A direct constraint-Trefftz FEM for analysing elastic...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2005; 63:1694–1718Published online 18 April 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1333

A direct constraint-Trefftz FEM for analysingelastic contact problems

K. Y. Wang1, Q. H. Qin1,2,∗,†, Y. L. Kang1, J. S. Wang1 and C. Y. Qu1

1Department of Mechanics, School of Mechanical Engineering, Tianjin University, Tianjin 300072, China2Department of Engineering, Australian National University, Canberra, ACT 0200, Australia

SUMMARY

A direct constraint technique, based on the hybrid-Trefftz finite element method, is first presentedto solve elastic contact problems without friction. For efficiency, static condensation is employed tocondense a large model down to a smaller one which involves nodes within the potential contactsurfaces only. This model can remarkably reduce computational time and effort. Subsequently, thecontact interface equation is constructed by introducing the contact conditions of compatibility andequilibrium. Based on the formulation developed, a general solution strategy, which is applicable tothe well-known three classical situations (receding, conforming and advancing) is developed. Finally,three typical examples related to the three situations mentioned are provided to verify the reliabilityand applicability of the approach. Copyright 2005 John Wiley & Sons, Ltd.

KEY WORDS: contact problem; direct constraint technique; hybrid-Trefftz finite element; staticcondensation; bisection method

1. INTRODUCTION

Contact phenomena occur frequently in engineering structures. The majority of these prob-lems are inherently non-linear due to changes in the actual contact zone during the loadinghistory. Historically, the first investigation of contact phenomena was carried out by Hertz[1] who provided some analytical solutions to linear elastic frictionless contact over verysmall regions. However, these solutions, which are based on the classical theory of elastic-ity, are limited to cases involving simple geometry and loading configurations [2]. To over-come these limitations, researchers have developed sophisticated numerical approaches whichcan efficiently solve certain practical contact problems [3–5]. For example, Qin and He [5]

∗Correspondence to: Q. H. Qin, Department of Engineering, Australian National University, Canberra, ACT 0200,Australia.

†E-mail: [email protected]

Received 7 June 2004Revised 5 January 2005

Copyright 2005 John Wiley & Sons, Ltd. Accepted 25 January 2005

A DIRECT CONSTRAINT-TREFFTZ FEM 1695

analysed the frictionless impact-contact between elasto-plastic bodies using mathematical pro-gramming. With the advent and development of the conventional finite element method (FEM)and boundary element method (BEM), a variety of different approaches have proved versatileenough to explore a far greater range of contact problems, as can be found in the literature[3–9].

The direct constraint technique, which can avoid the introduction of user-defined penaltyparameters [6–11] or artificial Lagrangian multipliers [12–15], imposes the compatibility ofdisplacements and equilibrium of forces at the contacting surfaces directly. Many researchershave successfully solved contact problems using this technique in conjunction with conventionalBEM or FEM. Among these researchers, Andersson et al. [16] was the first to apply BEM tothe solution of contact problems. The works of Ma [17], Karami [18], Huesmann and Kuhn[19], Martín and Aliabadi [20, 21] and Du et al. [22] should also be mentioned. However,relatively few contributions utilizing the direct constraint technique coupled with FEM can befound in the literature.

In recent years, on the other hand, the hybrid-Trefftz (HT) finite element (FE) model hasattracted considerable attention. This model, based on the Trefftz method [23], was devel-oped nearly three decades ago [24, 25]. In contrast to conventional FE models, the classof finite elements associated with the Trefftz method is based on a hybrid method whichincludes the use of an auxiliary inter-element displacement or traction frame to link theinternal displacement fields of the elements. As highlighted in several reports [26–28], theHT FE model preserves the advantages of the conventional FE and BE counterparts andavoids some of their drawbacks. As such, the concept of the Trefftz element has attracteda growing number of researchers into this field during the past decades. Trefftz elementshave successfully been developed and applied to many engineering problems, such as po-tential, elasticity, plate bending, shells, transient heat problems and others [28]. Since theHT FE formulation not only calls for integration along the element boundary only but alsoavoids the introduction of singular integral equations which may be very laborious to build,it is particularly more suitable to contact analysis than either conventional FEM or BEM.To the authors’ knowledge, however, there are very few reports of the application of HTFEM to contact problems in the literature. Wang et al. [11] developed a coupled HT-spring-element approach using the user interface UEL of ABAQUS 6.2. However, that paper [11]is only a preliminary exploration in this area. Although the coupled approach in our pre-vious work [11] can reduce researchers’ coding workload dramatically, the choice of anaccurate contact stiffness coefficient for the spring element is often an intractable issue,and there is as yet no rule to follow. On the other hand, the fact that one can controlthe cycles only within the UEL subroutine makes it inconvenient to deal with the contactstatus. To overcome these problems, the present work presents a direct constraint HT FEapproach for two-dimensional (2D) frictionless contact problems, in which a general solu-tion process is described. In the solution procedure, static condensation is efficiently em-ployed to condense a large model down to a smaller one which involves nodes within thepotential contact surfaces only. This model can remarkably reduce computational time andeffort. For verification purpose, an in-house developed HT FE analysis program was writtenin FORTRAN to demonstrate the applicability of the proposed approach. The results obtainedfrom three classical examples were observed to be in good agreement with the analytical orthe commercial FE software package ABAQUS solutions, except for minor discrepancies atsome points.

