Journal of Marne Scence and Technology, Vol., No., pp. -7 () DOI:.69/JMST--7- NLYSIS OF CONTCT PROBLEM USING IMPROVED FST MULTIPOLE BEM WITH VRIBLE ELEMENTS LENGTH THEORY Ha-Lan Gu, Qang L, Qng-Xue Huang, and Guang-Xan Shen Key words: fast multpole boundary element method (FM-BEM), coeffcent matrx, varable elements length theory, contact problem. BSTRCT Ths paper ntroduces a varable elements length theory to solve the change of contact area n contact problems. It avods recalculatng the whole coeffcent matrx. In ths theory, element length s changed nstead of element number n contact boundary problem. Ths measure keeps the contact area unchanged and saves computng tme. The teratve process s dscussed and the error estmate s gven. Fast multpole boundary element method (FM-BEM) s mproved by the varable elements length theory. The mproved FM-BEM s used to solve the contact problems of cubes and strp cold rollng process. The results of mproved FM-BEM are compared wth the results of tradtonal FM-BEM and expermental data. Numercal examples clearly demonstrate that the calculaton tme and accuracy are mproved by the mproved FM-BEM. Ths method s sutable for solvng contact problems and precson engneerng problems. I. INTRODUCTION Paper submtted //9; revsed /8/; accepted /7/. uthor for correspondence: Ha-Lan Gu (e-mal: guhl@yahoo.com.cn). Materal Scence & Engneerng Scence College, Tayuan Unversty of Scence and Technology, Tayuan, P.R. Chna. Technology Center, Tayuan Heavy Industry CO. LTD., Tayuan, P.R. Chna. College of Mechancal Engneerng, Yanshan Unversty, Qnhuangdao, P.R. Chna. Contact problem s a common problem n engneerng. The tradtonal numercal method s varatonal method, fnte element method (FEM) and boundary element method (BEM). Varatonal method s used to analyze elastoplastc frctonal contact problems wth fnte deformaton [], FEM s used to analyze crack-contact nteracton n two dmensonal ncomplete frettng contact problem []. However, the results solved by FEM are hard to satsfy the requrement of precson engneerng, especally analyss of pressure dstrbuton n contact boundary. The merts of BEM are smple calculaton and hgh accuracy, so t s wdely used n contact problems n recent years. Such as small dsplacement contact problems [], estmatng thermal contact resstance [4], and nonlnear nelastc unform torson of composte bars []. Wth the fast multpole method [,, 6, ] ntroduced, the new numercal method named fast multpole boundary element method (FM-BEM) s presented and used n engneerng problems, such as two dmensonal elastcs [], elastoplastc contact wth frcton [4], thn shell structures [, 6] and so on. In theory, fast IGMRES(m) method s appled n FM-BEM n order to mprove the calculaton speed [9]. dual boundary ntegral equaton s establshed for large-scale modelng based on FM-BEM [7, 8]. However, the calculaton accuracy of tradtonal FM-BEM s not very precse to solve precson contact problem. Ths s because that the contact boundary changes n loadng step. The results solved n n-th step s not used n (n)-th step. There are two methods for solvng ths problem. One method s the results solved n n-th step are modfed and be used n (n)-th step. It s dffcult to gve the modfed coeffcents accurately and t affects the calculaton accuracy. The other method s that the coeffcent matrx s recalculated n contact boundary. It saves computng tme. In ths paper, varable elements length s ntroduced to solve the contact boundary problem. In Secton II, boundary ntegral equaton (BIE) of contact problem s establshed; n Secton III, elements are analyzed n contact boundary. Varable elements length theory s presented and teratve algorthm s gven; n secton IV, the error analyss of mproved FM-BEM s gven; n Secton V, two numercal examples are gven. Through compare the results of mproved FM-BEM wth tradtonal FM-BEM and expermental data, t llustrates that mproved FM-BEM can mprove the computng tme and accuracy combned. Ths method s sutable for solvng precson engneerng problems. II. THE TRNSCEIVER STRUCTURE Consder a contact problem between body and body B.
