ( k )

x

^

B x f



¹

 

( k ) ( k ) ( k ) ﺎﻳ و

x  1  x  P  1 r

( k ) ( k )

r   b A x ما k ﻪﻠﺣﺮﻣ هﺪﻧﺎﻤﻴﻗﺎﺑ

صﺎﺧ ﺖﻟﺎﺣ :

ﻢﻴﻘﺘﺴﻣ ﻞﺣ P  A,N  0

Geomechanics

ﻲﻄﺧ يراﺮﻜﺗ يﺎﻬﺷور ﻲﺑﻮﻛاژ شور -

لﺪﻳﺎﺳ سوﺎﮔ شور - يزﺎﺳ دازآ شور -

تﻻدﺎﻌﻣ ﻞﺣ رد راﺮﻜﺗ شور

ﻲﻄﺧ

(15)

يﺮﻄﻗ يﺎﻫ ﻪﻔﻟﻮﻣ ﺲﻳﺮﺗﺎﻣ D A

ﺗﺎﻣ يﺎﻫ ﻪﻔﻟﻮﻣ ﻪﻨﻳﺮﻗ زا ﻲﺜﻠﺜﻣ ﻦﻴﻳﺎﭘ ﺲﻳﺮﺗﺎﻣ E A ﺲﻳﺮ

A ﺲ ﻳﺮﺗﺎﻣ يﺎﻫ ﻪﻔﻟﻮﻣ ﻪﻨﻳﺮﻗ زا ﻲﺜﻠﺜﻣ ﻻﺎﺑ ﺲﻳﺮﺗﺎﻣ F

ﻲﺑﻮﻛاژ شور -

j i n

i i i j j

i i j

x b a x , i ,. . . ,n.

a



 

 

  

 

 



1

1 1

j i n

( k ) ( k )

i i i j j

i i j

x b a x , i ,. . . ,n.

a







 

 

  

 

 



1

1 1

P  D, N     D A E F

 

A   D E  F

Geomechanics

زﺎﺳ دازآ ﺮﺘﻣارﺎﭘ ﻚﻳ ﻒﻳﺮﻌﺗ ﺎﺑ ﺮﺗ ﻲﻣﻮﻤﻋ ﺖﻟﺎﺣ ي

ﻲﺑﻮﻛاژ شور -



B

j

 D

^¹

( E  F )   I D

^¹

A

:Jacobi over-relaxation method

j i n

( k ) ( k ) ( k )

i i j

i j i

i i j

x b a x ( ) x , i ,. . . ,n .

a

 







 

 

    

 

 



1

1 1

j j

B

_

  B   ( 1  )I

( k ) ( k ) ( k )

x

^¹

 x   D

^¹

r ﺖﺳا رﺎﮔزﺎﺳ ﺮﻫ ياﺮﺑ شور ﻦﻳا ﺖﺳا ﻲﺑﻮﻛﺎﺟ شور نﺎﻤﻫ ﺖﻟﺎﺣ رد

  0

  1

JOR

Numerical Methods in Geomechanics 30

(16)

سوﺎﮔ شور -

ﻪﻠﺣﺮﻣ رد ﻪﻛ دراد توﺎﻔﺗ ﺮﻣا ﻦﻳا رد ﻲﺑﻮﻛﺎﺟ شور ﺎﺑ لﺪﻳﺎﺳ ما k+1

ﺮﻳدﺎﻘﻣ زا

ﺪﻧﻮﺷ ﻲﻣ هدﺎﻔﺘﺳا ﻞﺣ هار ندﺮﻛ زور ﻪﺑ ياﺮﺑ

لﺪﻳﺎﺳ سوﺎﮔ شور -

P   D E , N  F

( k )

x

i ^¹

j i n

( k ) ( k ) ( k )

i i j

i j i

i i j

x b a x ( ) x , i ,. . . ,n .

a

 







 

 

    

 

 



1

1 1

B

G S

 ( D  E )

^¹

F

Geomechanics

لﺪﻳﺎﺳ سوﺎﮔ شور -

زﺎﺳ دازآ ﺮﺘﻣارﺎﭘ ﻚﻳ ﻒﻳﺮﻌﺗ ﺎﺑ ﺮﺗ ﻲﻣﻮﻤﻋ ﺖﻟﺎﺣ

 ي

SOR : Seidel over-relaxation method

i n

( K ) ( K ) ( K )

( k )

i i i j j i j j i

j j i

i i

x [ b a x a x ( ) x

a



^ ^





  

  

¹ ¹

   

1

1 1

1

( k ) ( k )

( I   D

^¹

E )x

^¹

   ( 1   )I   D

^¹

F x     D

^¹

b

( k ) ( k ) ( k )

x

^¹

 x  (  1 D  E )

^¹

r ﺖﺳا رﺎﮔزﺎﺳ ﺮﻫ ياﺮﺑ شور ﻦﻳا ﺖﺳا لﺪﻳﺎﺳ سوﺎﮔ شور نﺎﻤﻫ ﺖﻟﺎﺣ رد

  0

  1

( , )

  0 1 under-relaxation

(17)

تﻻدﺎﻌﻣ ﻞﺣ رد راﺮﻜﺗ يﺎﻬﺷور ﻲﻄﺧ ﺮﻴﻏ

ﻦﺗﻮﻴﻧ شور -

f ( x )  0

Geomechanics

ﻦﺗﻮﻴﻧ شور ﻪﺑ لﺎﺜﻣ

ﻖﻴﻗد باﻮﺟ

0.35425 1.13264 x

y



1 1 2 2

2 2 4 det

y y

J J x

   

    detJ2 (2y x 4) 4y ¹₍₀₎ 1 2 2 2 3

J^ 10  

   

(18)

هﺪﺷ حﻼﺻا ﻦﺗﻮﻴﻧ شور -

1- Cyclic updating of the Jacobian matrix

keeping the Jacobian matrix (more precisely, its factorization) unchanged for a certain number,

2. Inexact solution of the linear systems

the maximum number of admissible iterations is fixed a priori 3. Difference approximations of the Jacobian matrix

Numerical Methods in Geomechanics 35

ﻲﻟاﻮﺘﻣ يزﺎﺳدازآ ﺶﻴﺑ -

The Successive-Over-Relaxation

(SOR) Method

(19)

لﻮﺼﻓ و ﻦﻳوﺎﻨﻋ ﺖﺳﺮﻬﻓ

Geomechanics

∆ ƒ

ƒ

∆ u u

1

iteration

2

NEWTON – RAPHSON ITERATION

∆ ƒ

ƒ

∆ u u

1 2

iteration

CONSTANT STIFFNESS ITERATION

3 4 56

Krylov Newton iteration

سﺎﻣﻮﺗ ﻢﺘﻳرﻮﮕﻟا -

Geomechanics