FUZZY BOUNDARY ELEMENT METHODS: A NEW MULTI-SCALE PERTURBATION APPROACH FOR SYSTEMS WITH FUZZY PARAMETERS

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1 MODELOWANIE INŻYNIERSKIE ISNN X 3, s , Gliwice 6 FUZZY BOUNDARY ELEMENT METHODS: A NEW MULTI-SCALE PERTURBATION APPROACH FOR SYSTEMS WITH FUZZY PARAMETERS JERZY SKRZYPCZYK HALINA WITEK Zakład Mechaniki Teoretycznej, Politechnika Śląska, Gliwice Abstract. The aim of the paper is to present applications of the new algebraic system with specifically strictly defined new interval nmbers. [8]. We present the Fzzy Bondary Integral Eqations where all operations are in the fzzy pertrbation sense. From now we assme that vales of bondary conditions, material properties, internal prescribed fields and the shape of a bondary are ncertain and we'll model these ncertainties sing the new methodology based on interval pertrbation nmbers. Illstrative examples from the potential theory are given to comment different aspects of the presented theory. Interval, triangle and trapezoidal - type fzzy bondary conditions are considered. To complete the presentation the potential problem in a fzzy domain is discssed. Presented methods give the complete methodology how to obtain good approximations of soltions of ncertain bondary problems with se of modern fzzy analysis. INTRODUCTION The bondary problem may not be known exactly and some fnctions i.e. the shape of a strctre, material properties, bondary conditions, external or internal excitations, soltions etc. may contain nknown parameters. Many different interpretations are possible for terminology of ncertain aspects of the Bondary Element Method (BEM) and we ll refer to these approaches as Fzzy Bondary Element Method (FBEM). Applications of the FBEM have been initiated in the 995 in papers by Skrzypczyk and Brczyński, cf. [4,]. The earliest applications sed the fzzy independent nmbers approach. From now we assme that vales of bondary conditions, material properties, internal prescribed fields and the shape of a bondary are ncertain and we'll model these ncertainties sing the new methodology based on interval pertrbation nmbers defined by Skrzypczyk, [5-8,]. The new methodology can be applied to very complicated problems with different ncertain parameters. To present possibilities of the new method we consider the bondary potential problem with fzzy bondary conditions q, ; fzzy internal sorces ξ ; fzzy fndamental soltion ( ) fzzy bondary. ;

2 434 J. SKRZYPCZYK, H. WITEK Let, q, ξ, ( ) and with the fzzy bondary, be fzzy fnctions. Bondary problems with sch complicated conditions are not nosidered, cf. Witek [].. -FUZZY BOUNDARY ELEMENT METHOD Formally we can write fzzy bondary eqation for the potential problem in the sal form and replace a bondary potential, flx q,, internal sorces ξ and a bondary by corresponding fzzy vales, q, ξ,. Ths we obtain fzzy bondary integral eqation in the form c ( ) ) )q x ( + q y ( d ( y ( d( + ξ ( d (, x, () where ( ) denotes fzzy fndamental soltion for the potential problem and all operations are in the fzzy sense, cf. [4,]. Eq. () is called fzzy bondary integral eqation in the fzzy domain with fzzy parameters. The Method of Fzzy Bondary Elements with se of -nmbers is called frther -FBEM.. -Fndamental soltions By the fndamental soltion ( ) of the fzzy Laplace eqation for isotropic media we call a fzzy soltion of the eqation n δ ( x y ), F(R), x, y R, () n where δ (, x R denotes the fzzy Dirac δ-distribtion. For simplicity n and y(ξ,η) R is an arbitrary point in the plane. In the present we have not niqe, sfficiently advanced theory of fzzy partial differential eqations for generalized fnction formlation, sch as eq. (). Frther we se the theory of intervals. Transform the Laplace operator to polar coordinates (r,ϕ), then we get ( ) r + r r ϕ (3) The generalized fzzy -δ-dirac fnction in polar coordinates takes the form δ (( x ξ, y η) ) δ πr (4) where symbol r denotes the distance between point (x, and (ξ,η). The Dirac implse excitation is radial symmetric and since we have the problem in infinite domain, we have not distrbaces from the bondary. After neglecting terms which are zero de to symmetry of the soltion, i.e. r δ r πr (5) where is the fzzy constant. Fther we get r dr δ π π H + C (6) Since r> then

