The interpolating boundary element-free method (IBEFM) is developed in this paper for boundary-only analysis of unilateral problems which appear in variational inequalities. The IBEFM is a direct boundary only meshless method that combines an improved interpolating moving least-square scheme for constructing interpolation functions with boundary integral equations (BIEs) for representing governing equations. A projection operator is used to formulate the BIEs and then the formulae of the IBEFM are obtained for unilateral problems. The convergence of the developed meshless method is derived mathematically. The capability of the method is also illustrated and assessed through some numerical experiments.

In the past two decades, considerable research effort in the field of computational science and engineering has been directed, both on theoretical and practical bases, into meshless (or meshfree) methods. Compared with the finite element method (FEM) and the boundary element method (BEM), meshless methods eliminate the difficulty of meshing and remeshing the considered computational structure via simply adding or deleting scattered nodes. The moving least square (MLS) [

Boundary integral equations (BIEs) are important and attractive computational tools as they can reduce the dimensionality of the considered problem by one. The MLS scheme has also been used in BIEs. Typical of them are the meshless local boundary integral equation (LBIE) method and the boundary node method (BNM) [

To avoid the shortcomings of the MLS scheme, Liu and Gu proposed the point interpolation method (PIM) [

To restore the delta function property of the MLS scheme, Lancaster and Salkauskas [

To overcome the problems in both the MLS scheme and the IMLS scheme, Wang et al. [

In the theory of variational inequalities [

The present paper develops and employs the IBEFM for boundary-only analysis of unilateral problems. Using a projection operator, the unilateral problem is reacted as a sequence of well-posed bilateral problems. Then, the IBEFM is used iteratively for solving bilateral problems. As a result, only the boundary of the problem domain is discretized by properly scattered nodes. The convergence of the developed meshless method is also derived mathematically.

The following discussion begins with the brief description of the IIMLS scheme in Section

Consider a two-dimensional domain

Let

For any

In (

The coefficient vector

The stationarity of

To generate the IIMLS shape functions, the

We list below some propositions of the IIMLS shape function

The influence domain of the node

We consider the unilateral problem in the bounded domain

We introduce the Sobolev space

In the unilateral problem (

The purpose of this paper is to develop a meshless method for the numerical solution of the unilateral problem (

The BIE for the solution of (

From the expression of the approximation function (

Substituting (

To obtain the unknowns at boundary nodes, boundary conditions needed to be imposed. For the convenience of discussion, we assume that the first

The IBEFM flow chart for the unilateral problem (

Choose a tolerance

Select

Assume initially that the boundary condition on the unilateral boundary

Determine

Obtain the linear algebra equations (

Solve (

Compute the relative error

Stop the algorithm if

In this paper, the allowable tolerance used is

The mathematical proof of convergence of some mesh-based numerical algorithms has been established [

Let

Using Green’s formula leads to

On the other hand, if

Under the conditions of Lemma

According to (

Let

In order to prove that

Theorem

In [

In this problem, Dirichlet boundary conditions are imposed on the outer circle corresponding to the exact solution, and unilateral boundary conditions are imposed on the inner circle as

As in [

Exact and numerical solutions for

Exact and numerical solutions for

To study the convergence, the

Convergence of the BEM and the IBEFM for Spann problem.

The corresponding number of iterations is shown in Table

Number of iterations for Spann problem.

| ||||||
---|---|---|---|---|---|---|

32 | 64 | 128 | 256 | 512 | 1024 | |

GBEM | 24 | 28 | 36 | 48 | ||

BE-LCM | 56 | 91 | 141 | 256 | 263 | 319 |

BE-PIM | 4 | 4 | 5 | 6 | 7 | 8 |

IBEFM | 5 | 6 | 6 | 7 | 8 | 8 |

CPU times (in seconds) for Spann problem.

| ||||||
---|---|---|---|---|---|---|

32 | 64 | 128 | 256 | 512 | 1024 | |

BE-LCM | 0.047 | 0.125 | 0.453 | 2.407 | 16.953 | 125.907 |

BE-PIM | 0.078 | 0.141 | 0.438 | 1.782 | 8.141 | 40.188 |

IBEFM | 0.062 | 0.094 | 0.203 | 0.609 | 2.484 | 11.218 |

The problem solved here is a groundwater flow problem related to percolation in gently sloping beaches [

We consider two cases of surface profiles given in [

The numerical results are plotted in Figure

Approximate solution of the groundwater flow problem with (a)

In order to investigate the convergence of the IBEFM, the problem is solved by choosing different numbers of boundary nodes. Since the exact solution of the problem is not known, the error is estimated by substituting the exact solutions with some selected approximate values which can be computed by more boundary nodes. In this work, we obtain these approximate values by using 4096 boundary nodes. Figure

Convergence of the IBEFM for the groundwater flow problem with (a)

A novel boundary-type meshless method, the IBEFM, is developed in this paper for solving unilateral problems arising from variational inequalities. The IBEFM is based on BIEs discretized with the IIMLS scheme that only uses a group of arbitrarily distributed boundary nodes. In this numerical algorithm, the nonlinear inequality boundary conditions are incorporated into the iterative scheme naturally, and boundary conditions can be imposed with ease. Convergence proof of this meshless method has been provided mathematically. The proof shows that this method generates a sequence of strongly convergent approximations.

Some numerical examples have been given to validate the capacity of the method. For all examples, a convergent solution has been gained. The numerical results have verified the accuracy, convergence and effectiveness of the IBEFM. The experimental convergence rate is very high and approximate to 2. The errors and CPU times were compared to those obtained by the BEMs. These comparisons show that the IBEFM provides monotonic convergence and high accuracy results with low computational expenses.

Initial numerical results from the IBEFM, applied to 2D unilateral problems, are encouraging. The numerical experiments indicate that the developed meshless method has not only the higher convergence rate and computational precision but also the less CPU times than the BEMs presented in [

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China under Grant no. 11101454, the Educational Commission Foundation of Chongqing of China under Grant no. KJ130626, the Natural Science Foundation Project of CQ CSTC under Grant no. cstc2013jcyjA30001, and the Program of Chongqing Innovation Team Project in University under Grant no. KJTD201308.