Thermal tomography on the basis of an information method

Transcription

1 ttp://d.do.org/0.6/qrt Termal tomograp on te bass of an nformaton metod *Karkov Natonal Unverst of Rado- Electroncs, Karkov, Ukrane Abstract B S. Melnk* A unfed (nformaton approac to te development of an optmum algortm of termal tomograp on te bass of te avalable pror nformaton as been desgned. Te essence of te new crteron s te mnmzaton of te complet of te descrpton of epermental results. It ncreases a resoluton of termal montorng b several folds. Te problem of te testng of delamnaton n a multlaer object s consdered as an eample of usage. Te capablt of detecton and determnaton of delamnaton borders s sown; f ts sze s muc smaller tan te dept and termal resstance s small.. Introducton Te development of termovson engneerng allows one to solve te problem of termal tomograp, n addton to termal non-destructve testng and defectometr problems. Currentl, te basc metod emploed n practce s te pulse termal tomograp. As a matter of fact, te one-dmensonal model of a eat flu s used n t. Onl one parameter - te dept of locaton of a not eat-conductng mperfecton s determned. Tus onl a small part of te nformaton about nteror structure of te montored object contaned n termovson flm s used. Te purpose of ts work s te development of a new unversal metod of solvng te termal tomograp problem. It s based on full usage of bot te termovson nformaton and pror nformaton on te object of montorng. Matematcal tecnques of algebrac nformaton teor ( te teor of complet [3 allow for solvng ts problem.. Te essence of te metod Te problem of te testng of delamnaton n a multlaer object s consdered as an eample of usage. An local non-unformt of termopscal propertes of te object of montorng (mperfecton can be represented b an equvalent eat source and eat dpole, dstrbuted accordngl n a volume and on a surface of ts nonunformt [. Te power of an equvalent eat dpole s proportonal to te tckness of an ar nterlaer and power of a eat flu. Tus, te termal tomograp of separatons s reduced to determnaton of a map of termal dpoles located on one of gven boundares of laers. In te elementar statonar case of two-dmensonal model, te temperature response to te surface of te object can be calculated as follows: k T( = = G ( q ( + δt (, ( s C.6.

2 ttp://d.do.org/0.6/qrt were G ( s te termal transfer functon of laers placed above te assgned equvalent termal dpole of power q ( ; δ T s ( s addtve nose wt a known spectrum; stands for convoluton. Tus te problem of te determnaton of q ( from known T ( and G ( s ll-posed. A major problem tat appears n te process of te soluton of te abovementoned nverse task s a coce between regularzaton crtera. Wt ts purpose, man known metodologes use a pror nformaton about a sgnal and a nose component. As a rule, ts component s te nose ampltude. In te case of a stocastc sgnal, ts can be nformaton about te spectrum of te sgnal and nose (.e., optmum spectral fltraton metod. We can state tat n tese and oter cases, te actuall mnmze complet of descrpton of epermental results, wc are set up at a defnte accurac. Hereb, a pror nformaton s requred at coce for descrpton- algortm. Hence, n partcular, follows a mnmzaton crteron for root-mean-square devaton for te case of normal dstrbuton of te nose component, Sannon s formula for random sgnal entrop evaluaton and man oters. For ts reason, te prncple of mnmzed complet of te descrpton of epermental results can be used as a unversal crteron for optmzaton of te soluton of te nverse task. (B te wa, ts metod s an ntegrated prncple of entrop mamum. Hereb, a pror nformaton on sgnals or noses can be consdered as addtonal prelmnarl obtaned epermental results. If a tested object s assumed to be a multlaer plate wt delamnaton, suc nformaton can nclude profle (probabl, rectangular, Gausse-tpe of delamnaton, dept of locaton for ever ndvdual laer, etc. Te practcal accessblt of dfferent estmatons of a termal montorng resoluton depends on te avalable pror nformaton. So, for eample, usage of te nformaton tat te defects are small local patces on gven dept (can be descrbed b δ - functons, allows to ncrease a resoluton of termal montorng b several fold. Te results of te computer eperments wt two small defects are sown n fgure. 3. Te pulse termal tomograp Te essence of te pulse termal tomograp s te armonc eat acton at te object surface and te measurement of te pase lag of ts temperature. Te results of te testng are te tckness of te plate or te dept of te delamnaton. Let's note tat te metod of pulse termal tomograp solves te problem of te delamnaton defectometr wen te szes of local separatons eceed ter dept, and te termal resstance of separaton eceeds termal resstances of above located laers. Te soluton of te eat transfer equaton for te unform plate b te eat transfer functon metod gves a constrant on te space tme spectrum of te temperatures T (, τ, T (, τ and eat flues q (, τ, q (, τ at te bot surfaces of te plate [ T ( w, w q ( w kλ, w T ( w, w, w cos( k sn( k = sn( cos(,, k k w w, ( q( w, w, w kλ C.6.

