Australia Joural of Basic Applied Scieces, 5(): 097-05, 0 ISSN 99-878 Mote Carlo Optimizatio to Solve a Two-Dimesioal Iverse Heat Coductio Problem M Ebrahimi Departmet of Mathematics, Karaj Brach, Islamic Azad Uiversity, Karaj, Ira Abstract: This paper establishes a umerical-probabilistic algorithm based o a discretizatio scheme a simulatio method for solvig a two-dimesioal iverse heat coductio problem with two uow boudary coditios At the begiig of the algorithm, alterative directio implicit scheme is used as a umerical method to discretize the problem domai the a approach of Mote Carlo method is employed as a probabilistic simulatio techique to solve the obtaied large sparse systems of liear algebraic equatios The rom search algorithm i Mote Carlo method for global optimizatio is adopted to fid the solutio of our iterest iverse problem Numerical results show that a excellet estimatio o the uow boudary coditios ca be obtaied withi a couple of miutes CPU time at petium IV-4 GHz PC Key words: Iverse heat coductio problem, Mote Carlo Optimizatio, Fiite differeces method, System of liear algebraic equatios, Mote Carlo simulatio AMS Subject Classificatio: Primary 35R30, 65C05; Secodary 65M06, 78M50 INTRODUCTION Iverse heat coductio problems (IHCP's) have a variety of applicatios i may idustries such as spacecraft, uclear reactors, supercoductive geerators, etc There have bee umerous applicatios of IHCP i various braches of sciece egieerig, such as the predictio of the ier wall temperature of a reactor, the determiatio of the heat trasfer coefficiets the outer surface coditios of a space vehicle, the predictio of temperature or heat flux at the tool-worpiece iterface of machie cuttig The estimatio for the boudary coditios i IHCP's has received a great deal of attetio i recet years To date various methods have bee developed for the aalysis of the IHCP's ivolvig the estimatio of boudary coditio or diffusio coefficiet from measured temperature iside the material (Shidfar, A et al, 006; Ebrahimi, M, 0; Che, HT et al, 00; Lai, CH, 00) Shidfar et al, have applied a umerical algorithm based o fiite differeces method least-squares scheme for solvig a oliear diffusio problem Recetly Ebrahimi (Ebrahimi, M, 0) has bee ivestigated a stochastic algorithm based o Feyma-Kac formula for a iverse heat coductio problem with uow diffusio coefficiet However, most aalytical umerical methods were oly employed to deal with oe-dimesioal iverse problems Few wors were preseted for two-dimesioal parabolic iverse problems because the difficulty of these problem was more proouced The literature reviews showed that Che et al, (00) have applied a umerical method for solvig a two-dimesioal parabolic iverse problem CH Lai et al, (00) have studied a two dimesioal oliear, parabolic IHCP for weldig of metals alloys I the preset study, a two-dimesioal liear parabolic iverse heat coductio problem is solved usig a umerical-probabilistic algorithm ivolvig the combied use of the fiite differeces method Mote Carlo simulatio based o rom samplig The fuctioal forms of the boudary coditios are uow priori The uow boudary coditios are approximated by the polyomial forms Mote Carlo optimizatio method is used for estimatio uow coefficiets of the polyomials Numerical experimets cofirm the accuracy efficiecy of the preset umerical-probabilistic algorithm for a parabolic iverse problem i a fiite regio Accordig to latest iformatio from the research wors it is believed that the solutio of the preset two dimesio IHCP with two uow boudary coditio based o umerical-probabilistic algorithm icluded the Mote Carlo optimizatio has bee ivestigated for the first time i the preset study Statemet of the Problem: Cosider a ifiitely log bar with costat thermal properties with a square cross sectio of uit sidethe adiabatic coditios are applied at the side of 0 The coditio o the side of 0 is isothermal the temperature is oe uit It is iitially at a uiform temperature the suddely two temperature fuctios,, are applied to the sides 0, respectively The mathematical formulatio of the two dimesioal liear parabolic problem cocered to the above metioed physical model ca be give as: Correspodig Author: M Ebrahimi, Departmet of Mathematics, Karaj Brach, Islamic Azad Uiversity, Karaj, Ira E-mail: moebrahimi@iauacir; Tel: (098)735406; Fax: (+98)774030 097
