U (x, y ) = : K (x i- x k) K (x i- x k, y j- y l), 2. 1

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1 22 5 V o l 22, N o TR IBOLO GY Sep, 2002,, (, ) :, (FFT )., 2 (Green ),,, 3,. ; ; FFT; ; TH A (2002) ,.,,,. L ub rech t [1 ]. (FFT ) [2 5 ]., [4, 5 ].,L iu [6 ] FFT., FFT, FFT, ;,. 1 1 p (x ) p (x, y ) 1, U (x ) U (x, y ) H ertz [7 ] : U (x ) = lngx - sgp (s) ds. (1) U (x, y ) = 2 ΠE κ 3 p (Ν, Γ) (x - Ν) 2 + (y - Γ) 2dΝdΓ. (2),, U (x i) = - U (x i, y j) = 4 3 K (x i- x k) p (x k). (3) k= 0 M k= 0 K (x i- x k, y j- y l) p (x k, y l). l= 0 : K (x i- x k) K (x i- x k, y j- y l), (4) k (k, l) i (i, j ) ( 1 ), K i k ( K i, j k, l),. (3 4), K 2,, , ( ) ; g, E2m ail: w angw z@po sṫ p im. tsinghua. edu. cn.,, 1971,,.

2 5 391 K i, j k, l= κ 8 h (x i- Ν, y j- Γ) n (Ν, Γ) dνdγ. (5) : 8 (x k, y l) ( 1 F ig 1 Influence coefficient 1 ), n (Ν, Γ),, h (x, y ), Green,, h (x ) = ln (x ) ;, h (x, y ) = 1g x 2 + y 2., K i, j k, l 2,, K i, j k, l,. n (Ν, Γ),,. n (Ν, Γ),, (6) K k, l= X k + x X k - x g2 + y h (Ν, Γ) dνdγ. - y h (Ν, Γ), K k, l= h (x k, y l) x y, Green,,,,. Green, (3, 4), 2,,., N,M a h H A, [8 ] : K k, l, (5) K k, l= X k + x X k - x g2 + y ck= h k- na n, k= 0, 1,,M + N - 2. (7) h (Ν, Γ) n (Ν, Γ) dνdγ. (6) n= 0 - y 2 N M, : X k N + M - 1, 2 N ( 1 ). a h h a, (6), n (Ν, Γ) :, n (Ν, Γ) h a= h a= c (0) c (1) c (2) c (3) c (4) c (5) c (6) c (0) c (1) c (2) c (3) 2 2,, 2, 2, = = ck= h < k- n> N an, k= 0, 1,,. (8) n= 0, 2 N N, < n> N n N.,, N = M = 4, h (0) h (1) h (0) 0 0 h (2) h (1) h (0) 0 h (3) h (2) h (1) h (0) 0 h (3) h (2) h (1) 0 0 h (3) h (2) h (3) h (0) h (3) h (2) h (1) h (1) h (0) h (3) h (2) h (2) h (1) h (0) h (3) h (3) h (2) h (1) h (0) a (0) a (1) a (2) a (3) a (0) a (1) a (2) a (3),,.. ; 2, h, h

3 , D F T {c (n) }= D F T {h (n) } D F T {a (n) }. (9) 2 2.,,,. h, a (n) h (n), N + M - 1 a (n) h (n), 2,. FFT FFT [ ], FFT, ; (3, 4)., (3 4), P N,, K 2,.,, 3N - 2,. 2 ( ), 2, F ig 2 P retreatm ent of influence coefficient (L 2computation dom ain,l 2extended dom ain) [ 6 ]. (1) L p (x ) N {P i}n, 2N - {PN i}2. 2 (L 2, L 2 ) 1 (2) 2, {K i}2,. : {K i}2 N - 1 K 1 K {KN i}2 KN N + 1 KN 2, { K i }2 N K N K 2 {KN i}2 N KN 1 KN N. (3) {PN i}2 {KN i}2, {P δ N i}2 {K δ N i}2. (4) {P δ N i}2 {K δ N i}2 {T δ i}2. {T δ i}2 {T i}2. (5) U i= T i, i= 1, 2,, N, {U i}n.,,., O (N 2 ) O (N lnn ). 3 FFT, 1,, (3 6). P m, P h H ertz (GPa). :,,,. FFT, H ertzian, M LM I FFT, 2.,. Pen tium g 400, V isual Fo rtran 6. 6, FFT IM SL. 2 2., FFT, 1g3

