روش ی ﺪدی ژ ﻮﻣﮑﺎ ﯿﮏ

(1)

Numerical methods in geomechanics

Hasan Ghasemzadeh

ﮏﯿﺎﮑﻣﻮژ یدﺪ ی شور

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

ﯽ ﻮ ﻦﺪ اﺮ ﻪ اﻮ ﯽ هﺎﮕﺸاد

ﻖﺤﻟا لﻮﻗ

ِزﺎﻨَﻣ ُهَرَّﺪَﻗو اًرﻮﻧ َﺮَﻤَﻘﻟاو ًءﺎﻴِﺿ َﺲﻤَّﺸﻟا َﻞَﻌَﺟ يﺬَّﻟا َو َل

َبﺎﺴِﺤﻟاو َﻦﻴﻨِّﺴﻟا َدَﺪَﻋ اﻮﻤَﻠﻌَﺘِﻟ

ﺲﻧﻮﻳ 5

(2)

ﻲﺑﺎﻳزرا

1 - مﺮﺘﻧﺎﻳﺎﭘ نﺎﺤﺘﻣا 70

3 - هژوﺮﭘ 20

4 - رﺎﻨﻴﻤﺳ و تﺎﻨﻳﺮﻤﺗ 10

5 - ﺖﺒﻴﻏ تارﺮﻘﻣ ﻖﺑﺎﻄﻣ –

Dr. Hasan Ghasemzadeh 3

ﻊﺎﻨ

 A first course in the fem(2010) Daryl Logan 5th ed.

 Fundamentals of engineering numerical analysis(2010) Parviz Moin 2en ed.

 Numerical Methods for Engineers and Scientific (2001) Joe D. Hoffma.

 Numerical Mathematics(2000) , Alfio Quarteroni, Riccardo Sacco, Fausto Saleri

 Finite Elements and Approximation, (1983) Univ. of Wales, Swansea, U. K.

(1983) O. C. Zien kiewicz & K. Morgan, John Wiley & Sons,

 The Boundary Element Method For Engineers, 1980, Brebbia C.A.

 Numerical Methods for Elliptic and Parabolic Partial Differential Equations, 2003, Peter Knabner Lutz Angermann

 The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media (1998) R.W. Lewis and B.W. Schrefler

 1380 ، داﮋﻧرﺪﺻ ﻦﻳﺪﻟاﺮﻴﻣاﺪﻴﺳ مﻮﺣﺮﻣ، دوﺪﺤﻣ ءاﺰﺟا شور ﺮﺑ يا ﻪﻣﺪﻘﻣ

 يدﺪﻋ يﺎﻬﺷور ﻪﺑ طﻮﺑﺮﻣ ﻲﻤﻠﻋ تﻻﺎﻘﻣ

 سﻼﻛ هوﺰﺟ

(3)

تﺎﻴﻠﻛ شور يﺎﻫ يدﺪﻋ رد ﻚﻴﻧﺎﻜﻣﻮﺋژ

– ﻞﻴﺴﻧاﺮﻔﻳد تﻻدﺎﻌﻣ

شور توﺎﻔﺗ دوﺪﺤﻣ

شور ياﺰﺟا دوﺪﺤﻣ

دﺮﺑرﺎﻛ شور ﻞﺋﺎﺴﻣ ﻞﺣ رد يدﺪﻋ يﺎﻫ ﻚـﻴﻧﺎﻜﻣﻮﺋژ

) ،ﺖـﺴﺸﻧ شﺮﺘـﺴﮔ ﺎﻬـﺸﻨﺗ

ـﭘ ، ﻲ يﺎـﻫ

، ﻞﺋﺎﺣ يﺎﻫراﻮﻳد ،ﺎﻫ ﻊﻤﺷ ،ﻲﺤﻄﺳ ،ﺖﻔﻧ نزﺎﺨﻣ

هﺎﭼ يﺎﻫ ﺖﻔﻧ (...

