FRICTIONLESS CONTACT WITH BEM USING QUADRATIC PROGRAMMING
By Srdan Simunovic' and Sunil Saigal2
ABSTRACT: The contact surface, with its accompanying load transfer, may well constitute the critical factor in a structural member. Thus it is essential to be able to perform contact stress analysis of a component accurately and efficiently. Considerable research effort is represented in the literature for contact analysis using finite elements. To obtain reliable results in the contact zone, it is necessary to provide a very fine discretization in that zone. Many distinct contact zones may exist, which may force the entire domain, and not just the contact zones, to be discretized finely. This generally leads to an excessive number of degrees of freedom (dof), resulting in an uneconomical, and sometimes intractable, analysis. The boundary element method (BEM), which deals with the discretization of only the boundary of the structure being analyzed, may be used to circumvent these difficulties and provide accurate, economical results. While a fine discretization of the contact zone is still unavoidable, the BEM leads to a smaller number of dof's because the rest of the boundary need not have a fine mesh. The problem of frictionless contact between an elastic body and a rigid surface is formulated as an optimization problem. Three distinct functions are defined in terms of the unknown variables (displacements and tractions) corresponding to the contact surface, and expressed as quadratic objective functions that are to be minimized. The solution is obtained using the standard quadratic programming techniques of optimization. A number of example problems with straight and curved contact boundaries were solved. The present formulations were validated through comparison of the test problems with existing alternative solutions.
INTRODUCTION
Contact problems are a common occurrence in engineering practice. Practical applications of such problems occur in linear-elastic, small-displacement range, such as in the case of operating mechanical components; as well as in the materially nonlinear, large-displacement, large-strain range, such as in the case of metal-forming operations. Some analytical solutions for the contact of objects with simple geometries exist, and a survey of such solutions may be found in, for example, Kalker (1977) and Gladwell (1980). For most practical cases, however, such analytical solutions are not possible, and numerical techniques must be employed. A number of numerical approaches for the solution of contact problems have appeared in the literature in recent years, including: (a) Incremental and iterative procedures; (b) the Lagrange multiplier method; (c) the penalty method; (d) the perturbed Lagrangian method; (e) the augmented Lagrangian method; and (f) methods based on the mathematical programming techniques of optimization theory. A substantial amount of progress has been made with these formulations using the finite element method (FEM). Surveys of linear and nonlinear contact analysis using the FEM may be found in, among others, Cheng and Kikuchi (1985), Johnson and Quigley (1989), and Eterovic and
'Res. Asst., Dept. of Civ. Engrg., Carnegie Mellon Univ., Pittsburgh, PA 15213-3890.
2Assoc. Prof., Dept. of Civ. Engrg., Carnegie Mellon Univ., Pittsburgh, PA. Note. Discussion open until February 1, 1993. To extend the closing date one
Bathe (1991). A comprehensive treatment of the subject of contact mechanics in elasticity is available in Kikuchi and Oden (1988). Recent works, such as Belytschko and Neal (1991), on the enhancement of contact algorithms for efficiency and for vectorization have also been reported.
In the recent years, the boundary element method (BEM) has become available as an alternative numerical procedure for structural analysis. This method possesses distinct advantages over the FEM for the solution of contact problems. Consider, for example, a loaded plate with multiple pins, where contact of the plate occurs with each pin under the applied load. Since numerical procedures require a fine discretization of the contact zone to yield accurate results, a fine FEM mesh at each of the pins will result in a fine discretization of the entire domain, leading to an uneconomical, if not intractable, analysis. The BEM, which deals with the discretization of only the boundary of the object, will allow a fine discretization of the contact zone and still provide accurate, economical results. Other advantages of the BEM for the analysis of contact problems have been noted by Andersson and Persson (1982), the chief of which is that the contact pressure, obtained as boundary traction, is a primary unknown quantity, solved with the same accuracy as the unknown displacements. Some literature concerning the analysis of contact problems using the BEM has appeared recently. Most of these papers are based upon the incremental and iterative class of procedures. A brief survey of such works may be found in, among others, Abdul-Mihsein et al. (1986). The contact problems dealt with in these works include frictionless and frictional thermoelastic and fracture mechanics (Kar-ami 1983, two-dimensional elastoplastic creep (Tsuta and Yamayi 1983), and coupling BEM contact with FEM analysis (Tanaka 1983).
