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A NOTE ON APPLICATIONS OF ADAPTIVE FINITEELEMENTS TO ELASTOHYDRODYNAMIC LUBRICATION

PROBLEMSS. R. WU AND J. T. aDEN

TleOM, University of Texas at Austin, Auslin, TX 78712, U.S.A.

SUMMARYAdaptive methods are applied to improve the finite element solutions for highly nonlinear elastohydrodynamiclubrication problem. A refinement scheme is developed which is based on the idea of equidistribution oflocal errors. Numerical experiments on the line contact problem show that the adaptive method is veryeffective in obtaining good approximations. Adaptive methods for point contact problems with light loadsare also examined.

INTRODUCfION

The solutions of many elastohydrodynamic lubrication problems exhibit large gradients in thepressure distribution. For improving the quality of approximations of the pressure, more nodescan be placed in the region where the pressures varies rapidly. Alternatively, one can use higherorder elements in that region. However, the structure of such regions may vary case by case andthe use of very fine uniform meshes is not feasible as it may require excessive computer storageand time. This is particularly true in the case of point contact problems. To obtain good approxi-mations in that case, the use of adaptive finite element methods to automatically produce a suitablerefinement is thus an attractive alternative.

Such schemes have been applied to many engineering problems in recent years (see, for example,References 1-3). The basic ideas and schenies of the adaptive method can be found in References4 and 5, and the recent advances and applications arc summarized in Reference 6. Here we shallapply h-methods, which involve adaptive mesh refinement, to the elastohydrodynamic lubricationproblem. .

PRELIMINARIES

For classical elastohydrodynamic lubrication problems, we have Reynolds' equation

j-'1'(h3 e-ap'1p) + 12µ{ld(ulil)/dx = -12µud(uJI2)fax,p > Din 01p = 0 in Do, plan = 0, D = D1UDo

where (referring to Figure 1)

h=IiI(P)+fi2

fil(p) = h 1(P) = 2hrE' f p(~) In (xo-~x-~r! d~ (line contact)== 217TE' f p(I;)/r(x,t;) dl; (point contact)

fi2 = ho + hzhz = R- J(RLx2) (line contact); R- J(R~-r-y2) (point contact)

(1)

(2)

(3)

(4)

0748-8025/87/060485-10$05.00© 1987 by John Wiley & Sons, Ltd.

Received February 1987

S. /(, wu A:>OU J. T. OU/:.:"

z

x

(4')

(a) Mechanism of Bearing

z

(b) Pressure and Deformation

Figure I. Mechanism of bearing

Here 110 is the refcrence thickness. Onc may also use the minimum film thickness II", in theformulation, where

1i,(P) = h,(P)-S(II)h2 = hm + hzS(h) = min (h\(p) + hz)

h,(P) is the change in film thickncss due to elastic dcformation and E' is the effective elasticmodulus; µo is the viscosity of thc lubricant under atmosphcric pressure. Ct is the exponentialparameter for the viscosity, and II is the rolling velocity. .

There is an undetermined boundary an, = DIUno, wherc p=O. Thus (1)-(4) characterize ahighly nonlinear free boundary problem. We shall take p'20 in D as a constraint and consider thefact that our Reynolds' equation is valid only in the contact region n,. We define the operator inReynolds' equation as

A:p--+-'1'(hJ(p)e-ap '1p) + 12µod(uh,(P»/dxand wc define the function

f = -12µoO(llli2)/dxA weak statcment for the problem in the form of a variational inequality is as follows:

(P): FindpEK = {qEViq'20a.e. inO}suchthat(A (P)-f, q-p}vxv' '20 VqEK

(5)

(6)

(7)

ADAI'IIVE HNllc LLL\ILNIS

Here we use the Sobolev space V=H'(l(n) as an appropriate domain for opcrator A: V-+V'= H-I(O), the dual space of V, and denote by C·) the duality on VxV'. K is the constraint set.

It is easy to show the following (see References 7 and 8).

LEMMA 1. If p is a solution of the variational inequality (7), then p is a solution of (1)-(4)in a distributional sense (a so-called weak solution).

