Numerical solutions of fuzzy partial differential equations and its applications in computational mechanics. Andrzej Pownuk1

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1 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs Abstract Andrz Pownk Car of Tortcal Mcancs Dpartmnt of Cvl Engnrng Slsan Unvrsty of Tcnology Calclaton of t solton of fzzy partal dffrntal qatons s n gnral vry dffclt. W can fnd t act solton only n som spcal cass. Fortnatly n most of ngnrng applcatons rlatons btwn t soltons and ncrtan paramtrs ar monoton w can assm tat wn t ncrtanty of t paramtrs s sffcntly small. In ts cas t act solton can b calclatd sng only ndponts of gvn ntrvals. In ordr to mprov t ffcncy of calclaton w can apply snstvty analyss. In ts papr a vry ffcnt algortm of solton was prsntd. Ts algortm s basd on fnt lmnt mtod or any otr nmrcal mtod of solton PDE lk for ampl FEM or BEM and snstvty analyss. Usng ts mtod w can solv ngnrng problms wt tosands dgr of frdom. Fzzy partal dffrntal qatons can b appld for modlng of mcancal systm strctrs wt ncrtan paramtrs. To constrct t fzzy mmbrsp fncton random sts can b appld. Ts tory contans fzzy sts and probablty tory as spcal cass. Usng algortms wc ar dscrbd n ts papr w can solv partal dffrntal qatons wt random and fzzy paramtrs. Kywords: fzzy sts random sts ntrval artmtc fzzy partal dffrntal qatons. E-mal addrss: pownk@zs.polsl.glwc.pl

2 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs Introdcton Fzzy nmbr s a fzzy st F of t ral ln R wc s conv and normal. Lt FR dnot t st of all fzzy nmbrs wc ar ppr smcontnos and av compact spport. If F FR µ : R µ [] R F w can also wrt µ.. : R F R F µ F [] R F Fzzy qaton Lt s consdr t followng qatons wt fzzy paramtr d d. 3 An analytcal solton of ts qaton s t followng. 4 T mmbrsp fncton of t fzzy solton F F R can b calclatd sng t tnson prncpl µ ξ sp. 5 F µ F : ξ

3 Random sts ntrprtaton of fzzy 3 Lt s consdr som partal dffrntal qatons wt vctor of fzzy paramtrs F k H... V k 6 wr V s som fnctonal spac. If w know t act solton of t problm 6 w can calclat t fzzy solton sng t tnson prncpl: µ F sp µ F 7 : T sam solton can b calclatd sng -lvl ct mtod Bckly Qy 99 Bckly Frng. T algortm s t followng: Algortm Calclat -lvl ct of fzzy paramtrs F ˆ { : µ F }. 8 Calclat t solton of partal dffrntal qatons wt ntrval paramtrs: ˆ { : ˆ }. 9 3 Calclat fzzy mmbrsp fncton of t solton: µ F sp{ : ˆ }. T most dffclt part of ts algortm s t stp.

4 4 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs 3 Random sts ntrprtaton of fzzy Lt s consdr probablty spac Ω P Ω Σ Ω and ntrval-vald random varabl: Hˆ : ˆ Ω Ω ω HΩ I R wr I R s a st of all ntrvals. Usng sc random varabl w can dfn ppr and lowr probablty Pl A P { : ˆ Ω ω H Ω ω A } Bl A P { : Hˆ Ω ω Ω ω A} 3 Lt s consdr dscrt random varabl wc satsfy t followng condton: Hˆ Ω ω Hˆ ω... Hˆ ω 4 Ω Ω n tn w can dfn fzzy mmbrsp fncton n t followng way: ˆ F PΩ { ω : H ω} 5 µ Ω Lt s consdr som mcancal systm and prformanc fncton g wc as t followng proprts: - f g tn t strctr s saf - f g < tn t strctr fald. Uppr probablty of falr of t strctr can b dfnd n t followng way: P ˆ f PΩ { ω : g HΩ ω } 6

