The Two-scale Finite Element Errors Analysis for One Class of Thermoelastic Problem in Periodic Composites

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1 7 Asa-Pacfc Engneerng Technology Conference (APETC 7) ISBN: The Two-scale Fnte Element Errors Analyss for One Class of Thermoelastc Problem n Perodc Compostes Xaoun Deng Mngxang Deng ABSTRACT The thermoelastc compostes are wdely used n some nds of complex envronments In ths paper, the two-scale fnte element method for one class of thermal mechancal problem n perodc structure s analyzed The two-scale fnte element method the approxmate errors for one class of couplng thermoelastc problem are obtaned INTRODUCTION In the process of developments of thermodynamcs, we usually need develop some equpment analyss for one class of thermal composte materals n perodc cell In recent years, the effectve physcal behavor of the thermal-mechancal phenomenon s focused by many scholars In mathematcs, many researchers have dscussed the numercal smulaton method for these compostes [-3] Owng to the small perodcty couplng effects, t s very dffcult to obtan the solutons for these problems, thus the effectve propertes of the thermal phenomenon are the man research drectons for scholars Up to now the correspondng results of approxmaton solutons for couplng thermoelastc problem are very few In ths paper, based on the two-scale asymptotc expansons, the two-scale fnte element method for one class of couplng thermal mechancal problem n perodc structure s analyzed The general lnear thermoelastc problem wth small perodc confguraton n mathematcs s gven by the followng partal dfferental boundary value problem, c h,, x x ah h u ahbh d u f x,, x x ( x) x, u x u x, on () Deng Xaoun,a, Deng Mngxang,b,* School of Management, Northwest Unversty of Natonaltes, 737 anzhou, Chna, School of Mathematcs Informaton Scences, Guangzhou Unversty, 56 Guangzhou, Chna a E-mal: dx64@6com, b E-mal: denggzhu@63com 4

2 n here s an unbounded perodc doman n R, s a perodc doman, satsfes the pschtz boundary condton; u x x denote the dsplacement vector the ncrement of the temperature as compared wth the reference temperature, respectvely; ah, bh are the stffness, thermal expanson thermal conductvty of materals, respectvely, these tensors satsfy, p, condtons n []; c d denote the st-order modfcaton of thermal conductvty the stffness; f x h x represent the nternal force nternal heat sources, respectvely For smplcty, t s assumed that f, h, u are suffcently smooth vector or scalar functons h u s the strans evaluated from the dsplacement u x THE TWO-SCAE FORMA ASYMPTOTIC EXPRESSION There are some tradtonal method to capture the effectve constants for ths system, such as averagng method homogenzaton method But these methods are faled to capture the local behavor By means of two-scale asymptotc method, we have the followng formal asymptotc soluton correspondng to Eq: x x x H H o( ), x x x u ( x) u ( x) ( x) u ( x) u ( x) N ( ) N ( ) M ( ) ( x) M ( ) o( ), x x x x x () where H, N, M are the -perodc scalar, matrx vector functons defned n unt cell, respectvely, they are solutons of some specal partal dfferental equatons defned n unt cube N ( ) m M ( ) could be determned n by the followng partal dfferental equatons: ah h m am,, N ah h ah bh,, M N ( ), m M,, N m, ( ) ah h N a m a h N hm Smlarly, M could be determned by the followng equatons: a h N hm ah,,, n,, ah h M a h h M ah M h a h bh H b,, N,,, M where a h ah alm lm h d, N (5) b ah h M ah bh d (6) (3) (4) 5