The mathematical programminng (MP) techniques that have been used with the FEM for the solution of contact problems with a great deal of success have not received much attention yet with the BEM. The advantages of these techniques over the iterative methods have been documented by, among others, Klarbring (1986). A textbook description of such techniques for use in conjunction with the FEM may be found in Haug and Arora (1979) and others. Several formulations concerned with the minimum principles on the boundary for unilateral contact problems have been presented by Panagiotopoulos and Lazaridis (1987). The application of a discretization procedure to these formulations, such as the FEM or the BEM, leads to a MP problem, requiring the use of optimization techniques. Both static (Panagiotopoulos and Lazaridis 1987) and dynamic (Misopoulos-Papasoglou et al. 1991) contact problems have been considered. Stavroulakis et al. (1991) noted that the MP formulations are considerably economical compared with the trial-and-error methods. Similar formulations, where the complementarity conditions lead to a mathematical programming approach, have been developed in the area of discrete plasticity by Maier and Novati (1983) and Maier (1971). Klarbring (1986) and Klarbring and Bjorkman (1988) recently provided MP formulations with the FEM for contact problems with friction including varying contact surfaces. Kwak and Lee (1988) derived an MP-based two-dimensional contact formulation with the BEM. Gakwaya and Lambert (1991) have presented a three-dimensional extension to this derivation without providing any numerical results. A systematic development of contact analysis using optimization techniques with BEM is not available thus far.
A simple formulation of the frictionless contact between elastic bodies and rigid surfaces using quadratic programming (QP) is presented in this
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paper using the BEM. The formulation is restricted to elastic bodies undergoing small displacements. The governing BEM equations and the Kuhn-Tucker conditions for contact are used to define the quadratic objective function as well as the linear constraints. Three distinct objective functions are formulated and studied in this paper. A variety of example problems, involving both straight and curved elastic boundaries coming in contact with both straight and rigid surfaces, are presented. A good agreement of the results with existing solutions is obtained.
THEORETICAL FORMULATION
Consider an elastic body occuping a volume ft and coming into contact with a rigid surface of arbitrary geometry, as shown in Fig. 1. The governing equations of elasticity and the corresponding boundary conditions may be expressed as
(X + |i,)w,-,y + wj, + ¥ , = 0; x £ ( l - • • (1)
K;(x) = M;(x); x £ T„ (2)
tfa) = f,(x); x E r, (3) r(x) = 0 for 5" > 0; x G Tc (4a)
t"(x) > 0 for 8" = 0; x e Tc (46)
r = 0; x e rc (5) with
8"(x) = e"(x) + U"(x) (6)
where X. and u. = Lame constants; w, and f, denote the displacements and tractions, respectively; ^ = the body forces; the overbar denotes prescribed values; T„ and T, = the portions of the surface T that bounds the volume ft on which the displacements and the tractions, respectively, are specified; Tc = the portion of T likely to come in contact with the rigid surface; t" and u" denote, respectively, the pressure on and the displacement of the
t
f t ELASTIC J ^ k BODY. T
FIG. 1. Elastic Body in Contact with Rigid Surface
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elastic body along the normal n to the rigid surface; V = the pressure on the elastic body along the unit tangent T to the rigid surface; e" = the initial gap or clearance between the elastic body and the rigid surface along normal n; and T = Fu + F, + Tc. It is noted that the boundary condition in (5) results from the fact that the contact between the elastic body and the rigid surface is frictionless.
