ement solution of the quasi-static two-body contact problem in the presence of finite ... to have gained wide acceptance in both research and commercial software .... denote the ordinates of a field at the left, right, and center node of element Ct, ...
Modelling Vol. 28, No. 4-8, pp. 373-384, 1998 @ 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0895-7177/98 $19.00 + 0.00 PII: SO8957177(98)00128-9 Mathl.
Comput.
A Lagrange Multiplier Method for the Finite Element Solution of Frictionless Contact Problems P. PAPADOPOULOS AND J. M. SOLBERG Department of Mechanical Engineering, University of California at Berkeley Berkeley, CA 94720, U.S.A.
Abstract-This article proposes a novel Lagrange multiplier-based formulation for the finite element solution of the quasi-static two-body contact problem in the presence of finite motions and deformations. The main idea rests in the interpretation of the two-body contact ss a composition of two simultaneous Signorini-like problems, which naturally yield geometrically unbiased approximations of the kinematics and kinetics of frictionless contact. A two-dimensional finite element is introduced that exactly satisfies the impenetrability constraint and allows for the direct computation of consistent pressure distributions on the interacting surfaces. @ 1998 Elsevier Science Ltd. All rights reserved. Keywords-Finite
element method, Contact mechanics, Lagrange multipliers. 1.
INTRODUCTION
The numerical simulation of the two-body contact problem poses a series of challenges owing to the strong nonlinearity of the governing equations of motion. In particular, the proper geometric identification of the contact surface and the subsequent enforcement of the impenetrability constraint in a discrete (e.g., finite element-based) setting are of critical importance in rendering a numerical procedure robust and convergent. Several formulations have been proposed in the literature of computational contact mechanics since the early works of Conry and Seireg [l], and Chan and Tuba [2]. In these, the Lagrange multiplier method [3] and its standard regularizations (e.g., perturbed and augmented Lagrangian methods) are frequently employed in connection with finite element approximations in the spatial domain, and appear to have gained wide acceptance in both research and commercial software environments. Here, the physical domain of the contact elements is intimately connected to the choice of numerical integration rule used for the work-like boundary integral term associated with the contact tractions in the weak form of the linear momentum equations. Application of nodal quadrature involving the contacting nodes of one or both surfaces yields the so-called oneor two-pass node-on-surface algorithms, respectively (41. Other integration rules, possibly more accurate, are generally applicable, provided there exists a continuous discretization of the contact interface [5,6]. Node-on-surface algorithms are not free of inconsistencies. A form of mesh locking is routinely encountered in two-pass algorithms when boundary nodes are not aligned with each other upon contact [7, p. 1651. This surface loclcing phenomenon can be trivially demonstrated in the case of bodies discretized using linear domain elements with arbitrarily positioned boundary nodes: here, The research work wss supported by the National Science Foundation under Contract Nr. MSS-9308339 with the University of California at Berkeley.
373
P. PAPADOPOULOSAND J. M. SOLBERG
374
global imposition of contact conditions essentially necessitates flatness of the contact surface. It should be noted that surface locking often goes unnoticed, as boundary nodes “release” from the contact surface, owing to the inequality form of the impenetrability constraint. The present paper makes use of an interpretation of two-body contact as a composition of two simultaneous Signorini-like problems [8], which serves as a background for the development of contact elements in connection with the Lagrange multiplier method. As in the case of finite element methods involving the incompressibility constraint, it will be shown that the choice of admissible pressure fields and the number of discrete constraints on the boundary displacement fields play a crucial role in the design of elements capable of bypassing surface locking and yielding stable and accurate numerical solutions. Elements of contact mechanics are presented in Section 2, with particular emphasis on the development of the background to the ensuing algorithmic developments. A two-dimensional contact element is proposed and analyzed in Section 3, and selected numerical simulations using this element are presented and discussed in Section 4. Concluding remarks are given in Section 5.