Copyright 2005 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2005; 63:1694–1718

1696 K. Y. WANG ET AL.

2. CONTACT KINEMATICS

Consider two linear elastic bodies involving contact, which are denoted in their referenceconfiguration (), = 1, 2 bounded by their boundaries () (see Figure 1). The boundaries

of each contacting body are composed of three disjoint parts: ()u where displacement is

prescribed, ()t where traction conditions are imposed, and ()

c which is the potential contactsurface taken to be sufficiently large to cover the actual contact area under loading.

The classical field equations of plane elasticity are governed by

L + b = 0 (1a)

= D (1b)

= LTu in () (1c)

together with the essential and natural conditions

u = u on ()u (1d)

t = A = t on ()t (1e)

where u, , and t are vectors of displacements, strains, stresses and boundary tractions,u, t and b designate prescribed displacements, boundary tractions and body forces, and

L =

⎡⎢⎢⎢⎣

x

0y

0y

x

⎤⎥⎥⎥⎦

T

(2a)

(2)Ω(2)tΓ

(2)uΓ

(2)

(1)uΓ

(1)tΓ

(1) (1)Ω

(1)cΓ(2)cΓ

Figure 1. Two bodies in contact.



D = E∗

1 − ∗2

⎡⎢⎢⎢⎣

1 ∗ 0

∗ 1 0

0 01 − ∗

2

⎤⎥⎥⎥⎦ (2b)

A =[

nx 0 ny

0 ny nx

](2c)

in which E∗ = E, ∗ = for plane stress analysis and E∗ = E/(1 − 2), ∗ = /(1 − )

for plane strain analysis. Here, E and are Young’s modulus and Poisson’s ratio, respec-tively, and nx, ny are direction cosines of the outward normal at a given point on theboundary.

For contact problems, Equations (1a)–(1c) should also be subject to the kinematical con-tact conditions. The potential contact region denoted by c = (1)

c ∩ (2)c is where the two

bodies meet, as shown in Figure 1. It is noteworthy that the determination of the commonnormal unit vector n on the contact region c plays a significant role in the contact analysis.For this purpose, the expression of n is presented in the deformation configuration below.The discretizations performed on both bodies in the potential contact zone match so that thecontact behaviour occurs just between each pair of opposite nodes i() (called contact nodepair, CNP for short), as shown in Figure 2. Unit normal vectors, n1 and n2, can be con-

structed through each contact node i() on bodies 1 and 2, to the straight element sides ()ci−−i

,respectively. Thus, the subtraction of n2 and n1 is then taken as an approximate commonnormal n21 [17].

When the nodes i() on opposite surfaces come into contact under loading, their projections

onto the common normal n21 should overlap. Obviously, the straight element sides i()− − i()

1n( )1iϕ

( )1i( )1Ω

( )1−i

( )2iϕ

( )2i( )2Ω

( )2−i

iϕ

1n−

2n21n

( )1iu

( )1i( )2i

( )2iu

X

Y

Figure 2. Definition of contact co-ordinate system.



(see Figure 2) pertinent to each particular body make the angles to X-axis equal to ()

i

()

i = arctg

∣∣∣∣∣∣Y

()

i − Y()

i−

X()

i − X()

i−

∣∣∣∣∣∣ (3)

Accordingly, the common normal n21 reads

n21 = n2 − n1 = ∑

(−1)n = ∑

cos(

2+ (−1)−1()

i

)e1 + sin

(

2+ (−1)−1()

i

)e2

= 2 cos1

2

∑

()

i

[∑

cos1

2

( +∑

(−1)−1()

i

)e1 + sin

1

2

( +∑

(−1)−1()

i

)e2

]

(4)

Then, the common normal unit vector n is conveniently given for CNP i by the expression

n = n21

|n21| = cos

(

2+ (2)

i − (1)i

2

)e1 + sin

(

2+ (2)

i − (1)i

2

)e2

= cos(

2+ i

)e1 + sin

(

2+ i

)e2 = cos ie1 + sin ie2 (5)

or in vector form

n = cos i sin iT (6)

where e1 and e2 represent, respectively, the unit vectors parallel with X- and Y -axes, and i

and i stand for, respectively, the angles from n to e1 and from e2 to n.Contact conditions considered normally for the frictionless problems studied herein may be

grouped into compatibility condition of normal displacements, and equilibrium condition offorces, as shown below.

(a) Compatibility condition:

(u(1)i − u(2)

i )T · n + g0ni = 0 (7)

where the normal initial gap at CNP i, g0ni , is defined in terms of the initial geometry, and

u()i = uiX uiY T (8)

g0ni = (X(1)i − X

(2)i ) cos i + (Y

(1)i − Y

(2)i ) sin i (9)

Substituting Equations (8) and (9) into (7) leads to

(u(1)iX − u

(2)iX )tg i − u

(1)iY + u

(2)iY =

gi (10)

wheregi = Y

(1)i − Y

(2)i − (X

(1)i − X

(2)i )tg i (11)



(b) Equilibrium condition:By the principle of action and reaction, the contact forces acting on CNP i must be equal

and opposite, namely

f (1)i + f (2)

i = 0 (12)

and in terms of their components

f(1)iX + f

(2)iX = 0 (13a)

f(1)iY + f

(2)iY = 0 (13b)

where

f ()i = fiX fiY T (14)

In addition, it is necessary that in the case of adhesive contact

f (1)i · = f (2)

i · = 0 (15)

or

f(1)iX cos i + f

(1)iY sin i = 0 (16a)

f(2)iX cos i + f

(2)iY sin i = 0 (16b)

where denotes the tangential direction at CNP i.