3 ttp://d.do.org/0.6/qrt ,, were T, T q q are obtaned wt Fourer transform; k = w a w w ; a = λ /( cρ. We can see tat te rato of te eat flu / and temperature spectrum at te dfferent depts s connected wt Eq. (3. q ( w w tg ξ = tg( ξ + k, were tg = ; =, kλt ( w, w ξ (3 For te one dmenson model and q 0 (non-eat-conductng delamnaton Eq. (3 transforms to te pase lag equaton T q = [ kλ tg( k δϕ δϕ, were k = w a. (4 Tat s just wat s used n te majort metodc of te pulse termal tomograp for te ( determnaton. We can allow for te eat ecange at te surfaces of te plate b te followng equaton: T kλ αtg( k δϕ = δϕ, (5 Q kλ( α + α ( k λ + αα tg( k were α and α are te effectve coeffcents of te eat transfer at te eated surface and te surface of te defect, respectvel. We can t use te one-dmenson appromaton n te general case. For eample, let us consder a tn delamnaton wt te varng tckness d (, δt were ( d ( T d = q( d( / λd ( = q ( R( δ s te temperature drop at te delamnaton; wt (, (6 R beng ts eat resstance. Te regon bend te delamnaton s assumed to be omogeneous and tck enoug. Ten te spectrum of te temperature as a functon of te eat flu spectrum at te boundar of ts regon s T δ q ( w, w / ( w, w Td ( w, w = ( kλ (7 d From Eqs. (, (6 and (7 we obtan δt ( F R( = δq ( + q ( [ ep( k F [ ep( k, (8 C.6.3

4 ttp://d.do.org/0.6/qrt were q ( = δt ( F [ kλ sn( δ s te eat dpole wc s equvalent to te delamnaton; δ s a comple k T( = T( Tnd ( valued temperature drop at te surface of te tested object, wc s caused b delamnaton, T nd ( s te temperature at te surface of te non-defect object. Te tasks of te ( δ and q ( T d δ determnaton are te nverse llposed problems. One of te regularzaton metods s used for tem. Te nformaton on te pase lag δϕ ( alone s not enoug n te two dmensonal problem. Tat s w te nonunformt of te surface s te man factor of te senstvt of te termal testng metod. In ts case te nverse task transforms to ( δt ( [ ε / ε ( T = F [ ep( k T ( 0 nd d, (9 or as n Eq. ( δt δε = G ( δtd ( T ( ε ( (0 For te statonar termal testng metod ( w = 0 we ave te famlar result G ( w, w = ep[ w + w. Te g frequences of te sgnal deca eponentall wt te dstance and te nformaton on te delamnaton sape s lost. 4. A pror nformaton as te part of te reconstructon metod Te new crteron s te mnmzaton of algortmc complet of descrpton of epermental data ncludng all avalable pror nformaton". Tus pror nformaton can be formalzed and represented b a set of codng algortms. Te are used as te components at buld-up of a procedure of te nverse problem soluton. At frst, let us consder te man optmzaton crteron for te ll-posed problem. Wt regard to q w, soluton of ( takes te form ( q ( w G ( w T( w ( w + α G ( w G( w =, δts ( w ( w = G( w q ( w α ( Now, we can see tat parameter α ( w s responsve to nfluence of noses, because n ter absence, t wll turn to zero. However, we cannot calculate ts parameter precsel unless we obtan te task soluton. In te evaluaton of te soluton accordng to te above formula, requred a pror nformaton s used. Te epresson [ + α ( w s known as te stablzng multpler. Its purpose s to neutralze an effect of soluton- nstablt, owng to growt of multpler at elevaton of frequenc. Tkonov as sown tat for convoluton- tpe equaton, a set of stablzng multplers of common form, suc as: mnmzaton of functonal equaton: p w = c w n α ( α leads to 0 n n= 0 C.6.4

