Engineering Analysis with Boundary Elements 29 (2005) 95–106 www.elsevier.com/locate/enganabound
Meshless solutions of 2D contact problems by subdomain variational inequality and MLPG method with radial basis functions J.R. Xiaoa,*, B.A. Gamaa, J.W. Gillespie Jra,b,c, E.J. Kansad a Center for Composite Materials, University of Delaware, Newark, DE 19716, USA Department of Materials Science and Engineering, University of Delaware, Newark, DE 19716, USA c Department of Civil and Structural Engineering, University of Delaware, Newark, DE 19716, USA d Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA 95616-5924, USA b
Received 29 September 2004; revised 28 October 2004; accepted 20 December 2004
Abstract A subdomain variational inequality and its meshless linear complementary formulation are developed in the present paper for solving two-dimensional contact problems. The subdomain variational inequality will be defined in detail. The meshless method is based on a local weighted residual method with the Heaviside step function as the weighting function over a local subdomain and radial basis functions as trial functions for interpolation. Three different radial basis functions (RBFs), i.e. Multiquadrics (MQ), Gaussian (EXP) and Thin Plate Splines (TPS) are examined and the selection of their shape parameters is studied based on 2D solid stress problems with closed-form solutions. The developed meshless/linear complementary method is applied to solve two frictionless contact problems. For the RBFs, it has been found that the TPS shape parameter is not sensitive to nodal distance and a value of 4 is found as a good choice for TPS from this research. q 2005 Elsevier Ltd. All rights reserved. Keywords: Variational inequality; Meshless method; Radial basis function; Contact problem; Linear complementary equation
1. Introduction The present study is concerned with the development of variational inequality/meshless method to solve frictionless contact problems. During the past few years, the idea of using meshless methods for numerical solution of partial differential equations (PDFs) has received much attention throughout the scientific community, and remarkable progress has been achieved on meshless methods. Among the available meshless methods, such as the Element-Free Galerkin (EFG) Method [1], the Reproducing Kernel Particle Method (RKPM) [2], hp-clouds method [3], the Partition of Unity Method (PUM) [4], the Meshless Local Petrov-Galerkin (MLPG) method [5–7], the MLPG method does not need any ‘element’ or ‘mesh’ for either field interpolation or background integration, and any
* Corresponding author. E-mail address:
[email protected] (J.R. Xiao). 0955-7997/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2004.12.004
non-element interpolation scheme such as the MLS, the PUM, or the RBFs can be used for trial and test functions. The flexibility in choosing the size and the shape of the local sub-domain leads to a convenient formulation in dealing with non-linear problems. Particularly, the local Heaviside weighted MLPG together with RBFs interpolation [8] showed great promise because only a regular boundary integral along the edges of subdomains is involved and no special treatment on imposing essential boundary condition is needed. The RBF MLPG method is used for the present study. Contact problems have been well formulated by variational inequalities and finite element approximations [9]. Such variational inequalities are usually defined within the whole problem domain. In order to apply the MLPG meshless method to contact problems, a local subdomain based variational inequality needs to be established to provide a mathematical foundation. In this paper, a set of subdomain variational inequalities have been developed to give an equivalence of the global
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variational inequalities for two-dimensional contact problems so that the MLPG type of meshless methods can be employed for approximation of the contact problems. It should be noted that the developed subdomain variational inequalities could be easily extended to three-dimensions. There are different approaches to solve the variational inequality problem. In this study, the variational inequality problem (constraint minimization problem) has been transformed into a linear complementary problem by using the natural physical properties of the contact constraint condition. Then, the corresponding linear complementary equation can be solved using mathematical programming