ISSN 746-7659, England, U Journal of Informaton and Computng Scence Vol., No. 3, 6, pp.69-79 Haar Wavelet Based Numercal Soluton of Elasto-hydrodynamc Lubrcaton wth Lne Contact Problems S.C. Shralashett *, M.H. antl, A. B. Desh Department of Mathematcs, arnatak Unversty, Dharwad,583, Inda, E-mal:shralashettsc@gmal.com ;Moble: +9 99863359; Phone: +9 836-5(O);Fa: +9 836-347884. (Receved February 7, 6, accepted June, 6) Abstract. In ths paper we present the haar wavelet based numercal soluton of the hghly nonlnear wth coupled dfferental equaton,. e., elasto-hydrodynamc lubrcaton wth lne contact problems. It s a new alternatve approach and we eplore ts perspectves and effectveness n the analyss of elasto-hydrodynamc lubrcaton problems. To confrm ts versatle features solutons obtaned, usng haar wavelet based method, are compared wth estng method. eywords: haar wavelet collocaton; elasto-hydrodynamc lubrcaton; non-lnear boundary value problem.. Introducton Wavelet analyss s capable of gvng rch and useful descrpton of a functon based on a famly of bass functons called wavelets. Recently, wavelet analyss has become an mportant tool n varous research areas. The wavelet transform s notable for ts ablty n tme frequency localzaton and mult-resoluton representaton of transent non-statonary sgnals. Some of the haar wavelet based technques has been successfully used n varous applcatons such as tme frequency analyss, sgnal de-nosng, non-lnear appromaton and solvng dfferent class of equatons arsng n flud dynamcs problems (Chen and Hsao [], Hsao and Wang [], Hsao [3], Lepk [4-6], Buurke et al. [7-9] and Islam []). Hghly nonlnear and sngularty n flud flow problems s a dffcult n numercal smulaton. In numercal weather predcton and numercal smulaton, the most common methods used are the fnte dfference method (FDM) on a unform grd and the spectral method. Snce the computatonal cost of the spectral method s rather large, the FDM s the preferable method at present. The grd space of a unform grd s restrcted to the mnmum scale of the synoptc processes concerned. In numercal smulaton of a hghly nonlnear and sngularty, a hgh resoluton s necessary to get a good accuracy. However, ths type of problems t s not reasonable to use a fne resoluton grd unformly across the whole doman (the storage and computatonal cost s very bg). To overcome ths, t requres the effcent method.e., haar wavelet method. The man am of ths paper s to present haar wavelet collocaton method () to solve elasto-hydrodynamc lubrcaton problems and t has been wdely appled n the feld of scence and engneerng numercal smulaton. The present work s organzed as follows, n secton, Wavelet Prelmnares are gven. Secton 3, dscusses about the method of soluton. Numercal eperments are presented n secton 4. Results and dscussons are gven n secton 5. Fnally, concluson of the proposed work dscussed s n secton 6.. Wavelet prelmnares.. Mult-resoluton analyss The understandng of wavelets s through a mult-resoluton analyss. Gven a functon f L ( ) a mult-resoluton analyss (MRA) of L ( ) produces a sequence of subspaces V, V,... such that the proectons of f onto these spaces gve fner and fner appromatons of the functon f as. A mult-resoluton analyss of L ( ) s defned as a sequence of closed subspaces V L ( ), wth the followng propertes Publshed by World Academc Press, World Academc Unon
