Error Analysis of Heat Conduction Partial Differential Equations using Galerkin s Finite Element Method

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1 Indan Journal of Scence and Technolog, Vol 9(36), DOI: /jst/06/v936/058, September 06 ISS (Prnt) : ISS (Onlne) : Error Analss of Heat Conducton Partal Dfferental Equatons usng Galern s Fnte Element Method S. M. Afzal Hoq,3 *, Erwn Sulaeman and Abdurahm Ohunov Department of Mechancal Engneerng, Internatonal Islamc Unverst, Malasa; esulaeman@um.edu.m Department of Scence and Engneerng, Internatonal Islamc Unverst, Malasa; abdurahmohun@um.edu.m 3 Department of Cvl Engneerng, Southern Unverst Bangladesh, Bangladesh; afzalhoqsu@gmal.com Abstract The present wor explores an error analss of Galern fnte element method (GFEM) for computng stead heat conducton n order to show ts convergence and accurac. The stead state heat dstrbuton n a planar regon s modeled b two-dmensonal Laplace partal dfferental equatons. A smple three-node trangular fnte element model s used and ts dervaton to form elemental stffness matrx for unstructured and structured grd meshes s presented. The error analss s performed b comparson wth analtcal soluton where the dfference wth the analtcal result s represented n the form of three vector norms. The error analss for the present GFEM for structured grd mesh s tested on heat conducton problem of a rectangular doman wth asmmetrc and mxed natural-essental boundar condtons. The accurac and convergence of the numercal soluton s demonstrated b ncreasng the number of elements or decreasng the sze of each element coverng the doman. It s found that the numercal result converge to the exact soluton wth the convergence rates of almost O(h²) n the Eucldean L norm, O(h²) n the dscrete perpetut L norm and O(h ) n the H norm. Kewords: Error Analss, Fnte Element Method, Galern s Weght Resdual Approach, Heat Conducton, Laplace Equaton, Partal Dfferental Equaton. Introducton Fnte Element Method (FEM) s one of the powerful numercal approaches to solve Partal Dfferental Equatons (PDE). FEM s commonl used n multdrectonal felds to solve partal dfferental equaton problems occurrng n sold mechancs, bomechancs, flud mechancs, electromagnetc, thermodnamcs etc. 8 The Galern s Fnte Element Method (GFEM) s one of the weght resdual methods. In ths weght resdual method, an approxmatng functon called tral or bass functon 9 satsfng elemental boundar condtons s substtuted nto the gven dfferental equaton to gve the resdual functon. The resdual s then weghted and the ntegral of the product, taen over the doman, s set to zero. In the GFEM method, the weghted functon s constructed based on the frst dervatve of the tral functon wth respect to the nodal varables. For ths reason, GFEM s perhaps the most approprate soluton of PDE usng wea formulaton. Therefore, n GFEM, the governng PDE s frst developed n the form of the wea formulaton. It s also called a varatonal formulaton of the problem or the method of weght resduals. Ths follows b the ntegraton of the resdual over the whole doman and, f necessar, the Green s ntegraton on the boundar. The ntegraton on the doman s performed b dscretzng geometrcall nto as man fnte elements as requred. Each element has ther nodal coordnates and nodal varables. For each of ths element, the Galern s approxmaton of the gven PDE s selected b tang nto account the nodal coordnates and varables. After that, the ntegraton of each element s performed resultng n the element matrx formulaton as well as the boundar condton vector matrx. Fnall, the sstem of lner equatons s solved to examne the qualt of *Author for correspondence

2 Error Analss of Heat Conducton Partal Dfferental Equatons usng Galern s Fnte Element Method the approxmaton solutons. The current wor attempts to mplement the GFEM procedure above to solve the Laplace partal dfferental equatons (tme ndependent wth no heat source) n a rectangular doman where the exact soluton s avalable. Consderng essental and natural boundar condtons for the soluton doman, the non-lnear partal dfferental equatons are solved. A smplfed stffness matrx that can be used for a homogenous rectangular doman problem that allow for a structured grd mesh generaton wth unform dstrbuton of element szes s presented where ts scheme can reduce sgnfcantl the CPU tme.. Mathematcal Formulaton We consder a stead state heat conducton/flow problem wth no heat source, n a homogeneous doman that leads to Laplace s equaton whch can be combned wth nhomogeneous Drchlet or eumann condtons as shown n Fgure 8. Fgure. Phscal doman of Ω bounded b Γ. The leadng partal dfferental equaton for a stead state heat conducton can be expressed as u u u= ( x, ) + ( x, ) nω x 0< x < a and 0< < b wth the boundar condtons: u= ub on Γe () u = fb on Γ n n where, u b and f b are the prescrbed essental / Drchlet and natural / eumann boundar condtons, respectvel. The strong formulaton of the weghted () resdual of the PDE and ts boundar of Eqs. () and () can be wrtten as u u u I = w + dω w dγ n Ñ (3) Ω x Γ where, w s the weghted functon formulated usng Galern approach. The wea formulaton of Eq. (3) can be performed b ntegraton b part to gve w u w u u I = w + dω+ w dγ x x n Ñ (4) Ω In the GFEM, the doman Ω s dscretzed nto fnte elements and the ntegraton s performed per element to form the so called stffness matrx. 3. Bass Functon and Stffness Matrx One of the smplest two dmensonal elements s the three-node trangular lnear elements. The basc element sutable for unstructured grd mesh s shown n Fgure (a). Assume that each trangular element has three nodes (x, ), (x, ) and (x 3, 3 ) and ther three nodal varables u, u and u 3 at the vertces of the trangle. The value of the varable u at arbtrar locaton (x, ) n the elemental trangular doman regon s approxmated b the nterpolaton/bass functon as follows : (, ) (, ) (, ) u= H xu+ H xu + H xu (5) 3 3 H, s the shape functon for lnear trangular element whch can be derved as functons of the three trangular geometr coordnates as follows: H = ( x3 x3) ( 3) x ( x3 x) A + + (6) H = ( x x 3) ( 3 ) x ( x x3) A + + (7) where, ( x ) H3 = ( x x) ( ) x ( x x) A + + Γ (8) and A s the trangular geometrc area whch can be evaluated as A= + x x + + x x + + x x ( )( ) ( 3)( 3) ( 3 )( 3 ) (9) Equatons (5) () show that the tral functon u represents all the three nodal varables at the vertces Vol 9 (36) September 06 Indan Journal of Scence and Technolog