3. SUBSTRUCTURE INTERFACE MODEL

3.1. Theory of 2D HT FE discretization

The basic idea of the HT FE model is to establish a finite element formulation whereby intra-element continuity is enforced on a non-conforming internal displacement field chosen so as tosatisfy, a priori, the governing differential equation of the problem under consideration [28].With this method the solution domain is subdivided into elements e, and over each element‘e’, two groups of independent displacement fields are assumed, as shown in Figure 3:

(a) A non-conforming intra-element displacement field

ue = ue +

m∑j=1

Nej cej = ue + Nece in e (17a)

(b) An exactly and minimally conforming auxiliary frame field

ue = Nede on e (17b)

being independently assumed along the element boundary in terms of nodal DOF de,where m denotes the number of Trefftz terms,

ue and Ne are, respectively, the particular

and homogeneous solutions to Equation (1a), ce stands for unknown parameters, Ne are



3

2

1

4

eee dNu~~ =

eee

m

jejejee c cNuNuu +=+=

=1

X

Y

Frame functions

Node 1

1

1

1−=ζ 1+=ζ0=ζ

( )ζ−12

1

( )ζ+12

1

1

~N

2

~N

Node 2

Figure 3. Typical four-node HT element in plane elasticity.

the shape functions (frame functions) defined in the customary way as in conventionalFEM. The tildes above the symbols in Equation (17b) allow the two fields to bedistinguished.

The corresponding stress field

e = e +

m∑j=1

Tej cj = e + Tece (18)

as well as the boundary tractions

te =

t e +m∑

j=1Qej cj =

t e + Qece (19)

can be easily deduced from e = DLTue and te = Ae, respectively.In the presence of constant body forces b, the particular displacements

ue can be conveniently

taken as

ue = 1

2G

bxy

2

byx2

(20)

The internal field of Equation (17a) requires knowledge of the homogeneous solutions whichare the so-called Trefftz functions. Using Muskhelishvili’s complex variable formulation, onecan obtain the following sequence [28]:

2GNej =

Re Z1k

Im Z1k

, Z1k = izk + kizzk−1 (21a)

2GNej+1 =

Re Z2k

Im Z2k

, Z2k = zk − kzzk−1 (21b)

2GNej+2 =

Re Z3k

Im Z3k

, Z3k = izk (21c)

2GNej+3 =

Re Z4k

Im Z4k

, Z4k = −zk (21d)



where

G = E∗

2(1 + ∗)(22a)

= 3 − 4∗ (22b)

in which z = x + iy, z = x − iy, i = √−1, the bar above the symbol denotes conjugatecomplex value, and (x, y) is the local co-ordinate system originating at the element centroidsubsequently defined in Equation (27).

As stated elsewhere [28, 29] the necessary but not sufficient condition for the element flexi-bility matrix He defined in Equation (39a) below to have full rank is

m > NDOF − 3 (23)

where NDOF is the number of nodal DOF on the element. It has been verified from manypractical applications that the choice of m = 7, 9, 11 and higher often yields robust and sta-ble results. For the best compromise between accuracy and computational effort, however, thevalue of m = 9 is selected in the present study. On the other hand, starting with k = 1and relations (21b), the sequence generates an independent displacement pattern. As a conse-quence, the explicit 2 × 9 matrix Ne involving the 9 homogeneous solutions can be obtainedas follows:

Ne = 1

2G

[( − 1)x y −x −2xy ( − 2)x2 − ( + 2)y2

( − 1)y x y ( + 2)x2 − ( − 2)y2 2xy

2xy y2 − x2 (3 − 3)x2y + ( + 3)y3 ( − 3)x3 − (3 + 3)xy2

x2 − y2 2xy ( + 3)x3 − (3 − 3)xy2 (3 + 3)x2y − ( − 3)y3

](24)

Using Equations (18)–(20), the corresponding matricese, Te,

t e and Qe can be givenexplicitly as

e =

⎧⎪⎨⎪⎩

0

0

bxy + byx

⎫⎪⎬⎪⎭ (25a)

Te =⎡⎢⎣

2 0 −1 −6y 2x 2y −2x −12xy −12y2

2 0 1 −2y 6x −2y 2x −12xy 12x2

0 1 0 2x −2y 2x 2y 6(x2 + y2) 0

⎤⎥⎦ (25b)

t e = (bxy + byx)

ny

nx

(26a)



Qe =[

2nx ny −nx 2nyx − 6nxy 2nxx − 2nyy 2nyx + 2nxy

2ny nx ny 2nxx − 2nyy 6nyx − 2nxy 2nxx − 2nyy

2nyy − 2nxx 6ny(x2 + y2) − 12nxxy −12nxy

2

2nyx + 2nxy 6nx(x2 + y2) − 12nyxy 12nyx

2

](26b)

In implementation of the HT FE model, the Trefftz dimensionless co-ordinate system

= x

a, = y

a(27)

together with

a = 1

n

n∑i=1

√x2i + y2

i (28)

must be exploited to replace x, y in matrices He, Te and Qe to prevent the element flexibilitymatrix He from overflow. Here, a denotes the average distance between the element nodes andits centroid.