2. ELEMENTS
OF CONTACT
MECHANICS
Let Xa be a typical material point of body f3”, (Y = 1,2, and denote by XQ and xQ the position vectors of Xa in the reference and current configurations, respectively. The reference configuration of the two bodies is identified in W3 with nonempty, open, and connected sets R” having smooth boundaries dG!*. The body motions xa are mappings on P U LW, such that at time t. The mapping xa is assumed smooth throughout its domain and X0 = x”(X”‘,t) invertible at least on R”. The image of R” in the current configuration is denoted by Q with boundary an; that possesses a unique outer unit normal nQ at each of its points. In addition, a displacement field uQ is defined on each body according to ua(XLI!, t) := x”(Xa, t) - XQ. The motions xQ are subject to the principle of impenetrability of matter [9, p. 2441, which implies that at all times n;nn;=s. (1) At any given time, the two bodies are said to be in contact along a subset C of their boundaries if, and only if,
aft:nanf:=c#s, sothat the boundary of each body can be uniquely decomposed according to
aq =r,ourpc, where classical Dirichlet and Neumann boundary conditions are enforced on r,O and rg, respectively. Although not explicitly noted, it should be clear from the above that rt,lYg, and C generally depend on time. In order to recast condition (1) in a more convenient form, let S(sZi U 0:) be the convex hull of the system of bodies and define a gap function gQ on the boundary of body CYas follows: for each x0 E H’lf, gQ : ail; x aSIf H W is given by
where
see Figure 1. Equation namely that
(2) implies that gap functions g(l) and g@) vanish identically on C, g(‘)(C) = g@)(C) = 0.
(3)
Lagrange
Figure
Multiplier
1. Definition
Method
375
of gap functions.
At the absence of inertial effects, the local form of the equations of motion for each body is written as divTa
+ pabO =
0,
UQ = fiQ, Tana
= i&),
in flp,
(4a)
on l?:,
(4b)
on ry,
(4c) where Ta is the Cauchy stress tensor, pQ the mass density, ba the body force per unit mass, V the prescribed boundary displacement, and %&, the prescribed traction vector on Ft. The Cauchy stress tensor T is related to the motion by a properly invariant constitutive law. Assuming that the principle of impenetrability holds for the reference configuration of the two bodies, it follows from (2) and (3) that condition (1) can be expressed in terms of the above gap functions 8s on an:. ga L 0, (44 A separate statement of linear momentum balance applies to the contact surface C. Recalling that, during persistent contact, C is a vortex sheet of order zero (9, Sections 185-1881, it follows that the local form of linear momentum balance on C takes the form [TaJnQ = 0, where [T”j := Tp - T*. Assuming frictionless contact, it is immediately concluded that t&, which implies that, since na = -no,
= -p”no,
balance of linear momentum on C is equivalent to p(l) = $2).
(44
Equations (4a)-(4e) constitute the strong form of the two-body contact problem. A weak form of the equations of motion is derived by means of the classical Gale&in method, according to which the displacement fields u” and the pressure fields pa satisfy
J
grad w” : T” dv -
fY
I OF
pow”
. b” dv -
J
T
we.
E&,
dy +
J
c= pawa - nQdr = 0, @a)
(5b)
1 c
r@-pa)dcy=O,
(54
P. PAPADOPOULOSAND J. M. SOLBERG
376
for all admissible functions w”, qQ, and T. The notation C” is used in the last integral term on the left-hand side of (5a) to underline that the common contact surface can be selectively viewed as part of either one of the two interacting bodies. Equations (5a), (5b) are identical in form to those of the Signorini problem for each of the two bodies, with coupling induced by (5~) via the pressure fields. Displacement fields Us belong to the spaces U” with U” :=
{uaE H’(V)
1ua =
iia on
I:},
and the weighting functions wQ belong to the spaces WQ defined as
W” := {WQ
E
H1(P)
1wQ = 0 on
rz}.
The admissible functions p” and qa belong to Pa, where Pa := {q” E
H-1’2(aR”)1qa
2 O},
while r E H’j2(C). Since, clearly paga = 0,
on Xl:,
and gQ vanishes on Ca, inequality (5b) implies that
I
qaga dy = 0,
(6)
cm
for all admissible functions qp E Pa. REMARK 2.1. The developments presented here can be readily extended to encompass the general multibody contact problem by reducing it to a set of individual two-body problems formulated as above.
3. A TWO-DIMENSIONAL
CONTACT
ELEMENT
The design of contact elements is subject to several restrictions and conventions. Typically, the admissible pressure fields should have relatively small support (i.e., retain the local piecewise polynomial form of traditional finite element functions), in order to minimize the additional computational effort during equation solving. Moreover, the resulting approximation should ideally preclude any bias in the treatment of surfaces C(l) and Cc2), as in the “master” and “slave” logic of the one-pass node-on-surface algorithm. Conditions on the admissible fields have been established in the special case of the Signorini problem, leading to formally convergent approximations [lo]. Although the above conditions may not be easily interpreted in the context of a general two-body problem, they serve as guidelines in identifying unacceptable choices of admissible fields which can produce surface locking (over-constraining) and/or instabilities in the form of oscillatory pressure fields [ll], at least in the limiting case as one of the interacting bodies becomes substantially stiffer than the other. 3.1. Finite
Element
Fields
In what follows, equations (5a), (5~)) and (6) are used in the finite element approximation of the twc+body contact problem in two spatial dimensions. To this end, a new contact element is introduced in conjunction with quadratic domain elements, such as the 6 and 7-noded triangles and the & and 9-noded isoparametric quadrilaterals. Following [8], surfaces C* are discretized according to
Lagrange
Figure 2. Collocation
Multiplier Method
of pressure field by projection
377
from the opposite surface.