3 S. M. Afzal Hoq, Erwn Sulaeman and Abdurahm Ohunov through the shape functon H. The Galern approach adopts the assumpton that the weght functon w n Eq. (4) s the dervatve of the tral functon wth respect to the nodal varables as u w = (0) u Fgure. Lnear trangular elements (a) For unstructured grd mesh (b) for structured grd mesh. such that, b consderng the frst ntegral of the wea formulaton gven n Eq. (4), the elemental stffness matrx can be derved as e w u w u K = e Ω + Ω x x () H H x e H H H H3 H H H H3 K = e Ω + Ω x x x x H 3 H 3 x () where, s the element doman. It can be shown that the elemental stffness matrx s a 3 b 3 smmetrc matrx as follow 3 e K = 3 4A () where the dagonal elements of the stffness matrx are ( ) ( ) = x x ( ) ( ) = x x ( ) ( ) 33 = + x x (3) and the off-dagonal elements of the stffness matrx are ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) = x x x x + = x x x x + = x x x x (4) wth the smmetrc propertes of the off-dagonal element as =, =, = (5) The element stffness matrx gven n Equatons (3) and (4) s applcable for unstructured grd mesh wth arbtrar geometr coordnates of the three vertces as shown n Fg. a. For a structured grd mesh, the trangular has one rght angle such as shown n Fg. b. For ths tpe of mesh, the stffness matrx can be further smplfed b tang nto account the advantage that the adjacent sdes of the rght angle are parallel to the x and coordnate axes. For example, for the trangular element shown n Fg. b, the dagonal element of the stffness matrx can be smplfed as 33 = d = b + d = b (6) and the off dagonal matrx elements can be smplfed to 3 3 = d = b (7) where the trangular element area s smpl A = b d For a homogenous rectangular doman where a structured grd mesh can be performed wth a unform dstrbuton of the length b and heght d of each element, the element stffness matrx gven n Equatons (6) and (7) s actuall the same for each element havng the same orentaton. Therefore, the stffness matrx needs to be calculated one tme onl for each tpe of orentaton. Ths scheme wll sgnfcantl reduce CPU tme especall f huge number of fnte elements s used snce no need to construct the stffness matrx for each element. Vol 9 (36) September 06 Indan Journal of Scence and Technolog 3

4 Error Analss of Heat Conducton Partal Dfferental Equatons usng Galern s Fnte Element Method 4. Performance Evaluaton and Result Dscusson Ths secton presents a number of numercal approxmaton examples to show the features of the GFEM based on trangular element and lnear nterpolaton/bass functon for Laplace partal dfferental equatons. The stead state heat conducton problem of rectangular doman shown n Fgure 3 s used. The length and wdth of doman are 0 and 5 respectvel. The Drchlet and euman boundar condtons mposed are as follows: u( 0, ) u( x,0) u( x,0) 0 sn( 0.p x) u ( 5, ) x 0 Γ e at x Γ e at Γ e at sn (0. π x) Ω 5 Γ n at x = 5 Fgure 3. Stead state heat conducton problem of rectangular doman. Ths problem has been presented n b usng trangular lnear element. In the present wor, the problem s solved usng the smplfed stffness matrx of the trangular element wth the modfed scheme of the stffness calculaton. The whole rectangular Cartesan mesh are refned or the elements szes wll be decreased as large error s usuall found wthn the elements 3. The convergence of the Galern method soluton and exact soluton are shown n Fgure 4. ote that the exact soluton to the problem wthn the rectangular doman s as follow. ( 0. p x) snh ( 0.p )/ snh ( p ) u 0 sn () x Fgure 4. Graphcal comparson between numercal soluton and exact solutons. In Table A,. means the usual L norm,. s the 0 usual sem H norm, and obvousl all norm are computed numercall subsequent to the mesh used 4 6. The quantt. s the dscrete nfnte norm L that shows the hghest of the absolute value of the nown functon at the nodes of the mesh. Where, H = u u % = (8) ( u ) u = (9) L = = L = max u % u (0) Table. L, L and soluton el Sze(h) H error of the Galern method u~ u 0 u~ u u ~ u Vol 9 (36) September 06 Indan Journal of Scence and Technolog

