Determining of temperature field in a L-shaped domain

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1 Available online at Advances in Applied Science Research, 0, (:-8 Determining of temperatre field in a L-shaped domain Oigo M. Zongo, Sié Kam, Kalifa Palm, and Alione Oedraogo ISSN: CODEN (USA: AASRFC Département de Physiqe, UFR-SEA, Université de Oagadogo, Oagadogo, Brkina Faso Institt de Recherches en Sciences Appliqées et Technologies (IRSAT, Oagadogo, Brkina Faso _ ABSTRACT This paper deals with solving Dirichlet s problem with nonhomogenos bondary conditions, sing the method of large singlar finite elements for Laplace s eqation in a L-shaped domain. This method is particlarly sitable for solving singlar problems since the analytical form of singlarities has been integrated in the approximate soltion. It is niqe in that the eqations are exactly verified except on those areas where internal segments are approached with great precision. Reslts are compared with those obtained throgh finite elements method by sing the COMSOL software. They dealt with soltion vales and those of its first derivatives. Both methods give reslts that align qite well everywhere, except near singlarities where significant differences exist. Keywords: Singlar points, least sqares, large elements, finite elements. _ INTRODUCTION The general theory of soltions to Laplace's eqation is known as potential theory. The soltions of Laplace's eqation are the harmonic fnctions, which are important in many fields of science, notably the fields of electromagnetism [] and flid dynamics [], becase they can be sed to accrately describe the behavior of electric, gravitational, and flid potentials. In the stdy of heat condction [, ], the Laplace eqation is the steadystate heat eqation. Nmerical solving of Laplace s eqation still remains very difficlt and sal methods of finite elements and finite differences give nsatisfactory reslts when they are sed in their standard form. These methods as evidenced by varios athors [-] can be significantly improved if they take into consideration the analytical form of the soltion near of the singlarities. We therefore apply the method of the large singlar finite elements [] to calclate the temperatre field in polygon L-shaped with Dirichlet non-homogenos bondary conditions. This gives good reslts all over the stdy domain while the finite element method gives good reslts only on areas located far from singlarities. This shows the power, efficiency and accracy of the method of large singlar finite elements for a limited nmber of conserved coefficients than the finite elements method. MATERIALS AND METHODS Either to determine the stationary temperatre field in a L-shaped polygon comprising eight segments of nit length kept at varios constant temperatres. In redced variables, the problem consists in solving Laplace s eqation with Dirichlet bondary conditions with mps. The L-shaped domain [] of side and bondary conditions are given in figre. The problem as posed is singlar de to discontinities in limit conditions on borders and the six smmits of the domain where there is discontinity in the external normal in the border of the domain. This is solved sing method of large singlar finite elements.

2 Oigo M. Zongo et al Adv. Appl. Sci. Res., 0, (:-8 Figre -L-shaped domain. Dirichlet bondary conditions. Figre - L-shaped domain. Smmary of local coordinates. We solve the Laplace s eqation on a polygonal domain sing MATLAB software. Bondaries Data can combine conditions of Dirichlet or Nemann. The method of large singlar finite elements comprises three steps []:

3 Oigo M. Zongo et al Adv. Appl. Sci. Res., 0, (:-8 Step : Decomposition of the domain. Althogh the domain L-shaped is geometrically symmetrical, it is not possible to replace the problem by an eqivalent problem in a redced domain becase conditions on limits are not symmetrical. The polygon mst therefore be taken as a whole. The first step of the method consists in dividing the domain into eight sb-domains identified in Figre, separated by ten sb-borders. Step : Solving axiliary problems The nmber of problems mst be eqal to the nmber of sb-domainsω. Therefore, to each sb-domain Ω is associated with an originσ, which is a singlarity, α the opening angle and a local system with polar coordinates, ( i i r (figre. i i i i First Axiliary Problem r = 0 ( r ( r,0 = 0 ( Ω π ( r, = 0. Second Axiliary Problem ( r = 0 ( r Ω ( r,0 = 0. ( r, π = 0 Third Axiliary Problem ( r, = 0 ( r, Ω ( r,0 = 0. π ( r, = 0. Forth Axiliary Problem ( r = 0 ( r Ω ( r,0 = 0. π ( r, = 0. Fifth Axiliary Problem ( Ω r, = 0 ( r, ( r,0 = 0. π ( r, = 0. Sixth Axiliary Problem ( r = 0 ( r Ω ( r,0 = 0. π ( r, = 0. Seventh Axiliary Problem ( r, = 0 ( r, Ω (-a (-b (-c (-a (-b (-c (-a (-b (-c (-a (-b (-c (-a (-b (-c (-a (-b (-c (-a