Aust J Basic & Appl Sci, 5(): 097-05, 0,0,0,0, (),,,, 0, 0, () 0,, 0, 0, 0, (3), 0,,, 0, 0, (4),, 0, 0, 0, (5),,00,0,0 (6) The direct problem cosidered here is cocered with the determiatio of the medium temperatures whe the surface temperatures,,, the iitial boudary coditios o the boudaries of square plae,:0,0 are ow For the iverse problem, the surface temperatures,, are regarded as beig uow I additio, temperature measuremets tae at some grid locatios time o the square plae are also cosidered available I fact, to estimate the uow fuctios,, the additioal iformatio of discrete temperature measuremets is required Therefore, for the iverse problem of fidig,,, measuremets of,, are assumed to be available o a grid of size Let the temperature measuremets tae at these grid poits over the time period be deoted by:,,,,,,, (7) where is the fial time for temperature measuremets represets the umber of grid i directios The measured data, could be obtaied as,,,, where is average rom error that may be is cosidered withi -05 to 05, obtaied from the solutio of the direct problem ()-(6) We ote that the measured temperature, should cotais measuremets errors Therefore the iverse problem ca be stated as follows: by utilizig the above-metioed measured temperature data,, estimate the uow fuctios,, over the etire space time domai Overview of the Numerical-Probabilistic Algorithm: At first, we will cosider a umerical problem as a mappig :, where is some Baach space We will call is a solutio operator The elemets of are the data, for which the problem has to be solved for Υ, Υ is the exact solutio For a give Υ we wat to compute (or approximate) Υ Now, the applicatio of the preset umerical-probabilistic algorithm to fid the solutio of the iverse problem ()-(7) ca be divided ito the followig steps Step : Fiite Differeces Method: Let us cosider the direct problem ()-(6) The discrete problem ca be cosidered as a mappig of fuctio Υ, oto I fact, the purpose of this sectio is to describe the implemetatio of the solutio operator (or discrete operator) that maps,,, as a iput, oto a output,, For a give Υ, we wat to compute (or approximate), Sice the problem ()-(6) is a well-posed problem, the liear operator, mappig the data,,, 0, oto,,, is well defied Now, solvig the iverse problem ()-(7), is thus equivalet to solvig the operator equatio, Υ By discretizig the direct problem ()-(6) we ca compute a approximatio of the operator I the preset wor we use fiite differeces method a equidistat grid o Therefore, defie the co-ordiates,, of the mesh poits of the solutio domai by,, where, are positive itegers Also, 098
Aust J Basic & Appl Sci, 5(): 097-05, 0 deote the values of,, at these mesh poits by,,,, Now, We use Alterate Directio Implicit (ADI) method for discretize equatio () cetral-differeces approximatio for equatios (3) (5) We assume the solutio is ow for time the replacig oly oe of the secod-order derivatives, say, by a implicit fiite differeces approximatio i terms of uow pivotal values of from the time level The other secod-order derivative,, beig replaced by a explicit fiite differeces approximatio Therefore the equatios,,,,,,,,,, (8),,,,,,,,,,,, (9),,,,, 0, (0),,,, 0, (),,,,,, () U p,q,0 = 0, r=0, (3) where 0, are used to advace the solutio from the to the time step Now, we cosider the equatios (8)-(3) may be writte i the followig matrix form:,, (4) where B ( ) ( ) ( ), ( ),,,,,,,,,,,,,,,,,,,, 0 0,,,, 099