Following the standard BEM procedures, (1) can first be expressed in the boundary integral form as Somigliana's identity (Banerjee and Butter-field 1981), and then upon discretization of the boundary using boundary elements, the final equations may be expressed in the matrix notation as
Fu = Gt + f (7) where F and G = the BEM system matrices; u and t = the vectors of nodal displacements and tractions, respectively; and f = a vector incorporating such volume effects as body forces and thermal loads, etc. The details for obtaining (7) and for the numerical evaluation of various matrices may be found in standard textbooks [e.g., Banerjee and Butterfield (1981)]. After the assembly process, matrix entries corresponding to the degrees of freedom at the contact region Tc are transformed into their respective local coordinate systems. This is done to express the contact relations in terms of components along the normal and the tangent to the contact surface as
p1 trans
I"1 trans
T ,=
= TfF,-Tfc (8a)
= T/Ge/T, (8b)
cos a, —sin a, sin a, cos a,
(8c)
where T, denotes the transformation matrix, i and e denote the node and the element being transformed, respectively. The same procedure may be done at the level of the kernel evaluation proposed by Andersson and Persson 1982). For the purposes of the contact formulation presented in this paper, it is convenient to partition (7) as
F F* FT„ F *• nu
F„, F„ FT, F„
F F 1 n F F
F " F„ F F *• nn.
H u. K
We)
Glm
Gtu
GT„ .G„„
G„, G„ GT, G„,
GUT
Gtr
GTT
Gr„T
Gun G,„ GT„ G„„_
ft„l tt
t ;
U?J
• + •
lsA f< f T i f"
(9)
where the subscripts u, t, and c refer to the quantities corresponding to the boundaries T„, T„ and rc, respectively; the superscripts T and n refer, respectively, to the quantities along the tangent and the normal to the rigid surface; Vc = tT(x), x G Tc and similarly for t", uj, and u". The boundary conditions given by (2) and (3) are now applied to (9). Additionally, all the unknown variables (displacements and tractions) corresponding to T„ and r„ and not Tc, are collected together into a vector to obtain
"•aa
with
A. M = I"8- B«l (b4 + M (10)
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Xfl = I u" 1 (11a)
b„ = J t" I (life)
t.= if" | (He)
Applying the condition of frictionless contact in (5) and rearranging (10), we obtain
A,x, = »h (12)
where
Ai = L"cc — A c aA a a Aa c , \ca\aa Dac — BCCJ (13)
* = { ? ! } •• <14> and
b t = [Bc„ - A ^ A - ^ J b , + (f? - A ^ A ^ f J (15)
The matrix relation in (12) incorporates (1), (2), (3), and (5), which define the contact problem. Eq. (4) may be restated as (Haug et al. 1977).
r 8 " = 0 (16)
f > 0 (17)
S" > 0 (18)
The complete solution of the contact problem requires the simultaneous consideration of (12), (16), (17), and (18). This may be accomplished by casting these equations as a QP problem in the optimization theory. For this purpose, three different formulations to reduce (12) and (16)-(18) into a QP problem were considered, as described in the following. To accomplish this, it is first necessary to express, in the quadratic form, the function Fu given as
F1 = f f-8- dYc (19) JTc
Express the variables u", t", and e" in terms of the shape functions as
NNODE
W = 1 ha)u1 (20a) f = i
NNODE
t" = 2 ht(Ot? (206)
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•B- = 2 ha)*? (20c) ( = 1
where NNODE = the number of nodes per element; ht = the shape functions; the superscript i denotes the value of the variable at node /; and £ = a nondimensional coordinate along the length of the element. Substituting the relations in (20) into (19), we get the quadratic form of Fj as
F, = cfXl + | xfHf-Xj (21)
with
Cl = { Je
ll! = 0 J 0 0
(22)
(23)
NCELM
j = E r (24)
H = J_7 hty dk (25)
where / = V(3x/d£)2 + (du/d£)2; E" = the vector of initial gap values e" at the nodes; NCELM = the number of elements used for discretizing Tc; the summation sign in (24) denotes assembly of element matrices; the superscript sym denotes the symmetric part of the matrix; and dx/d£, and dyl d£ are obtained by first expressing x and v using shape functions, similar to the form in (20), and then taking the appropriate derivatives. It is noted that the quadratic term in (21) results from using the property that xrHx = xrHsymx. A simplified expression for the quadratic function resulting from the condition in (16) may be obtained by enforcing that condition at discrete nodal locations. The quadratic function is then of the form
NCNOD
Fx = 2 t?S? (26)
where NCNOD = the total number of nodes on the boundary Tc; and the superscript i refers to the value of the quantity at node i. Eq. (26) may be expressed in the matrix form similar to that shown in (21)-(25), except now Jfi = b,jle where 8,7 = the Kronecker delta and le = the corresponding element length. The quadratic function shown in (21) is now used to develop the statements QP1 and QP2.