5 S. M. Afzal Hoq, Erwn Sulaeman and Abdurahm Ohunov Generated general regresson form Error = E b a h b = ah () where, h s the elements sze, and a and b are two constants to be determne from the actual values of the nterpolaton and soluton errors for each case. For error analss wth lnear regresson we can see that the data n Table. Obe..990 u% u 0.69 h () u% u h, (3).938 u% u h (4) whch ndcates that the result u h converge to the exact soluton wth the convergence rates of almost O(h²) n the L norm, O(h) n the H norm and O(h²) n the dscrete perpetut norm. Fgures 5(a) and 5(b) llustrate that naccurac results for the growng number of the elements and decreasng sze of the elements. Fgure 6. Surface plot for D heat conducton PDEs. (a) (a) (b) Fgure 7. Contour plot usng b (a) present method and (b) Anss. Fgure 6 presents a surface plot for the D heat conducton. Fgure 7(a) depctng the contour plot of the present soluton and Fgure 7(b) depctng the contour plot of ASYS solutons show that the heat temperature dstrbuton n the whole regon. It s clear from the contour plot that results obtaned b the present wor are ver close to the ASYS soluton. (b) Fgure 5. Error analss for (a) ncreasng of the elements number, (b) decreasng of the elements sze. 5. Concluson The Galern fnte element method usng three-nodes trangular element models constructed n the present Vol 9 (36) September 06 Indan Journal of Scence and Technolog 5

6 Error Analss of Heat Conducton Partal Dfferental Equatons usng Galern s Fnte Element Method wor shows a convergence result b ncreasng the number of elements or decreasng the sze of elements. The accurac of the present model s demonstrated b comparson wth analtcal soluton as well as ASYS result. 6. Acnowledgement The supports of the Mnstr of Hgher Educaton, Malasa under the grant FRGS s gratefull acnowledged. 7. References. Redd J. An ntroducton to the fnte element method. 3 rd edn, McGraw Hll Internatonal Edton; Hoffman JD. umercal methods for engneers and scentsts. nd ed, Marcel Deer, Inc; Antonopoulou DC, Plexousa M. Dscontnuous Galern methods for the lnear Schrödnger equaton n nonclndrcal domans. umersche Mathemat. 00; 5(4): Manas KD, Babusa IM, Oden JT. Soluton of stochastc partal dfferental equatons usng Galern fnte element method. Computer Methods n Appled Mechancs and Engneerng. 00; 90: Kreszg E. Advanced engneerng mathematcs. 0 th edn, John Wle & Sons: ew Yor; Schotzau D, Schwab C. Tme dscretzaton of parabolc problems b the HP-verson of the dscontnuous Galern fnte element method. SIAM Journal on umercal Analss. 000; 38(3): Rchter T, Vexler B. Effcent numercal realzaton of dscontnuous Galern methods for temporal dscretzaton of parabolc problems. umersche Mathemat. 03 Ma; 4(): L D, Zhang C. Error estmates of dscontnuous Galern methods for dela dfferental equatons. Appled umercal Mathematcs. 04; 8: Gl S, Saleta ME, Toba. Expermental stud of the eumann and Drchlet boundar condtons n two-dmensonal electrostatc problems. Amercan Journal of Phscs. 00 ov; 70(): Burden RL, Fares JD. umercal Analss. 7 th ed., Thomson Broos/Cole; 00.. Gocenbach SM. Partal dfferental equatons, analtcal and numercal methods, The Socet for Industral and appled Mathematcs, Unted States; 00.. Kwo YW, Bang H. The fnte element method usng Matlab. CRC Press, Boca Raton London, ew Yor Washngton: D.C; Gerald CF, Wheatl PO. Appled numercal analss, 6th Ed, Pearson Educaton (Sngapore) Pte. Ltd; Answorth M, Oden JT. A posteror error estmaton n fnte element method. Computer Methods n Appled Mechancs and Engneerng.997 Mar; 4( ): Y L. An L -error Estmate for the h-p verson contnuous Petrov-Galern method for nonlnear ntal value problems. East Asan Journal on Appled Mathematcs. 05 ov; 5(4): He X, Ln T, Ln Y. Interor penalt blnear IFE dscontnuous Galern methods for ellptc equatons wth dscontnuous coeffcent. Journal of Sstems Scence and Complext. 00 Jul; 3(3): Vol 9 (36) September 06 Indan Journal of Scence and Technolog