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IVERSE HEA CODCIO I AISOROPIC AD FCIOALLY GRADED MEDIA BY FIIE IEGRAIO MEHOD J. JI, J.L. ZHEG,. HAG, J.J. YAG, H.S. WAG, P.H.WE, J.M. LI 3 School of Communcaton and ransportaton Engneerng, Changsha nverst of Scence and echnolog, Chna School of Engneerng and Materals Scence, Queen Mar nverst of London, K 3 Dept of thermal Engneerng, snghua nverst, Bejng, Chna ABSRAC Based on the recentl developed Fnte Integraton Method (FIM) for solvng one and two dmensonal ordnar and partal dfferental equatons, ths paper etends FIM to both statonar and transent heat conducton nverse problems for ansotropc and functonall graded materals wth hgh degree of accurac. Lagrange seres appromaton s appled to determne the frst order of ntegral and dfferental matrces whch are used to form the sstem equaton matr for two and three dmensonal problems. Sngular Value Decomposton (SVD) s appled to solve the ll-condtoned sstem of algebrac equatons obtaned from the ntegral equaton, boundar condtons and scattered temperature measurements. he convergence and accurac of ths method are nvestgated wth two eamples for ansotropc meda and functonall graded materals. Ke words: fnte ntegral method, Lagrange seres, ntegraton matr, nverse heat conducton.. IRODCIO Inverse Heat Conducton Problems (IHCP) are used to estmate unnown quanttes on the unreachable boundares based on temperature and heat flu measurements n the analss of phscal problems n thermal engneerng. As an eample, the nverse heat conducton problem s used to compute the unnown temperature and heat flu at an unreachable boundar from scattered temperature measurements ether at reachable nteror ponts or boundar of the doman. In order to obtan the mamum nformaton regardng the phscal problem under stud, IHCP enables a much closer collaboraton between epermental and theoretcal researchers. Snce the solutons ma become unstable as a result of errors nherent the measurements the dfcultes encountered n the soluton of IHP should be recognzed. IHCP s mathematcall classfed as ll-posed n a general sense and s man reason that nverse problems were ntall taen as not of phscal nterest. radtonal computatonal methods ncludng the fnte element method, boundar element method and meshless method for well-posed drect problems fal to produce acceptable solutons to these nds of nverse problems. Several technques have been proposed for solvng a one-dmensonal IHCP [-5]. Applcatons wth dfference technques can be found b the fnte dfference [8,9], boundar element [6,7] and fnte elements [,] for two-dmensonal IHCP. More recentl, the method of fundamental soluton [,3] and meshless methods have been developed to IHCP n two and three dmensons such as meshless local Petrov-Galern method [4,5] whch have recentl attracted great attenton n scence and engneerng communtes for numercall solvng tme-dependent problems. - -
Despte the great success of the fnte element method, boundar element method and meshless methods, there s stll a need for developng numercal schemes for mult-dmensonal IHCP. Recentl, Wen and hs colleagues [6,7,8] developed the Fnte Integraton Method (FIM) for solvng one and two dmensonal, dfferental equaton problems and demonstrated ts applcatons to non-local elastct problems. It has been shown that the FIM gves much hgher degree of accurac than the Fnte Dfference Method (FDM) and the Pont Collocaton Method (PCM). In ths paper, the FIM s further etended to solve IHCP for ansotropc meda and functonall graded materals.. FORMLAIO OF IVERSE HEA CODCIO PROBLEMS We assume that n engneerng, the thermal conductvtes λ () are drectonall dependent propertes of an ansotropc materal. All the propertes of the materal could be dependent on the spatal coordnates n a non-homogeneous materal. he governng equaton for stead heat conducton n ansotropc and contnuousl non-homogeneous meda s gven b the followng partal dfferental equatons wth spatall varable propertes: q, ( ) = w( ), (, ) Ω q ( ) = λ ( ) ( ) (), α ( ) + βq ( ) n = g( ), (, ) Γ where denotes the temperature dstrbuton and w () represents the heat source term, the frst term s the rate of nternal energ wth tme, the second term q () s the heat flu vector, λ () s the thermal conductvt tensor, coeffcents α and β are constants, n s the unt outward normal to the boundar. In addton, ( ) s a gven ntal temperature n the doman and g(, t) gves the boundar value. For nverse heat conducton problem, t s convenent to assume that the boundar Γ conssts of three parts Γ = ( Γu Γq Γ σ ) and the boundar condtons are gven as ( ) = ( ) Γ u, P (Drchlet b.c. and Pnt temperature) (a) q( ) = q ( ) n ( ) = q( ) Γ (eumann b.c.) (b) q where () and q() are temperature and heat flu specfed on the boundar. he IHCP to be nvestgated n ths paper s to determne the temperature on the unreachable boundar Γ from ω gven Drchlet data on Γ u, eumann data on Γ q and scattered temperature measurements P() at some ponts P ether n the doman Ω or on the eumann data boundar Γ. q 3. FIM SIG LAGRAGE SERIES he smplest computaton scheme for ntegraton was ntroduced n [6,7] and named as the Ordnar Lnear Approach (OLA). For convenence of analss n the followng sectons, coordnates and are used to replace and, respectvel, and subscrpts or presents the nodal number. Consder an ntegral and appl trapezodal ntegral rule - -
u( ) d = = ( ) = ξ ξ a u( ) (3) For trapezodal rule, the coeffcents are a =, h, =, (4).5h, =,3,...,, a =.5h, =,, >, where = ( ) h, h = a /( ), =,,...,, are nodal ponts n [,a], and =, = a. ote that (3) can be wrtten n a matr form as = Au (5) where =,,...,, u = [ u, u,..., u ], and the frst order of ntegraton matr / / / /, (6) A = ( a ) = / /.................. / / and = ( ), u = u( ) are the values of ntegraton and the ntegral functon respectvel at each nodes. he frst order of ntegral matr A n Eq. (6) can be evaluated usng the Lagrange nterpolaton seres. he ntegral functon s appromated as [7] u ( ) (7) = = c where the coeffcents {..................... c } = are defned b c u c u =......... c u or the ntegral functon can be wrtten as ( ) u( ) = u (9) j= = ( j ) j j (8) - 3 -
herefore, we have ( ) h c u( ) dξ = = = = c ( ) = ξ ( ) h =,,..., () For dfferent boundar condtons, the frst and second order of dervatves should be consdered. hen for an ponts ncludng two ends, from Eq. (9), we have du = ( ) c and d u 3 = d ( )( ) c. () = d = 3 From Eq. (9), the frst order dervatve can also be wrtten as du = u' ( ) = u j ( j ) ( ) = d j ( ) u () j d j= =, j = =, j, j= Smlar to the ntegraton matr, the frst order dervatve can be wrtten, n matr form, as D = Bu (3) where D = [ u ', u',... u' ], u ' = u'( ) and matr B s determned from Eq. (). hereafter, consder a mult-ntegral for one-dmensonal problem () ( ) ξ = u( ξ ) dξ dξ, [, a] (4) () Applng the above OLA technque agan for ntegral functon ( ), we have () ( ) = ξ j = j = = () u( ξ ) dξ dξ = a a u( ) = a u( ). (5) In the same wa for the frst order ntegraton Eq.(5), the mult-ntegral above can also be wrtten n a matr form as () () = A u = A u (6) herefore, we can etend ths concluson to an mult-laer ntegrals as ( m) ( m) m = A u = A u (7) In the same wa, for hgher order dervatve to the functon u, we have ( m) ( m) m D = B u = B u (8) 4. FIM FOR WO DIMESIOAL PROBLEMS For two-dmensonal problems, consder a unform dstrbuton of collocaton ponts as shown n Fgure, the ntegraton matr s defned as (, ) = u( ξ, ) dξ, (, ) = u( ξ, ) dξ (9) and the total number of ponts s = ( j ) +, where and j denote the number of column and the number of row, respectvel. hs numberng sstem s called the global number sstem. We can also epress each nodal value of ntegraton Eq. (9) n a matr form as = A u () - 4 -