Statement QP1
minimize Fx = cfxj + | x f H r * ! (27)
subjected to
AiXj = bx (28a)
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x ^ l , (286)
where
>i = 1 ~ o e " J" < 2 9 > The inequality condition above ensures that the elastic body does not penetrate the rigid surface, and that the pressure at the contact surface is non-negative.
Statement QP2 The composite function
F2 = cfxx + | x f H r * i + - (AlXl - bx)2 (30)
is now minimized. In the classical quadratic form, upon collecting linear and quadratic terms separately, we have
minimize F2 = c$xx + - x1rHfmx1 (31)
subjected to
Xi a I, (32)
with
c2 = d - - Afbx (33)
Rsym = Hsym + 1 A r A (34)
U.
where ix = a penalty parameter. In this study fx = 1 was employed.
Statement QP3 To obtain this statement, consider first the deformation Aj, of the elastic
body without the presence of the rigid contacting body, i.e., t? = 0. Substituting this condition in (10), then eliminating Xa and rearranging, we obtain
Ai = u"c\tr° (35)
or in the expanded form
^ l — \&cc ~~ A c a A a a Aac) [\**ca ~~ A c a A a a aaa)Da + ^ t c — l\ca\aa la)\
(36)
Consider next the deformation, A2, of the elastic body in the absence of the external applied tractions t„ i.e., tr = 0, but subjected to the contact tractions tn
c. Again, (10), after some manipulation, leads to
A2 = u?|t,=0 = A3t? (37)
where
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A3 = (Acc - AcaA-SAj-KKcc ~ A ^ A i ' B J (38)
For this case, the contact displacements may be obtained using (35) and (37). Following Chand et al. (1976), for linear-elastic behavior, the contact solution A may be obtained as A = Ax + A2. The resulting gap between the elastic body and the rigid surface is then given as
8" = E" + Aj + A2 (39)
The expression for 8" in (39) is now substituted into either (19) or (26) to obtain the quadratic expression to be minimized in order to obtain the solution. The QP statement can now be written as
minimize F3 = cjx3 + - x^Hf^Xa (40)
subjected to
A3x3 & b3 (41a)
x3 a 0 (416)
where
x3 = t£ (42a)
c3 = J(e" + A0 (426)
H3 = A3 j (42c)
b3 = - ( e " + A0 (42d)
It is noted that, in this case, the unknown vector is in terms of the contact tractions only, unlike the previous two cases where the contact displacements are additionally involved.
The functions Ft in (27) and the function F2 in (30) constituting the statements QP1 and QP2, respectively, of the QP problem both involve the matrix Hfm in their quadratic terms. Since Hfm is an indefinite matrix, as seen from (23), the objective functions Fx and F2 are both nonconvex. Inertia-controlling methods (Gill et al. 1988) must be employed to control the indefiniteness of the projected Hessian matrix for these functions. These methods converge in a finite number of steps, provided no degenerate stationary points exist in the feasible region. Degenerate stationary points may be caused by the presence of linearly dependent active constraints. Since the constraints in (28) and (32) for the QP1 and QP2 formulations, respectively, are linearly independent, a local minimum is ensured in a finite number of steps. However, if the value of the objective function is nonzero, i.e. different than the global minimum, the results obtained from these approaches do not represent the solution to the problem. The QP1 and QP2 formulations may then be used with caution.
The quadratic term in the objective function F3 for the statement QP3 rises from the portion fTc f?A2 dTc of the integral fFc t?bn dFc. It is noted that this integral is performed over the contact surface, and it represents the work done by the contact tractions, tn
c, on the deforming contact surface, as seen from (37). No additional external influences are included in this expression. Since this work done is always positive, the resulting quadratic term is positive definite, and thus the function F3 is convex.
In the present study, the subroutine QPSOL given by Gill et al. (1984)
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was employed. The details of the algorithm used are available in Gill et al. (1981). To improve the behavior of the method, the procedure of scaling by a diagonal matrix, explained by Gill et al. (1981), was also employed in the present implementation.