هزﺎﻓ ﻪﺳ ود يﺎﻬﻄﻴﺤﻣ يدﺪﻋ ﻞﺣ ﺖﻛﺮﺣ ﻞﺣ جﻮﻣ

ﻞﺨﻠﺨﺘﻣ يﺎﻬﻄﻴﺤﻣ رد

يدﺪﻋ يﺎﻫﻮﮕﻟا –

يرﺎﺘﺧﺎﺳ تﻻدﺎﻌﻣ -

ﺎﻬﺠﻨﺳاﺮﻓ ﻲﺑﺎﻳزرا

سرد ﺐﻟﺎﻄﻣ ﺖﺳﺮﻬﻓ

- رد تﻻدﺎﻌﻣ ﻚﻴﻧﺎﻜﻣﻮﺋژ

يدﺪﻋ يﺎﻬﺷور تﺎﻴﻠﻛ - يدﺪﻋ يﺎﻬﺷور عاﻮﻧا - ﻞﺣ يروآدﺎﻳ - تﻻدﺎﻌﻣ

يدﺪﻋ يﺮﻴﮔ لاﺮﮕﺘﻧا - يژﺮﻧا لﻮﺻا -

تﺎﻴﻠﻛ شور يدﺪﻋ يﺎﻫ ﻚﻴﻧﺎﻜﻣﻮﺋژ رد

(4)

كﺎﺧ رد شواﺮﺗ

2

0

2 2

2





 



z k h x

k

_x

h

_z

شواﺮﺗ

كﺎﺧ رد شواﺮﺗ

z K D z

z

t 





 



 ( () )

 

 

 h

K D( ) ( )

(5)

كﺎﺧ رد ﻲﮔدﻮﻟآ

pollution

groundwater mining subsidence

كﺎﺧ رد ﻲﮔدﻮﻟآ

2 2

c c c

D C

x x t

    

  

(6)

كﺎﺧ رد تراﺮﺣ رﺎﺸﺘﻧا

Nuclear waste disposal

كﺎﺧ رد تراﺮﺣ رﺎﺸﺘﻧا

T

2

t  T

  



(7)

كﺎﺧ رد ﻢﻴﻜﺤﺗ

2

2 v

u u

z c t

  

 

2 2 2

2

u w

t C z

w u

t D z

  

  

 

 

  



كﺎﺧ رد ﻞﻜﺷ ﺮﻴﻴﻐﺗ و ﺶﻨﺗ ﻞﻴﻠﺤﺗ

لدﺎﻌﺗ ﻪﻟدﺎﻌﻣ )

ﻪﻟدﺎﻌﻣ ﻪﺳ (

ﺶﻧﺮﻛ يرﺎﮔزﺎﺳ ﻪﻄﺑار -

ﻪﻄﺑار ﺶﺷ (

يرﺎﺘﺧﺎﺳ ﻪﻄﺑار

(8)

يﺪﻌﺑ ﻚﻳ جﻮﻣ ﻪﻟدﺎﻌﻣ E

c_p  xx

p

tt

c u

u 

²

يﺪﻌﺑ ﻪﺳ ﻂﻴﺤﻣ رد جﻮﻣ ﻪﻟدﺎﻌﻣ تﻻدﺎﻌﻣ ﻴﻣﺎﻨﻳدﻮﺘﺳﻻا ﻚ

كﺎﺧ رد ﻲﻟﻮﻃ جاﻮﻣا رﺎﺸﺘﻧا

 

^v ² x ²₂^x

u u

x t

  ^    ^

 

 

^v ² y ²₂^y

u u

y t

  ^    ^

 

 

^v ² z ²₂^z

u u

z t

  ^    ^

 

First‐Order PDEs

 First‐order linear wave equation (advection eq.)