5 mrcal mtods of solton of partal dffrntal qatons 5 If t strctr as fzzy paramtrs F tn w can calclat ppr probablty of falr n t followng way: P sp µ 7 f : g < Uppr probablty of falr of t strctrs wt random n X Ω : Ω ω XΩ ω R and fzzy m µ F : R µ F R paramtrs.. y g can b calclatd sng t followng formla wr f F P P { } E { µ } 8 µ F Ω Ω F µ F sp µ F 9 : g < 4 mrcal mtods of solton of partal dffrntal qatons Many problms n ngnrng can b dscrbd sng partal dffrntal qatons partclarly: - statc and dynamc of strctrs - bomcancs - at and mass transfr - lctromagntc flds - mtorology tc. T most poplar mtods of solton of sc qatons ar: - fnt lmnt mtod FEM - bondary lmnt mtod BEM - fnt dffrnc mtod FDM.

6 6 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs Most nvrsal and poplar s fnt lmnt mtod Carlt 978. T FEM algortm as t followng stps: Formlat bondary val problm L f V wr L. s dffrntal oprator f s som fncton and V s a fnctonal spac.g. Sobolv spac. Formlat varatonal qaton of t problm v V a v l v 3 Dscrtz doman Ω sng fnt lmnts Ω Ω a solton spac V and bld V R... wr n s a matr of sap fnctons.... n An appromat solton as t followng form: Ω.. 3 Ω T appromat solton satsfs t followng varatonal qaton: v V a v l v 5

7 mrcal mtods of solton of t fzzy partal dffrntal qatons 7 5 Vctor of nodal solton can b calclatd as a solton of t followng systm of lnar qaton K Q 6 wr K stffnss matr and t vctor Q ar dfnd n t followng way: K a 7 Q 8 l 6 If w know t solton of t systm of qaton 6 vctor of nodal solton w can calclat t val of appromat solton btwn t nods sng qatons 3 4. Fnt lmnt mtod was mplmntd n many commrcal ngnrng programs.g.: - ABACUS ttp:// - ADIA ttp:// - ASYS ttp:// - tc. Usng ts mtod w can solv problms wt tosands or vn mllons dgr of frdom.

8 8 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs 5 mrcal mtods of solton of t fzzy partal dffrntal qatons To solton of t fzzy partal dffrntal qatons w can also apply algortm of t fnt lmnt mtod. In ts cas qaton 6 as t followng form: or n nonlnar problms K Q ˆ 9 K Q ˆ 3 If w know t solton û of paramtr dpndnt systm of qaton 9 ˆ { : ˆ ll K Q } 3 t symbol ll S dnot t smallst ntrval wc contan t st S w can calclat a fzzy mmbrsp fncton of t fzzy nodal solton n t followng way: F µ F sp{ : ˆ } 3 T act solton st { : K Q ˆ } of t problm 9 s vry complcatd bcas of ts n applcatons w s only t smallst ntrval wc contan t act solton.. ll { : K Q ˆ } Klpa t all 998. T fzzy solton btwn t nods F can b calclatd sng t followng formla: µ F sp{ : ˆ } 33

9 Systms of algbrac qatons wt ntrval paramtrs 9 wr ˆ s dfnd as follows.. ˆ { : ˆ ll } 34 ˆ { : ˆ ll K Q } 35 T fnt dffrnc mtod and t bondary lmnt mtod can b also appld to calclaton of nmrcal solton of t fzzy partal dffrntal qaton. Wt all mntond mtods w can apply t followng gnral algortm. Algortm Calclat -ct of fzzy paramtrs ˆ { : µ F }. Solv systm of paramtr dpndnt systm of qaton ˆ { : ˆ ll K Q } 36 3 Calclat fzzy nodal solton F µ F sp{ : ˆ } 37 4 Calclat fzzy solton btwn t nodal ponts F µ F sp{ : ˆ } 38