NUMERICAL EXAMPLES
The preceding formulations were applied for the solution of a number of two-dimensional example problems dealing with the contact of elastic bodies with rigid surfaces. The elastic object was modeled using three-noded quadratic boundary elements (Banerjee and Butterfield 1981), in all the examples considered. A state of plane strain was assumed in each of the cases. The present results were compared with the analytical results or with results obtained using alternative solution methods, when available. The numerical examples were selected to include both straight and curved contact boundaries. The following numerical results demonstrate the accuracy of the present formulations. All computations reported here were performed on a DEC 3100 workstation at Carnegie Mellon University. For each of the example problems attempted in this study, the positive tolerances, which define the maximum permissible violation in each constraint in order for a point to be considered feasible (Gill et al. 1981), were chosen to be equal to the machine precision of 10~15. The subroutine, QPSOL, employed in these problems for the solution of the quadratic programming problems, requires an initial guess for the solution vector. In each of the cases reported, a null solution vector was prescribed as the initial guess. It is noted that the nonconvex QP is far more computationally expensive than its convex counterpart. The computational expense depends on the inertia-controlling algorithm employed, as well as the prescribed initial guess.
Cantilever Beam in Contact with Curved Rigid Surface The contact of a shallow cantilever beam, shown using dashed lines in
Fig. 2, with a curved rigid surface was analyzed. The beam was modeled using five and 10 equally spaced boundary elements, respectively, along each of the longitudinal and transverse faces of the beam. The numerical data used for this problem were: modulus of elasticity, E = 2* 106 kN/cm2; Poisson ratio, u, = 0; cantilever length L = 100 cm and height h = 10 cm. The beam was first loaded with a uniformly distributed load of q1 = 50 kN/ cm2, in the absence of the contact surface. The deformed contour of the lower face, AB, of the beam was taken as the outline of the rigid surface for performing the contact analysis. The unloaded beam was now loaded with q2 = 100 kN/cm2, in the presence of the rigid surface defined as described. For a frictionless contact between the thin beam and the rigid surface, a uniform transverse traction of 50 kN/cm2 must develop at the contact surface. The contact surface for the beam and the corresponding tractions obtained from the present study are shown in Fig. 2. The contact traction results corresponding to each of the three statements QP1, QP2, and QP3, respectively, are compared with the analytical result of a uniform Taction of 50 kN/cm2 in Fig. 2. The three statements yield nearly identical esults. The number of iterations required to satisfy the prespecified con-ergence tolerance using the quadratic programming subroutine QPSOL ire also shown in Fig. 2. For all the subsequent examples studied, the QP1 statement was used for obtaining the solutions.
A. • - B - ^ - f f l ^ f l r ^ W ^ w g i " / ^ j g mi$Jfc^ Pftl*
60 80 100 X COORDINATE [cm]
FIG. 2. Contact of Cantilever Beam with Curved Rigid Surface
Infinite Elastic Cylinder on Rigid Foundation The problem of an infinitely long circular cylinder resting on a flat rigid
foundation and subjected to a uniform pressure acting on the top of the cylinder, P = 810 kN/cm, was studied. This problem is commonly known as the Hertz problem after Hertz (1882), who first provided an analytical elasticity solution for this problem. The numerical data used for this problem were: E = 1,000 kN/cm2, LL = 0.3, and R = 8.0 cm. This problem may be modeled using only a quadrant of the cylinder under the uniform pressure, q = P/2R, acting on the center plane as explained by Kikuchi and Oden (1979) and shown in Fig. 3. For the present study, the cylinder quadrant was modeled using 24 boundary elements. The number of elements used for discretizing each face of the cylinder is shown in the square boxes in Fig. 3. A graded mesh was used, with more elements provided at the sharp corner in the model as well as at the expected contact region. The contact region obtained using the present approach along with the traction distribution along the contact zone are shown in Fig. 3. The analytical solution for this problem has been given by Hertz (1882) and is shown in Fig. 3 for comparison. A good correlation of the present results with the analytical solution was observed.