To further the study and to implement an effective numerical scheme. we introduce a penaltyterm to regularize the variational inequality (7). Penalty mcthods have been successful in dealingwith certain classes of contact problems and other kind of free boundary problems. For a generalreference, see Oden and Kikuchi.9

Let p- = min (P,0)=(P-lpl)/2. Introduce a penalty operator $: /il(l(O)-+H-1 as

ct>(p) = p-/E with E>O (8)

We then construct a penalty problem,

(P.) : ForE>O, find p.E V=HIll(O) such that

(A(P.),q) + ((p.),q) = if, q) VqEV(9)

The validity of the penalty method is confirmed by the following theorem (for the details of theproof, see References 7 and 10).

TJzeorem 2. Denote by P. the solutions of (9) corresponding to E>O. As E-+O, there exists asubsequence of {p.} which converges weakly to some p in V, where p is a solution of the variationalinequality (7). and, therefore, also a weak solution of (1)-(4). Moreover, this sequence alsoconverges to p strongly in V.

STRATEGY FOR ADAPTIVE REFINEMENT

In a previous work II, we derived an a priori error estimate for the finite clement solutions for thepenalized problem (9) (for details, see Refcrcnce 7 or the forthcoming paper, Reference 8).

IIerrorllu' s ChklP./,I HWe shall use the number Jzk/p' l,.,rk + 1 as an error indicator for each element and pick the elementswith larger indicators for refinement. The idea is to attempt to distribute the error evenlythroughout the mesh.

For Lagrangian finite elements, the functions interpolated with shape functions are continuous,but may have jumps in the derivatives across the inter-elemcnt boundaries.

For one-dimensional linear elements (the k=l case), the first derivatives of the shape functionsare step functions. When we calculate the second-order semi-norm, we obtain &-functions. Thus,

(1P12)2 = f(d2p/dx2)2d X""" L; ([dpldxDlxi)2For two-dimensional problems, the rectangular bilinear elements have the same feature, exceptthat the jumps on the boundaries x=x; (or y=yJ) are still functions of y (or x, respectively), so

(lPbP - J([a2plax2)2 + (a2play2)2JdO"""I. f([apldxDlx'P dy + Ii J ([dplay]Iyi)2 dx

For simplicity, we use rectangular elements with shape functions of a fixed polynominal degree(generally k = 1 or 2) for the two-dimensional point contact problem. When we refine an elementinto four elemcnts, if the ncighbouring element is not to be refined, it will have extra nodes onits boundary. To maintain consistency, these new nodes should be constrained until thc neighbour-ing element is refined. Meanwhile, we employ the one-node rule, which means that on one sideof a linear element at most one constrained node (a pair for quadratic elements) is allowed. Thus,when a secono node on the side is to be constrained, thc neighbouring element should be refinedinstead.

S. R. WU AND 1. 1'. aDEN

0.4

0. j

0. jI')

IL'

6 0.2......I-::JlD......:= o. 2III....a

~ O. I::JIIIIIIWQ::

Q.. O. I

0.0

O~O~ 010 0.010

Figure 2. Pressure distributions (one-dimensional adaplive linear clements, /lll=O'()

APPLICATIONS TO LINE CONTACf PROBLEMS

Figure 2 shows thc prcssure distributions solved on a series of adaptively refined lincar elementmeshes starting with a uniform lO-element mesh, for a light load case (Ho=O·O, W=0·304E-5).It is shown that the solutions from the fourth (28 elements) and the fifth (33 elements) refinedmeshes are very close. In fact, the relative difference of pressure 11P4-PsII/llPsll is 5·0 percent in theHI-norm and 0·75 percent in the U-norm. The relative difference of load is 0·40x 10-3. For thefilm thickness, it is 0·96 percent in the HI-norm and 0·41 x 10-3 in the U-norm.

Figure 3 shows the adaptive solutions with quadratic element, starting with a uniform 5-elementmesh. The relative difference of pressure between the solutions from the fourth (17 elemcnts) andthe fifth (20 elements) refined meshes is 2·0 per cent in the HI-norm and 0·45 per cent in the U-norm.