Journal of Marne Scence and Technology, Vol., No. () Ω and Ω B respectvely denote the doman of and B, = C D denotes the boundary of, where D and C respectvely denote the non-contact boundary and contact boundary of. B = BC denotes the boundary of B, where and BC respectvely denote the non-contact boundary and contact boundary of B. Usng the weghted resdual method, the BIEs of and B are gven. () cu = u pd p u d u bdω j j j Ω where NC denotes the element number n contact boundary, ND denotes the element number n non-contact boundary. Usng boundary couplng condtons n contact boundary B u = u and p p B =, BIE can be obtaned D D U P D C D C C C U = P BC BC H H G G H H -G G U P (7) () cu = u pd p u d u bdω B B B B j B j B j Ω where c s a geometry functon, n smooth boundary, c = δ, u B and u are components of dsplacement fundamental solutons, p and p B are components of stress fundamental solutons. In order to smplfy dscusson, the body force s defned zero. Dscretzed boundary of and B, the dscrete BIE can be wrtten as where H KD KD = ( H ) and H KC KC = ( H ) respectvely denote dsplacement coeffcent matrx of body K (K =, B) n noncontact boundary and contact boundary, U KD and U KC respectvely denote dsplacement matrx of body K (K =, B) n non-contact boundary and contact boundary, G KD KD = ( G ) and G KC KC = ( G ) respectvely denote stress coeffcent matrx of body K (K =, B) n non-contact boundary and contact boundary, P KD and P KC respectvely denote stress matrx of body K (K =, B) n non-contact boundary and n contact boundary. () c u = ( u p p u ) d ( u p p u ) d D D j j C j j C (4) B B B B cu = ( u p p u ) d ( u p p u ) d j j BC j j BC where D and C respectvely denote the non-contact boundary and contact boundary of body K (K =, B). Let: H = p d G = u d D * D * D D H = p d G = u d C * C * C C H = p d G = u d H = p d G = u d BC BC BC BC Eqs. () and (4) can be wrtten as NC ND C C D D = j= j= () cu ( G p H u) ( G p H u) NC ND BC BC = j= j= (6) cu ( G p H u) ( G p H u) III. PROCESSING TECHNIQUES Iteratve calculaton s always used n contact problem and the contact boundary may be changed from (n-)-th step to n-th step. In ths state, the dsplacement coeffcent and stress coeffcent n contact matrx must be recalculated. t the same tme, the results solved n (n-)-th step are not use n n-th step. Ths s very waste for solvng problems. We must fnd a method to make sure the contact coeffcent matrx s constant. In ths paper, varable elements length theory s presented to make sure the contact coeffcent matrx n contact boundary s nvarant. Supposed the elements s nvarant n contact boundary, the contact boundary change s expressed by element length change. Let the element total number s k n contact boundary. t (n-)-th step, the real contact element number s m, the element length s l, =,,, m, so the boundary ntegral n m contact boundary s d = l. ccordng to hypothess, n = the element number s stll m at n-th step, but the boundary m ' ntegral n contact boundary s changed nto d = l. The boundary ntegral term also be changed: (n-)-th step: G = u d G = u d Cn * BCn C n n BC n n H = p d H = p d n = Cn * BCn C n n BC n n
H.-L. Gu et al.: nalyss of Contact Problem Usng FM-BEM n-th step: G = u d G = u d Cn * BCn C n n BC n n H = p d H = p d Cn * BCn C n n BC n n Supposed the ntal coeffcent matrx are: H = p d KC KC KC KC K* G = u d KC where H (K =, B) denotes the component of dsplacement KC coeffcent n contact boundary at ntal step, G (K =, B) denotes the component of stress coeffcent n contact boundary at ntal step. ccordng to the varable elements length theory, the n-th step can be wrtten as: K* H = p d = p d p d K K K KC n n KC KC n n = H H (8) KC KC n G = u d = u d u d K K K KC n n KC KC n n = G G (9) KC KC n where H (K =, B) denotes the component of dsplacement