where ntegral value = [,,... ], nodal value functon u = [ u, u,..., u M ] and M s the total number of collocaton ponts ( M = ) for grd shown n Fgure ). For a rectangular doman, the frst order ntegraton matr s A... A () A =............ A n whch, A s the ntegraton matr for one-dmensonal problem gven n Eq. (5) wth dmenson. Followng the same procedure, the ntegraton along as (, ) = u(, η ) dη, (, ) = u(, η ) dη () can be wrtten n the matr form as = A' u (3) n the local sstem for the collocaton ponts, where = ( ) + j. he frst order ntegraton matr n the local sstem s A... A (4) A' =............ A n whch A s the ntegraton matr for one-dmensonal problem gven n Eq. (5) wth dmenson. B a smple re-arrangement of the number of the nodes from the local number sstem to global number sstem, Eq. (3) can be rewrtten, n the global sstem, as = Au. (5) For mult-ntegraton n the two-dmensonal problem n a rectangular doman, we consder the followng ntegral wth respect to coordnate () (, ) ξ = u( ξ, ) dξ dξ, [, a], [, b] (6) where a b s ntegral doman. sng the same procedure for one-dmenson, we have () (, ) = ξ u(, ) dξdξ = = j = ξ ( a ) ( a ) u (7) or n matr form () = A u (8) where j j - 5 -
A = A A A... A =............ A (9) Fgure. nform dstrbuton of collocaton ponts and the number of nodes. () Smlarl, we have mult-ntegraton (, ) wth respect to coordnate () (, ) = η u(, ) dηdη = = j = η ( a ) ( a ) u (3) and () = A u. (3) In addton, ths method can be etended to mult-laers ntegraton wth two coordnates and as follow: ( mn) (, ξ η ) =...... u( ξ, η) dξ... dξ m dη... dη n m laer n-laer j j [, a], [, b] (3) and the nodal values of the above ntegraton are obtaned n the matr form as ( ) mn m n = A A u. (33) hs method can be easl etended to mult-laer dfferentals wth two coordnates and as ( mn) m n D = B B u. (34) - 6 -
5. APPLICAIO OF FIM O WO DIMESIO Eample. Stead state heat conducton n an ansotropc square plate Consder a square plate of wdth a = b =, as shown n Fgure (a), wth thermal conductvt tensor: λ = λ = λ and λ = λ =. λ. o heat source s consdered,.e. w =. he analtcal soluton [5] s gven as * (, ) = + 5 (35) whch satsfes the statonar counterpart of governng equaton () for the consdered thermal conductvt tensor. hen the boundar condtons are gven as (,) = +, (,) = + 5, q(, ) = 4 5 (36) and the scattered temperature measurement ponts are collected ether on boundares or n the doman as shown n Fgure (b). he unreachable boundar s = ( Γω ). sng ntegraton matr ntroduced n sectons 3 or 4, we have followng governng equaton n matr form n terms of nodal values [ A + λa A + λa ] = XΨ f + Ψ f + YΨ g + Ψ g λ (37),,..., M X = {,,..., M }, = {,,..., M }, where = { }, g = { g, g g },..., p Y f = { f, f,..., f },, (=,), q and p are numbers of ponts to be used for nterpolaton of functons f ( ) and g( ) collocated on and aes, respectvel, Ψ and Ψ are shape functon matrces of one-dmensonal shape functons wth respect to coordnates and, respectvel. Integral functons f ( ), f( ), g ( ) and g ( ) can be nterpolated n terms of the nodal values descrbed n [7] wth followng two steps: () Determne the regons of functons f ( ) and g( ),.e., ],, ], and unforml [ q dstrbuted ponts n these regons; () Determne one-dmensonal shape functon matrces Ψ and Ψ f ( ) = f < < or < < g( ) = f < < n m or n m q < < p [ p otherwse n n f ( ) = fn + fn f n < < n n n n n (39) m m g( ) = gm + gm f m < < m m m m m For a regular dstrbuton of nodes, we select = =, M =, q = p =. hen the shape functon matrces Ψ = Ψ = I. It s obvous that the number of equatons from boundar condtons s L = 4 4. hus the numbers of unnown nodal values s M and the numbers of nodal values of ntegral functons f () and g () ( p + q) = 4. SVD scheme s used to solve a set of lnear algebrac equatons [7]. he average error s defned as q (38) - 7 -
ε = M ω Mω = * where M s the number of nodes on the unreachable boundar as shown n Fgure (b). he ω nteror and boundar ponts are selected as shown n Fgure (a) wth = = M ω = and number of pont measurements M P =. he average errors wth dfferent gaps between the unreachable boundar and measurement ponts for dfferent nose levels are presented n Fgure 3. Apparentl n the cases when σ =, where σ ndcates the magntude of nose, the numercal solutons are all close to the analtcal solutons wth mamum error of 3 (%). For the cases of wthout nose, we tae the measurements of temperature b (, * P P ) = (, ) sn( ) P P + σ π. It s seen that the solutons are stable n P the regon of D / a. 4. When D / a >. 