4 Oigo M. Zongo et al Adv. Appl. Sci. Res., 0, (:-8 ( r,0 = 0 (-b π ( r, = 0. (-c Eighth Axiliary Problem 8 ( r8 8 = 0 ( r8 8 Ω8 (8-a ( r 8,0 = 0. (8-b 8 8 ( r 8, π = 0 SOLUTIONS AND DISCUSSION General soltions of the eight axiliary problems are written taking into accont varios bondary conditions specified in figre []: i ( r = a ir sin i (9-a π i= π = ( r = ( a r sin k ( r = ( ak r sin π k = k m ( r = ( amr sin π m= ( r = ( π n= a n m r m m sin p ( r = + a pr sin π p= q ( r = aqr sin π q= = q p 8 8 ( r8 8 = ( a8 r8 sin 8 π (8-c (0-a (-a (-a (-a (-a (-a (-a Step: Connecting axiliary soltions The third step of the method consists in connecting these axiliary soltions by imposing the continity of the fnction and its normal derivatives all along the varios sb-borders Γ separating two adacent sb-domains Ωk and Ω. If conditions on interfaces Γ are met, therefore the exact soltion to the initial problem will be fond. l Actally, since we cannot solve an infinite system, we mst generally keep to approximate soltions. The approximation reslt on the one hand, the fact that we mst limit the development (9-a to (-a to a finite nmber of terms, and secondly, that 'we mst content orselves, with rare exceptions, a connection imperfect. Approximate soltions have been obtained by limiting the series sed in general soltions. The nmber of terms kept in each of the sms is chosen according to the principle proposed by Desclox and Tolley [] aiming at representing the approximate soltions sing fnctions of degrees as niform as possible. This is achieved by keeping more coefficients for sb-domains with larger openings. Approximate axiliary soltions are all of N degrees (N being the nmber of coefficients retained for the sqare sb-domains which opening angle is π /. Finally, the total nmber of nknown parameters a, which vale can be freely chosen will be: ( + + N = N

5 Oigo M. Zongo et al Adv. Appl. Sci. Res., 0, (:-8 This allows getting the following approximate soltions: N i ( r = + a ir sin i (9-b π i= N ( r = ( + a r sin π = ( k N k r = ( + ak r sin π k = ( m N m r = ( + amr sin π m= n= m N m ( r = ( + anr sin π ( p N p r = + + a pr sin π p= ( q N q r = + aqr sin π q= N 8 8 ( r8 8 = ( + a8 r8 sin 8 π = (0-b (-b (-b (-b (-b (-b (-b We are therefore connecting the soltions of axiliary problems according to least sqares method, i.e. we calclate coefficients I ( a mn = a that allow minimizing the fnction: ( ( i a a l ik ( i ( aik ( a l ds ni n < + + Γ i i These coefficients are soltion of the linear algebraic system positive definite sqare matrix of order N with N nknowns classically called normal eqations of Gass: I ( amn = O a m, k =,,..., 8; n, l =,.... N (8 The accracy of approximate soltions is directly linked to the qality of the connection of soltions to axiliary problems. It is therefore natral to characterize its precision by measring the imperfections of continity conditions. We will se the overall connecting error definite by (9: = ( + + ds η (9 Where k l k l k l S < Γ k l υ υ ds is the element which arch length is Γ, S its length and k i υ and ( υ l the normals to the sb-border separating both adacent sb-areas. If the overall error is nll, the approximate soltion got aligns with the exact soltion. Mode of convergence of the method of large singlar finite elements is exponential. Indeed, the base 0 logarithm of the overall error decreases linearly with N as shown in Figre. Moreover, this crve of the overall error allows s to say, for example, that by keeping 88 coefficients a the nmerical reslts are obtained, with an absolte accracy of less than 0.