10 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs 6 Systms of algbrac qatons wt ntrval paramtrs T most dffclt part of scond algortm s t stp. It can b sown tat fndng t solton of systm of a lnar ntrval qaton s P-ard Krnovc t all 998. Bcas of tat t ntrval solton û wc s dfnd n t qaton 36 can b fond only n spcal cass. 6. Applcaton of monoton fnctons Many nmrcal ampls sows tat t rlaton btwn t solton and ncrtan paramtrs s monoton McWllam oor t all Pownk. Lt s consdr fncton y and t ntrval ˆ [ ]. If fncton s monoton tn t trm vals of t fncton ovr t ntrval ĥ can b calclatd sng only t ndponts. wr ˆ y [ y y ] [ mn{ } ma{ }] 39 y nf y sp. 4 : ˆ : ˆ Lt s assm tat t fncton dpnds on m paramtrs m n... m.. : R R wc blong to t m ntrvals ˆ ˆ... m.. ˆ. If ts fncton s monoton tn t trm vals can b fond aftr calclaton all combnaton of ndponts of t mltdmnsonal ntrvalĥ.

11 Systms of algbrac qatons wt ntrval paramtrs mn{ : Vrtˆ} ma{ : Vrtˆ}. 4 4 wr Vrt ˆ s a st of all vrt of t ntrval ĥ. ow w can wrt ˆ n n [ ] [ ]... [ ] ll ˆ 43 To calclaton of t vctor û w av to calclat t val of t m fncton tms. In t sam way w can calclat t solton of t problm 36 mn{ : K Q Vrtˆ} ma{ : K Q Vrtˆ} Unfortnatly ts mtod as vry g comptatonal complty and cannot b appld to problms tat ar mor complcatd. 6. Applcaton of snstvty analyss Lt s consdr a fncton : R R. If t drvatv as constant sgn tn trm vals can b calclatd sng t followng formlas If > tn 46 If < tn 47

12 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs wr ˆ md. In mltdmnsonal cas n ordr to calclat trm vals of fncton w can compt t sgn vctor T sgn sgn S m T [ S... Sm] S S 49 T wr m m... mdˆ. If fncton s monoton tn ppr bond can b calclatd sng t followng pont wr ppr ppr ppr [... ] T m 5 k f S > tn k f S < tn ppr k k 5 ppr k k. 5 ppr In ts cas. In t sam way w can constrct lowr bond of t fncton. wr lowr lowr [... ] T lowr m 53

13 Systms of algbrac qatons wt ntrval paramtrs 3 k f S > tn k f S < tn ppr k k 54 ppr k k. 55 and fnally lowr ppr 56 ˆ m [ ]... [ m ]. 57 W s tat trm vals of t fncton can b calclatd sng t sgn vctors S and t ndponts of ntrval ĥ ppr lowr lowr ppr S ˆ S ˆ 58 lowr ˆ ppr S S ˆ ] ˆ [ ] [ 59 T vctor S av to b calclatd for ac coordnat of t vctor.. n tms. From t dfnton of t vctors S ars tat lowr ˆ lowr S S ˆ 6.. t sgn vctor ppr ˆ ppr S S ˆ 6 S gnrat t sam lowr and ppr bond as vctor S. From comptatonal pont of vw t s convnnt to fnd ndpndnt sgn vctors wc gnrat dffrnt lowr and ppr bonds of t solton.