Closing of Elliptical Crack in Rectangular Panel The plane strain problem of the closing of a crack in an elastic body was
considered next. This problem was treated earlier by Kikuchi and Oden (1979) using a finite element-based formulation. The numerical data used for this problem were: E = 1,000 kN/cm2, |x =. 0.3, h = 10 cm, b = 8 cm, and e = 0.3 cm. The crack was assumed to be elliptical, with its geometry defined by v = 0.1V9 - x2. The panel is subjected to a uniform axial
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i_J__iBJ__M
i. 'so tn z o p 150 O
2
§ ' "
30
0.0 0.6 1.0 1.5 2.0 2.5 X COORDINATE [cm]
FIG. 3. Infinite Elastic Cylinder on Rigid Foundation
compression of q = 75 kN/cm2. A complete description of the problem can be found in Kikuchi and Oden (1979). Due to double symmetry of the problem, only a quarter of the panel was modeled, using 14 boundary elements, as shown in Fig. 4. The number of elements for each side of the panel is shown in the figure in square boxes. In addition, the mesh was graded following the pattern for the finite element mesh used by Kikuchi and Oden (1979), with more elements provided at the crack tip. Under the applied load of 75 kN/cm2, all but two boundary element nodes lying next to the crack tip did not close. A similar observation was made by Kikuchi and Oden (1979). The traction distribution for this case along the closure obtained from the present study is shown in Fig. 4. The corresponding traction distribution presented by Kikuchi and Oden (1979) using the FEM approach is shown in Fig. 4 for comparison, and a general agreement of the two results, both in trend as well as magnitudes, is apparent. The BEM results for tractions along the closure were also obtained by doubling the number of boundary elements along the elliptical crack. The solution obtained from this finer discretization agrees closely with that obtained from the previous coarser discretization, and is also shown in Fig. 4. This problem was also analyzed using the FEM program ABAQUS (ABAQUS 1988). These results are also shown in Fig. 4 for comparison.
Rigid Bolt in Thick Plate The contact of a rectangular, thick plate with a rigid bolt placed in the
hole was studied. The center of the hole lies at the middle of the plate
HERTZ O THIS STUDY
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Ul-l-l-i-0
a a
O 0 N = 4, THIS STUDY D D N = B, THIS STUDY
KIKUCHI & ODEN (1979) o o ABAQUS
NUMBER OF ELEMENTS FOR THE ELLIPTICAL CRACK
•1.2 -0.7 -0.2 X COORDINATE [cm]
FIG. 4. Closing of Thin Elliptical Crack in Rectangular Panel
height. The plate is loaded with a uniform tensile force t, acting on the right vertical face. A detailed description of the problem may be found in Kikuchi and Oden (1988). Due to the symmetry of the structure and the loading, only a half of the plate was modeled, using 14 boundary elements, with three elements along the left quarter of the circle, which was considered as the possible contact region. The geometry and the loading of the plate are shown in Fig. 5. The numerical data used for this problem were: E = 1,000 kN/cm2, LL = 0.3, q = 25 kN/cm2, a = 6 cm, D = 4 cm, h = 6 cm, and e = 0.2 cm. The analysis was also performed by doubling the mesh to check the convergence of the solution. The traction distributions along the contact region for both these analyses are shown in Fig. 5. The contact tractions obtained using ABAQUS are plotted in Fig. 5 for comparison. A good agreement of results was obtained.
Cantilever Beam in Contact with Flat Rigid Surface The contact of a cantilever beam with a rigid surface at a distance, e,
below the bottom horizontal face of the beam, as shown in Fig. 6, was studied. An uniform transverse pressure,/?, acts on the horizontal top face of the beam. The numerical data used for this problem were: E = 2*106
kN/cm2, |x = 0.29, L = 50 cm, h = 5 cm, t = 1 cm, q = 3.6 kN/cm2, and e = 1 cm. The number of elements used for discretizing each segment of the beam is shown in square boxes in Fig. 6. The problem was also solved using twice the number of elements in the contact region, which was esti-
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D-e e
O o N»3, THIS STUDY D D N = 6, THIS STUDY
40 60 60 70 80 80 ANGLE 6
FIG. 5. Rigid Bolt in Thick Plate
M M tgi-rrT i [g a—[T}J?
