Mathematical modeling of uncertainty in computational mechanics

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Mathematical modeling of uncertainty in
computational mechanics

• Andrzej Pownuk
• Silesian University of Technology Poland
• andrzej_at_pownuk.com
• http//andrzej.pownuk.com

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Schedule

• Different kind of uncertainty
• Design of structures with uncertain parameters
• Equations with uncertain parameters
• Overview of FEM method
• Optimization methods
• Sensitivity analysis method
• Equations with different kind of uncertainty in
  parameters
• Future plans
• Conclusions

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Rod under tension

• Differential form of equilibrium equation

E Young modulus. A area of cress-section. n
distributed load parallel to the rod, u
displacement
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Different kind of uncertainty
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Floating-point and real numbers
- parameter
e.g.
Floating-point numbers
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Uncertain parameters

• Taking into account uncertainty using
  deterministic corrections.
• Control problems
• Gregorian and Julian calendar vs astronomical
  year (common years and leap years)

steering wheel is necessary
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Uncertain parameters

• Semi-probabilistic methods

This method is currently used in practical
civil engineering applications (worst case
analysis)
- safety factor
Some people believe that probability doesn't
exist.
- partial safety factor
Law constraints
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Uncertain parameters

• Random parameters

Using probability theory one can say that
buildings are usually safe ...
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Uncertain parameters

• Bayesian probability

Cox's theorem - "logical" interpretation of
probability
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Uncertain parameters

• Interval parameters

Interval parameter is not equivalent to
uniformly distributed random variable
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Uncertain parameters

• Set valued random variable
• Upper and lower probability

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Uncertain parameters

• Nested family of random sets

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Uncertain parameters

• Fuzzy sets

Extension principle
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Uncertain parameters

• Fuzzy random variables
• Random variables with fuzzy parameters

Etc.
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Design of structures with uncertain parameters
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Design of structures

• Safety condition

P load, A area of cross-section s stress
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Safe area
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Design of structures with interval parameters
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Design of structures with interval parameters
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More complicated cases
- design constraints
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Design constraints
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Geometrical safety conditions
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Applications of united solution set

• In general solution set of the design process is
  very complicated.
• In applications usually only extreme values are
  needed.

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Different solution sets

• United Solution Set
• Controllable Solution Set
• Tolerable Solution Set

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Example
United Solution Set
Tolerable Solution Set
Controllable solution set
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Example

• United solution set
• Tolerable solution set
• Controllable solution set

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Safety of the structures
- true but not safe
- unacceptable solution
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Safety of the structures

• Definition
• of safe cross-section

or

• Definition
• of safe cross-section

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More complicated safety conditions
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It is possible to check safety of the structure
using united solution sets
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Equations with uncertain parameters
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Equations with uncertain parameters

• Lets assume that u(x,h) is a solution of some
  equation.

How to transform the vector of uncertain
parameters through the function u in the point
x?
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Transformation of uncertain parameters through
the function ux
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Transformation of interval parameters
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Transformation of random parameters
Transformation of probability density functions.
- the PDF of the uncertain parameter h is known.
PDF of the results
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Transformation of random parameters
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Main problem

• The solution ux(h) is known implicitly and
  sometimes it is very difficult to calculate the
  explicit description of the function uux(h).

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Analytical solution

• In a very few cases it is possible to calculate
  solution analytically. After that it is possible
  to predict behavior of the uncertain solution
  ux(h) explicitly.
• Numerical solutions have greater practical
  significance than analytical one.

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Newton method
or
Etc.
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Continuation method

• Continuation methods are used to compute solution
  manifolds of nonlinear systems. (For example
  predictor-corrector continuation method).

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Many methods need the solution of the system of
equations with interval parameters
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Interval solution of the equations with interval
parameters
- smallest interval which contain the exact
solution set.
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Methods based on interval arithmetic

• Muhannas method
• Neumaiers method
• Skalnas method
• Popovas method
• Interval Gauss elimination method
• Interval Gauss-Seidel method
• etc.