14 4 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs IndS * k* { S... S } 6 * * * * * * S S IndS S S S S 63 Accordng to my prnc nmbr of t sgn vctors * S s mc lowr tan nmbr of t vctors S Pownk. ow w can apply snstvty analyss mtod to solton of t problm 36. Wol algortm as t followng stps Algortm 3 Formlat paramtr dpndnt systm of qaton wt ntrval paramtrs n t form 36. And calclat K ˆ Q md 64 For...m calclat wr K Q K For...n n nmbr of dgr of frdom calclat t sgn vctor S. sgn... sgn 66 m * k* 4 Calclat ndpndnt sgn vctors IndSgn { S... S } sng condton 63 and crat vctor U sc tat

15 * S 5 For...k calclat ntrval solton Systms of algbrac qatons wt ntrval paramtrs 5 S wr U 67 * ˆ lowr * ˆ ppr * S S ˆ ] ˆ * [ 68 IndSoltn {ˆ... ˆ } 69 6 Calclat trm ntrval solton û. For...n * k* * * ˆ [ ] wr U 7 Comptatonal complty of ts algortm: - stp solton systms of qatons - stp m solton systms of qatons - stp 5 k solton systms of qatons k n. In prsntd algortm w av to calclat a systm of qaton btwn m and m n tms. 6.3 Calclaton of t solton btwn t nods Somtms w wold lk to know t ntrval solton ˆ of t bondary val problm n t pont Ω btwn t nodal ponts. If w assm tat t fncton s monoton for fd Ω tn to calclaton of trm vals snstvty analyss can b appld. Frst w av to calclat snstvty vctor S sgn S... sgn. 7 m

16 6 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs From t qaton 4 ars tat: q q q 7 T vctors wr calclatd n t algortm 3 b- cas of tat to calclaton of any systm of lnar qaton. ow w av to cck f t sgn vctor W assm tat t sgn vctor If t sgn vctor q S s nq f w don t av to solv S s nq. * * * S IndSgn S S S S 73 S s not nq.. p* p* p* S IndSgn S S S S 74 tn trm solton can b calclatd sng t followng formlas lowr p* * ˆ p S 75 ppr p* * ˆ p S 76 S s nq tn w av to calclat a nw n- If t sgn vctor trval solton ˆ ˆ [ ˆ ] 77

17 Calclaton of t val of fzzy fncton 7 lowr ˆ ppr S S ˆ ] ˆ k * [ 78 nt * S S k 79 k* IndSgn : IndSgn { S } 8 k * IndSolton : IndSoton {ˆ } 8 Etrm solton can b calclatd sng t followng formlas lowr k * * ˆ k S 8 ppr k * * ˆ k S 83 ˆ ˆ [ ˆ ] 84 7 Calclaton of t val of fzzy fncton In tcncal applcatons vry oftn w av to calclat t val of fncton wc dpnds on t solton of fzzy partal dffrntal qatons.g. y f. Etrm vals of fncton f can b calclatd sng snstvty analyss: f f S sgn... sgn 85 wr m

18 8 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs f f f 86 ow w can apply smlar procdr lk n prvos paragrap. If t sgn vctor S s not nq tn trm solton can b calclatd sng t followng formlas S s nq tn calclat a nw ntrval sol- If t sgn vctor ton nt p* lowr p* f ˆ f S 87 p* ppr p* f ˆ f S 88 lowr ˆ ppr S S ˆ ] ˆ k * [ 89 k * S S 9 k* IndSgn : IndSgn { S } 9 k * IndSolton : IndSoton {ˆ } 9 Etrm vals of t fncton f can b calclatd sng t followng formlas k * lowr k * f ˆ f S 93 k * ppr k * f ˆ f S 94

19 mrcal ampl plan strss problm n tory of lastcty 9 8 mrcal ampl plan strss problm n tory of lastcty Lt s consdr t followng partal dffrntal qatons E E ββ β β ρf ν ν β * Ω 95 * σβ nβ t Ωσ wr E s a modl of lastcty ν s a Posson s rato ar dsplacmnts ρ s a mass dnsty f ar mass forcs σ β ar strss n coordnat of t nt vctor wc s normal to t bondary Ω t * ar bondary tracton. Ts ar qlbrm qatons of t plan strss lastcty problm. W can wrt ts qatons n t varatonal form Ω σ δε dv ρfδdω t δds 96 Ω Ω wr ε s a stran tnsor. If w tak nto accont t constttv qatons and gomtrc qatons w can dfn t blnar form σ 97 Cklε kl ε 98