Aust J Basic & Appl Sci, 5(): 097-05, 0 We ote that i equatio (4), U 0 U U By the Gerschgoris circle theorem, matrix method [5], we cocluded that the fiite differeces scheme (8) is ucoditioally stable Now, we assume the solutio is ow for time the advacemet of the solutio to the time level is obtaied by replacig by a implicit fiite differeces approximatio, by a explicit oe Hece the equatio,,,,,,,,,, (5),,,,,,, I cojuctio with the equatios (9)-(3), while 0, are used for advacemet the solutio from the to the time step I this case, if we cosider the may by we use the followig matrix form:, 0, (6) where B ( ) ( ) ( ), ( ),,,,,,,,,,,,, Ad, 0 0 We ote that i equatio (6), U,,,,, U U Note that based o Gerschgoris circle theorem, ucoditioally stability of the fiite differeces scheme (5) will be obtaied, easily Step Mote Carlo Simulatio: we assume, 00
Aust J Basic & Appl Sci, 5(): 097-05, 0, Therefore from equatio (4) we obtai, (7) where B B To solve the liear system (7), we cosider the followig iterative method, (8) where i,, N (0,] This is called the Jacobi overrelaxatio iterative method with relaxatio parameter (0,] Equatio (8) may be writte i the followig matrix form U ( ) LU ( ) f,,, (9) where D diag( B ( ) ( ) ( ) t ( u,, un is the -th iterative solutio of (8), L I _ DA, f DY U ),, B N, N ),, is a diagoal matrix I fact, we covert the system (7) ito a equivalet system of the followig form U LU f (0) Now, we cosider the Marov chai X x x x x () 0 trasitio matrix P pi, j, i, j,, N Let P( x0 i) pi, P( x N j x N i) pij, where p i p ij are the iitial distributio the trasitio probabilities of the Marov chai, respectively The weight fuctio W, for Marov chai () with m N states, is defied by usig the recursio formula with state space,,, N l xm, m W0, Wm Wm, m,, p xm, m Now the followig rom variable is defied H [ H ] p x0 x0 m0 W m c x m, x 0 which is associated with the sample path x x x t where is a give iteger umber H h,, h ) is a give vector We also cosider the problem of fidig the ier product ( N 0
Aust J Basic & Appl Sci, 5(): 097-05, 0 H, U hu h N u N, where U is the solutio of problem ()-(6) The solutio operator for the above formulated problem ca be writte i the followig form: K ( F, G) H, U * * Now, we defie a adjoit operator K try to fid the umerical solutio of K ( F, G) U Theorem : The mathematical expectatio value of the rom variable [H ] is equal to the ier product E H ( ), U hu h u ( ) [ H ] H, U H, U N N,, ie, Proof see referece (Rubistei, RY, 98) To estimate h u h u ( ) ( ) ( ) N N, We simulate N rom paths ( s) ( s) ( s) ( s) x0 x x x, s () M, Each with the legth of, evaluate the sample mea [ H ] N [ H ] E ( ) [ H ] H, U ( ) ( ) I fact, from Theorem we coclude tha [H ] is a ubiased estimator of the ier product H, U t ( ) ( ) It is readily see that by settig H (0,,0,,0,,0 ) we obtai H, U u j, j,, N Hece j [H ] is a ubiased estimator of the ( ) u j Step 3 Mote Carlo Optimizatio Techique: I this wor the polyomial form proposed for the uow fuctios F ( y, G ( x, before performig the iverse calculatios Therefore F ( y, G ( x, are approximated as F app G app ( y, ( a ( x, ( b a ( a 3 a4 y ( a 5 a6 y ( a a b ( b3 b4 x ( b5 b6 x ( b b where ( a,, a ) ( b,, b ) are costats which remai to be determied simultaeously These uow coefficiets ca be determied i such a way that the followig fuctioal is miimized: x y 0
Aust J Basic & Appl Sci, 5(): 097-05, 0 tf N N cal mea (,,,,, ) [ pq, ( p, q, ) pq, ( p, q, )], t 0 p q cal Here, p q ( x p, yq J a a b b U x y t U x y t dt U, are the calculated temperatures o the pla at the grid locatios ) These quatities are determied from the solutio of the direct problem give previously by usig a approximated F app ( y, G app ( x, for the exact F ( y, G ( x,, respectively The estimated values of a j, j,, b i, i,, are determied util the value of J ( a,, a, b,, b ) is miimum The computatioal procedure for estimatig uow coefficiets a j b i are described as follows: Cosider the followig determiistic optimizatio problem * * mi J( A) J( A ) J, ( ) AR () where J (A) is real-valued bouded fuctio defied o A,,,,, It is assumed that achieves its maximum value at a uique poit A The fuctio J (A) may have may local maximum i but oly oe global maximum Rom Search Algorithm: For solvig problem () we