Using a similar technique to that in the conventional FE model, the overall frame functionmatrix Ne can be directly defined and assembled in positions appropriate to the node numbersof the element. For example, a simple interpolation function on the side 1–2 of a particularelement (Figure 3) can be given in the form

N12 =[

N1 0 N2 0

0 N1 0 N2

](29)

where

N1 = 12 (1 − ), N2 = 1

2 (1 + ) (30)

To obtain the expression of Ne, introduce a new function ij () [28] such that

ij () =

1 when ∈ side i − j

0 otherwise(31)

and then Ne can be given as

Ne =[

N∗1 0 N∗

2 0 N∗3 0 N∗

4 0

0 N∗1 0 N∗

2 0 N∗3 0 N∗

4

](32)

where N∗1 = 12()N1 + 41()N2, N∗

2 = 23()N1 + 12()N2, N∗3 = 34()N1 + 23()N2

and N∗4 = 41()N1 + 34()N2.

Substituting Equations (17a), (17b) and (19) into the modified variational functional below[28]

= ∑e

e = ∑e

[∫ ∫e

1

2Te De d −

∫eu

teue d −∫

et

(te − te)ue d −∫

eI

teue d

](33)



one can obtain the form

e = 12 cT

e Hece − cTe Ge de − cT

e he + dTe ge + terms without ce or de (34)

where e represents the element complementarity energy, e = eu ∪ et ∪ eI , while eu =u ∩ e, et = t ∩ e, and eI is the inter-element boundary of element ‘e’.

Taking the vanishing variation of Equation (34) with respect to ce results in

e

ce

= 0 ⇒ ce = H−1e (Gede + he) (35)

and then the element stiffness equation can be obtained in a similar way with respect to de

e

de

= 0 ⇒ Kede − Pe = 0 (36)

where

Ke = GTe H−1

e Ge (37)

is the element stiffness matrix, and

Pe = ge − GTe H−1

e he (38)

is the vector of the equivalent nodal forces. The auxiliary matrices He, Ge, he and ge areexplicitly expressed as

He =∫

e

QTe Ne d =

∫e

NTe Qe d (39a)

Ge =∫

eI +et

QTe Ne d (39b)

he = − 1

2

∫e

NTe be d − 1

2

∫e

(QTe

ue + NT

e

t e) d +∫

eu

QTe u d (39c)

ge =∫

et

NTe (te −

t e) d −∫

eI

NTe

t e d (39d)

Once the element stiffness matrix Ke and the equivalent nodal forces Pe for all the elementshave been evaluated, the structural stiffness equation can be assembled in the following matrixform analogous to that adopted in conventional FEM [30]:

Kd = P (40)

3.2. Contact interface equation

According to Equation (40) the structural HT FE stiffness equation for each contacting bodycan be formulated as follows:

K()d() = P() + f () (41)



where f () are contact forces. In most contact problems the contact region is small comparedto the domain of the structure. Further, the subdomains () associated with the HT FE modelcan be viewed as substructures. Therefore, static condensation [31], which is used to condensethe model down to a smaller one which involves nodes within the potential contact surfacesonly, would be efficient for contact analysis. After introducing the prescribed displacementsinto Equation (41) and splitting the unknowns into two parts, one (denoted by subscript c)

containing unknowns associated with the potential contact surfaces, and the other (denoted bysubscript r) containing the remaining unknowns, one arrives from (41) at the following formin terms of submatrices for each contacting body⎡

⎣K()rr K()

rc

K()cr K()

cc

⎤⎦⎧⎨⎩d()

r

d()c

⎫⎬⎭ =

⎧⎨⎩P()

r

P()c

⎫⎬⎭+

0

f ()c

(42)

The solution of the first submatrix equation for d()r yields

d()r = K()−1

rr (P()r − K()

rc d()c ) (43)

Since d()r only indirectly depends on f ()

c , it can be eliminated by substituting (43) into thesecond submatrix equation of (42). This leads to a much smaller system of equations

K∗()d()c − f ()

c = P∗() (44)

where the following notations are used:

K∗() = K()cc − K()

cr K()−1

rr K()T

cr (45a)

P∗() = P()c − K()

cr K()−1

rr P()r (45b)

Here K∗() and P∗() are obtained at an intermediate step of the Gaussian elimination process,which would result in a significant reduction in computational effort. The advantage of staticcondensation is that only a smaller system of equations has to be solved during the contactanalysis. Also, the calculation of matrices K∗() and P∗() needs to be done only once, so thattheir results can be used many times in the contact solution process.