3 are the homogenzed constants correspong to a h b h H could be defned smlarly n unt cell ( u x, ( x)) s the homogenzaton soluton correspondng to ( u x, ( x)), t can be determned by the followng homogenzaton equatons wth homogenzaton coeffcents that reflect the effectve propertes of compostes hh u a d u f ( x) b,,, n, x, x x c h( x), x, x x u x u x, ( x) ( x), x, where H ( ) l d l, (8) d d d, c c d, f x f d h x h d, (9) () (7) THE TWO-SCAE FINITE EEMENT APPROXIMATION AND ESTIMATES The asymptotc approxmatons of the frst tems for u ( x) ( x ) n Eq are gven as follows: ( ) l l ( x) ( x) ( x) H ( ) D ( x), l l (a) ( ) l l l l u ( x) u ( x) u ( x) N ( ) D u + M ( ) D l l l l For these asymptotc solutons, we obtaned the followng asymptotc error theorem Theorem Assumng that ( x ), u ( x) are the wea solutons of Eq, 3, u H ( ), h f H ( ), then the followng estmatons hold, ( ) () c 5/ h, () H ( ) H ( ) H ( ) u u c ( h u f ), () () 5/ 5/ H ( ) H ( ) H ( ) H ( ) H ( ) where c c are postve constants ndependent of 6

4 The two-scale fnte element approxmate solutons of the frst tems for u ( x) ( x ) n Eq are desgned as follows: (, h, h ) (, h ) l h l (, h ) ( x) ( x) H ( ) D ( x), l l (, h, h ) (, h ) l h l (, h ) l h l (, h ) u ( x) u ( x) N ( ) D u + M ( ) D l l l l (3) (, h) ( h) where ( u, ) s the fnte element soluton correspondng to homogenzaton soluton ( u x, ( x)) h h h, Nm, M H are the fnte element solutons of N M H In general, m, Now some mportant lemmas theorems of errors between solutons of Eq two-scale fnte element solutons of Eq are presented emma For N ( ), ( ) H ( ), we have: m M N ( ) C N ( ) C N c, (4) m 3 3 m M ( ) C M ( ) C M c, (5) H ( ) C H ( ) C H c, (6) 5 6 where c C,,, 6 are postve constants ndependent of Now we presented the man error result of ths paper Theorem et W ( ), u ( W ( )) n be the soluton of Eq7, 3 3 H W ( ), ( W ( )) n M N ( ( )) n m W be the solutons of Eq3-Eq4, more assumng that h h, then (,h,h ) c ( h h )max H },,h W 4 W W u u c ( u ) ( h h h ) M (, hh, ), h W W W ( h u h u h u ) max N } m, where c m W c are postve constants ndependent of hold, CONCUSION In ths paper, the two-scale fnte approxmaton for one couplng thermal mechancal problem n perodc structure s dscussed Because the dsturbance of the stffness, thermal-expanson thermal conductvty of materals are relatvely large complcated, t s very dffcult to get the analytc solutons for these equatons n mathematcs We try to compute the two-scale fnte element numercal solutons smulate them In order to mprove the accuracy of the calculaton, we use the two-scale method to analyze the asymptotc expansons two-scale fnte 7

5 element method to smulate ther numercal solutons In addton, some approxmaton error estmates are gven, the results could be regarded as the reference for evaluate the effectve physcal mechanc thermal behavor More, our results could be useful for computng the numercal solutons for those couplng systems n multple physcal felds ACKNOWEDGMENT The research s supported by the Scence Technology Proect for the Colleges Unverstes of Guangzhou Educaton Bureau [Grant: 4365] REFERENCES [] Feng YP, Deng MX, Guan XF, Cu JZ, A Two-scale fnte element analyss of the thermo-elastc effects n compostes, Internatonal Journal of Computatonal Method (4) 3566 [] Chen JR, Cu JZ, Two-scale fnte element method for non-self-adont ellptc problems wth rapdly oscllatory coeffcents, Appl Math Comp 5(4)585-6 [3] Du R, Mng PB, Convergence of the heterogeneous mult-scale fnte element method for ellptc problem wth non-smooth mcrostructure, Multscale Model Smul 8() [4] Han F, Cu JZ, Yu Y, The statcal second-order two-scale method for thermo-mechancal propertes of statstcally nhomogeneous materals, Computatonal Materals Scences 46(9) [5] Cao, Cu JZ, Homogenzaton method for the quas-perodc structures of composte materals, Math Numer Snca (999)

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