4
-I-40 41 42 43
O N = 5, THIS STUDY • N = 10, THIS STUDY O ABAQUS
45 46 4? 48 49 50 X COORDINATE [cm]
FIG. 6. Cantilever Beam in Contact with Flat Rigid Surface
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mated using the coarser model. The traction distributions along the contact region for each case are shown in Fig. 6. The contact tractions were also obtained using ABAQUS. A close agreement between the present results and the FEM results was seen. The contact distance, d, may be analytically computed using the beam theory as
, . l72Efc d = " = 45.7 cm (43)
A distance d = 42.5 cm was obtained using the present developments, and a distance d = 42 cm was obtained from ABAQUS. This may be attributed to the two-dimensional effects that are not considered in the classical beam theory.
CONCLUSIONS
A computational formulation for the frictionless contact of an elastic body with a rigid surface, treated as an optimization problem, is presented. The contact conditions, along with the equilibrium equations of the elastic solid expressed as boundary integral equations, are employed together to define the objective functions and the corresponding constraints. Three distinct objective functions are considered, and each is reduced to a quadratic form. The final equations are written in terms of the unknown contact displacements and tractions only. The solution is obtained using standard quadratic programming algorithms in optimization. A number of numerical examples involving contact regions of arbitrary geometric configurations are presented. For some cases, the examples are compared with analytical solutions to demonstrate the accuracy of the present formulations. The current formulations are able to predict sharp contact tractions as well as lift-off of the elastic body from the rigid surface. The boundary discretization employed in this study is especially advantageous in the case of elastic bodies, in which more than one contacting region exists in different parts of the body. For such cases, a fine discretization of the various contact regions may force the entire domain of the body to be discretized with a fine mesh when using the finite element method, resulting in an expensive, if not intractable, solution. Such is not the case for boundary elements, since the boundary regions away from the contact zone may still be discretized using a mesh of usual density and a discretization of the domain is not required. Thus the present formulations will be especially useful for the economical solution of large contact problems with multiple contact regions.
ACKNOWLEDGMENTS
This work is based upon research supported by the National Science Foundation Presidential Young Investigator award, under Grant MSS 9057055 to Carnegie Mellon University. The initial discussions with Xiaogang Zeng regarding this problem are also acknowledged.
APPENDIX. REFERENCES
Abdul-Mihsein, M. J., Bakr, A. A., and Parker, A. P. (1986). "A boundary integral equation method for axisymmetric elastic contact-problems." Comput. Struct., 23(6), 787-793.
Andersson, T., and Persson, B. G. A. (1982). "The boundary element method
1889
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nloa
ded
from
asc
elib
rary
.org
by
WA
SHIN
GT
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UN
IV I
N S
T L
OU
IS o
n 03
/12/
13. C
opyr
ight
ASC
E. F
or p
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righ
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ved.
applied to two-dimensional contact problems." Progress in boundary elements, C. A. Brebbia, ed., 2, Pentech Press, London, U.K.
Banerjee, P. K., and Butterfield, R. (1981). Boundary element method in engineering science, McGraw-Hill Book Co., London, U.K.
Belytschko, T., and Neal, O. M. (1991). "Contact-impact by the pinball algorithm with penalty and Lagrangian methods." Int. J. Numer. Meth. Engrg., 31(3), 547-572.
Chand, R., Haug, E. J., and Rim, K. (1976). "Analysis of unbonded contact problems by means of quadratic programming." / . Optim. Theory Appl., 20(2), 171-189.
Cheng, J.-H., and Kikuchi, N. (1985). "An analysis of metal forming processes using large deformation elastic-plastic formulations." Comput. Meth. Appl. Mech. Engrg., 49(1), 71-108.
Eterovic, A. L., and Bathe, K. J. (1991). "On the treatment of inequality constraints arising from contact conditions in finite element analysis." Comput. Struct., 40(2), 203-209.
Gakwaya, A., and Lambert, D. (1991). "A boundary element and mathematical programming approach for frictional contact problems." Advances in boundary elements, C. A. Brebbia and J. J. Connors, eds., Computational Mechanics Publications, Southampton, U.K.
Gill, P. E., Murray, W., and Wright, M. H. (1981). Practical optimization. Academic Press, San Diego, Calif.
Gill, P. E., Murray, W., Saunders, M. A., and Wright, M. H. (1984). "User's guide for QPSOL (version 3.2)." Tech. Report SOL 84-6, Department of Operations Research—SOL, Stanford, California.