It is interesting to make a comparison with the solutions from uniform fine meshes. Figure 4(a)shows the comparison between the solutions from the fifth refined linear element mesh (33elements) and a uniform fine mesh (120 elements). The comparison for solutions with quadraticelements is shown in Figure 4(b). Figure 4(c) compares the linear and quadratic adaptive solutions.It is observed that the adaptive solutions and those from fine uniform meshes are very close, andthe adaptive solutions with linear elements and quadratic elements ate also very close. Thcseresults thus show that the adaptive method is quite efficient and reliable in obtaining an accurateapproximation.

A second examplc is for a heavy load case (Ho=-0'16E-3, W=0·233E-4). Figure 5 shows thepressure distributions solved on a series of adaptive-refined linear element meshes, starting witha lO-element uniform mesh. The computation for quadratic elements i$ shown in Figure 6. In theproccss, the scheme at first appeared to be divergent at the load level; the first two refinementsare performed at a lower loading level. Then we increased the load again and continued therefinement procedures.

U. 4

U. J

U.J,.,!.!

6 0.,2.....I-:::IlDH

cr O. 2t-V!Ho

~ O.:::IV!V!Wcrll.. O. I

0.0

O-:O~010

,... ,-/ "\

\\\\\\\\\\\

J._--0.005 :J.OOO 0.005

CONTACT REGIotl0.010

Figure 3. Pressure distribulions (one-dimensional adaplive quadralic elements. Ho=O·O)

0.4

,.,w.z O. Jo~,f-:::IlDH:= O. 2V!...o!.!cr~ O. IV!Wcrll..

o. UCONTA.CT R[GION CONl A.CT REGION CONTACT REGlON

Figure 4. eomparisons with fine uniform meshes (one·dimensional. H ..=O·O)

4':1lJ S. IL WU ANU J. l. UUl:.N

3. 5

3.0

• 2. 5zoHI-

~ 2. 0HIt::I-VlH

01•5wIt::::JVlVl

~ I. 0ll..

O. 5

(}~O~010 -0.005 O. 000 O. 005CONTACT REGION

0.010

Figure 5. Pressure distributions (one·dimensional adaptive linear elemenls. fl,,=-Q'16£-3)

J. 0 ~ "", .. .

z 2. 5' . '0......I-::J(IJ

:;; 2. aI-Vl......0

~ 1.5::JVlVlW

~ t. a

o. 5

o..:o~ 010 -0.005 0.000 0.005 0.010CONTACT REGION

Figure 6. Pressure distributions (two-dimensional adaptive quadratic elements. flu= -0·16£- 3)

...,w• 2. 0

zoH

I-~ 1.5Ha::l-V)

8 I. 0

0.00.000

CONTACT REGION0.000

CONTACT REGIONG. GOO tJ. tJl0

CONTACT REGION

Figure 7. Comparison with fine uniform mesh (one·dimensional, HII=-0'16£-3)

V) ... 5.0E-05·-V)wZl£U>-IJ:...... 2. 5E-05:::;..JH

lL.

...0. uE +00-0. G1a -0.005 0.000 0.005

CONTACT REGION0.010

Figure 8. Film shape near contact cerllre (one-dimensional, H,,= -0'16£-3)

Figure 7 shows a comparison of the adaptive solutions with the fine uniform mesh solutions.Figure 8 shows the film shape near the contact centre.

These experiments and the comparisons show that adaptive methods can be very effective forthese types of nonlinear problems. .

APPLICATION TO POINT CONTACT PROBLEM

For a light load case (Ho=O'5E-5, W=0·680E-8), computed with bilinear elements, after threerefinements the adaptive solutions become quite close to each other. The pressure profiles, aty=O, of the adaptive solutions are shown in Figure 9. Figure 10 exhibits the sixth refined meshgenerated by the adaptive method.

Figure 11 shows the result for biquadratic elements. Also, after several refinements, the solutionsbecome very close to each other. Figure 12 shows the sixth refinement mesh patterns.

The numerical results for light-load cases are encouraging and suggest that further studies ofadaptive techniques for these types of problems may be warranted.