where CF belongs to the boundary of element e, which, in turn, is assumed to lie on the contact region of body cr. The case of elements whose surfaces are only in partial contact can be accommodated without serious algorithmic complications and is omitted for the sake of brevity. With reference to the standard local isoparametric coordinate 5,” E [-1, 11,the pressure field pg at a given time t is defined on C,O as PWYIc,o
= P”(GY) = c
J52,iWP?Y
(7)
i=l ,r.c
where Lz,i are the Lagrangian interpolation functions of degree 2, and subscripts I, f, and c denote the ordinates of a field at the left, right, and center node of element Ct, respectively. Correspondingly, the gap function ghQis defined as
Pressure variables pf and p; in (7) are unknown parameters to be determined by the global solution, while pz is locally defined by a point-collocation method intended to weakly enforce the momentum balance equation (5~). Specifically, let Ez be the set of points on CF obtained by projection from corner nodes of Cp, namely
and define c: E EF as the point found on Cf by projection from xf, for which
see Figure 2. Subsequently, equation (5~) is weakly enforced by means of a collocation method, according to Pa (CT> = Pf (Jq
7
63)
so that, invoking (7) and (8), p: is obtained as pF
=
La
cl(t-@
[pa
(x”>
-
L2,l
(CT)
PP - L27 (c7
PF] .
If 2; is empty, p: is directly set equal to (pp + p3/2 and the admissible pressure field on CF reduces to a piecewise linear function of pp and p:. In this case, a third independent constraint
378
P. PAPADOPOULOS AND J. M. SOLBERG
with an associated pressure parameter could be introduced for element Cf at the cost of increasing the complexity of the method. In case outer nodes on each of the two surfaces align with each other, a collocation method is used on the respective pressure variables in order to satisfy equation (5c), thus resulting in a standard node-on-node formulation. This identification is essential in the proposed method, as node-on-node contact yields only one independent constraint condition for each pair of nodal points. The above choice of admissible pressure fields is essentially dictated by the given admissible displacements. Indeed, the order of the pressure interpolation (7) within C,Q is restricted by the three displacement degrees-of-freedom in the direction normal to CF. Contrary to the case of the Signorini problem, a general piecewise quadratic pressure interpolation produces surface locking, as it requires vanishing of the gap at three distinct and fixed points in each element. This, in turn, forces the contacting surfaces to match each other at a number of points which is sufficient to ab initio restrict their global geometric form and, thus, artificially after the underlying boundary-value problem. The proposed element is free of this pathology, because it assumes a quadratic pressure interpolation in which only two of the three pressure parameters are obtained by satisfying the impenetrability constraint on a gives surface, while if ZF is nonempty, a third one is merely sampled from the opposite surface. REMARK 3.1. In general, since each member of E: represents an impenetrability constraint condition originating on the opposite surface, the total number of points where the gap vanishes on Cp is equal to the number of members in ZF increased by two (in order to account for the additional constraints imposed on the edges of Ct). REMARK 3.2. The pressure fields determined by the above method are continuous along interelement boundaries. The same conclusion applies to surface tractions due to contact, provided that a simple weighted-averaging scheme is employed for the determination of a unique normal no to surface Co at all boundary nodes that are shared between contiguous elements.
As in the node-on-surface methods, balance of linear momentum across C is enforced by a collocation method. However, here collocation applies selectively to the pressure fields rather than to the resultant tractions. REMARK
3.3.
3.2. Suggested
Solution
Procedures
Let the discrete admissible fields matrix notation as
ug, wg, p:,
and qhQ,and the gradient of wt be written using
u;(x, t) = N,a(x)P(t), w;(x, t) = N,O(x)\irQ(t),
PX(X,t) = N;(x)tiQ(t), qha(xvt) = N;WIQ@), gradwz(x,
and
t) = Bff(x)Ga(t),
respectively, where Q”, Go E W2Na and p”, 4” E WMP . In the above equations, N,Q and NF contain the interpolation functions for Q” and $* and BP = gradNE. Switching to matrix notation, equation (5a) takes the form ,+itbT
BQTTa dv -
J
paN,aT b” dv -
fv
J
Nu”‘“&aj dr +
CT
while equation (6) becomes
J
ACYT
q
C”
NzTgU dr = 0.