solvers such as the Lemke’s algorithm [10] and the conjugate gradient projection method [11]. A more detailed discussion about the advantages of mathematical programming and linear complementary formulations can be found in [12–14]. Similar developments have been conducted successfully for one-dimensional contact problems of thin and thick beams based on the MLPG with polynomial interpolations [15,16]. Radial basis functions (RBFs) [17] have been frequently employed in solving partial differential equations (PDEs) using collocation meshless methods [18,19], Galerkin meshless methods [20–22], and Petrov-Galerkin meshless methods [7,8], and show excellent interpolation properties and great promise in meshless methods due to their effectiveness in interpolating multivariate scattered data in many science and engineering problems. The widely used RBFs are Hardy’s multiquadrics (MQ) gðrÞZ ðr 2 C c2 Þ0:5 , inverse multiquadrics gðrÞZ ðr 2 C c2 ÞK0:5 , Gaussian (EXP) 2 2 gðrÞZ eKc r , and Duchon’s thin plate splines (TPS) 2 gðrÞZ r log r. In this study, the shape parameters for MQ are the exponent of the square of the Euclidean distance, r2C c2, and c that can be interpreted as a distance in the (dC1) dimension, where d is the space dimension of the problem under investigation. For EXP, c2, is the shape parameter that can be considered the inverse of the variance of a Gaussian distribution. While TPSs have no adjustable shape parameters, the power of r will be considered adjustable. It should be noted that the RBFs satisfy the delta function property that enables one to impose essential boundary conditions easily. The above three type of RBFs are globally supported that usually leads to dense system matrices and exceptionally high computational costs. However, Kansa and Hon [23] showed that domain decomposition is effective in circumventing ill-conditioning, and Ling and Kansa [24] developed a preconditioner, that when combined with domain decomposition [25] permits efficient iterative solution methods for a variety of large scale PDE problems. However, when the classical RBFs are incorporated into the EFG [21] or MLPG [8] schemes, the above-mentioned problems are removed because their interpolation of field variables is performed in localized (compactly supported) domains supported by a predefined interpolation domain size. There is another problem in using classical RBFs: their shape parameters may be sensitive to nodal distance
and may have different behaviors when used in different schemes [8,26,27]. In the present study, the MQ, EXP and TPS radial basis functions are first examined in two-dimensional stress analysis problems to select their shape parameters. The Duchon’s TPS has been modified as gðrÞZ r h log r with h as a shape parameter and an optimal value (other than 2) could be expected. After the shape parameters are selected, the present meshless/linear complementary method is applied to solve two contact problems. Implementation details and numerical examples are presented to demonstrate the convergence and efficiency of the developed method.
2. Variational inequalities of contact problems Consider a frictionless contact problem of two elastic bodies Ue (eZ1, 2) bounded by surfaces GeZGe Z Gc g Geu g Get (eZ1, 2), in which Gc is a candidate contact surface between two bodies with prescribed displacements along Geu and tractions applied on Get (eZ1, 2). It should be noted that the contact function surface is not known in advance. If the gap along the contact surface is denoted by g, the contact condition can be described in the form of inequalities as Ftc Z 0
(1)
gR 0
(2)
gFnc Z 0
(3)
Fnc % 0
(4) Ftc
Fnc
and are the tangential and normal contact where force density on the interfaces. For each body, displacement field and stress field satisfy sij;j ðue Þ C bei Z 0 in Ue
ðe Z 1; 2Þ
(5)
where sij(ue) is the stress tensor, which corresponds to the displacement field ue, bei is the body force vector, and ( ),j denotes vðÞ=vxj . The boundary conditions are given as follows tie Z sij ðue Þnej Z tei uei Z u ei
on the natural boundary Geu
on the essential boundary Get
(6) (7)
in which nej is the outward unit normal to the domain Ue, and u ei and tei denote the prescribed displacements and tractions, respectively. One defines the solution spaces as CðUe Þ Z fve 2H1 ðUe Þjve Z uei on Geu ; gR 0 on Gc ðe Z 1; 2Þg
(8)
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where C(Ue) is a subset defined in the Sobolev space H1(U). Then the corresponding variational problem is 8 2