20 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs Ω Ω Ω Ω d v C d a l k kl δε σ v 99 and t lnar form Ω Ω Ω Ω d v t d v f l ρ v T varatonal qatons of t tory of lastcty can b wrttn n t followng form v v v l a V ow w can solv ts qatons sng FEM mtod. T local stffnss matr can b wrttn n t followng form Ω Ω d T B D B K wr B 3 E ν ν ν ν D 4

21 mrcal ampl plan strss problm n tory of lastcty and ar sap fnctons wc wll b dscrbd latr. T load vctor can b calclatd from t followng qatons: Q ρ fdω tds 5 Ω T In calclaton w wll b s tranglar lmnt wc s sown n Fg Ω T Ω 4 3 Fg. Dsplacmnt n t lmnt can b dscrbd n t followng way:

22 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs T Sap fnctons of t lmnt satsfy t followng condtons: δ 7 If w assm tat t sap fnctons ar lnar.. c b a 8 tn t fncton as t followng form: wr 3 3 tc. Lt s consdr strctr wc s sown n Fg..

23 L q mrcal ampl plan strss problm n tory of lastcty 3 E 3 4 L L Fg. k In calclaton w assm tat L [m] q ν. 3. m Tabl. Fzzy Yong s modls E ˆ [89 3][GPa] [GPa] E ˆ [89 3][GPa] [GPa] 3 E ˆ [89 3] [GPa] [GPa] 4 E ˆ [89 3] [GPa] [GPa]

24 4 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs Aftr applyng algortm 3 w gt t followng nmrcal rslts: Tabl. Fzzy strss ˆ σ y [ ] [kpa].9763 [kpa] ˆ σ y [ ] [kpa].894 [kpa] ˆ 3 σ y [ ] [kpa] [kpa] ˆ 4 σ y [ ] [kpa].9 [kpa] Tabl 3. Fzzy dsplacmnts o. ˆ [m] [ ] [ ] 3 [ ] 4 [ ] 5 [ ] 6 [ ] 7 [ ] 8 [ ] 9 [ ] [ ] [ ] [ ] Ts problm as 8 dgr of frdom.

25 mrcal ampl plan strss problm n tory of lastcty 5 9 mrcal ampl plan strss problm n tory of lastcty T qlbrm qatons of a rod as t followng form d d V d EA d n. wr E s an Yong s modl A s an ara of cross scton n s a load and V s som fnctonal spac. W can formlat t problm n t varatonal form: wr v V a v l v L d dv a v EA d 3 d d L l v nvd... 4 To solton of t problm w can apply fnt lmnt mtod. Local stffnss matr n local coordnat systm as t followng form: wr T E A K B D B d 5 V L

26 6 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs B 6 L L D [ E ] 7 Local stffnss matr n global coordnat systm as t followng form: T T K C B D B C d 8 wr rotaton matr as t followng form V 4 y 3 y cos Fg. 3 sn sn sn cos C 9 cos sn cos

27 mrcal ampl plan strss problm n tory of lastcty 7 Local load vctor n global coordnat systm: Q C nd V T T T strctr s sown n t Fg. 4. P 3 P P Fg. 4 mrcal data ar as follows P [k] L [m] ν. 3 t Yong s modls s t sam lk n t prvos ampl. Intrval aal forcs ar sown n tabl 4.