coeffcent n contact boundary at n-th step, G KC n (K =, B) denotes the component of stress coeffcent n contact boundary at n-th step. So n Eqs. () and (6), the coeffcent matrx n contact boundary can be wrtten as: N N N KC H u = H u H u j= j= j= () N N N KC G p = G p G p j= j= j= () D Cn D C H H H H = BCn BC H H H H D Cn D C G G G G = BCn BC G G G G C n G BC n G () Combned wth fast multpole method, equaton system can be solved n Krylov space, the dsplacement and stress n boundary can be obtaned. IV. ERROR NLYSIS Varable elements length theory s used n FM-BEM, called mproved FM-BEM. It s not only reduce the computatonal complexty of coeffcent matrx, but also avods the error accumulaton. fter usng the varable elements length theory, the error analyss about dsplacement s gven as follow. TH: To the Drchlet problem of three dmensonal Laplace equaton, there s: σ ( x) u( y) = ds, y R 4 π x y x, (4) Supposed u h denote the approxmate soluton, σ h( x) uh( y) = ds 4 π h x y where σ h( x) V h ( h ), y R. The dstance from pont y to boundary s d( y, ) δ >, h s suffcently small, the error estmate can be wrtten as: Cd( y, ) C σ ( d( y, )) k u( y) uh( y) { h, m σ m, k, k, h u u h u u } () Cd( y, ) C ( d( y, )) k u( y) uk ( y) { h σ, m σ m, k, k, h u u h u u } (6) Because the error of boundary element method (BEM) comes from dscrete elements and boundary functon approxmaton, t s not relatonshp wth the number of elements and nodes. So the calculaton accuracy s not affected by varable elements length theory. When the boundary precson s satsfed, the above error estmate can be wrtten as: hx C n H BC n H () (7) m m u( y) uh( y) C( ) h σ l m, l= ( d( y, ))
4 Journal of Marne Scence and Technology, Vol., No. () P = 8 MPa C B stress n X drecton (MPa) 4 - -6 6 8 8 6 4 X coordnaton (mm) - - - - Fg.. Computed result of contact problem usng tradtonal FM-BEM. Fg.. Computng model of contact problem. -. - C B stress n X drecton (MPa) - -6 4 6 8 8 6 4 X coordnaton (mm) -. - -.. - Y Z X Fg.. Dscrete model of contact problem. (8) m m u( y) uh( y) C( ) h σ l m, l= ( d( y, )) Through the error estmate, we can know that the varable elements length theory effects the calculaton accuracy only O(h m ). But the elements are always subdvded n contact boundary, the elements length are very small, so the change of elements have even smaller mpact to accuracy. V. NUMERICL EXMPLES. Contact of Three Elastc Cubes Consder the contact problem of three elastc cubes. The sde length of cubes are respectvely mm, mm, 8 mm. The computng model and dscrete model show as Fgs. and. Body s fxed constrant on the bottom surface and body C s forced a unform load P = MPa on the upper surface. For three bodes, the Young s modulus s E = GPa, the Fg. 4. Computed result of contact problem usng mproved FM-BEM. Posson s rato s ν =., the frctonal coeffcent s f =., the contact lmted s. mm. Ths model s solved by tradtonal FM-BEM and mproved FM-BEM. ll computatons are run on a Wndows XP computer equpped wth one.8 GHz Intel Pentum 4 unt and MB of core memory. We manly dscuss the stress and pressure dstrbuton n contact boundary. The calculaton tme of these methods are respectvely 9.4 seconds and 9. seconds. In order to llustrate the stable and calculaton accuracy, stress and pressure dstrbuton are dscussed about body B. Fgs. and 4 show that the stress n X drecton solved by tradtonal FM-BEM and mproved FM-BEM. Fgs. and 6 show that the pressure dstrbuton on the upper surface n Z drecton. From the fgures, we know that the stress and pressure contnuty obtaned by mproved FM-BEM s better than tradtonal FM-BEM. The result of tradtonal FM-BEM has uneven dstrbuton n center on the surface. Ths s because that the teratve error reduced usng varable elements length. The coeffcent matrx s constant n non- contact boundary.