The incorporation of the potential contact surfaces ()c is of the form

[K∗(1) −I 0 0

0 0 K∗(2) −I

]⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

d(1)c

f (1)c

d(2)c

f (2)c

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

=

P∗(1)

P∗(2)

(46)

where nc refers to the current number of CNPs along the contact zone, K∗() is 2nc × 2nc

matrix, I the identity 2nc × 2nc matrix, and P∗(), d()c , f ()

c are 2nc column vectors. Sofar, a unique solution to (46) cannot be obtained because the number of unknowns is 8nc



while the number of equations is 4nc only. Therefore, it is necessary to provide 4nc morecontact conditions as additional equations to (46) for the solution. For conciseness, four of theindependent compatibility and equilibrium conditions, for example, Equations (10), (13a), (13b)and (16a), at each CNP may be rephrased in matrix notation as

⎡⎢⎢⎢⎣

tg i −1 0 0 −tg i 1 0 0

0 0 1 0 0 0 1 0

0 0 0 1 0 0 0 1

0 0 cos i sin i 0 0 0 0

⎤⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

u(1)iX

u(1)iY

f(1)iX

f(1)iY

u(2)iX

u(2)iY

f(2)iX

f(2)iY

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

gi

0

0

0

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(47)

After introducing the contact conditions (47) into (46), one obtains the resulting contactinterface equation in the manner

SX = C (48)

where S is the contact coefficient matrix and X are the unknowns including contact forces andassociated displacements. Since zero appears on the diagonal of S, which poses some difficultiesin the solution process, the above system can be solved using Gaussian elimination withpartial pivoting [32]. Then the contact forces and displacements are computed simultaneously.Accordingly, the contact stress for each particular CNP i is readily given as

in = (f(1)iY cos |i | − f

(1)iX sin |i |)/Ai (49)

where Ai represents the contact subarea dominated by CNP i.

3.3. Solution strategy

The distinct characteristic of the present algorithm is that the problem can actually be analysedby only one total load step in the case of frictionless contact. In order to determine theactual contact boundary, the bisection method [32] is adopted in the iterative process. When[f (1)

n ]A(1) > TOL and [f (1)n ]B(1) < −TOL, where TOL designates a user-specified tolerance, the

actual contact boundary is somewhere (denoted by C()) between A() and B() as shown inFigure 4. The left CNP may be considered the lower limit of the contact boundary and theright is the upper limit. For the actual contact boundary, the position of B() can be adjustedfollowing the bisection method and Equation (48) is resolved until [f (1)

n ]B(1) > −TOL.The corresponding overall solution algorithm is summarised as follows:

Step 0: Initialize nc and other information.Step 1: Compute (3) for each CNP, and calculate (41) for each contacting body.Step 2: Calculate (45a), (45b) for each contacting body.



( )1A( )1B

( )2B

( )2A

( )1C

( )2C

Figure 4. Determination of actual contact boundary.

Step 3: Construct (48) and solve it using Gaussian elimination with partial pivoting.Step 4: Check the normal force of f

(1)nc

at the last CNP nc.

IF |f (1)nc

| < TOL THENgo to Step 6

ELSEIF f

(1)nc

> 0 (compressive) THENnc = nc + 1, return to Step 2

ELSEgo to Step 5

END IFEND IF

Step 5: Adjust the position of the last CNP nc using the bisection method, return to Step 1.Step 6: Output the results.

4. NUMERICAL EXAMPLES

In this section, a series of 2D problems covering the three conceptually different contactsituations, namely, receding, conforming and advancing, is considered as a benchmark to assessthe applicability of the direct constraint HT FE approach and to make comparisons withanalytical solutions as well as with results from the commercial FE software ABAQUS 6.2(element type: CPE4H; normal behaviour: ‘Hard’ contact). The plane strain state is assumedand four-node HT elements are used in all examples.

4.1. Compression of a layer on a foundation

The layer problem, as graphically depicted in Figure 5, pertains to a receding contact withoutfriction, the final size of the contact zone being smaller than the original but independent ofthe magnitude of load applied. Due to the symmetry about the central line, only the right halfof the domain needs to be modelled with the geometry, loads and boundary conditions.

The HT FE model for this problem consists of 1252 elements (1371 nodes) and a finermesh has been uniformly distributed near the centre of the contact zone. A potential contactlength of 20 mm is chosen on both bodies, and contact surfaces on both bodies are discretisedby 21 equidistant nodes, with interval size 1 mm. To enable direct comparison with analytical



100mm

P/2

X

Y

80mm

10m

m

( )1Ω

( )2Ω

The layer

The foundation

P=5N/mm

200m

m

Figure 5. HT FE model of a layer on a foundation.

solution [33], the Dunders’ parameter

= E(2)(1 − (1)2) − E(1)(1 − (2)2

)

E(2)(1 − (1)2) + E(1)(1 − (2)2

)(50)

is set to be 0.229. The parameter can be viewed as an index for the mismatch in the uniaxialcompliances of the two materials. For identical materials, or = 0, the contact stresses areseen to be completely independent of the elastic constants. In the case of = 0.229, onecan obtain many combinations of the elastic constants. In this example Young’s modulusE(1) = 4000 MPa, E(2) = 7412.28 MPa and Poisson’s ratios (1) = (2) = 0.35 are takenfor two bodies, respectively. Figure 6 shows the results of contact stresses obtained by thepresent approach and comparison is made with the results obtained by the analytical solutionas well as with the solution obtained by ABAQUS. It is clearly observed that the stressesdetermined by the present approach coincide well with those found by ABAQUS, but areobviously discrepant with analytical solution at some CNPs. The reason for the discrepanciesmay be that the foundation domain in this example is finite, while the analytical solution ofKeer et al. [33] holds true for a half space only. The size of contact zone used by both the



0 2 4 6 8 10 12 14 160.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

Con

tact

str

ess

(MP

a)

Contact zone (mm)

Analytical solution (Keer et al) ABAQUS 6.2 Present approach

Figure 6. Contact stresses along the contact zone for the elastic case.