28 8 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs Tabl 4. Intrval aal forc o. Aal forc [] [ ] [ ] 3 [ ] 4 [ ] 5 [ ] 6 [ ] 7 [ ] 8 [ ] 9 [ ] [ ] [ ] [ ] 3 [ ] 4 [ ] 5 [ ] 6 [ ] 7 [ ] 8 [ ] 9 [ ] [ ] [ ] [ ] 3 [ ] 4 [ ] 5 [ ] 6 [ ] 7 [ ] 8 [ ] 9 [ ] 3 [ ] 3 [ ] 3 [ ] 33 [ ] 34 [ ]

29 35 [ ] 36 [ ] 37 [ ] 38 [ ] 39 [ ] 4 [ ] 4 [ ] 4 [ ] 43 [ ] 44 [ ] 45 [ ] 46 [ ] 47 [ ] 48 [ ] 49 [ ] 5 [ ] 5 [ ] 5 [ ] 53 [ ] 54 [ ] 55 [ ] 56 [ ] 57 [ ] 58 [ ] 59 [ ] 6 [ ] 6 [ ] 6 [ ] 63 [ ] 64 [ ] 65 [ ] 66 [ ] 67 [ ] 68 [ ] 69 [ ] mrcal ampl plan strss problm n tory of lastcty 9

30 3 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs Qas-analytcal mtod Lt s consdr bondary val problm wt fzzy paramtrs L f V F If w dffrntat t bondary val problm wt rspct to w gt a nw bondary val problm f H V F Lt s assm tat w know t solton of t bondary val problm for mdˆ... ow w can sbsttt problm H v and w gt a nw bondary val f v v V 3 Problm 3 can b solvd sng any mtod. If w know t solton of t qatons 3 w can calclat trm solton by sng t algortm 3. Lt s consdr t followng ampl d d T act solton s t followng 4 5

31 Qas-analytcal mtod 3 If w calclat drvatv of t bondary val problm 4 wt rspct to w gt d d 6 or v d dv 7 wr v 8 t solton of t qaton 7 s t followng v 9 If > tn v> and t fncton s monoton for fd and trm solton can b calclatd n t followng way: ] [ ˆ 3 wr 3 3

32 3 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs It sold b notd tat f w know t nmrcal solton problm w cold not calclat drctly. of t Fnt dffrnc mtod T drvatv can b calclatd drctly sng t fnt dffrnc T val s a solton of t bondary val problm L f V. 34 Otr mtods of calclaton of t snstvty can b fnd n t book Klbr 997.

33 Pont monotoncty tsts 33 Pont monotoncty tsts. Frst ordr monotoncty tsts If drvatv of t fncton as constant sgn tn w can assm tat t fncton s monoton. T val of t fncton can b appromatd by sng t lnar fncton: m 35 An ntrval fncton s an ntrval-vald fncton of on or mor ntrval argmnts. Consdr a ral-vald fncton f of ral varabls... n and an ntrval fncton fˆ of ntrval varabls ˆ... ˆ n. T ntrval fncton fˆ s sad to b an ntrval tnson of f f wr... D fˆ... f n f n n D f s a doman of t fncton f. Tat s f t argmnts of fˆ ar dgnrat ntrvals.. ˆ tn f... ˆ s a ˆ ˆ n dgnrat ntrval qal to f... n. From proprts of ntrval tnsons mar 99 ars tat f ˆ ˆ tn ˆ. 37 and w can assm tat t fncton s monoton.

34 34 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs. Hg ordr monotoncty tsts W can also appromat drvatv of t fncton sng g ordr polynomals 3 k k m m k k m 38 If p ˆ ˆ tn ˆ p. 39 and w can assm tat t fncton s monoton. 3 mrcal ampl - dsplacmnt of t sll strctr T qlbrm qatons of sll strctrs can b wrttn n t followng form: Ω Ω p n M ds d n M p n M b n T b M b T b M b T 3 3 β β β β β βγ γ β β ββ β β β β βγ γ β β τ 4 wr