J
c” NETpQnQdy
= 0,
(9)
(10)
Lagrange Multiplier Method
379
Taking into account the arbitrariness of %P and au, scalar equations (9) and (10) yield a system of Cz,,(2N” + MQ) nonlinear equations expressed as
J J
paNETnQ d-y = 0,
+
and
(11)
CP
fp” :=
N,aTga dr = 0.
(12)
cm
The integrals in equations (11) and (12) are generally evaluated using numerical techniques. In the proposed method, the boundary integrals on Ca are computed using Simpson’s rule. It follows from the integral over Co in (11) that the contribution made by contact forces to the balance of linear momentum is evaluated exactly in the case of flat contact under constant pressure. This is a consistency condition which has been identified and discussed in (121. Within the context of a classical incremental formulation, consider a finite time interval (tn,tn+i] and restrict attention to the determination of the unknown displacements ti:= [ti(1)Q(2)]T and pressures @:= [@(‘)fi(2)]T at &+I. This can be attained by solving (11) and (12) iteratively using Newton’s method, according to the recursive formula
(13) where f, := [$)f~2)]T and fP := @)@)lT, DU(.) and DP(.) denote the G&teaux derivatives of (.) in the directions of C and @, respectively, and Ati) := (.)(i+l) - (.)(“I. The initial guess to the solution at tn+i is typically provided by the converged solution at t,. The submatrix DUfz which appears in the left-hand side of (13) can be singular, even when the full matrix is nonsingular (e.g., due to rigid-body modes which are eventually suppressed by the contact boundary conditions). In this case, a direct solution (e.g., using Gauss elimination) requires pivoting, which, in turn, destroys the banded profile of the matrix and increases the required computational effort. In order to circumvent this difficulty, the system of (13) can be recast in an equivalent form as D,f,, + W(DUfJTDUfp D&
i=l,2,...,
(14)
where w > 0. Assuming that the constitutive law renders D,,f,, symmetric positive semidefinite (e.g., as in certain cases of hyperelasticity with dead loading [13]), where the remaining singularity is due exclusively to rigid-body modes which are eliminated when C # 0, it can be seen that D,f,, + w(D,fP)TDUfP is necessarily positive definite. Indeed, if the above does not hold, then there is a vector v E W2(N(‘)+N(a)) such that vTD,fUv
= 0,
D,f,v
= 0.
and
(15)
Since D,f,, is symmetric, there exists an orthonormal set of eigenvectors vk [14, p. 1011 associated with real eigenvalues &, such that 2(N(‘)+N(*)) v=
c k=l
akvk.
380
P. PAPADOPOULOS AND J. M. SOLBERG
From (15) and the above decomposition,
it is seen that 2(lv(‘)+iv(9
vTD,fUv
=
&a;
c
= 0,
k=l
which immediately implies that v is spanned by the eigenvectors associated with the rigid-body modes of body a, hence D,f,,v = 0. (16) Subsequently, let (Ati)ti, A(i)p) be the (unique) solution of system (13). Then, it follows from (16) that (A%i+v, A(i)jj) is also a solution, which contradicts the original assumptions, thus proving the positive-definiteness of DUfU + W(DUfp)TDUfp. An alternative solution procedure is based on a penalty-like regularization of the pressure fields, which, with reference to (7), takes the form
where c > 0 is a penalty parameter and (a) denotes the Macauley bracket. As with the Lagrange multiplier method in Section 3.1, gf , g,” are computed directly on CQ, while g,* is determined by a point-collocation method from the opposite body P.
4. NUMERICAL
SIMULATIONS
The proposed Lagrange multiplier method has been implemented within the environment of FEAP, a general-purpose nonlinear finite element program partially documented in [15,16]. Selected numerical simulations have been conducted using a compressible isotropic neo-Hookean hyperelastic model with strain energy functional W defined as W = iIL(rc - 3) - /I In IIIg2
+ fX (11Ip
- q2 )
where C is the right Cauchy-Green deformation tensor, Ic = tr C, 111~ = det C, and X, p are the Lame constants. All deformable bodies are spatially discretized using nine-noded quadrilateral isoparametric elements.
Figure 3. Contact between deformable cylinders; discretization in the reference configuration.
Lagrange Multiplier Method
I
E300
2
I
I
I
I
381
1
I
I
-
uppercylinder
_--
lower cylinder -
I
. . . . . . He& 200 -. . . . . . . . . . . ...*...._
50 -
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
x,-coordinate Figure 4. Contact between deformable cylinders, pressure fields at times t = 5.0, 10.0, and 15.0. ~......“....“..........“...‘................................