0 4 8 12 16 20 24-0.0030

-0.0025

-0.0020

-0.0015

-0.0010

-0.0005

Ver

tical

dis

plac

emen

t (m

m)

Contact zone (mm)

ABAQUS 6.2 (The layer) ABAQUS 6.2 (The foundation) Present approach (The layer) Present approach (The foundation)

Figure 7. Vertical displacements along the contact zone for the elastic case.

present approach and ABAQUS is 12 mm, which is a little bit larger than that of the analyticalsolution. The vertical displacement distributions along the contact zone are plotted in Figure7. Only the ABAQUS results are provided here for comparison because the correspondinganalytical results are not available in the literature. Hollow and filled circles separate eachother from X = 12 mm. This indicates that CNPs with X<12 mm are in contact while oneswith X>12 mm are open. Furthermore, X = 12 mm is the contact boundary used by presentapproach, which coincides with the value as illustrated in Figure 6.

Figures 8 and 9 show the distributions of contact stresses and associated displacements,respectively, when one of the contact bodies becomes rigid. As can be observed from the



0 2 4 6 8 100.0

0.1

0.2

0.3

0.4

0.5

Con

tact

str

ess

(MP

a)

Contact zone (mm)

ABAQUS 6.2 Present approach

Figure 8. Contact stresses along the contact zone for the almost rigid case.

0 2 4 6 8 10 12 14 16-0.0001

0.0000

0.0001

0.0002

0.0003

0.0004

Ver

tical

dis

plac

emen

t (m

m)

Contact zone (mm)

ABAQUS 6.2 (The layer) ABAQUS 6.2 (The foundation) Present approach (The layer) Present approach (The foundation)

Figure 9. Vertical displacements along the contact zone for the almost rigid case.

two figures, the stresses obtained from the present approach agree well with those derived byABAQUS. It can be clearly observed from Figures 6 and 8 that the size of contact zone inthe case of elastic foundation is larger than that in the case of rigid foundation whereas thereverse results appear for the maximum contact stress.

To examine the convergent performance of the method four more meshes are used over thesolution domain (Figure 10). The potential length of contact zone is assumed to be the sameas that given in Figure 5 and details of other data for each mesh are listed in Table I. Itis evident from Figure 11 that the numerical results converge to the analytical solution along



Figure 10. Four more meshes for example 4.1.

Table I. Different meshes for example 4.1.

No. of Interval size Computationalelements No. of nodes (mm) time (s)

Figure 10(a) 107 141 4 9Figure 10(b) 176 217 2.59 13Figure 10(c) 296 351 2 35Figure 10(d) 525 595 2 110Figure 5 1252 1371 1 165

with increases in mesh density. Moreover, the accuracy of the results depends largely on thedensity, namely the length of contact element, of CNPs distributed in the potential contactzone. When X = 4 mm no difference can be observed between the numerical and analyticalresults. For X < 4 mm numerical results are lower than the analytical solution, whereas thenumerical results is higher than the analytical solution when X > 4 mm. Form the analysis, itcan be seen that the numerical results can converge to analytical solution if the FE mesh issufficiently fine and the interval size is sufficiently small.

The HT FE code for contact problems has been executed in DELL workstation. The cor-responding computational time for each mesh is also listed in Table I. Because the efficiencydepends greatly on the choice of the initial number nc of CNPs it is of importance to preset aproper value for nc at the beginning of solution. It is noted that the listed time for each meshis related to the last iteration.



0 2 4 6 8 10 12 14 160.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Con

tact

str

ess

(MP

a)

Contact zone (mm)

Analytical solution (Keer et al)

Fig.10(a) Fig.10(b) Fig.10(c) Fig.10(d) Fig.5

Figure 11. Convergence for different meshes.

4.2. Compression of a rectangular punch on a foundation

The punch problem, as graphically depicted in Figure 12, pertains to a conforming contactwithout friction, the final size of the contact zone, independent of the magnitude of loadapplied, being coincident with the original. Due to the symmetry about the central line, onlythe right half of the domain needs to be modelled with the corresponding geometry, loads andboundary conditions.

The HT FE model for this problem consists of 1394 elements (1503 nodes) and the out-of-contact zone in the vicinity of the corner is discretised by the same mesh, mirrored by thecorner. To obtain more accurate values in the vicinity of the corner, a local refinement of thegrid has been employed to take into account the stress singularity. In a similar manner to thatin the former example, 19 nodes are spaced on both contact surfaces, where the interval sizeof the 11 nodes near the end is 1 mm while that of others is 5 mm.

