In this work, the localized method of fundamental solution (LMFS) is extended to Signorini problem. Unlike the traditional fundamental solution (MFS), the LMFS approximates the field quantity at each node by using the field quantities at the adjacent nodes. The idea of the LMFS is similar to the localized domain type method. The fictitious boundary nodes are proposed to impose the boundary condition and governing equations at each node to formulate a sparse matrix. The inequality boundary condition of Signorini problem is solved indirectly by introducing nonlinear complementarity problem function (NCP-function). Numerical examples are carried out to validate the reliability and effectiveness of the LMFS in solving Signorini problems.

In 1933, Signorini studied the frictionless contact problem between a linear elastic body and a rigid body. The related boundary condition is expressed by inequality constraints or complementary forms, which is also called the ambiguous boundary [

Based on variational inequality and finite element method (FEM), a heuristic asymptotic formula is derived and applied to solve a percolation in gently sloping beaches [

In recent years, the meshless numerical methods have been popular in solving partial differential equations (PFEs). In general, the meshless numerical methods can be divided into boundary-type and domain-type methods according to the nodes distributions. The boundary type methods only need nodes on the boundary, such as the method of fundamental solution (MFS) [

Recently, a localized boundary type method has been proposed based on the fundamental solutions. It combines the advantages of both the domain type method and the boundary type method. The LMFS approximates the field quantity at each node by a linear combination of the field quantities of the local nodes. The idea of the LMFS is similar to the localized domain type method [

In this work, the LMFS is further applied to the Signorini problem, where the inequality boundary constraints are treated as the nonlinear complementarity problems (NCPs). By utilizing the Fischer-Burmeister NCP-function, the LMFS yields a system of nonlinear algebraic equations [

The structure of this paper is organized as follows.

The Signorini problem normally includes an inequality boundary constrain. The details of the 2D Signorini problem with solution

where _{D} and Γ_{N}, respectively. _{s},

where

Signorini inequality boundary can be classified according to different situations of Γ_{s} as the following:

Considering

_{s} boundary for

or

From the above formulations, the Signorini problem can be transferred into a nonlinear problem given as

In this section, the LMFS discretization is presented based on a simple Laplace problem. As shown in _{b1} is the number of nodes along Γ_{D}, and _{b2} is the number of nodes along Γ_{N}.

In the LMFS, the numerical solution of the ^{th} node in

where ^{th} sub-domain, as shown in _{s}. The number of the source nodes is the same as the local nodes. The radius of the circular artificial boundary ^{th} node. Previous work in [

The numerical solutions of the local nodes in the sub-domain can be obtained by considering

where

Thus, the numerical solution at ^{th} node can be expressed in the following form:

where ^{th} node and ^{th} node and the other

In the traditional MFS, the derivative of unknown coefficients can be directly solved with ^{th} node can be derived using similar procedures in

where

are the vectors of the derivative form of the fundamental solution at the ^{th} node,

Since the fundamental solution satisfies the governing equations, the following form is introduced to enforce the internal nodes to numerically satisfy governing equations by considering

For the boundary nodes located at Γ_{D}, the Dirichlet boundary conditions can be given directly as

similarly, the boundary nodes on the Γ_{N} satisfy the Neumann boundary conditions, and the third system of linear algebraic equations can be obtained

where ^{th} node. The numerical solutions of the LMFS are obtained by considering

where

By using the transformation of NCP-Function in ^{−6}.

Here, we consider a well-known steady-state shallow dam problem with respect to infiltration on a gently sloping beach [

where

Totally, _{in} = 400 nodes are uniformly distributed on the boundary and in the inner domain, respectively. Five source nodes are used in the LMFS to evaluate the numerical results by considering three different surface profiles in

A different number of boundary nodes

0.60 | 0.65 | 0.70 | 0.75 | 0.80 | 0.85 | 0.90 | |
---|---|---|---|---|---|---|---|

_{1}( |
−0.6400 | −0.7025 | −0.7600 | −0.8125 | −0.8600 | −0.9025 | −0.9400 |

MFS ( |
−0.6599 | −0.7148 | −0.7664 | −0.8125 | −0.8601 | −0.9027 | −0.9417 |

_{1}( |
−0.6400 | −0.7025 | −0.7600 | −0.8125 | −0.8600 | −0.9025 | −0.9400 |

MFS ( |
−0.6599 | −0.7148 | −0.7664 | −0.8125 | −0.8601 | −0.9027 | −0.9417 |

BIE ( |
−0.6395 | −0.7024 | −0.7600 | −0.8125 | −0.8600 | −0.9025 | −0.9400 |

BIE ( |
−0.6404 | −0.7025 | −0.7600 | −0.8125 | −0.8600 | −0.9025 | −0.9400 |

BIE ( |
−0.6400 | −0.7025 | −0.7600 | −0.8125 | −0.8600 | −0.9025 | −0.9400 |

SBM ( |
−0.6401 | −0.7025 | −0.7600 | −0.8125 | −0.8600 | −0.9025 | −0.9400 |

SBM ( |
−0.6400 | −0.7025 | −0.7600 | −0.8125 | −0.8600 | −0.9025 | −0.9400 |

SBM ( |
−0.6400 | −0.7025 | −0.7600 | −0.8125 | −0.8600 | −0.9025 | −0.9400 |

LMFS ( |
−0.6400 | −0.7025 | −0.7600 | −0.8125 | −0.8600 | −0.9025 | −0.9400 |

LMFS ( |
−0.6400 | −0.7025 | −0.7600 | −0.8125 | −0.8600 | −0.9025 | −0.9400 |

LMFS ( |
−0.6400 | −0.7025 | −0.7600 | −0.8125 | −0.8600 | −0.9025 | −0.9400 |

28 | 48 | 64 | 84 | 128 | 256 | |
---|---|---|---|---|---|---|

BPIM [ |
25 | 25 | 24 | 24 | 25 | 25 |

MFS [ |
72 | 119 | 893 | 946 | ||

LMFS | 6 | 7 | 7 | 6 | 8 | 8 |

The well-known electroplating problem [

In this case, a Signorini problem of a square field (−0.5, 0.5)

where _{s} constitutes the workpiece to be painted. The exact solution of this problem is given in [

_{s} when 80 boundary nodes and 400 interior notes are used. Different values of _{s} are presented with a curve of arc-length

As shown in

where

We take _{in} = 400 for the LMFS, and compare the numerical results obtained from the LMFS with the numerical results of the MFS [

In this paper, the LMFS is further applied to the Signorini problem. The stability of the LMFS is validated by considering several different numerical examples. The numerical results show that the number of source nodes and the initial value have little effect on the number of iterations steps. Comparing the numerical results obtained by LMFS with other methods, the LMFS can get more accurate results with the same number of nodes. The LMFS is more efficient, stable and has a higher convergence rate in dealing with Signorini problem. In our future work, more complicated problems, such as three-dimensional Signorini problems, Seepage problems, obstacle problems will be considered.