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Methods based on interval arithmetic

• These methods generate the results with
  guaranteed accuracy
• Except some very special cases it is very
  difficult to apply them to some real engineering
  problems

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Overview of FEM method
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Finite Element Method (FEM)
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Real world truss structures
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Truss structure
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Boundary value problem
E Young modulus A area of cross-section u
displacement n distributed load in x-direction
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Potential energy
N axial force L length
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Finite element method
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Truss element 1D
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Truss element 2D
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Truss element 3D
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Variational equations
Frechet derivative
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Variational equations
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Galerkins method
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Ritzs method
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Parameter dependent system of equations
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Optimization methods
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These methods can be applied to the very wide
intervals
The function
doesn't have to be monotone.
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Numerical example
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Numerical data
Analytical solution
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Interval global optimization method
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Other optimization methods
DONLP2 and AMPL
COCONUT Project http//www.mat.univie.ac.at/neum/
glopt/coconut/
Till today the results in some cases are
promising however sometimes they are very
inaccurate and time-consuming.
Main problems time of calculations, accuracy
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Sensitivity analysis method
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Monotone functions
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Sensitivity analysis
If
, then
If
, then
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Truss structure example
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Accuracy of sensitivity analysis method (5
uncertainty)
Accuracy in Accuracy in
0 1,04E-02
0 0,00E00
0,003855 0,00E00
0 0,00E00
0 0,00E00
0 0,00E00
0 1,89E-03
0 5,64E-01
0,026326 0,00E00
0 4,87E-03
0 1,21E-03
0 0
18 interval parameters
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Extreme value of monotone functions
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Complexity of the algorithm, which is based on
sensitivity analysis
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Vector-valued functions

In this case we have to repeat previous algorithm
m times. We have to calculate the value of
m(n2) functions.
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Implicit function
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Sensitivity matrix
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Sign vector matrix
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Independent sign vectors
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Complexity of the whole algorithm.
1 - solution
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All sensitivity vector can be calculated in one
system of equations
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Sensitivity analysis method give us the extreme
combination of the parameters

• We know which combination of upper bound or lower
  bound will generate the exact solution.We can
  use these values in the design process.

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Example
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Analytical solution
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Sensitivity matrix
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Sign vectors
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Independent sign vectors
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Lower bound- first sign vector
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Upper bound- first sign vector
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Lower bound second sign vector
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Upper bound second sign vector
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Interval solution
The solution is exact
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Taylor expansion method
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The function uu(h) is usually nonlinear
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Accuracy of two methods of calculation (20
uncertainty)
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Accuracy of two methods of calculation (50
uncertainty)
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Comparison 50 uncertainty
 Sensitivity method  Sensitivity method Taylor method   Taylor method    Comparison  Comparison
           
-0,03 -1,19 43,01 -48,34 143466,7 3962,185
-37,1 -0,39 -11,27 -46,95 69,62264 11938,46
-1,53 -0,24 28,41 -44,04 1956,863 18250
-0,25 -4,3 -41,91 21,75 16664 605,814
-0,29 -0,28 43,11 -47,35 14965,52 16810,71
-0,33 -0,04 -45,43 38,26 13666,67 95750
0 -1,97 31,88 -45,78 inf 2223,858
-13,59 -15,68 -32,33 -30,86 137,8955 96,81122
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Time of calculation(endpoints combination method)
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Time of calculation(First order sensitivity
analysis)
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Time of calculation(First order Taylor expansion)
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Comparison
Time in seconds
Number of interval parameters Sensitivity Taylor
105 2 0,02 9900
410 452 1,22 36949
915 15 208 16,64 91294
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APDL description
(description of the nodes)

• N 1 0 0
• N 2 1 0
• MP 1 EX 210E9
• F 3 FX 1000
• R 1 0.0025

(material characteristics)
(forces)
(other parameters cross section)
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Interval extension of APDL language

• MP EX 1 5
• F 3 FX 5
• R 1 10

(material characteristics)
(forces)
(other parameters cross section)
Uncertainty in percent
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Web applications
http//andrzej.pownuk.com/interval_web_application
s.htm
Endpoint combination method
Sensitivity analysis method
Taylor method
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Automatic generation of examples
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The APDL and IAPDL code
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The results
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Calculation of the solution between the nodal
points
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Extreme solution inside the element cannot be
calculated using only the nodal solutions
u. (because of the unknown dependency of the
parameters)
Extreme solution can be calculated using
sensitivity analysis
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Calculation of extreme solutions between the
nodal points.
1) Calculate sensitivity of the solution. (this
procedure use existing results of the
calculations)
2) If this sensitivity vector is new then
calculate the new interval solution. The extreme
solution can be calculated using this solution.
3) If sensitivity vector isnt new then
calculate the extreme solution using existing
data.
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Use of existing commercial FEM software
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Use of existing commercial FEM software
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Papers related to sensitivity analysis method
Pownuk A., Numerical solutions of fuzzy partial
differential equation and its application in
computational mechanics, Fuzzy Partial
Differential Equations and Relational Equations
Reservoir Characterization and Modeling (M.
Nikravesh, L. Zadeh and V. Korotkikh, eds.),
Studies in Fuzziness and Soft Computing,
Physica-Verlag, 2004, pp. 308-347
Neumaier A. and Pownuk, A. Linear systems with
large uncertainties, with applications to truss
structures(submitted for publication).
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Monotonicity tests