The case of an elastic punch on an elastic foundation is first investigated. For this purpose,the same linear elastic material properties characterised by Young’s modulus of 4 × 103 MPaand Poisson’s ratio of 0.35 are taken for both bodies, respectively. The comparison betweenpresent approach and ABAQUS presented in Figure 13 shows that the contact stresses obtainedfrom the two methods above agree very well except for those near the edge of the punch(X = 49.9 mm, 50 mm). A singularity exists at the end of contact zone which, in practice,results in abrupt increase of contact stress. As demonstrated in Figure 14, although the verticaldisplacement distributions obtained by the two methods do not coincide, the maximum relativeerror is only 1.97% which can be negligible in practical engineering.

Consider next the case that the punch becomes rigid. In order to maintain fidelity with theassumptions of the analytical solution, a Young’s modulus 105 times stiffer is taken for thepunch. Figures 15 and 16 show, respectively, the contact stresses and associated displacementsof two bodies along the contact zone. Accordingly, an analytical stress distribution for this case



100mm

The foundation ( )2Ω

( )1ΩThe punch

q=1.2MPa

200m

m

X

50mm q

Y

100m

m

Figure 12. HT FE model of a punch on a foundation.

0 10 20 30 40 50 600.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Con

tact

str

ess

(MP

a)

Contact zone (mm)

ABAQUS 6.2 Present approach




0 10 20 30 40 50 60-0.036

-0.034

-0.032

-0.030

-0.028

-0.026

-0.024

-0.022

Ver

tical

dis

plac

emen

t (m

m)

Contact zone (mm)

ABAQUS 6.2 (The punch) ABAQUS 6.2 (The foundation) Present approach (The punch) Present approach (The foundation)

Figure 14. Vertical displacements along the contact zone for the elastic case.

0 10 20 30 40 50 600

2

4

6

8

10

12

14

Con

tact

str

ess

(MP

a)

Contact zone (mm)

Analytical solution ABAQUS 6.2 Present approach


is expressed by the equation [2]

n(X) = P

√

l2 − X2(51)

where l is the half length of the contact zone and X denotes the distance from the centre.For X 40 mm, stress distributions by both present approach and ABAQUS agree well withanalytical solution, while for 40 mm < X < 50 mm, a slight difference was observed betweennumerical and analytical results. At X = 50 mm (singularity point), unlike in the analytical



0 10 20 30 40 50 60-0.036

-0.034

-0.032

-0.030

-0.028

-0.026

-0.024

Ver

tical

dis

plac

emen

t (m

m)

Contact zone (mm)

ABAQUS 6.2 (The punch) ABAQUS 6.2 (The foundation) Present approach (The punch) Present approach (The foundation)

Figure 16. Vertical displacements along the contact zone for the almost rigid case.

method, either present approach or ABAQUS can only provide a finite value. Unfortunately, thevertical displacements from the present method cannot be verified by the analytical solution, asthe analytical one holds for the half-space foundation and is given up to an arbitrary point only.In Figure 16 displacements obtained by the present approach are lower than that of ABAQUSand the highest relative error is only 2.44%.

4.3. Compression of two identical cylinders

The Hertz problem of two identical cylinders, as graphically depicted in Figure 17, pertains toan advancing contact without friction, a final size of the contact zone increasing progressivelyas the magnitude of load increases. Due to the symmetry about the central line, only the righthalf of the domain needs to be modelled with the geometry, loads and boundary conditions.

The HT FE model consists of 1908 elements (1792 nodes) and a potential contact arcof 5.74 is chosen on the surfaces of both bodies, and each contact surface is divided into26 potential contact nodes, with interval size 0.23. Figure 18 illustrates the contact stressdistributions when both contact bodies are elastic, with material properties taken to be thesame as those in example 4.2. Figure 20 shows the corresponding stresses when one of thetwo contact bodies becomes rigid. For reference purpose, the analytical (Hertz) solutions [2]calculated from

n(X) = 2P

l2

√l2 − X2 (52)

together with

l2 = 4PR

E (53a)

1

R = 1

R(1)+ 1

R(2)(53b)



P/2

( )1ΩThe upper cylinder

100m

m10

0mm

Y

( )2ΩThe lower cylinder

P=1000N/mm

X

Figure 17. HT FE model of two identical cylinders.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00

20

40

60

80

100

120

140

160

180

200

Con

tact

str

ess

(MP

a)

Contact zone (mm)

Analytical solution (Hertz) ABAQUS 6.2 Present approach




0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0-1.24

-1.20

-1.16

-1.12

-1.08

-1.04

-1.00

-0.96

-0.92

-0.88

-0.84

-0.80

-0.76

Nor

mal

dis

plac

emen

t (m

m)

Contact zone (mm)

ABAQUS 6.2 (Upper) ABAQUS 6.2 (Lower) Present approach (Upper) Present approach (Lower)

Figure 19. Normal displacements along the contact zone for the elastic case.

0.0 0.5 1.0 1.5 2.0 2.5 3.00

50

100

150

200

250

Con

tact

str

ess

(MP

a)

Contact zone (mm)

Analytical solution (Hertz) ABAQUS 6.2 Present approach


1

E = 1 − (1)2

E(1)+ 1 − (2)2

E(2)(53c)

are also plotted in these figures. Here R and E are, respectively, the relative curvature ratioof two cylinders and the effective modulus. Additionally, the numerical results obtained byABAQUS are again plotted in the figures. It is evident that a good agreement is achievedamong the results of Hertz, ABAQUS and the present approach (Figures 18 and 20). For thesimilar reason as in the previous two examples, the analytical normal displacements on thecontact surfaces are not available in the literature and therefore a comparison between the twomethods cannot be made in Figures 19 and 21.