6 Oigo M. Zongo et al Adv. Appl. Sci. Res., 0, (:-8 Figre - Evoltion of the overall error according to the nmber of coefficients a retained. In order to compare reslts obtained sing the method of large singlar finite elements with those provided by the method of conventional finite elements in its standard form, the problem was solved sing COMSOL software. The comparison of both methods is smmarized in few graphics shown in figres, and. In figre, approximations of the fnction and its derivatives were presented all along circles centered on singlarity (figre. The fll lines have been obtained sing the method of large singlar finite elements by keeping 88 coefficients a and the small circles are the reslts from the finite elements method for a range of, degrees of freedom. Figre is similar to the previos one, bt only data on fnction for the for circles are there presented. Finally, the bi-logarithmic chart in figre gives the maximal gaps between reslts obtained sing the method of large singlar finite elements and the finite elements method all along varios circles centered inσ according to their radis. Reslts obtained by both methods are in line with the whole domain of calclation. However, if we examine the nmerical vales obtained near the singlar points (discontinity points of the fnction, it appears that the method of large singlar finite elements is significantly well accrate enogh. If gaps between both methods are arond 0 over a circle which radis is.0, it increases to 0 for a circle which radis is0, then to 0 if the radis of the circle is 0. σ

7 Oigo M. Zongo et al Adv. Appl. Sci. Res., 0, (:-8 Figre Comparing the vales of (ble, (red and (black obtained throgh the method of large x y singlar finite elements (fll lines and finite elements method (circles. 8

8 Oigo M. Zongo et al Adv. Appl. Sci. Res., 0, (:-8 Figre - Comparing the vales of obtained sing the method of large singlar finite elements (fll lines and finite elements method (circles. 9

9 Oigo M. Zongo et al Adv. Appl. Sci. Res., 0, (:-8 Figre - Maximal local gap between the vales of and its derivatives calclated sing the method of large singlar finite elements and finite elements method all along the varios circles centered on the singlarity σ CONCLUSION Reslts obtained by the method of large singlar finite elements and finite elements method are in line all over the domain. However, by checking the nmerical vales obtained near the singlar points, this shows that the method of large singlar finite elements is well accrate enogh. The method of large singlar finite elements takes the existence of singlarities into accont, while trying to find analytical soltions near them; this allows therefore getting withot frther formlation, derivative vales. Moreover, the mode of convergence of this method is exponential. Acknowledgements The athors wold like to thank CUD, Belgim, for the grant (CIUF, P, Physics given to Mr. Zongo and for the provision of MATLAB Software. REFERENCES [] 0. Marice, A. Reineix, P. Hoffmann, B. Bernard and P. Poligen, Adv. in Appl. Sci. Res. 0, (, 9. [] D. S. Chahan and V. Kmar, Adv. in Appl. Sci. Res. 0, (,. [] R. Saravana, S. Sreekanth, S. Sreenadh and S. Hemadri Reddy. Adv. in Appl. Sci. Res. 0, (,. [] V. Sri Hari Bab and G.V. Ramana Reddy, Adv. in Appl. Sci. Res. 0, (, 8. [] R.E Barnhill and J. R. Whiteman, Singlarities de to re-entrant bondaries in elliptic problem. in L. Collatz (ed., I.S.M.N. 9, Birkhaser-Verlag, Basel, 9. [] A. F. Emery, Trans. ASME (C, 9, 9,. [] G. J. Fix, Math. Mech., 99, 8,. [8] Z.C. Li and T.T. L, Math. Compt. Model., 000,, 9. [9] H. Motz, Qart. Appl. Math., 9,,. [0] G. Strang and G.Fix, Analysis of the finite element method. Prentice Hall, 9. [] R. Wait and A. R. Mitchell, J. Compt. Phys. 9, 8,. [] Z. C. Li, T. T. L, H. Y. H and A. H. D. Cheng. Eng. Anal. Bond. Elem., 00, 9, 9, 80

10 Oigo M. Zongo et al Adv. Appl. Sci. Res., 0, (:-8 [] L. Fox, P. Henrici and C. B. Moler, SIAM, J. Nmer. Anal. 9,, 99. [] M. D. Tolley, PhD Thesis, Université Libre de Brxelles, (Brxelles, Belgim, 9. [] J. Desclox and M. D. Tolley, Compt. Method. Appl. Mech. Eng, 98, 9 (,. 8