35 mrcal ampl - dsplacmnt of t sll strctr 35 γ β β β β 4 βγ β l g g l k gkl. g k. l 4 k g s a mtrc tnsor. Lt s consdr sll strctr wc s sown n Fg. 5. Fg. 5 In calclaton w assm t followng nmrcal data 5 5 E [.. ][ MPa] ν [..3] L.63 [m] 3 r.6 [m] F444.8 [] t.38 [ m ]. W wll b lookng for an ntrval dsplacmnt n drcton of t forc F. Usgn frst ordr monotoncty tst w can cck monotoncty of t solton. Bcas t fncton E ν s monoton tn trm vals of t solton can b fond sng only t ndponts of gvn ntrvals. T ntrval solton s as follows: m 43 : [ ] [ ]

36 36 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs : -.4 [m]. 44 Usng pont monotoncty tst w can calclat t ntrval solton only n som slctd ponts. In ts ampl profssonal FEM program Ansys was appld. 4 Taylor modl of t solton If t solton s sffcntly smoot tn w can appromat tm by sng Taylor srs m. 45 Etrm vals of t solton can b appromatd drctly by sng qaton 45 and ntrval artmtc ˆ ˆ ˆ ˆ m. 46 Ts mtod as vry low comptatonal complty m systm of qatons Akapan t all. Unfortnatly t qaton 46 gvs only appromat solton.

37 Intrval monotoncty tsts 37 5 Intrval monotoncty tsts 5. Lnar qatons Lt s consdr t problm 36 and assm tat w know t solton of t followng systms of lnar ntrval qaton K ˆ ˆ ˆ ˆ Q. 47 ˆ ˆ K Qˆ ˆ ˆ ˆ K ˆ ˆ. 48 wr ˆ ˆ ˆ ˆ ˆ ll K Qˆ. 49 If ˆ ˆ ˆ Qˆ Kˆ ˆ K ˆ ˆ ˆ ˆ ll 5 tn t solton of t problm 36 s monoton wt garantd accracy.

38 38 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs 5. mrcal ampl at transfr Lt s s consdr at transfr problm R r R r R < r < R : r d dr r r dt : - dr : T T dtr r Q dr t T r T b. 5 In calclaton w assm t followng nmrcal data W o R.5 [m] R R T b 3 C m K o W W Tt 37 C Q 45 λ [..3]. 3 m m K mrcal soltons ar sown n t tabl 5. Tabl 5. Intrval tmpratr o [ o C] T [ o C] T

39 Intrval monotoncty tsts onlnar qatons Somtms systm of algbrac qatons s nonlnar F ˆ. 5 From mplct fncton torm ars tat F F...m. 53 Eqaton 53 s a systm of lnar qaton wt nknown bcas of tat F F F F Fn Fn Fn F... n F F n... Fn n 54 F F n. 55 From qaton 55 arrays tat f t followng dtrmnats F F n. 56

40 4 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs av constant sgn tn t drvatv as also constant sgn and t fnctons ar monoton. From proprts of t dtrmnats and Darbo torm ars tat f ˆ n F 57. ˆ F t Jacoban matr ar rglar tn t fnctons ar monoton. From proprts of ntrval artmtc ars tat ˆ ˆ ˆ ˆ F F 59 and ˆ ˆ ˆ ˆ n n F F. 6 W can s tat f t ntrval Jacoban matrcs 59 6 ar rglar tn t fnctons ar monoton.

41 Intrval monotoncty tsts mrcal ampl fram strctr T qlbrm qatons of bam s as follows: d d d EJ d q V. 6 If w apply t fnt lmnt w gt qlbrm qatons n t followng form: K Q. 6 Lt s consdr a strctr wc s sown n t Fg. 6. L q q 8 9 q q P P P P q q 3 q 7 q 5 q 6 q H q q 4 H Fg. 6 In calclaton w assm t followng data E [ ][ GPa] J [ m ] A [.5.55 ][ m ] LH [m] P [k].