H.-L. Gu et al.: nalyss of Contact Problem Usng FM-BEM pressure (MPa) pressure (MPa) 8 6 X coordnaton (mm) 4 8 6 4 Fg.. Calculaton result of tradtonal FM-BEM. 8 X coordnaton (mm) 6 4 4 6 8 Fg. 6. Calculaton result of mproved FM-BEM. 9 8 7 6 4 9 8 7 6 4 Table. Condtons of rollng process. Calculaton parameters Dameter (D) Strp wdth (B) Strp thckness (H) Parameter value 7 mm mm mm Depress rato (ε) % Shear frcton coeffcent (m).7 Young s modulus (E) GPa Yeld stress (σ S ) MPa Sldng frcton coeffcent ( f ). Yeld hardenng coeffcent (h). Posson rato (ν). ngle ncremental ( θ).ra Rollng speed (ν).4 m s - Z Y Z Y X B X Fg. 8. Element dvson of strp. Fg. 7. Calculaton model of cold rollng process.. Contact Problem n Strp Cold Rollng Process In ths case, varable elements length theory s appled n engneerng problems, such as the strp cold rollng process. Consder the smplfed model as Fg. 7, s work roll and B s strp. Supposed the zero dsplacement constrants are forced n the thck and horzontal surface, no tenson force, namely the former and latter tenson s zero. Frcton condton obeys the Coulomb frcton low, and the frcton coeffcent s constant. The calculaton parameters of rollng process are gven n Table. Fgs. 8 and 9 show the element dvson of strp and work roll. In contact boundary, elements are dvded accordng to the rule of fast algorthm, namely strp elements n contact boundary are dvded nto l levers, l =,,.. Work roll elements are dvded nto ll levels. The nteracton of partcles Fg. 9. Element dvson of work roll. c are Z d, whch d represents the number of element, c represents the levels of element. The rollng process s smulated by four-hgh reversng cold Z X Y
6 Journal of Marne Scence and Technology, Vol., No. () dsplacement n Y drecton/mm -......4 -. -. -.4 -. -.6 FEM -.7 mproved FM-BEM -.8 Expermental data -.9 dstance to center/mm Fg.. Dsplacement n Y drecton. Fg.. The laboratory mll. dsplacement n Y drecton -. - dstance to the center 4 4 4... wdth n contact area mm -. -. -. -.4 -. -.6 -.7 -.8 Fg.. Dsplacement n Y drecton. Fg.. The workng status of mll. rollng mll n Tayuan Unversty of scence and technology. Fg. shows that the laboratory four-hgh reversng cold rollng mll, Fg. s the workng status of mll. Through solved by mproved FM-BEM, the dsplacement n Y drecton shows n Fg.. Compared the results solved by mproved FM-BEM wth the results of FEM and expermental data measured n laboratory mll, t s concluded that the results solved by mproved FM-BEM s more close to the expermental data than FEM and mproved FM-BEM s more sutable for solvng contact problems. It can provde stable and accuracy smulaton for engneerng problems. Fg. shows the three dmensonal mage about dsplacement n Y drecton solved by mproved FM-BEM. VI. CONCLUSIONS Based on fast multpole boundary element method (FM- BEM), varable elements length theory s presented to solve the contact problem. In ths theory, the number of elements s supposed constant, the element length s changed to llustrate the contact boundary change. The coeffcent matrx of dsplacement and stress are only calculated n contact boundary nstead of calculatng whole coeffcent matrx. Combned varable elements length theory wth FM-BEM, two numercal example about contact problems are solved by mproved FM-BEM. Through analyss of the results, the calculaton tme and accuracy are mproved. The