Figure 5. Contact between deformable cylinders; deformed configuration at t =
4.1. Contact
between
Two Deformable
15.0.
Cylinders
Let two cylinders of radii & = 8.0 and R, = 18.0 be pressed on each other by two flat rigid surfaces, as in Figure 3. A uniform vertical displacement a,(t) = -t is imposed on the upper rigid surface, while the lower rigid surface remains stationary. The problem is solved under plane strain conditions and the material properties of the cylinders are X = 2.88 x 102,
/A= 1.92 x 102.
The pressure fields pa developed between the two cylinders at times t = 5.0, 10.0, and 15.0 are plotted in Figure 4 against the zr-coordinate. The computed pressure fields are very smooth and nearly identical on the two surfaces, except at the end of contact where edge effects appear. For reference, the computed pressure profiles are compared to the approximate linear elastic solution
382
P. PAPADOPOIJLOSAND
Figure 6. Stretching 85 Z
of an elastic sheet; discretization
I
I
7-
J. M. SOLBERG
I
I
in the reference configuration.
I
I
I
sheet
- - -
z.
cylinder
/f 6-
--~
t=1 .o
_-...-
..---
/ \
54t=o.5 .___.__..__.__-.--+
_-.._..-
3 EC.2-
1
l0
0.0
I 0.5
I 1.0
I 1.5
I
I
k
2.0
2.5
3.0
I 3.5 x,-coordinate
4.0
Figure 7. Pinching of an elastic ring; pressure fields at times t = 0.5 and 1.0.
Figure 8. Stretching
of an elastic sheet; deformed configuration
at t = 1.0.
Lagrange Multiplier Method 2
Et
20
8
,9 _
a
16 -
I
I
I
I
I
...........‘...... two-point, Lagrange multipliers _-two-point penalty, e=106 -. - three-pointpe.nalty.e=1W .-.a- threepointpenalty,E=106
14 _
.’
..-.
6 /’
‘\
..\
6 &.-.__
.*.*.-..-*..*/
**...-..
I
: : ! I i i i ; i i .i i
**\
, ?.-.--_.-.+_,,$~
I .? i;
12 10 -
383
i
‘.\
4-
A:
\..,!
2n ‘b.0
I 0.5
I 1.0
I 1.5
I 2.0
Figure 9. Pinching of an elastic ring; a comparison cylinder at time t = 1.0.
I 2.5
3.0
I 3.5 x,-coordinate
of pressure distributions
4.0
on the
due to Hertz [17], as presented in [18]. The plots demonstrate that the deviation of the computed pressure fields from the respective Hertzian fields is increasing with 02, as expected. The current configuration of the contacting bodies is shown in Figure 5 at time t = 15.0. 4.2. Stretching of an Elastic Sheet An elastic sheet of thickness T = 1.0 is wrapped around a significantly stiffer hollow elastic cylinder of inner radius & = 1.0 and outer radius R, = 3.0, ss see in Figure 6. The interior surface of the hollow cylinder is assumed fixed, while the ends of the sheet are subject to a symmetric uniform vertical displacement a,(t) = -t. The analysis is conducted under plane strain conditions and the material properties of the sheet and cylinder are A, = 1.0 x 103,
ps = 5.0 x 10’
x, = 1.0 x 103,
/.& = 5.0 x 102’9
and
respectively. The pressure developed on the interface between the cylinder and the sheet is depicted in Figure 7 for times t = 0.5 and 1.0, as a function on the zi-coordinate. Also, the deformed configuration at t = 1.0 is shown in Figure 8. The above problem is also solved using two distinct penalty regularization procedures based on (17), where the impenetrability constraint is approximately enforced at two or three points in each contact element, depending on whether g,” is obtained by point-collocation from the opposite surface (as suggested in Section 3.2) or computed directly on P. Figure 9 illustrates that the pressure fields obtained by the latter penalty procedure fail to converge as the penalty parameter increases. This is a direct consequence of surface locking, as discussed in Section 3.1. In contrast, the penalty regularization procedure of Section 3.2 is stable and, for relatively large values of the penalty parameter, yields pressure fields that virtually coincide with those of the Lagrange multiplier method.
5. CONCLUSIONS The geometrically nonlinear two-body contact problem is resolved into two simultaneous Sign+ rini-like problems and analyzed using a Lagrange multiplier method. This formulation is employed in the development of a robust two-dimensional contact finite element, where the admissible fields are constructed so that balance of linear momentum is satisfied in a weak sense along the contact surface, while at the same time, over-constraining in the form of surface locking is eliminated.
384
P. PAPADOPOULOS ANDJ. M. SOLBERG
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