4 Table 1 The computation cases Case Load Dom aingmm Loading fo rm Case 1 U nifo rm - a x a L ine loading Case 2 T riangle - a x a L ine loading Case 3 U nifo rm 2a 2b Po int loading Case 4 H ertzian p ressure C ircle w ith R a Po int loading F ig 3 D efo rm ations due to a unifo rm p ressure ( line loading) 3 F ig 4 D efo rm ations due to a triangle p ressure ( line loading) 4 F ig 5 D efo rm ations due to a unifo rm p ressure on a rectangle area 2a 2b 5 2a 2b F ig 6 D efo rm ations due to a H ertzian p ressure app lied to a circular region 6 H ertz 2 Table 2 Time needed for FFT-based method and MLM I tgs FFT2based m ethod M LM I < Table 3 Relative errors to closed form solution Relative erro r g% FFT2based m ethod M LM I D irect summ ation

5 ,. 2 Green, 3, FFT M LM I, ( )., FFT. 4 a. 2 2,. b., FFT M LM I, M LM I 3. c.,. [ 1 ] L ubrech t A A, Ioannides E. A Fast So lution of the D ry con2 tact P roblem and A ssociated Surface Stress F ield, U sing M ul2 tilevel Techniques [ J ]. A SM E Journal of T ribo logy, 1991, 113: [ 2 ] H u Y Z, Barber Gary C. N um erical A nalysis fo r the E lastic Contact of Real Rough Surfaces[J ]. STL E T ribo logy T rans, 1999, 42: [ 3 ] N ogi T, Kato T. Influence of a hard surface layer on the lim it of elastic contacṫ Part 1: A nalysis using a real surface model [J ]. A SM E Journal of T ribo logy, 1997, 119: [ 4 ] Ju Y, Farris T N. Spectral A nalysis of Two2D im ensional Con2 tact P roblem s [ J ]. T ransactions of the A SM E, 1996, 118: [ 5 ] Po lonsky IA, Keer L M. Fast m ethods fo r so lving rough con2 tact p roblem s: a comparative study[j ]. A SM E Journal of T ri2 bo logy, 2000, 122: [ 6 ] L iu S, W ang Q, L iu G. A V ersatile M ethod of D iscrete Con2 vo lution and FFT (DC2FFT ) fo r Contact A nalyses[j ]. W ear, 2000, 243: [ 7 ] Johnson K L. Contact m echanics[m ]. L ondon: Cam bridge U 2 niversity P ress, [ 8 ] Burrus C S, Park s T W. D FT gfft and Convo lution A lgo2 rithm s theo ry and imp lem entation [M ]. N ew Yo rk: R ice U ni2 versity, 1984, [ 9 ] Co lin F, L ubrech t A A. Comparison of FFT 2M LM I fo r E lastic D efo rm ation Calculations [ J ]. A SM E Journal of T ribo logy, 2001, 123: Fast Com putation of Surface D eformation in L ubr icated Con tact W AN G W en2zhong, W AN G H u i, HU Yuan2zhong (S tate K ey L aboratory of T ribology, T sing hua U niversity, B eij ing , Ch ina) Abstract: Based on the exam ination of calcu lation p rocess of su rface defo rm ation and discrete convo lu tion, an FFT 2based m ethod fo r calcu lation of su rface defo rm ation w as derived and described. T he m ethod requ ires a p retreatm en t fo r the discrete data of influence coefficien ts and p ressu re distribu tion so as to get rid of the pe2 riodic erro rs. N um erical examp les are given to validate the m ethod and to compare the resu lts w ith tho se from o ther fast app roacheṡ T he resu lts show that the FFT 2based m ethod is very accu rate and efficien ṫ It has a great po ten tial in app lication to the num erical analysis of m echan ical param eters, such as su rface defo r2 m ation s and temperatu re riseṡ Key words: influence coefficien t; su rface defo rm ation; FFT; m u lti2level m u lti2in tegration; discrete convo lu2 tion Author: W AN G W en2zhong, m a le, bo rn in 1971, Ph. D. studen t, E2m a il: w angw z@po sṫ p im. t singhua. edu. cn.

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