A new method of boundary treatment in Heat conduction problems with finite element method

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1 Avalale onlne at Proceda Engneerng 6 (11) Frst Internatonal Smposum on Mne Safet Scence and Engneerng A new method of oundar treatment n Heat conducton prolems wth fnte element method Yuepng QIN a, Hongo LIU a, *, Yanhu ZHANG a, huan QIN a, Shengzhu ZHANG a a hna Unverst of Mnng & Technolog (Bejng), Bejng 18, hna Astract In regards to the defect of respectvel handlng three condtons, ths paper puts forward a new unfed epresson fttng for dfferent oundar condtons. It also otans the parameter values of epresson n dfferent oundar condtons. Take the nfnte square clnder puttng nto cold lquor as an eample, t works out a program usng the mproved new oundar treatment to smulate clnder temperature feld. Ths paper gets dstrutons of temperature feld at four dfferent tmes. The results of smulaton not onl meet the phscal stuaton, ut also have hgh precson compared wth the theoretcal solutons. Snce eng eas to deal wth comple oundar condtons, the new wa makes the FEM useful agan n engneerng calculaton and research. 11 Pulshed Elsever Ltd. Open access under BY-N-ND lcense. Selecton and/or peer-revew under responslt of hna Academ of Safet Scence and Technolog, hna Unverst of Mnng and Technolog(Bejng), McGll Unverst and Unverst of Wollongong. Kewords: fnte element method; oundar treatment; square clnder; unstead temperature feld 1. Introducton ourant at frst put forward the concept of element n 194, and then Fnte Element Method was appled n sold mechancs prolem n the 195s. Along wth the development of computer technolog, FEM has sustantal theor ass and comprehensve appled prospect n varous knds phscs feld of engneerng [1]. The method of fnte element s essentall a mathematcal method. The fnte element method s a dscrete process to appromatel solve contnuaton functon n connected areal ased on * orrespondng author. Tel.: ; fa: E-mal address: cumt.hong@gmal.com Pulshed Elsever Ltd. do:1.116/j.proeng Open access under BY-N-ND lcense.

2 676 Yuepng QIN et al. / Proceda Engneerng 6 (11) varaton prncple. It also means that usng fnte comnaton of small smple shapes to replace comple areal whch need to solve. The fnte element method as a numercal analss tool whch has sold theoretcal ass has comprehensve applcaton n solvng varous knds of phscal felds n recent twent ears. The fnte element method has a lot of advantages, such as havng drect phscal sgnfcance, sutng for comple oundar condtons, havng hgh precson and so on. But the fnal fnte element equatons don t have unfed epresson equaton and need to respectvel manage three class oundar condtons. Ths makes t hard for programmer to develop software to solve practcal prolems and wastng computer resources. Amng at defects of fnte element method handlng oundar condtons, ths paper put forward mproved method and makes fnte element method have more comprehensve applcaton n engneerng computng and scentfc research. Nomenclature α convectve heat transfer coeffcent T f temperature of flud T, T j temperature of pont and j. Improvement of oundar treatment Temperature feld has three tpes of oundar condtons. The frst tpe oundar condton alread known s ts node temperature value, so t needn t to estalsh node energ conservaton equatons. The second tpe oundar condton s known as heat-flow denst of oundar and the thrd tpe s known as the relatonshp etween heat-flow denst and temperature. Reference [] has made deep research of the fnte element method applcaton n heat transfer prolems. It has resolved plane transent temperature feld usng weghted resdual method. The computng area s dssect quadrlateral and otaned asc computng equaton of plane temperature feld. Ths paper emphaszes on the second tpe and thrd tpe oundar condtons. It analzes nodes energ conservaton equaton when the elong to second tpe oundar, thrd tpe oundar or comne pont of the two tpes. As shown n Fg.1 (a), node s related to one unt and two oundares, so t can estalsh energ conservaton equatons on the area of quadrlateral AB. As shows n Fg.1 (), node s related to two unts and two oundares, so t can estalsh energ conservaton equatons on the area of quadrlateral ABDE where oundar AE s dvded to two sectons E and A. As shows n Fg.1 (c), node s related to three unts and two oundares, so t can estalsh energ conservaton equatons on the area of quadrlateral ABDEFG. m B (e) A k j l n D E m (e) B A k j 6 7 A 1 B (1) 5 G 1 () () F D E 4 (a) () (c)