Propagation of wave with speed c

Advection of passive scalar with speed c

 First‐order nonlinear wave equation (inviscid Burgers’s

equation) gas dynamics and traffic flow

x 0 c u t

u 



 



x 0 u u t

u 



 



(9)

Second‐Order PDEs

•Advection‐diffusion equation (linear)

•Burger’s equation (nonlinear)

2 2

  

 

  

u u u

t  x u x

2 2

    

  

T T T

t  x c x

Other Common PDEs

•Korteweg‐de Vries (KdV) equation

•Laplace and Poisson’s equations

x 0 u x

u u t u

3





 



 



^Nonlinear

dispersive wave

waves on shallow water surfaces



 





 



 



 

 : Poisson

Laplace :

) ,

( f 0

0 y f

x y f

u x

u u

₂

2 2 2 2

(10)

Other Common PDEs

•Helmholtz equation

•Tricomi equation

Time-dependent harmonic waves Propagation of acoustic waves

0 u y k

u x

u

₂

2 2 2

2

 



 





 





 



 



hyperbolic

:

elliptic

:

0 y

0 0 y

y u x

y u

₂

2 2

2 Mixed-type

transonic flow

Other Common PDEs

•Wave equation

•Fourier equation (Heat equation)

x 0

c u t

u

2 2 2 2

2





 



2 2

x T t

T



 



 

(11)

Navier‐Stokes Equations

•Navier‐Stokes equation

•Vorticity / stream function formulation

 



 



 

 







 



 



 



 









2 2 2

2 2

y y x

x v t u



 





Navier‐Stokes Equations

•Navier‐Stokes equation

•Primitive variables













 







 



 



 

 

 



 





 







 



 



 

 

 



 



 

 



2 2 2 2

y v x

v y

p 1 y

v v x u v t v

y u x

u x

p 1 y

v u x u u t u

y 0 v x u

 

(12)

RANS Equations: Turbulent Flows

•Reynolds‐Averaged Navier‐Stokes equation





















 







 



 



 

 

 



 











 







 



 



 

 

 



 



 

 



y vv x uv y

V x

V y

P 1 y V V x U V t V

y uv x

uu y

U x

P 1 y V U x U U t U

y 0 V x U

2 2 2 2

 

















 











 

 

 



 









 











 

 

 



 



) (

 



 

 







1 2

2 2 2 2 t

C G k C y y x

x V t U

y G k x

k y

V k x U k t k

•Linear second‐order PDE in two independent variables (x,y), (x,t), etc.

•A, B, C, …, G are constant coefficients (may be generalized)

0 G y Fu

E u x D u y C u y x B u x

A u ₂

2 2

2

2   



 



 



 



 



Classification of PDEs

 The equation types are coordinate invariant, i.e., coordinate transformation will not change the type of equations

 Physical processes are independent of coordinates

(13)

Coordinate Transformation

•Physical plane Transformed plane

Physical Plane

x y









Transformed Plane



 

 





) , (

) ,

(  



 

y y

x x y

x y x

Classification of PDEs

•The classification depends only on the highest‐order derivatives (independent of D, E, F, G)

•For nonlinear problems [A,B,C = f(x,y,u)],the discriminant can still be used.

0 G Fu Eu

Du Cu

Bu

Au_xx  _xy  _yy  _x  _y   



















hyperbolic :

parabolic :

elliptic :

tion Classifica

0 AC 4 B

2 2 2

(14)

Classification of PDEs

•General form of second‐order PDEs (2 variables)

(1) Hyperbolic PDEs (Propagation)

0 G Fu Eu

Du Cu

Bu

Au

_xx



_xy



_yy



_x



_y

  











 

 



 

 



order) (second

0

order) -

(first 0

2 2 2 2 2

x c u t

u x u u t Advection equation u

Wave equation

hyperbolic :

0 4

4 ²

2  AC  c 

B

Classification of PDEs

(2) Parabolic PDEs (Time‐ or space‐marching)

0 G Fu Eu

Du Cu

Bu

Au

_xx



_xy



_yy



_x



_y

  











 



 



 



2 2

x t

ν x

u x t

 



 Burger’s equation 

Fourier equation

Diffusion / dispersion

2 4 0 : p bolic B A C ara

(15)

Classification of PDEs

(3) Elliptic PDEs (Diffusion, equilibrium problems)

0 G Fu Eu

Du Cu

Bu

Au

_xx



_xy



_yy



_x



_y

  













 

 



 

 



 

 



0 y c

x

y x y f

x

y 0

x

2 2 2 2 2

2 2 2 2

 