contnuty of stress and pressure are also stable than FEM. So t llustrates that varable elements length theory s n favor of contact problems and mproved FM-BEM s more sutable for solvng precson engneerng problems. CKNOWLEDGMENTS Fnancal supports for the project from the Natonal Natural Scence Foundaton of Chna (44), the Basc Research Project n Shanx Provnce (9-) and Doctor Research Foundaton of Tayuan Unversty of Scence and Technology (8) are gratefully acknowledged. REFERENCES. Blázquez,. and París, F., On the necessty of non-conformng algorthms for small dsplacement contact problems and conformng dscretzatons by BEM, Engneerng nalyss wth Boundary Elements, Vol., No., pp. 84-9 (9).. Cheng, H., Greengard, L., and Rokhln, V., fast adaptve multpole algorthm n three dmensons, Journal of Computatonal Physcs, Vol. 6,
H.-L. Gu et al.: nalyss of Contact Problem Usng FM-BEM 7 pp. 9-7 (997).. Gll, J., Dvo, E., and Kassab,. J., Estmatng thermal contact resstance usng senstvty analyss and regularzaton, Engneerng nalyss wth Boundary Elements, Vol., No., pp. 4-6 (9). 4. Gner, E., Tur, M., Vercher,., and Fuenmayor, F. J., Numercal modelng of crack-contact nteracton n D ncomplete frettng contacts usng X-FEM, Trbology Internatonal, Vol. 4, No. 9, pp. 69-7 (9).. Greengard, L., Spectral ntegraton and two-pont boundary value problems, Socety for Industral and ppled Mathematcs, Vol. 8, pp. 7-8 (99). 6. Greengard, L. and Rokhln, V., Fast algorthm for partcle smulatons, Journal of Computatonal Physcs, Vol. 7, pp. 8 (987). 7. Lu, Y. J., new fast multpole boundary element method for solvng large-scale two-dmensonal elasto statc problems, Internatonal Journal for Numercal Methods n Engneerng, Vol. 6, pp. 86-88 (6). 8. Lu, Y. J. and Shen, L., dual BIE approach for large-scale modelng of -D electrostatc problems wth the fast multpole boundary element method, Internatonal Journal for Numercal Methods n Engneerng, Vol. 7, pp. 87-8 (7). 9. Mu, Y. F. and Yu, C. X., Fast IGMRES(m) method based on the frctonal contact FM-BEM and ts convergence, Second Internatonal Conference on Innovatve Computng, Informaton and Control, ICICIC 7, Kumamoto, Japan, p. (7).. Rokhln, V., Rapd soluton of ntegral equatons of classcal potental theory, Journal of Computatonal Physcs, Vol. 6, pp. 87-7 (98).. Sapountzaks, E. J. and Tspras, V. J., Nonlnear nelastc unform torson of composte bars by BEM, Computers and Structures, Vol. 87, Nos., pp. -66 (9).. Wan, J., Tang, G. J., and L, D. K., Shape desgn senstvty analyss of elastoplastc frctonal contact problems wth fnte deformaton, Chnese Journal of Theoretcal and ppled Mechancs, Vol. 4, No. 4, pp. -7 (9).. Yamada, Y. and Hayam, K., multpole boundary element method for two dmensonal elastcs, Techncal Reports, Department of Mathematcal Engneerng and Informaton Physcs, Faculty of Engneerng, The Unversty of Tokyo (99). 4. Yu, C. X. and Shen, G. X., Program-teraton pattern Fast Multpole BEM for elasto-plastc contact wth frcton, Chnese Journal of Computatonal Mechancs, Vol., No., pp. 6-7 (8).. Zhao, L. B. and Yao, Z. H., The fast multpole-bem about -D elastc problem appled n thn shell structure, Postdoctoral Report n Tsnghua Unversty, pp. (). 6. Zhao, L. B. and Yao, Z. H., Fast multpole BEM for -D elasto statc problems wth applcaton for thn structures, Tsnghua Scence and Technology, Vol., pp. 67-7 ().