http://andrzej.pownuk.com Adaptive Taylor Series and its Place - - PowerPoint PPT Presentation

▶
http andrzej pownuk com
SMART_READER_LITE
LIVE PREVIEW

http://andrzej.pownuk.com Adaptive Taylor Series and its Place - - PowerPoint PPT Presentation

Aug 29, 2022 291 likes •923 views

The University of Texas at El Paso Andrzej Pownuk http://andrzej.pownuk.com Adaptive Taylor Series and its Place photo here Applications in the Interval Finite Element Method Equation with uncertain parameters = b = ax b x a Example x =

slide-1
SLIDE 1

Adaptive Taylor Series and its Applications in the Interval Finite Element Method

The University of Texas at El Paso

Place photo here

Andrzej Pownuk http://andrzej.pownuk.com

slide-2
SLIDE 2

Equation with uncertain parameters

ax b =

Example

[1,2] [1,4] x =

? x =

= b x a

slide-3
SLIDE 3

Algebraic solution

[1,2] [1,4] x = [1,2] x =

because

[1,2][1,2] [1,4] =

slide-4
SLIDE 4

Algebraic solution

[1,4] [1,4] x = [1,1] 1 x = =

because

[1,4] 1 [1,4]  =

slide-5
SLIDE 5

Algebraic solution

[ 1 , 8] [ 1 , 4] x =

? x =

Algebraic solution do not have physical interpretation.

slide-6
SLIDE 6

United solution set

[1,2] [1,4] x =

1 ,4 2   =     x

because

{ : , [1,2], [1,4]} x ax b a b = =   x

slide-7
SLIDE 7

Web applications (Java)

slide-8
SLIDE 8

Monte Carlo method 3D solution set

slide-9
SLIDE 9

Monte Carlo method 3D solution set

slide-10
SLIDE 10

Solution Set

( ) { ( ): ( ) ( ) 0, }

i i

u p A p u f p p  = − = u p p [ ( ) , ( ) ] { ( ): ( ) ( ) 0, }

i i i

u p A p u u u f p p = − =  p p p

slide-11
SLIDE 11

Monotonicity of the solution

  • Monotone solution
  • Non-monotone solution

1 1 2 2 2

1 1 1 1 p u p u p       =       −           +

1 1 1 2 2

, 2 2 p p p u u = + =

4 2

u p − =

2 2 1 2

, u p u p = − =

slide-12
SLIDE 12

Interval solution for monotone functions (gradient descent method)

u p   

If then

,

m i n m ax

p p p p = =

u p   

If then

,

m i n m ax

p p p p = =

m ax

( ) , ( )

m i n

u u p u u p = =

slide-13
SLIDE 13

Monotonicity of the solution

slide-14
SLIDE 14

Plane stress

slide-15
SLIDE 15

Interval solution Verification of the results by using search method with 3 intermediate points

slide-16
SLIDE 16

Interval solution

slide-17
SLIDE 17

Interval solution

slide-18
SLIDE 18

Interval solution

slide-19
SLIDE 19

Interval solution

slide-20
SLIDE 20
slide-21
SLIDE 21

Truss example

slide-22
SLIDE 22

Truss structure

d du EA n dx dx   + =    

slide-23
SLIDE 23

Interval solution Verifications of the results by using search method with 5 intermediate points

slide-24
SLIDE 24

Interval solution

slide-25
SLIDE 25

Interval solution

slide-26
SLIDE 26

Verification of the monotonicity by using second order monotonicity test

slide-27
SLIDE 27

More examples

  • A. Pownuk, Monotonicity of the solution of the interval equations of

structural mechanics - list of examples, The University of Texas at El Paso, Department of Mathematical Sciences Research Reports Series Texas Research Report No. 2009-01, El Paso, Texas, USA [ Download ] 817 pages

slide-28
SLIDE 28

Example

slide-29
SLIDE 29

Automatic generation of examples

slide-30
SLIDE 30

Scripting language

slide-31
SLIDE 31

Results of the calculations

# ***************** # Postprocessing # ***************** print_interval_displacements print_Jacobian print_Jacobian_binary print_parameters print_dof all print_number_DOF print_time print_number_of_simulations

slide-32
SLIDE 32

2D interval solution

# ***************** # Postprocessing # ***************** print_interval_displacements print_interval_stress print_interval_Mises_stress print_interval_Mises_stress_to_matlab print_interval_stress_to_matlab export_model_to_ansys print_time

slide-33
SLIDE 33

Uncertainty in geometry

slide-34
SLIDE 34

Andrzej Pownuk, Behzad Djafari-Rouhani, Naveen Kumar Goud Ramunigari, Finite Element Method with the Interval Set Parameters and its Applications in Computational Science AMERICAN CONFERENCE ON APPLIED MATHEMATICS (AMERICAN-MATH '10) University of Harvard, Cambridge, USA, January 27-29, 2010. ISBN: 978-960-474-150-2, ISSN: 1790-2769, pp. 310-315.

slide-35
SLIDE 35

Uncertainty in geometry

slide-36
SLIDE 36

Time dependent solution

slide-37
SLIDE 37

Time dependent solutions

slide-38
SLIDE 38

Interval solution

slide-39
SLIDE 39

Combinatoric solution

slide-40
SLIDE 40

Damped vibrations

slide-41
SLIDE 41

Combinatoric solution

slide-42
SLIDE 42

Numerical integration

slide-43
SLIDE 43

Numerical integration

slide-44
SLIDE 44

Interval solution

slide-45
SLIDE 45

Nonmonotone example

slide-46
SLIDE 46

Response surface method (Hermitte interpolation)

slide-47
SLIDE 47

Time dependent solution

slide-48
SLIDE 48

Numerical values

slide-49
SLIDE 49

Big uncertainty

slide-50
SLIDE 50

Big uncertainty – numerical values

slide-51
SLIDE 51

Small uncertainty

slide-52
SLIDE 52

Small uncertainty – numerical values

slide-53
SLIDE 53

Adaptive Taylor series

slide-54
SLIDE 54

Numerical values

slide-55
SLIDE 55

Numerical values

slide-56
SLIDE 56

Midpoint and upper and lower bound

slide-57
SLIDE 57

5 solutions

slide-58
SLIDE 58

Vibrations with the interval parameters

slide-59
SLIDE 59

Sensitivity analysis

slide-60
SLIDE 60

Interval solution after sensitivity analysis

slide-61
SLIDE 61

Sensitivity vs midpoint solution

slide-62
SLIDE 62
  • M.V.Rama Rao, Andrzej Pownuk, Stefan Vandewalle, David Moens

Transient Response of Structures with Uncertain Structural Parameters. Journal of Structural Safety (submitted for publication).

slide-63
SLIDE 63

Conclusions

  • Using adaptive Tylor series it is possible to get the interval

solution.

  • The procedure is very effective.
  • It is possible to get error estimation and control the accuracy
  • f the calculations.