Stress analysis of elastic bi-materials by using the localized method of fundamental solutions

: The localized method of fundamental solutions belongs to the family of meshless collocation methods and now has been successfully tried for many kinds of engineering problems. In the method, the whole computational domain is divided into a set of overlapping local subdomains where the classical method of fundamental solutions and the moving least square method are applied. The method produces sparse and banded stiffness matrix which makes it possible to perform large-scale simulations on a desktop computer. In this paper, we document the first attempt to apply the method for the stress analysis of two-dimensional elastic bi-materials. The multi-domain technique is employed to handle the non-homogeneity of the bi-materials. Along the interface of the bi-material, the displacement continuity and traction equilibrium conditions are applied. Several representative numerical examples are presented and discussed to illustrate the accuracy and efficiency of the present approach.


Introduction
The multi-layered materials containing single or multiple layers have been widely synthesized, designed and utilized in industrial application to improve machining performance [1][2][3][4]. The wellestablished and widely applied finite element (FEM), finite difference (FDM) and boundary ), (4) where () x , 1, 2 j = , stand for the outward unit normal vector, i u and i t represent the prescribed boundary conditions. Here and in the following, the classical Einstein's notation for summation over repeated subscripts is employed.  (6) where G is the shear modulus of the material and ij  denotes the well-known Kronecker delta. The boundary tractions (4) are related to stresses by:

The LMFS formulation for linear elasticity problems
In the LMFS, a cloud of points is firstly scatted inside the entire computational domain (see Figure 1). And the method defines a set of overlapping local subdomains and matches the solutions in each of the subdomain by using the classical MFS approximation. For each node ( )    (10) are fundamental solutions for displacements [5,55,56]. It is noted that the radius s R of the artificial circle is a parameter which should be manually determined by the user. Substituting the coordinates of 1 s N + points inside s  to Eqs (8) and (9) will result in the following system of equations: 11 or for briefly: ,..., , , , ,..., ,  The boundary points should satisfy the given boundary conditions. For node with displacement conditions, the following linear system of equations can be obtained: Similarly, for points with traction boundary conditions, another system of linear algebraic equations can be obtained [55]. By combining the above systems of equations, the following spare and banded linear algebraic equations can be formed: is the vector of unknown displacements at every node, and 21 N  B is the vector of the corresponding boundary conditions. Both the direct matrix inverse method and the moving least square approximation can be used to solve the final system of equations. Here, the direct matrix inverse method is used. On solving this system of equations, the numerical solutions of displacements at every node inside the entire domain can be obtained. Once all displacements are solved, the stresses can be obtained by replacing the displacement fundamental solutions with these for stresses, we refer the interested readers for Refs. [55,57] for further details.

The multi-domain LMFS for bi-materials
The aforementioned numerical procedures are derived for homogeneous materials, a multidomain technique [53,54] is used here to solve the bi-material problems. As shown in Figure 2, the bimaterial considered here is divided into two subdomains along the interface, which are respectively homogeneous and isotropic, with the upper layer as the subdomain 1  and the lower as the subdomain 2  . The final system of equations can be formed by assembling equations written for each subdomain, based on the displacement continuity and traction equilibrium conditions along the interface of the bi-material. Application of the LMFS formulation to both subdomains will result in the following matrix equations:  (22) for the subdomain 2  . In the above equations, the subscript 'I' denotes the interface of the bi-material.
By solving the above equations, the displacements at any point inside the domain and along the boundary can be determined. More equations will be added into the equation system in a similar way for other possible subdomains.

Numerical results and discussions
Two benchmark numerical examples are examined in this section to verify the accuracy and efficiency of the present multi-domain LMFS method for the stress analysis of elastic bi-materials. In order to evaluate the performance of the present method, the relative error defined below is employed:  M on the overall accuracy of the LMFS method has been discussed in Refs. [49,55]. It was observed that the numerical solutions were relatively insensitive to these two parameters.

Test problem 1: stress analysis of a finite bi-material plate
The first example considered here is the stress analysis of a finite bi-material plate, where L=1 m is the length of the bi-material,  Figure 3. Similar to Ref. [54], the following exact solutions (26) for the layered coating and (27) for substrate are employed. A Chebyshev collocation scheme proposed by Bai et al. [58] is applied for calculating the particular solutions for the given elastic problem. The Poisson's ratio and elastic modulus of the bi-material are taken to be 0.25 For the numerical implementation, a total of 3600 points are discretized inside the whole computational domain.  Figures 4 and 5 show the relative errors of the calculated displacements and stresses at points along the interface of the bi-material. We can observe that the results calculated by using the present LMFS method are in excellent agreement with the corresponding analytical solutions. The LMFS model with only 3600 collocation points are quite accurate for this example, the size of the final system of equations is, therefore, quite small.

Test problem 2: stress analysis of a circular shaft with two layers of coatings
Here, the stress analysis of a circular shaft with two layers of coatings is considered (see Figure 8). The two coatings consist of two different materials where the Young's modulus of outside coating/Young's modulus of inner coating=1/2 and Poisson ratio of outside coating = Poisson ratio of inner coating 0.2 = . The substrate and the two layered coatings have radii 12   To investigate the convergence of the present multi-domain LMFS method, the relative errors of the radial ( r  ) and tangential (   ) stresses at points A(5.5, 0) and B(6.5, 0) are provided in Figure 9, as the number of the LMFS nodes increases from 800 to 15,000. As can be seen from Figure 9, the present LMFS results converge towards their corresponding analytical solutions as the number of nodes increases. In addition, radial and tangential stress results calculated at points along the line 0  = are illustrated in Table 1.

Discussion
This paper makes the first attempt to apply the localized method of fundamental solutions (LMFS) for the stress analysis of two-dimensional (2D) elastic bi-materials. A multi-domain LMFS formulation is proposed to handle the non-homogeneity of the bi-materials. On the subdomain interface, compatibility of displacements and equilibrium of tractions are imposed. Two benchmark examples are well-studied to clarify the accuracy, efficiency and convergency of the present multi-domain LMFS approach. Further analyses are required in order to fully explore the applicability of the new method, including the detailed convergence order analyses of the method as well as the optimal choice of many different parameters. It must be pointed out that the proposed LMFS has many inherent shortcomings compared with the FEM. For example, the method cannot be used for problems whose fundamental solution is either not known or cannot be determined. The method is also not applicable to nonlinear problems for which the principle of superposition does not hold. The method also offers great promise in the analysis of many other problems, including wave propagations, flow problems, non-linear problems. Some work along these lines is already underway.