7 S.C. Shralashett et al.: Haar Wavelet Based Numercal Soluton of Elasto-hydrodynamc Lubrcaton wth Lne Contact Problems... V V V.... () () The spaces V satsfy V s dense n L ( ) and V. () If f ( ) V, f ( ) V,.e. the spaces V are scaled versons of the central space V. (v) If f ( ) V, f ( k) V.e. all the V are nvarant under translaton. (v) There ests V such that ( k); k s a Resz bass n V. V The space s used to appromate general functons by defnng approprate proecton of these functons onto these spaces. Snce the unon of all the V s dense n, so t guarantees that any functon n L ( ) defned lke L ( ) can be appromated arbtrarly close by such proectons. As an eample the space then the scalng functon J V W V W W V... W V h ( ) and dlaton. For each the space nclude all the functons n V V can be generates an MRA for the sequence of spaces V, by translaton W set of functons whch form bass for the space.. Haar wavelets serves as the orthogonal complement of that are orthogonal to all those n W V are called wavelets []. The scalng functon h ( ) for the famly of the Haar wavelets s defned as for, h( ) otherwse The Haar wavelet famly for, s defned as h ( ) V n V. The space W under some chosen nner product. The k k.5 for, m m k.5 k for, () m m otherwse l In the above defnton the nteger, m, l,,..., J, ndcates the level of resoluton of the wavelet and nteger k,,..., m s the translaton parameter. Mamum level of resoluton s J. The nde n Eq. () s calculated usng, m k. In case of mnmal values m, k, then. The mamal value of s J. Let us defne the collocaton ponts p.5 p, p,,...,, dscretze the Haar functon h ( ) and the correspondng Haar coeffcent matr H (, p) ( h ( p )), whch has the dmenson. The followng notatons are ntroduced () JIC emal for contrbuton: edtor@c.org.uk
Journal of Informaton and Computng Scence, Vol. (6) No. 3, pp 69-79 7 PH ( ) h ( ) d, n, n, (3) PH ( ) PH ( ) d, n, 3,... (4) These ntegrals can be evaluated by usng Eq. (), frst and n th operatonal matrces are as follows, k k k.5 for, m m m k k.5 k PH, ( ) for, m (5) m m otherwse and n k k k.5 for, n! m m m n n k k.5 k.5 k for, PHn, ( ) n! m m m m (6) n n n k k.5 k k for, n! m m m m otherwse We also ntroduce the followng notaton CH PH, ( ) d (7) Any functon f( ) whch s square ntegrable n the nterval (, ) can be epressed as an nfnte sum of Haar wavelets as f ( ) a h ( ) (8) The above seres termnates at fnte terms f f( ) s pecewse constant or can be appromated as pecewse constant durng each subnterval. 3. Method of soluton Consder the second order dfferental equaton y'' f (, y, y') wth the dfferent boundary condtons, the method of soluton s as follows: Assume that y ''( ) a h ( ) Case : Consder the ntal condtons: y() A and Integratng Eq. (9) wth respect to from to, we obtan Integratng agan Eq. () we get (9), y'() B y '( ) B a PH ( ) () y( ) A B a PH, JIC emal for subscrpton: publshng@wau.org.uk
7 S.C. Shralashett et al.: Haar Wavelet Based Numercal Soluton of Elasto-hydrodynamc Lubrcaton wth Lne Contact Problems y( ) A B a PH (), y'() A Case : Consder the med boundary condtons: Integratng Eq. (9) wth respect to from to, we obtan Integratng agan Eq. () we get Put = we get From (3) and (4) we get, and y '( ) A a PH ( ) y() B () y( ) y() A a PH (3), (4) B y() A a C y() B A a C y( ) B A a C A a PH, Case 3: Consder the Drchlet boundary condtons: Integratng Eq. (9) wth respect to from to, we obtan Integratng agan Eq. (6) we get Put = we get a ' s (5) y() A 3, and y '( ) y '() a PH ( ) y() B (6) y( ) A y '() a PH (7) 3, B A y '() a C y '() B A a C 3 3 3 3 (8) From (7) and (8) we get y( ) A3 B3 A3 ac aph, (9) y( ) A3 B3 A3 ac aph, () By substtutng the values of y( ), y'( ) and y''( ) n gven dfferental equaton, we get the haar wavelet coeffcents. Fnally, substtute these coeffcents n y () we get the requred soluton of gven dfferental equaton. In order to vew sold assessment of the accuracy of for dfferental equatons arsng n flud dynamcs, we use the two knds of errors as, mamum absolute error and root mean squared error are 4. Numercal eperments E y( ) y( ),,,..., abs L L ma( E abs ) / Eabs. rms In ths secton, we apply to solve dfferental equatons arsng n flud dynamcs. Eample 4. Consder the Vander pol equaton, y '' y' y y y' sn sn cos, () 3 JIC emal for contrbuton: edtor@c.org.uk