42 4 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs Tabl 6. Intrval dsplacmnts o. q [m] q [m] Sbdvson T ntrval tnson of t Jacoban matr may bcom snglar vn for vry narrow ntrvals ĥ. In ts cas w can dvd ts ntrvals and rpat procdr agan.

43 Optmzaton mtods 43 6 Optmzaton mtods 6. Dscrpton of t algortm If t ntrvals ĥ ar vry wd tn w cannot apply mtods wc wr dscrbd blow. In sc staton optmzaton mtods can b appld. V ma V mn f L f L ˆ ˆ. 63 Appromat solton can b dfnd as follows: Q K Q K ˆ ˆ ma mn mrcal ampl dsplacmnts of bam Lt s consdr bam strctr wc s sown n Fg. 7. q L L Fg. 7

44 44 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs T qlbrm qaton as t followng form d d EJ q d d L 3L d d d d. 65 3L In calclatons w assm tat E [.. ][ MPa] J [ m ] L [.999.][ m] q [ 9.9.][ k]. mrcal rslts ar sown n Fg y Fg. 8

45 Rfrncs 45 7 Conclsons Calclaton of t soltons of t fzzy partal dffrntal qatons s n gnral vry dffclt P-ard. In ngnrng applcatons t rlaton btwn t solton and t ncrtan paramtrs s sally monoton. 3 Usng mtods wc ar basd on snstvty analyss w can solv vry complcatd problms of comptatonal mcancs vn wt tosands dgr of frdom. 4 If w apply t pont monotoncty tsts w can s rslts wc wr gnratd by t stng ngnrng softwar. 5 Rlabl mtods of solton of t fzzy partal dffrntal qatons ar basd on t ntrval artmtc. Ts mtods av g comptatonal complty. 6 In som cass.g. f w know analytcal solton t optmzaton mtod can b appld. 7 In som spcal cass w can prdct t solton of t fzzy partal dffrntal qatons. 8 T fzzy partal dffrntal qaton can b appld to modlng of mcancal systms strctrs wt ncrtan paramtrs. Rfrncs [] Akapan U.O. Koko T.S. Orsamol I.R. Gallant B.K. Practcal fzzy fnt lmnt analyss of strctrs Fnt Elmnt n Analyss and Dsgn vol. 38 pp. 93- [] Bckly J.J. Qy Y. 99 On sng -cts to valat fzzy qatons Fzzy Sts and Systms vol. 38 pp.39-3 [3] Bckly J.J. Frng T. Fzzy dffrntal qatons Fzzy Sts and Systms vol. pp [4] Carlt P.G. 978 T Fnt Elmnt Mtod for Ellptc Problms ort- Holland w York [5] Klbr M. 997 Paramtr Snstvty n onlnar Mcancs Tory and Fnt Elmnt Comptatons Jon Wlly and Sons w York [6] Krnovc V. Lakyv A. Ron J. Kal P. 998 Comptatonal Complty Fasblty of Data Procssng and Intrval Comptatons Klwr Acadmc Pblsrs Dordrct

46 46 mrcal soltons of fzzy partal dffrntal qatons and ts applcatons n comptatonal mcancs [7] Klpa Z. Pownk A. Skalna I. 998 Analyss of lnar mcancal strctrs wt ncrtants by mans of ntrval mtods Comptr Assstd Mcancs and Engnrng Scncs vol. 5 pp [8] McWllam S. Ant-optmzaton of ncrtan strctrs sng ntrval analyss. Comptrs and Strctrs vol. 79 pp.4-43 [9] mar A. 99 Intrval mtods for systms of qatons Cambrdg Unvrsty Prss w York 99 [] oor A.K. Starns J.H. Ptrs J.M. Uncrtanty analyss of compost strctrs Comptr mtods n appld mcancs and ngnrng vol. 85 pp [] Pownk A. Applcaton of fzzy sts tory to assssmnt of rlablty of cvl ngnrng strctrs n Pols P.D. Dssrtaton Slsan Unvrsty of Tcnology