3 FUZZY BOUNDARY ELEMENT METHODS: A NEW MULTI-SCALE PERTURBATION C (7) π r r ln + C ln + C (8) π where C i C are integration constans. We can prove that C and C is some arbitrary potential so we can assme that C. Tab.. -Fndamental soltions of D and 3D Laplace eqation D 3D ln π 4π r n. -FBEM in pertrbation formlation π r 4π r The fzzy soltion of Eq. () is formlated in the conditional sense, i.e. we assme that the fzzy bondary F(M). For arbitrary, nonfzzy eq. () can be formlated as the family of -cts of the form ) c ( x ( + q ( d ( q ( d ( + We define the conditional soltion set as ( R ( x ) ξ ( d (,,, (9) x, [,] c( ( + q ( d ( + ξ ( d ( q( d (, : (, q) : () ( z) ( z), q ( z) q ( ), ( ) ( ) z z ξ z ξ z z z, q q x, y R R x, y R R for M. Methodology of fzzy eqations is based on the theory of intervals and methods described in [8]..3 e-interval Bonday Eqations - methodology of calclations Remember that eq. () is analysed in the conditional sense, with the assmption that the fzzy contor F(M). For arbitrary eqs. (9) mst be solved with sfficient qality. Mathematical problems with fzzy differential eqations for generalized fnctions force s to se the new algebraic methodology based on -intervals. We se that theory to solve the family of eqations (9) for M, and to obtain pper approximations for (). Assme that we looking for -type intervals for --cts in the form ( ( ( + rad( ( ), x, M, () where. Let

4 436 J. SKRZYPCZYK, H. WITEK ( ( ( + rad( ( ), x q ( ( q ( + rad q(, x ξ ( x ) ξ ( ( + rad ξ (, x, c ( x ) c ( ( + rad( c ( ), x ( ), ( ) and for --cts of -fndamental soltions ( ( x ) ( + rad ( q ( q ( + rad q (, x R ( ( ), ( ) k, k Since -cts of any fzzy nmber are intervals we write eqs. (9) in the -interval form c ( ( + q ( d ( q ( d ( + ξ ( d(, () (3) (4), (5) x, [,] where all operations are in the -interval sense, cf. [8,] and integrals are -extensions of srface integrals. Consider now how we can discretize eq. (5) to obtain -interval algebraic eqations for bondary vales. For simplicity we assme that the domain is D, the bondary is divided into N elements. Let M U N j j, where j is the bondary of j-element. -interval vales and q are considered as -constant/linear/ qadratic over each element.. ALGEBRAIC METHODOLOGY OF -CALCULATIONS If we assme that bondary points are nmbered between to N we get from eq. (5) the system of N -interval algebraic eqations in the -matrix form H U GQ + V, (6) where H and G are two NxN -matrices and U, Q, V are -vectors of lenth N, ],]. Notice that some N -fzzy vales of and N -vales of q are known on the bondaries i respectively, so we have only -nknowns in the system (6). We have to rearrange system (6) to obtain the standard system of -interval algebraic eqations A X F,, (7) where X is -interval vector of nknown -cts and q.eqs. (7) are very similar to the classic linear eqations over the field of real nmbers and we can easy obtain niqe Σ A, F. That family of -interval soltions is called -ALGEBRAIC -interval soltion ( ) INTERVAL SOLUTION - -AIS. REMARK. -interval soltion is an abstract object for any ],]. To obtain real interval soltion we have to sbstitte for any real small nmber and make necessary interval operations. We get approximate soltion which for every ],] is the I-order approximation Σ A, F. We can prove that accracy of that of interval eqation, in the same notion ( ) approximation is of order o( -δ ), where δ> is some arbitrary small constant. We say that the family Σ ( F ) A, of -AIS, dla M, generates -CONDITIONAL r x, x,with the membership ALGEBRAIC FUZZY SOLUTION (-CAFS) ( ) M fnction µ N ( y; r ( x ): Σ ( A, F ), x M, y R U. (8) [,]