) , ( Laplace equation

Possion’s equation Helmholtz equation

elliptic :

0 4

2 4AC   B

Classification of PDEs

(4) Mixed‐type PDEs

0 G Fu Eu

Du Cu

Bu

Au

_xx



_xy



_yy



_x



_y

  







 



 



 

supersonic

:

subsonic

: )

( M 1

1 0 M

n M s

1 ₂

2 2 2

2

 

Steady, compressible potential flow

1 0

4 4 ) 1

(

4  ²   ²   

 M M M











parabolic :

1

elliptic :

1 M

M

(16)

Classification of PDEs

(5) System of Coupled PDEs

0 G Fu Eu

Du Cu

Bu

Au

_xx



_xy



_yy



_x



_y

  

Navier-Stokes Equations













 











 



 

 

 



 





 











 



 

 

 



 



 





2 2 2 2

y v x

v y

p 1 y v v x u v t v

y u x

u x

p 1 y v u x u u t u

y 0 v x u

 

ﺖﻔﻧ هﺎﭼ فاﺮﻃا شواﺮﺗ

(17)

ﺖﻔﻧ هﺎﭼ فاﺮﻃا ﻞﻜﺷﺮﻴﻴﻐﺗ

تﻻدﺎﻌﻣ ﻞﺣ يﺎﻫ شور

شور يﺎﻫ ﻲﻠﻴﻠﺤﺗ شور 

يﺎﻫ يدﺪﻋ

تﻻدﺎﻌﻣ هدﺎﺳ

و فوﺮﻌﻣ

تﻻدﺎﻌﻣ ﻲﻄﺧ

ﻂﻳاﺮﺷ يزﺮﻣ هدﺎﺳ

باﻮﺟ ﻖﻴﻗد عاﻮﻧا

تﻻدﺎﻌﻣ تﻻدﺎﻌﻣ ﻲﻄﺧ

و ﺮﻴﻏ ﻲﻄﺧ

ﻂﻳاﺮﺷ يزﺮﻣ هﺪﻴﭽﻴﭘ

باﻮﺟ ﻲﺒﻳﺮﻘﺗ

يﺎﻫزﺎﻴﻧ 

ﻲﺳﺪﻨﻬﻣ نﺎﻣز

ﻞﺣ ﺐﺳﺎﻨﻣ

ﺖﻗد ﺐﺳﺎﻨﻣ

ﻦﺘﺴﻧاد دوﺪﺣ ﺐﻳﺮﻘﺗ

(18)

ﻪﺑ هدﺎﺳ شﺮﮕﻧ دﺮﻜﻠﻤﻋ

يدﺪﻋ يﺎﻫ شور

يدﺪﻋ يﺎﻬﺷور رد ﻞﻴﺴﻧاﺮﻔﻳد تﻻدﺎﻌﻣ

ﺮﺑ ﻢﻛﺎﺣ ﻂﻴﺤﻣ ﻜﻳ ﻪﺑ ﻪﻟﺎﺴﻣ يﺮﺴ

ددﺮﮔ لﺪﺒﻣ هدﺎﺳ يﺮﺒﺟ تﻻدﺎﻌﻣ .

ﺑ باﻮﺟ ﻪﻠﺻﺎﺣ تﻻدﺎﻌﻣ يدﺪﻋ ﻞﺣ ﺎﺑ ﺎ

ددﺮﮔ ﻲﻣ ﻞﺻﺎﺣ ﺎﻄﺧ يرﺪﻗ .

ﻪﺑ هدﺎﺳ شﺮﮕﻧ دﺮﻜﻠﻤﻋ

يدﺪﻋ يﺎﻫ شور

يدﺪﻋ ﺎﻬﺷور عاﻮﻧا دوﺪﺤﻣ فﻼﺘﺧا

دوﺪﺤﻣ ي اﺰﺟا‐ يزﺮﻣ ي اﺰﺟا‐ مﺎﺠﺣا‐ دوﺪﺤﻣ

اﺰﺠﻣ ياﺰﺟا‐ ﻪﻜﺒﺷ نوﺪﺑ‐

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