cosider the followig rom search algorithm: Geerate dimesioal rom variables Z, from a -dimesioal ormal, Z distributio with zero mea covariace matrix C that is Z ~ N (0, C) Z Z, Select a iitial poit J( A ) 3 Compute 4 Set 5 If, go to step 8 6 Set A A 7 Go to step 0 8 J ( A ) Compute i Z i 9 If J ( Ai Z i ) J ( Ai ) the (where 0) set A A Z else set A A 0 If the stoppig criterio is met, stop; otherwise, set Go to step 5, Z Numerical Experimet: I this sectio, we are goig to demostrate some umerical results for determie ( U ( x, y,, F(, y,, G( x,0, ) i the iverse problem ()-(7) All the computatios are performed o the PC However, to further demostratig the accuracy efficiecy of this method, the preset problem is ivestigated the followig example is cosidered Example: Cosider ()-(7) with F app ( y, 0 04t 03y 07 yt 0y 0 0y t G app ( x, 05 03t 0x 04xt 05x 0 x t Table shows the values of U i x px, x, t rt, whe x y 00 I Table, f 03
Aust J Basic & Appl Sci, 5(): 097-05, 0 U (0,08, t ), U (0, 0, t ), U ( 0 5,08, t ), U ( 09,0, U (09,0 8, betwee the exact results the preset umerical results Tables 3 shows the exact umerical values of F (, y, G( x,0, respectively Figures shows the graphical results of Tables 3, respectively Table : Results for U with t f 04 t 005 0004 000 0000 05 00070 000 00034 05 000504 000085 00006 0003 00070 00060 00003 000004 0000444 000006 00006 000038 00003 00006 00005 000090 00008 00007 00060 00030 00030 Table : Results for F (, y, with x y 00 t=005 y Numerical Exact 0 0543 0545 0 097 090 03 090 095 04 0704 0700 05 038 035 06 03574 03570 07 04033 04035 08 0458 0450 09 0503 0505 0 05557 05550 Table 3: Results for G(x,0, with x y 00 t=005 y Numerical Exact 0 0536 053 0 05590 05594 03 05963 05969 04 0644 06446 05 0707 0705 06 0770 07706 07 0848 08484 08 0937 09374 09 0365 036 0 453 450 t=05 Numerical Exact 008 005 0453 0450 0904 0905 0338 03380 0387 03875 0439 04390 049 0495 05483 05480 06053 06055 06650 06650 t=05 Numerical Exat 05664 05663 05978 0598 06503 06407 06933 06938 0757 07575 0834 0838 0960 0967 04 0 86 83 358 350 t=05 Numerical Exat 0473 0485 0994 0990 0358 0355 04064 04060 046 0465 059 050 0588 0585 06446 06440 07083 07085 0775 07750 t= 05 Numerical Exat 06003 06005 06376 06370 0684 06845 0743 07430 089 085 08937 08930 09486 08485 0874 0870 008 005 357 350 Fig : Results for F (, y, with x y 00
Aust J Basic & Appl Sci, 5(): 097-05, 0 Fig : Results for G ( x,0, with x y 00 Coclusio: I this paper a umerical-probabilistic algorithm ivolvig the fiite differeces method i cojuctio with the Mote Carlo optimizatio is employed to solve a two-dimesioal parabolic iverse problem, successfully the followig results are obtaied: Mote Carlo methods are preferable for solvig large sparse systems, such as those arisig from approximatios of partial differetial equatios Oe of the most importat advatages of Mote carlo methods is that they ca be used to evaluate oly oe compoet of the solutio or some liear form of the solutio This advatage is of great practical iterest 3 From the illustrated example it ca be see that the proposed method is efficiet accurate to estimate the uow boudary coditios i a two-dimesioal IHCP REFERENCES Che, HT, SY Li LC Fag, 00 Estimatio of surface temperature i two-dimesioal iverse heat coductio problems Iteratioal Joural of Heat Mass Trasfer, 44: 455-463 Ebrahimi, M, 0 Estimatio of diffusio coefficiet i gas exchage process withi huma respiratio via a iverse problem Australia Joural of Basic Applied Scieces, (to appear) Faroosh, R M Ebrahimi, 007 Mote Carlo method via a umerical algorithm to solve a parabolic problem Applied Mathematics Computatio, 90(): 593-60 Lai, CH, CS Ierotheou, CJ Palasuriya KA Pericleous, 00, A domai decompositio algorithm for iverse weldig problems Computig Visualizatio i Sciece, 4: 05-09 Rubistei, RY, 98 Simulatio the Mote Carlo method, Wiley, New Yor Shidfar, A, R Pourgholi M Ebrahimi, 006 A umerical method for solvig of a oliear iverse diffusio problem Computers Mathematics with Applicatios, 5: 0-030 05