a method for outer interval solution of parametrized systems of linear equations
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A method for outer interval solution of parametrized systems of linear equations. Iwona Skalna Department of Applied Informatics Cracow, Poland. Parametrized systems. Consider a family of linear algeraic systems of the following type. where. and. Solution set.

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a method for outer interval solution of parametrized systems of linear equations

A method for outer interval solution of parametrized systems of linear equations

Iwona Skalna

Department of Applied Informatics

Cracow, Poland

NFS workshop on Reliable Engineering Computing 2004

parametrized systems
Parametrized systems

Consider a family of linear algeraic systems

of the following type

where

and

NFS workshop on Reliable Engineering Computing 2004

solution set
Solution set

This family of linear algebraic systems is usually

written in a compact form

and is called parametrized system of linear interval

equations. The solution set of such system is defined

as

NFS workshop on Reliable Engineering Computing 2004

new method
New method

Presented here method is based on the following

inclusion

where  denotes an interval hull,

NFS workshop on Reliable Engineering Computing 2004

algorithm
Algorithm

NFS workshop on Reliable Engineering Computing 2004

results
Results

In this presentation there is only one example

shown, however in the paper some other

examples may be found.

The results produced by the described method are

compared with the results of the method called

Random Sampling of Parameter Intervals which

gives an inner enclosure of the solution set.

NFS workshop on Reliable Engineering Computing 2004

example plane truss
Example – plane truss

y

30kN

30kN

30kN

(0, 9)

(4, 9)

(8, 9)

(12, 9)

(16, 9)

1

4

2

3

5

8

6

7

9

(16, 6)

(4, 6)

(8, 6)

(12, 6)

10

11

(16, 3)

(12, 3)

12

(16, 0)

x

NFS workshop on Reliable Engineering Computing 2004

example description
Example - description

The exemplary plane truss structure is subjected

to downward forces of 30 [kN] at nodes 2, 3, 4.

All the bar elements have the same Young’s

modulus E=7e10 [Pa], and the same cross-section

area A=0.003 [m2]. The lengths (in meters) of the

bars are shown in the figure.

Now the stiffness of some of the bars (denoted

in the figure with thick red lines) is assumed to

be uncertain by 5%.

NFS workshop on Reliable Engineering Computing 2004

results x coordinates
Results – x coordinates

NFS workshop on Reliable Engineering Computing 2004

results y coordinates
Results – y coordinates

NFS workshop on Reliable Engineering Computing 2004

conclusions
Conclusions

The problem of solving parametrized systems

of linearinterval equations is very important

in practicalapplications. Well known classical

methods (e.g. Gauss Elimination) fail since they computeenclosure forthe solution set of nonparametrized systems which is generally

muchlargerthen solution set of the parametrized ones. A direct method for solvingparaterized systems of linear interval equationswas proposed and checked to be usefullin structuremechanics. The method produced tightenclosure for the solutions set of parametrizedsystemsfor exemplary truss structures.

NFS workshop on Reliable Engineering Computing 2004

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