Gill, P. E., Murray, W., Saunders, M. A., and Wright, M. H. (1988). "Inertia-controlling methods for quadratic programming." Tech. Report SOL 88-3, Department of Operations Research—SOL, Stanford, Calif.
Gladwell, G. M. L. (1980). Contact problems in the classical theory of elasticity. Sijthoff and Nordhoff, Germantown, MD.
Hug, E. J., and Arora, J. S. (1979). Applied optimal design. John Wiley & Sons, New York, N.Y.
Haug, E. J., Chand, R., and Pan, K. (1977). "Multibody elastic contact analysis by quadratic programming." 7. Optim. Theory Appl., 21(2), 189-198.
Hertz, H. (1882). "On the contact of elastic solids." J. F. Math, 92, 156-171. ABAQUS, User's Manual. (1988). Hibbitt, Karlsson & Sorensen, Inc., Providence,
R.I. Johnson, A. R., and Quigley, C. J. (1989). "Frictionless geometrically non-linear
contact using quadratic programming." Int. J. Numer. Meth. Engrg., 28(1), 127-144.
Kalker, J. J. (1977). "A survey of the mechanics of contact between solid bodies." ZAMM, 57(5), T3-T17.
Karami, G. (1983). "Boundary integral equation method for two-dimensional contact problems," PhD thesis, Imperial College, University of London, London, U.K.
Kikuchi, N., and Oden, J. T. (1979). "Contact problems in elasticity." Tech. Report 79-8, Texas Institute for Computational Mechanics, University of Texas at Austin, Austin, Tex.
Kikuchi, N., and Oden, J. T. (1988). Contact problems in elasticity: A study of variational inequalities and finite element method, SIAM, Philadelphia, Pa.
Klarbring, A., and Bjorkman, G. (1988). "A mathematical programming approach to contact problems with friction and varying contact surface." Comput. Struct., 30(5), 1185-1198.
Klarbring, A. (1986). "A mathematical programming approach to three-dimensional contact problems with friction." Comput. Meth. Appl. Mech. Engrg., 58(2), 175-200.
Kwak, B. M., and Lee, S. S. (1988). "A complementarity problem formulation for two-dimensional frictional contact problems." Comput. Struct., 28(4), 469-480.
Vlaier, G., and Novati, G. (1983). "Elastic-plastic boundary element analysis as a linear complementarity problem." Appl. Math. Model., 7(2), 74-82.
1890
J. Eng. Mech. 1992.118:1876-1891.
Dow
nloa
ded
from
asc
elib
rary
.org
by
WA
SHIN
GT
ON
UN
IV I
N S
T L
OU
IS o
n 03
/12/
13. C
opyr
ight
ASC
E. F
or p
erso
nal u
se o
nly;
all
righ
ts r
eser
ved.
Maier, G. (1971). "Incremental plastic analysis in the presence of large displacements and physical instabilizing effects." Int. J. Solids Struct., 7(4), 345-372.
Misopoulou-Papasoglou, E., Panagiotopoulos, P. D., and Zervas, P. A. (1991). "Dynamic boundary integral 'equation' method for unilateral contact problems." Engrg. Anal. Bound. Elements, 8(4), 192-199.
Panagiotopoulos, P. D., and Lazaridis, P. P. (1987). "Boundary minimum principles for the unilateral contact problems." Int. I. Solids Struct., 23(11), 1465-1484.
Stavroulakis, G. E., Panagiotopoulos, P. D., and Al-Fahed, A. M. (1991). "On the rigid body displacements and rotations in unilateral contact problems and applications." Comput. Struct., 40(3), 599-614.
Tanaka, M. (1983). "Application of the boundary element method to machine elements (stress analysis of threaded connections and gears)." Fifth Int. Conf. on Boundary Elements, C. A. Brebbia, T. Futagami, and M. Tanaka, eds., Springer-Verlag, Berlin, Germany.
Tsuta, T., and Yamayi, S. (1983). "Boundary element analysis of contact thermo-elastoplastic problems with creep and the numerical technique." Fifth Int. Conf. on Boundary Elements, C. A. Brebbia, T. Futagami, and M. Tanaka, eds., Springer-Verlag, Berlin, Germany.