5 FUZZY BOUNDARY ELEMENT METHODS: A NEW MULTI-SCALE PERTURBATION Ths -FUZZY ALGEBRAIC SOLUTION (-FAS) r (, x is defined as follows µ µ N y; r x : sp ; ;, R, M y r x y x, (9) ( ( ) ( ( )) ( ) ( ) µ 3. EXAMPLE In the simple example, details abot the contor see Fig.. in Skrzypczyk J., Mlti-Scale Pertrbation Methods In Mechanics, (this jornal). Uncertainties are introdced into bondary conditions and into bondary, see Figs. and 3. Uncertainties are of the fzzy - triangle type (9,3,3) and of the fzzy-trapezoidal type, see Fig.3 for nodes 3-6. Fzzy reslts for temperatre of nodes and are illstrated by α-cts of their membershipfnctions at Fig.. The deformation of the triange shape of membership fnction to the trapezoidal one is forced by the fzzy character of the bondary, see Fig. 4. ALFA Y TEMP. 4 6 X Fig.. α-cts of temperatre in and nodes Fig.. Uncertainty of the bondary ALFA TEMP. Fig. 3. Membership fnction of fzzy temperatre in 3-6 nodes Fig. 4. Membership fnction of fzzy temperatre in 4 and 9 nodes 4. CONCLUSIONS With the new -Fzzy Bondary Element Method we get a set of very simple and sefl mathematical tools which can be easy sed in analytical and comptational parts of analysis of complex technical problems with ncertain parameters.

6 438 J. SKRZYPCZYK, H. WITEK Advantages of the new algebraic system are as follows: we can omit all complex analytical calclations, which are typical for expanding approximated vales of soltions in infinite series. It works for expanding nknown vales - soltions as well as for pertrbed coefficients of the problem; we get a great simplification of all arithmetic calclations which appear in analytical formlation and analysis of the problem; most of known nmerical algorithms can be simply adapted for the new algebraic system withot any serios difficlties. REFERENCES. Brebia C.A., Domingez J., Bondary elements an introdctory corse, comptational mechanics pblications, London, 989. Brebbia, C., Telles J.C.F., Wrobel L.C., Bondary element techniqes - theory and applications in engineering, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, Brczyński T., Wspomaganie kompterowe CAD CAM, Metoda elementów brzegowych w mechanice, WNT, Warszawa, Brczyński T., Skrzypczyk J., Fzzy aspects of the bondary element method, engineering analysis with bondary elements, Special Isse: Stochastic Bondary Element Methods 9, 997, Skrzypczyk, J.: Pertrbation methods I- algebra, fnctions, linear eqations, eigenvale problems: new algebraic methodology. Proc. of International Conference New Trends In Statics And Dynamics Of Bildings, October 4, Faclty of Civil Engineering SUT Bratislava, Slovakia, Bratislava, 4, Skrzypczyk J., Metody pertrbacyjne - Nowa arytmetyka, Zeszyty Nakowe Katedry Mechaniki Stosowanej Politechniki Śląskiej 3, Gliwice 4, str Skrzypczyk, J.: Pertrbation methods - new algebraic methodology. Proc. of CMM-5 Compter Methods in Mechanics, Jne -4, 5, Częstochowa, Poland, Częstochowa, 5 8. Skrzypczyk J., Pertrbation methods for systems with interval parameters, Proc. of International Conference New Trends In Statics And Dynamics Of Bildings, October 5, Faclty of Civil Engineering SUT Bratislava, Slovakia, Bratislava, 5 9. Skrzypczyk J., Mlti-scale pertrbation methods in mechanics, Proc. of Symposim Modelling in Mechanics, Wisła Febrary 6 - March 6, Poland 6. Skrzypczyk, J. Brczyński, T.: Theoretical and comptational aspects of the fzzy bondary element methods. Klwer Academic Pblishers, IUTAM/IACM/IABEM Symposim on Advanced Mathematical and Comptational Mechanics Aspects of the Bondary Element Method, May 3 - Jne 3, Cracow 999, Dordrecht/Boston/London,, Skrzypczyk, J. Winkler, A.: Pertrbation methods ii- differentiation, integration and elements of fnctional analysis with applications to pertrbed wave eqation. Proc. of International Conference New Trends In Statics And Dynamics Of Bildings, October 4, Faclty of Civil Engineering SUT Bratislava, Slovakia, Bratislava, 4, Witek, H.: Metoda rozmytych elementów brzegowych w analizie konstrkcji bdowlanych o parametrach niepewnych. PhD Dissertation, Civil Eng. Faclty, Silesian Univ. of Technology, Gliwice 5