~ 0.15.Il.

oII

>-to-4:

~ 0.10HlL.oll::Il.

Wll::~VIVIW~ 0.05

(I;./

\

\

. I'. \

',I.\I',II

-0.005 0.000CONTACT REGION-X

0.005

Figure 9. Pressure profiles of adaptive solutions at y=O (two·dimensional bilinear elemenls, Hu=O'5£-5)

0.015

0.010

f--

f t

O. 005

0.000

Cd)-0.020

Fit ._-0.015 -0.010 -0.005 0.000

Figure 10. Refinement pattern for Figure 9.

~ 0.15•

Q,

o

">-I-«

~ 0.10.....[",acra...wcr::JVIVIWg: O. 05

\

\

\

\, \

, \

" \

" \

,\

-0.005 0.000CONTACT REGION-X

Figure 11. Pressure profiles of adaplive solutions at y=O (two·dimensional biquadratic clements. lIo=O·5£-S)

O. 015

O. OlD

0.OD5

H- tItI +H-I0.000

-0.020 -0. 0'5 -0.0' 0 -0.005 0.000

Figure 12. Refinement pallern for Figure I I

ACKNOWLEDGEMENT

Support from the Air Force Office of Scientific Research (AFSC) under cOntract No,F49620-84-C-0024 is gratefully acknowledged.

REFERENCES

1. L. Demkowicz. P. Devloo and J.T. Oden. 'On an }Hype mesh refinement strategy based on minimizalion ofinterpolation errors. eomp. Melhs. Appl. Meclr. Eng .. 53(1),67-90 (1985). \

2. L. Demkowicz, J.T. Oden and T. Strouboulis. 'Adaptive finite element methods for flow problems with movingboundaries. Part I: Variational principles and a'posleriori estimates', eomp. Met/IS. Appl. Mech. Ellg .. 46(2),217-251(1984).

3. J.T. Oden, T. Strouboulis and P. Devloo. 'Adaptive finite element methods for the analysis of inviscid compressibleflow: t. Fast refinementlunrefinement and moving mesh methods for unstructure meshes. TleOM Report 86-3. Univ.of Texas at Austin (t986).

4. I. l3abuska, 'The selfadaptive approach in the finile element method. in The Mathelllalics of Fillite Elemellls andApplications 2, MAFELAP 1975. J.R. Whiteman, Ed.). Academic Press. New York, t975. pp. 125-142.

5. I. Babuska and W.C. Reinboldt, 'Reliable error estimate and mesh adaptation for the finite element method', ineomputatiollal Melhods ill NOlllillear Mechanics. (J.T. Oden. Ed.), North-Holland. Amsterdam. 1980, pp. 67-108.

6. I. Babuska, O.c. Zienkiewicz, l.R. de S.R. Gago and A. Oliveira, Adaptive Methods and Error Refinement ill FiniteElemem eomputalioll, Wiley, ehichesler. 1986.

7. S. R. Wu, 'Oualilative and numerical analyses of elastohydrodynamic lubrication problems with penally method',Ph.D. dissert., Univ. of Texas at Austin (1986).

8. S.R. Wu and J.T. Oden. 'Convergence and error estimates for finite clement solutions of elaslohydrodynamiclubrication' eomp. Math. w. Appl. (to appear).

9. l.T. Oden and N. Kikuchi, 'Theory of variational inequalities with applications to problems of flow through porousmedia', 1m. J. Eng. Sci .. 18(IO-A). 1I7~1284 (1980).

10. S.R. Wu, 'A penally formulation and numerical approximation of the Reynolds-llertz problem of elastohydrodynamiclubrication'. Int. J. Eng. Sci., 24(6), 1001-1013 (1986),

It. J.T. Oden and S.R. Wu, 'An a'priori error estimate of finite elemenl solutions for elastohydrodynamic lubrication',To be presented at The First World Congress on Compulational Mechanics, 22-25 Sept. 1986. at the Univ. of Texasat Austin.

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page10titlesSupport from the Air Force Office of Scientific Research (AFSC) under cOntract No,