0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5-0.20

-0.18

-0.16

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

Nor

mal

dis

plac

emen

t (m

m)

Contact zone (mm)

ABAQUS 6.2 (Upper) ABAQUS 6.2 (Lower) Present approach (Upper) Present approach (Lower)

Figure 21. Normal displacements along the contact zone for the almost rigid case.

5. CONCLUSIONS

A new direct constraint approach for the HT FE analysis of frictionless contact problems hasbeen developed in this paper. The computational time may be remarkably reduced by performingstatic condensation on the potential contact surfaces of each contacting body. Additionally, thecurrent research can directly guarantee the imposition of the active contact conditions and avoidthe introduction of user-defined penalty parameters or artificial Lagrangian multipliers.

A novel feature of the advocated algorithm is that the nodal tractions and displacementson the contact interface can be simultaneously calculated with the aid of Gaussian eliminationwith partial pivoting. Moreover, the total load can be applied in just one load step for thethree situations of frictionless elastic contact analysis described. For more exact definition ofthe contact zone, the bisection method can predict its size accurately. Three classical examplespertinent to the above situations have been examined to prove the reliability of the algorithmand the applicability to a wide family of contact problems.

The present approach opens up a promising way of using the HT FE method for the solutionto contact problems. Extension of this work is possible, such as inclusion of frictional effectsin the contact interface model. This work is underway.

REFERENCES

1. Hertz H. Miscellaneous Papers on the Contact of Elastic Solids (Translated by Jones DE). McMillan:London, 1896.

2. Johnson KL. Contact Mechanics. Cambridge University Press: Cambridge, 1985.3. Böhm J. A comparison of different contact algorithms with applications. Computers and Structures 1987;

26:207–221.4. Mijar AR, Arora JS. Review of formulations for elastostatic frictional contact problems. Structural and

Multidisciplinary Optimization 2000; 20:167–189.5. Qin QH, He XQ. Variational principles, FE and MPT for analysis of non-linear impact-contact problems.

Computer Methods in Applied Mechanics and Engineering 1995; 122:205–222.6. Stadler JT, Weiss RO. Analysis of contact through finite element gaps. Computers and Structures 1979;

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7. Landenberger A, El-Zafrany A. Boundary element analysis of elastic contact problems using gap finiteelements. Computers and Structures 1999; 71:651–661.

8. Luo JW. Penalty finite element method for the frictionless elastic contact problem. Chinese Journal ofMechanical Engineering 1984; 20:82–86.

9. Asano N. A virtual work principle using penalty function method for impact contact problems of two bodies.Bulletin of JSME 1986; 29:731–736.

10. Barlm D, Zahavi E. The reliability of solutions in contact problems. Computers and Structures 1999; 70:35–45.

11. Wang KY, Qin QH, Kang YL, Lin XH, Wang JS. Solving elastic frictionless contact problems by couplinghybrid-Trefftz finite element method and spring elements, submitted.

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13. Wriggers P, Zavarise G. Application of augmented Lagrangian techniques for non-linear constitutive laws incontact interfaces. Communications in Numerical Methods in Engineering 1993; 9:815–824.

14. Dostál Z, Friedlander A, Santos SA. Analysis of semicoercive contact problems using symmetric BEM andaugmented Lagrangians. Engineering Analysis with Boundary Elements 1996; 18:195–201.

15. Rebel G, Park KC, Felippa CA. A contact formulation based on localised Lagrange multipliers: formulationand application to two-dimensional problems. International Journal for Numerical Methods in Engineering2002; 54:263–297.

16. Andersson T, Fredriksson B, Allan-Persson BG. The boundary element method applied to two-dimensionalcontact problems. In New Developments in Boundary Element Methods, Brebbia CA (ed.). CML Publications:Southampton, 1980; 247–263.

17. Ma SY. The solution of two-dimensional frictionless elastic contact problems using boundary element method.Journal of Hebei Institute of Technology 1984; 2:11–21.

18. Kararmi G. Boundary element analysis of elastoplastic contact problems. Computers and Structures 1991;41:927–935.

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20. Martín D, Aliabadi MH. Boundary element analysis of two-dimensional elastoplastic contact problems.Engineering Analysis with Boundary Elements 1998; 21:349–360.

21. Martín D, Aliabadi MH. A BE hyper-singular formulation for contact problems using non-conformingdiscretization. Computers and Structures 1998; 69:557–565.

22. Du LH, Deng LJ, Chen HJ, Ye JQ. Direct constraint procedure to solve contact problems in hydrostructures.Journal of Tsinghua University 2003; 43:1534–1537.

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26. Jirousek J, Guex L. The hybrid-Trefftz finite element model and its application to plate bending. InternationalJournal for Numerical Methods in Engineering 1986; 23:651–693.

27. Jirousek J, Wroblewski A. T-elements: state of the art and future trends. Archives of Computational Methodsin Engineering 1996; 3:323–434.

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