Journal of Informaton and Computng Scence, Vol. (6) No. 3, pp 69-79 73 wth the boundary condtons, y ( ), y'() and eact soluton y( ) cos. The soluton s obtaned by the procedure as follows: We assume that Eq. () s ntegrated twce from to, we get Substtute Eqns. () - (4) n Eqn. (), we have y ''( ) a h ( ) () y '( ) a PH ( ) (3), y( ) a PH (4) a h ( ) a PH ( ) a PH a PH a PH ( ) (5),,,, sn sn cos Solve Eqn. (5), we get haar wavelet coeffcents for 6, a [-.8, -.4,.36, -.477,.389, -.8, -.543, -.9,.54,.35,.8, -., -.6, -.36, -.4, -.486] and then substtutng these values n Eqn. (4), we obtan the based numercal soluton of the gven problem () and results are presented n Fgure. The error analyss of the eample 4. s presented n Table.,.95.9 Eact Numercal soluton (y).85.8.75.7.65.6.55.5...3.4.5.6.7.8.9 Fgure. Comparson of soluton wth Eact soluton for 8 of Eample 4.. JIC emal for subscrpton: publshng@wau.org.uk
74 S.C. Shralashett et al.: Haar Wavelet Based Numercal Soluton of Elasto-hydrodynamc Lubrcaton wth Lne Contact Problems Table. Error norms versus dfferent values of L L rms 4 6.8E- 3.9E- 8 7.66E-.534E- 6 7.659E-.95E- 3 7.99E-.398E- 64 8.35E-.4E- 8 8.97E- 7.57E-3 of Eample 4.. Eample 4. Consder nonlnear dfferental equaton wth varable coeffcent [3] 3 6 y ''( ) y'( ) y ( ) 4, (, ) (6) wth ntal condtons y ( ), y'() and eact soluton of the Eqn. (6) s. Usng the method eplaned n secton 3 (case-), we obtan the based numercal soluton of the gven problem (6), results are presented n Fgure and error analyss gven n Table. y( ).9.8 Eact Numercal soluton (y).7.6.5.4.3.....3.4.5.6.7.8.9 Fgure. Comparson of soluton wth Eact soluton for 8 of Eample 4.. Table. Error norms versus dfferent values of of Eample 4.. L 4.588E-9.94E-9 8.97E-8.35E-8 6.9E-5 5.74E-6 3 3.44E-5 6.84E-6 64.99E-4.64E-5 Eample 4.3 Consder the electrohydrodynamc flow of a flud n an on drag confguraton n a crcular cylndrcal condut was frst revewed by Mcee [4]. A full descrpton of the problem was presented n whch the governng equatons were reduced to the nonlnear boundary value problem (BVP) [5] as L rms JIC emal for contrbuton: edtor@c.org.uk
Journal of Informaton and Computng Scence, Vol. (6) No. 3, pp 69-79 75 y y '' y ' Ha -, (7) - y subect to the boundary condtons y'(), y() (8) where s the flud velocty, s the radal dstance from the center of the cylndrcal condut, Ha s the Hartmann electrc number, and the parameter s a measure of the strength of the nonlnearty. Homotopy analyss method, Perturbatve and numercal solutons to Eqs. (7) and (8) for small/large values of were provded n [5, 6] proved the estence and unqueness of a soluton to Eqs. (7) and (8), and n addton, dscovered an error n the perturbatve and numercal solutons gven n [4] for large values of. Here, we use above eplaned procedure n secton 3 (case-). We obtan the based numercal soluton of the gven problem (7), results are presented n Fgure 3 and resdual error analyss gven n Table 3. These numercal tests demonstrated that the for varous values of the relevant parameters. To facltate the error analyss of eample 4.3, we substtute ~ y ( ) (appromate soluton) nto Eqs. (7) to obtan the resdual functon. y R( ) y '' y ' Ha - (9) - y y (), Ha.7.5.6.45.4 Ha= Ha=5 Numercal soluton (y).5.4.3.. Ha= Ha= Ha=5 Ha=5 Ha= Ha=4 Numercal soluton (y).35.3.5..5..5 Ha= Ha=5 Ha=4 Ha=...3.4.5.6.7.8.9...3.4.5.6.7.8.9 (a).5 (b) Numercal soluton (y).35.3.5..5..5 Ha= Ha=4 Ha= Ha= Ha=5 Ha=5 Numercal soluton (y).8.6.4...8.6.4. Ha= Ha= Ha=4 Ha=5 Ha=5 Ha=...3.4.5.6.7.8.9...3.4.5.6.7.8.9 (c) (d) 5 Fgure 3. Comparson of and soluton for 8 of Eample 4.3. JIC emal for subscrpton: publshng@wau.org.uk