• Taylor expansion of derivative
• Interval methods

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Monotonicity tests
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High order monotonicity tests
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Exact monotonicity tests based on the interval
arithmetic
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Finite difference method
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Slightly compressible flow- 2D case
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Example
Injection well
Production well
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Interval solution (time step 1)p_upper(t) -
p_lower(t)
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Single-region problems
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Multi-region problems
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More constraints less uncertainty
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Multi-region case
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Data file
alpha_c 5.614583 / volume conversion factor
/ beta_c 1.127 / transmissibility
conversion factor / / size of the block / dx
100 dy 100 h 100 / time steps / time_step
15 number_of_timesteps 10 reservoir_size 20 20
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Interval solution (time step 5)
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Comparison Single region - Multi-region
0,55 psi
0, 390 psi
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Sensitivity in time-dependent problems
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Sensitivity
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Calculation of total rate and total oil
production
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Interval total rate
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Interval total oil production
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Exact value of total rate
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Equations with different kind of uncertainty in
parameters
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Combination of random and interval parameters
r random parameter h interval parameter
or
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Combination of random and fuzzy parameters
r random parameter h fuzzy parameter
or
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Combination of random and random sets parameters
r random parameter h random set parameter
(set valued random variable)
etc.
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Calculation risk of cost using Monte Carlo method
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Interval web applications
http//andrzej.pownuk.com/interval_web_application
s.htm
Node NumberOfNode 0 NumberOfChildren 2 Children 1
2 IntervalProbability 0.05 xMinMin 1 xMinMax
1.1 xMidMin 2.0 xMidMax 2.0 xMaxMin 6 xMaxMax
6.11 NumberOfGrid 1 ProbabilityGrids
2 DistributionType 3 End
Node NumberOfNode 1 NumberOfChildren 1 Children
2 xMinMin 1 xMinMax 1.1 xMidMin 3 xMidMax
3.11 xMaxMin 6 xMaxMax 6.11 NumberOfGrid
2 DistributionType 2 End
Node NumberOfNode 2 xMinMin 1 xMinMax
1.1 xMidMin 3 xMidMax 3.11 xMaxMin 6 xMaxMax
6.11 NumberOfGrid 3 DistributionType 1 End
Results Xmin 0 Xmax 10 NumberOfSimulations
2000 NumberOfClasses 10 NumberOfGrid
2 DistributionType 2 End
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Future plans
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Future plans -uncertain functions
Equivalent of random fields
Different kind of dependences not only interval
or random constraints. Time series with interval,
fuzzy, random sets parameters.
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Future plans - software (web applications)
http//andrzej.pownuk.com/download.htm
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Future plans - design and optimization under
uncertainty
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Taking into account economical constraints
- real cost
- assumed cost
- investment risk
- acceptable risk level
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Cooperation with commercial companies

• ChevronTexaco http//www.chevrontexaco.com/
• Commercial FEM companies

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Conclusions
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Conclusions

• In cases where data is limited and pdfs for
  uncertain variables are unavailable, it is better
  to use imprecise probability rather than pure
  probabilistic methods.
• Using interval methods we can create mathematical
  model which is based on very uncertain
  information.

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• Presented algorithms are efficient when compared
  to other methods which model uncertainty, and can
  be applied to nonlinear problems of computational
  mechanics.
• Sensitivity analysis method gives very accurate
  results.
• Taylor expansion method is more efficient than
  sensitivity analysis method but less accurate.

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Conclusions

• It is possible to include presented algorithms in
  the existing FEM code.
• In calculations it is possible to use different
  kind of uncertainty (crisp numbers, intervals,
  random variables, random sets, fuzzy sets, fuzzy
  random variables etc.)

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Thank you