5, as epected, the error ncreases sgnfcantl snce the measurement ponts are far awa from the unreachable boundar. From Fgure 3, t can be observed that more accurate solutons could be obtaned wth smaller gaps. In addton, the convergence s observed wth respect to the dstrbuton denst of nodes n the doman and on the boundar. When the nteror and boundar ponts are selected as = 7, the dfferences between the numercal results for the two node dstrbutons are seen to be ver small. (4) Fgure. Dstrbuton of collocaton ponts: (a) nternal and boundar ponts; (b) boundar condtons for nverse heat conducton problems, unreachable boundar; eumann and Drchlet boundar; measurement ponts P n doman, where D/a ndcates the gap. - 8 -
Fgure 3. Average error versus the gap between unreachable boundar and measurement ponts wth dfferent levels of nose at measurement ponts. Eample. wo dmensonal stead state heat conducton problem Consder stead state heat conducton n a quarter of ds (D) wth sotropc meda, as shown n Fgure 4(a). he governng equaton () n polar coordnate sstem s + + = r r, θ π/ (4) r r r r θ wth the boundar condtons ( r,) = ( r, π / ) = ; (, θ ) = / 4sn θ ; (, θ ) = 4sn θ. * 4 he eact soluton s obtaned as ( r, θ ) = r / 4sn θ. B coordnate transform, Eq. (4) can be normalsed as 4 + ( ) + + = + (4) + ( + ) ( + ) π where transformed coordnates = r and = πθ / and the doman s transformed nto a square as shown n Fgure 4(b). sng the ntegraton matr n secton 4, we have followng governng equaton, n matr form, n terms of nodal values 4 A + A A + A A R + A R = A A w + XΨ f + Ψ f + YΨ g + Ψ g (43) π where R = dag{/( + }, R = R, w = dag{( + ) } (44) n whch the ntegral functons on the rght hand sde are defned n Eq. (37). Same nodes dstrbuton n Eample are used. he average relatve error s defned as M * * * ε /, * re = ma ma = (, π / 4) = 4 (45) M = - 9 -
Varaton of the logarthms of the average relatve error ε re wth dfferent numbers of grd s shown n Fgure 5. For comparson the numercal solutons obtaned b the fnte dfference method (FDM) s also gven n Fgure 5. It s seen that the accurac of FDM solutons ncreases lnearl wth ncreasng the number of nodes. As epected, the accurac of the numercal solutons obtaned b b FIM s much hgher than that b FDM even for a small number of grd. Secondl, we consder that the unreachable boundar Γ ω s collocated at nner surface ( = ). B changng M to M n Eq. (45) and selectng the number of grd ω =, Fgure 6 shows the average errors over the unreachable boundar for dfferent gaps D and dfferent levels of nose at the ponts of measurement. o observe the effect of * measurement nose, the measurements of temperature are taen b ( P, P ) = ( P, P ) + σ sn( π ). Apparentl n the regon of P D / a. 6, t s seen that the temperature obtaned on unreachable boundar s stable. However, when D / a >. 7, the error ncreases sgnfcantl wth nose. Fgure 4. (a) Quarter of rng; (b) transformaton of coordnates. 6. COCLSIOS he fnte ntegraton method wth Lagrange seres epanson has been demonstrated for two and three dmensonal nverse heat conducton problems for the frst tme. Ansotropc meda and functonall graded materals are consdered. Appromaton b Lagrange seres s used to determne the frst order ntegraton and dervatve matrces. he Sngular Value Decomposton (SVD) s appled to solve the ll-posed lnear sstem of algebrac equatons, derved from the governng equaton, boundar condtons and scattered temperature measurements n the present wor. he stablt and accurac were observed for nverse heat conducton problems n three eamples. It has been shown that the fnte ntegraton method provdes an effcent algorthm for solvng nverse heat conducton problems. However, f the Drchlet condton (measurements) and eumann condton are specfed at the same ponts of - -
the boundar, the predcton of the temperature on the unreachable boundar becomes unstable wth certan noses from measurement. Fgure 5. he logarthms of the average relatve error versus the segment number. Fgure 6. he average errors n the doman versus the gap between unreachable boundar and measurement ponts for dfferent levels of nose. - -
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