76 S.C. Shralashett et al.: Haar Wavelet Based Numercal Soluton of Elasto-hydrodynamc Lubrcaton wth Lne Contact Problems Table 3. Mamum value of resdual error norms versus dfferent values of Ha.5 Ha Ha 4 Ha Ha of Eample 4.3. Ha 4 4 6.89E-4.493E-4 9.54E-5.978E-4 8.76E-5 5.445E-6 6 4.89E-5.4E-7 6.733E-6 7.548E-7 3.76E-8.473E-7 3 9.3E-7 3.5E-8.639E-8.98E-8 4.86E- 6.4E-9 64.87E-9 5.633E- 6.878E-.477E-.4E-.785E- 8 5.367E- 5.3E- 3.47E-3 4.3E-.46E-.587E-4 Eample 4.4 The EHL lne contact problem models the lubrcant flows between two cylnders rotatng under an appled load. The physcal problem s descrbed by the couplng of Reynolds equaton, for the flow of lubrcant, and elastc deformaton equaton of cylnders. For a steady sothermal lubrcant flow wth smooth surfaces the dmensonless form of pressure P s descrbed by Reynolds equaton [7]. d dp d ( P) H, ( 4, ) d d d (3) where H 3, P() and H() are unknown pressure and flm thckness, a dmensonless speed parameter. H() satsfes the ntegral equaton H ( ) H H s the central offset flm thckness, c log ' P( ') d ' (3) defnes the undeformed contact shape, thrd term pertans H to elastc deformaton of the contactng surfaces. s determned ndrectly by the load balance equaton, gven n non-dmensonal form, c P ( ) d. (3) The nondmensonal forms of densty (P) and vscosty (P), whch are functons of pressure, are gven by the relatons [7]..59e 9.34Pph ( P) (33).59e 9 Pph vald for both mneral and synthetc lubrcants, and z p Pp h ( P) ep (34) z p (Roelands relaton [8])respectvely. z s the vscosty nde (z =.6), p ambent pressure ( p.98e 8 ), the pressure vscosty relaton (.65e 8), p h s the mamum Hertzan pressure( p h 5.8e 8). The physcal non-dmensonal parameters characterzng the EHL lne contact problems are velocty (U), load (W) and elastcty (G) parameters. The correspondng boundary condton are dp P ( ) P( c ) and at c (35) d JIC emal for contrbuton: edtor@c.org.uk
Journal of Informaton and Computng Scence, Vol. (6) No. 3, pp 69-79 77 c where corresponds to cavtaton pont [9]. Snce EHL problems are nonlnear the haar wavelet collocaton method s approprate. The based numercal soluton of pressure (3) and flm thckness (3) followng steps as, Step-: We begn wth an ntal guess for P, and the cavtaton pont. Step-: Evaluate H (3) from the fnte dfference appromate we have H ( ) H SP( ) (36) where S log log (37) c for,,..., and,,...,, and. Step-3: Evaluate densty ( ) and vscosty ( ). Step-4: To solve Reynolds equaton (3) for P as eplaned n secton 3 (case-3). Step-5: Update usng the force balance equaton (3). Move the cavtaton pont based upon the value of dp d H at the cavtaton pont. H Step-6: Whle not converged go to Step-. We obtan the based numercal soluton of the gven Eqns. (3) and (3), results are presented n Fgure 4 and resdual error s gven n Table 4. c.4 Numercal soluton of pressure (P)..8.6.4. FDM Numercal soluton of flm thckness (H) 8 6 4 FDM -4-3 - - - -4-3 - - Fgure 4. Comparson of and FDM soluton for 64 of Eample 4.4. Table 4. Mamum value of resdual error norms versus dfferent values of Eample 4.4. FDM 6 4.36E- 6.3E-3 3 3.498E-.97E-3 64.34E- 9.74E-4 8 9.9E- 7.5E-4 56 5.3E-3.595E-4 5 3.768E-3 8.58E-5 5. Result and dscussons JIC emal for subscrpton: publshng@wau.org.uk
78 S.C. Shralashett et al.: Haar Wavelet Based Numercal Soluton of Elasto-hydrodynamc Lubrcaton wth Lne Contact Problems Here, we present numercal and graphcal results obtaned from appled to dfferent types of nonlnear dfferental equatons arsng n flud dynamcs along wth the study of electrohydrodynamc flow model and EHL wth lne contact model. The algorthm s mplemented n MATLAB software. In order to assess the accuracy of the method n terms of nfnty norm, root mean square error norm and resdual norm verses number of grd level of the and are shown n Tables. In eample 4. and eample 4., we consder nonlnear problem and the haar