یدﺪ ی شور ﮏﯿ ﺎﮑﻣﻮ ژ

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Numerical methods in geomechnics

Hasan Ghasemzadeh

یدﺪ ی شور ﮏﯿﺎﮑﻣﻮژ

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

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ﺎﻄﺧ ناﺰﻴﻣ -

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يدﺪﻋ تﺎﺒﺳﺎﺤﻣ يروآدﺎﻳ -

ﺎﻄﺧ ناﺰﻴﻣ - ﺮﻠﻴﺗ ﻪﻴﻀﻗ - يژﺮﻧا لﻮﺻا -

Numerical Methodes in Geomechnics

Methods for solving Nonlinear Equations

o Bisection Method

o Newton-Raphson Method

o Secant Method

o Fixed-Point Method

solution analytic

0 No 5

2

²

9













x

e x

x x

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5

Methods for solving Nonlinear Equations

o Bisection Method

o Newton-Raphson Method

o Secant Method

o Fixed-Point Method

solution analytic

0 No 5

2

²

9













x

e x

x x

( ) 0 ( )

f x   g x  x

1 ( )

i i

x_  g x

1 2

Cramer's Rule can be used to solve the system

3 1 1 3

5 2 1 5

1, 2

1 1 1 1

1 2 1 2

But Cramer's Rule is not practical for large problems.

To solve N equations in N unknowns we need (N 1)(N 1)N!

multiplications.

x   x  

 

To solve a 30 by 30 system, 2.3 10 multiplications are needed. 35

Solution of Systems of Linear Equations

1 2

1 1 3

1 2 5

x x 

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Methods for solving Systems of Linear Equations

o

Naive Gaussian Elimination

o

Gaussian Elimination with Scaled Partial pivoting

o

Algorithm for Tri-diagonal Equations

Curve Fitting



Given a set of data



Select a curve that best fit the data. One choice is find the curve so that the sum of the square of the error is minimized.

x 0 1 2

y 0.5 10.3 21.3

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Interpolation



Given a set of data



find a polynomial P(x) whose graph passes through all tabulated points.

xi 0 1 2 yi 0.5 10.3 15.3

table in the

is )

( i i

i P x if x

y 

Methods for Curve Fitting

o

Least Squares

o Linear Regression

o Nonlinear least Squares Problems

o

Interpolation

o Newton polynomial interpolation

o Lagrange interpolation

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Methods for Numerical Integration

o

Upper and Lower Sums

o

Trapezoid Method

o

Romberg Method

o

Gauss Quadrature

2

0

? no analytical solutions

a

e

^x

dx 



Solution of Partial Differential Equations

Partial Differential Equations are more difficult to solve than ordinary differential equations

) sin(

) 0 , ( , 0 ) , 1 ( )

, 0 (

0

2 2

2

x x

u t

u

t u x

u





 

 



Numerical Solution ?

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ﺎﻄﺧ ناﺰﻴﻣ -

ﺮﻠﻴﺗ ﻪﻴﻀﻗ - يژﺮﻧا لﻮﺻا -

Normalized Floating Point Representation

 Standard Representations

 Normalized Floating Point Representation

 Advantage Efficient in representing very small or very large numbers

1 2 3 4

0. 10

sign mantissa exponent

d d d d n

 

3 1 2 1 2 . 4 5 1 5 sign integral part fraction part



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Significant Digits

 Significant digits are those digits that can be used with confidence.

48.9

Accuracy and Precision

 Accuracy is related to closeness to the true value

 Precision is related to the closeness to other estimated values

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Rounding and Chopping

 Rounding: Replace the number by the nearest machine number

 Chopping: Throw all extra digits

Error Definitions

if the true value is known

100 value *

true

ion approximat value

true

Error Relative

Percent Absolute

ion approximat value

true

Error True

Absolute

t

 





 E

t

True Error

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Error Definitions

When the true value is not known

100 estimate *

current

estimate prevoius

estimate current

Error Relative

Percent Absolute

Estimated

estimate prevoius

estimate current

Error Absolute Estimated

 





a

E

a



Estimated error

Notation

the estimate is correct to n decimal digits if

the estimate is correct to n decimal digits roundedif

n

 10 Error

n



 10

2

Error 1

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ﺮﻠﻴﺗ ﻪﻴﻀﻗ -

يژﺮﻧا لﻮﺻا -

Taylor Series

( )

0 0

0

( )

0 0

0

1 ( ) ( )

!

if the series converge we can write

( ) 1 ( ) ( )

!

k k

k

k k

k

Taylor Series f x x x k

f x f x x x

k









 



( ) ^x ( ) ?

f x e  f  

-3 -2 -1 0 1 2 3 4

x

x-x³/3!

x-x³/3!+x⁵/5!

sin(x)

3 5 7

sin( ) ....

3! 5! 7!

x x x

x  x    ^0.5

1 1.5 2 2.5 3

1 1+x 1+x+0.5x² exp(x)

!

k

x x

e







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Convergence of Taylor Series

(Observations, Example 1)

 The Taylor series converges fast (few terms are needed) when xis near the point of expansion. If |x-c| is large then more terms are needed to get good approximation.

 How many terms are needed to get good approximation???

These questions will be answered using Taylor Theorem

Taylor Theorem

h x x where

n h h f

k x h f

x f

x c h x x

c x where

c x n f

c f k x

c x f

f

n n n

k k k

n n n

k k k



 











 







 



 





and between is

)!

1 (

) (

! ) ) (

(

, and between is

) )! (

1 (

) ) (

! ( ) ) (

(

) 1 1 ( 0

) (

) 1 1 ( 0

) (



Error

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Example 1

  0 . 2 8 . 14268 05 )!

1 (

) )! (

1 (

) (

1

≥

)

≤

( )

(

1 4 2

. 0 1

) 1 1 ( 1

2 . 0 )

( )

(





 



 



 



E n E

E e

c n x

E f

k for e

f e

x f

n n

n n n

k x

k



How large is the error if we replaced ( ) by the first 4 terms (n 3) of its Taylor series expansion

about 0 0.2?

f x ex

c when x



 