3 Yuepng QIN et al. / Proceda Engneerng 6 (11) Fg.1 (a) One unt has two sdes on oundar; () The frst tpe of one unt has one sde on oundar; (c) The second tpe of one unt has one sde on oundar In a word, there have een unfed epressons whch are shown n equaton (1) sutale for all knds of oundar nodes. e T q F q V V q F ( e ) ne ( ) ( e ) ve ( ) ( e ) p ( e ) t (1) Where, summaton smol wth suscrpt e stands for the sum of unts related to node ; summaton smol wth suscrpt stands for the sum of oundares related to node. From the vew of unt related to oundar, t can alwas respectve computng the contruton from oundares to nodes, although the unt mae has one sde or two sdes on the oundar. The contruton computng process shown n equaton (1) has no dfference wth nternal unt computng process, the computng equatons and results can fnd n reference []. So ths paper emphasze on the contruton from oundares to nodes. Take the contruton from sde j to node as an eample shows n Fg.1 (a). Assumng temperature has lnear change on the oundar j, ecause onl secton A has effect on heat alance of ths area, t can get equaton () when the oundar elongng to second tpe and heat-flow denst s constant. q F sq () Where, s stands for the length of oundar j; q stands for heat-flow denst on outer normal drecton of oundar j; 1 qf s q qj () Where, q, q j are heat-flow denstes of and j on oundar j n the drecton of outward normal. When the oundar elongs to thrd tpe, t can get equaton (4). 1 qf s T Tj Tf (4) In the same wa, t can compute the contruton from oundar j to node j. hoosng the same smol as used n fnte element method, t can get equaton (5). J T q F u u T v j J q u T jfj j ujj j vj T j (5) When the oundar elongs to the second tpe and heat-flow denst s constant, the parameter values of equaton (5) are shown n equaton (6).

4 678 Yuepng QIN et al. / Proceda Engneerng 6 (11) qs u uj = uj ujj =; v vj = (6) When heat-flow denst changes n lnear, the parameter values of equaton (5) are shown n equaton (7). 1 u u = u u =; v v = s q q j j jj j j (7) When the oundar elongs to the thrd tpe, the parameter values of equaton (5) are shown n equaton (8). 4s s st u ujj = ; uj = uj ; v vj = 9 9 The fnal comnng equaton shows n equaton (9). e J T e J ( 1,, n) T f (8) (9) Where, the frst term of left stands for summaton of varaton temperature T of nternal nodes; the second term of left stands for summaton of varaton temperature T of the second tpe oundar and the thrd tpe oundar.. Applcaton eample There s a rght angle square clnder whose secton s wdth m and heght m. Its orgnal temperature s 1. Put the clnder nto lquor wth temperature 1 to make the clnder coolng. The coeffcent of heat conductvt of clnderλ s 5W/(m ( ). Thermal dffusvt of clnder a s also 4m/s. The coeffcent of heat convecton etween surfaces of square clnder and lquor h s 8W/(m^ ). Request to know the temperature dstruton at. second,.1 second,. second and.4 second A B 1 Fg. (a) Square clnder wth nfnte length ; () Grddng method

5 Yuepng QIN et al. / Proceda Engneerng 6 (11) Ths paper grds the resolvng area accordng the method of fnte element. Because of the smmetr of the graphc and smplf of the process, t can choose the ottom rght of the secton to analze along. As t shows n Fg. (), sde AB and B elong to the thrd tpe of oundar condton, sde A and should deal wth as nternal nodes. Usng the method of oundar treatment ntroduced aove, ths paper work out a program to solve ths prolem. The result of stmulaton of temperature feld shows n Fg. (a), (), (c) and (d). The result of the numercal smulaton has lttle error compared wth theor soluton whch analzed n reference [4]. So usng the unfed oundar condtons epresson not onl can get relatvel precse result, ut also can smplf the process of programmng a c d Fg. (a) dstruton of temperature feld of tme. second; () dstruton of temperature feld of tme.1 second; (c) dstruton of temperature feld of tme. second; (d) dstruton of temperature feld of tme.4 second.

6 68 Yuepng QIN et al. / Proceda Engneerng 6 (11) onclusons Ths paper analzes the process of oundar treatment of the method of fnte element and put forward mproved new oundar treatment. Then prove t was rght through applcaton eample. The man fndngs of ths paper have three ponts. Frstl, ths paper puts forward the new unfed epresson of oundar treatment and gets the parameter values of three dfferent oundar condtons. Secondl, t resolves the comnng process of fnte element method and the reccle desgn of dfferent oundar condtons. Thrdl, t works out the eample smulaton program and got the unstead state dstruton of temperature feld when nfnte square clnder puttng nto the cold lquor. Acknowledgement The authors gratefull acknowledge the contrutons of The Natonal Basc Research Program of hna (11B1) and Fundamental Research Funds for the entral Unverstes (1YZ1) for fundng ths stud. References [1] hen H T, Yao X Y. The dscrete fnte volume method on quadrlateral mesh. Journal of Suzhou Unverst 5, 4(): 6 1. (n hnese) [] Kenneth H. Huener. The Fnte Element Method for Engneers. New York : A Wle-Inter scence Pulcaton; [] Kong X Q. Applcaton of FEM n heat transfer ssues. Bejng: Scence Press; (n hnese) [4] Zhang X M. Heat transfer theor. Bejng: hna archtecture & uldng press;7. (n hnese) [5] L Z X. Fnte volume method on unstructured trangular meshes and ts applcatons on convectve phenomena. Mathematcs. In Practce and Theor ; 4(8): [6] Mn Y, Numercal Analss of Some Fnte Volume Element and Fnte Volume Schemes. Master s Thess, Shandong Unverst; 5. (n hnese) [7] Nguen D H, Tran T N. Analss of cracked plates and shells usng "mets" fnte element model. Fnte Elements n Analss and Desgn 4; 6(4):855 一 878. [8 Wang J. Fnte ulk-fnte element means on solvng convecton-dffuson prolem. Journal of Xnang Teachers ollege 4; 1(5):1. (n hnese)