wavelet solutons are presented n Fgure and Fgure respectvely. The accuracy of the method ncreases consderably by ncreasng the level of grd ponts N. Numercal convergence of the algorthm n terms L and are presented n Table and Table respectvely. In eample 4.3, we takng sngular wth nonlnearty of electohydrodynamc flow problem. The numercal fndngs are presented n Fgure 3 wth dfferent relevant parameters. The accuracy of the method n L rms, Ha terms of resdual and square resdual errors are presented n Table 3 for dfferent values of Ha, Ha N,, whch shows the convergence of the soluton by ncreasng. Eample 4.4, consdered for the applcaton of the, whose results are presented n Fgure 4 and Table 4. 6. Concluson The s appled for the numercal solutons of a dfferental equatons arsng n flud dynamcs (nonlnear, electrohydrodynamc flow and elastohydrodynamc lubrcaton wth lne contact problem). It has been found that the provdes a smple applcablty and a fast convergence of the haar wavelets provde a sold foundaton for usng these functons n the contet of numercal appromaton ustfed through the numercal eperments. 7. References C.F. Chen, C.H. Hsao, Haar wavelet method for solvng lumped and dstrbuted-parameter systems, IEEE Proc. Pt. D. 44: (997) 87 94. C.H. Hsao, W.J. Wang, Haar wavelet approach to nonlnear stff systems, Math. Comput. Smu. 57 () 347 353. C.H. Hsao, Haar wavelet approach to lnear stff systems, Math. Comput. Smu. 64 (4) 56 567. U. Lepk, Numercal soluton of dfferental equatons usng haar wavelets, Math. Comput. Smu. 68 (5) 7 43. U. Lepk, Numercal soluton of evoluton equatons by the haar wavelet method, Appl. Math. Comput. 85 (7) 695 74. U. Lepk, Applcaton of the haar wavelet transform to solvng ntegral and dfferental equatons, Proc. Estonan Acad. Sc. Phys. Math. 56: (7) 8 46. N.M. Buurke, S.C. Shralashett, C.S. Salmath, An applcaton of sngle-term haar wavelet seres n the soluton of nonlnear oscllator equatons, J. Comput. Appl. Math. 7 () 34-44. N.M. Buurke, C.S. Salmath, S.C. Shralashett, Numercal soluton of stff systems from nonlnear dynamcs usng sngle-term haar wavelet seres, Nonlnear Dyn. 5 (8) 595-65. N.M. Buurke, S.C. Shralashett, C.S. Salmath, Computaton of egenvalues and solutons of regular Sturm- Louvlle problems usng haar wavelets, J. Comp. Appl. Math. 9 (8) 9-. S. Islam, I. Azz, B. Sarler, The numercal soluton of second-order boundary-value problems by collocaton method wth the haar wavelets, Math. Comput. Model. 5 () 577-59. I. Daubeches, Ten Lectures on Wavelets, MA, SIAM, Phladelpha, 99. I. Daubeches, Orthonormal bases of compactly supported wavelets, Commun. Pure Appl. Math., 4 (988) 99 996. M. Razzagh, Y. Ordokhan, Soluton of dfferental va ratonalzed haar functons, Maths. Comps. Smu. 56 () 35-46. S. Mcee, Calculaton of electrohydrodynamc flow n a crcular cylndrcal condut, Z. Angew. Math. Mech. 77 (997) 457 465. J.E. Paullet, On the solutons of electrohydrodynamc flow n a crcular cylndrcal condut, Z. Angew. Math. Mech. 79 (999) 357 36. A. Mastroberardno, Homotopy analyss method appled to electrohydrodynamc flow, Commun. Nonlnear Sc. Numer. Smulat. 6 () 73-736. and JIC emal for contrbuton: edtor@c.org.uk
Journal of Informaton and Computng Scence, Vol. (6) No. 3, pp 69-79 79 D. Dowson, G. R. Hggnson, Elasto-Hydrodynamc lubrcaton. The Fundamentals of Roller and Gear Lubrcaton. Pergaman Press, Oford, Great Brtan, 966. C. J. A. Roelands, Correlatonal aspects of the vscosty-temperature-pressure relatonshp of lubrcatng ols. Ph. D. Thess, Technsche Hogeschool Delft, V.R.B., Gronngen, The Netherlands, 996. H. Lu, M. Berzns, C. E. Goodyer, P.. Jmack, Hgh order dscontnuous galerkn method for elastohydrodynamc lubrcaton lne contact problems, Comm. Num. Methods n Eng. () -6. JIC emal for subscrpton: publshng@wau.org.uk