5 ttp://d.do.org/0.6/qrt M α ( q, T = ( G q T d + α ( w q( w dw, ( were te former component s proportonal to R.M.S. devaton and te latter s to compensate for large values of ampltude and dervatves from te obtaned soluton. In ts manner, coce of algortm for regularzaton of non- correct task- soluton s based on a pror- nformaton on smootness and lmts of desrable soluton. Tus ts algortm of regularzaton can be represented as a partcular case of te new unversal crteron. We can take account not onl of te spectrum propertes of te sgnal and te nose but of an pror nformaton on tem also wt te new crteron. We calculate te mnmzed functonal equaton as M comp [ T = Comp[ q, δ q + Comp[ G q T, δq, δt ( q, T = Comp (3 Te functon of te complet calculaton Comp s vared wt te knd of te pror nformaton. Te accurac of q ( descrpton s te vared parameters of te functon Comp, but te accurac of T( descrpton s an nput parameter. Te requrement of uncancellablt of descrpton complet of nput data for a drect problem s avalable to a mnmum of nformaton crteron (n practce ts requrement s equvalent to lack of unaccounted regulartes n descrpton of real montored object propertes. Te procedure of te soluton of a termal tomograp problem b nformaton metod ncludes te followng ponts: - To formalze te soluton of a drect problem - matematcal model of non destructve testng process (te prevous part of ts report for eample - To develop te algortms of optmum algortmc descrpton ( Comp functons of all data necessar for te soluton of a drect problem (as requred parameters of object of montorng, and parameters of te defects, and nose; - To develop te mnmzaton algortm of complet of te termal montorng data; - To develop te algortm of te termal testng parameters optmzaton. 5. Results of te numercal smulaton Te capablt of te detecton and determnaton of delamnaton borders and wdt as functons of coordnates s llustrated n fgure (compared wt oter metods. References Storozenko V.A., Melnk S.I., Orel R.P. Te New Algortms of Termal Defectometr. // 0t Internatonal THERMO Conference, 8-0 June, 997, Budapest, Hungar.. Storozenko V.A., Melnk S.I. Development of te Termal Defectometr on te Base of te Transfer Functon Metod //Abstracts of te 8- t Internatonal Conference THERMO-93,Hungar, Budapest, Kolmogoroff A.N..Logcal bass for nformaton teor and probablt teor.-ieee Trans. Inform. Teor, 968, vol. IT-4, p C.6.5

6 ttp://d.do.org/0.6/qrt a b c Fg. Te resoluton of termal nondestructve testng a te raw regstered sgnal makes.7 of te dept of local defects b te result of te nverse problem soluton b Tonov regularzaton metod makes 0.6 of te dept of local defects c te soluton obtaned n vew of te nformaton on te sape of te defects makes 0. of te dept of local defects (a nose level s equal to 5 %. a b c d Fg. Te restoraton of te sape of te delamnatng Te wdt of te delamnatng s equal to ts dept. a Nose level s equal 0. Te results of te restoraton b Tonov regularzaton and Comp metods. b Nose level s equal 5%. Te result of te restoraton b R.M.S. mnmzaton metod. c Nose level s equal 5%. Te result of te restoraton b Tonov regularzaton metod. d Nose level s equal 5%. Te result of te restoraton b Comp metod. C.6.6