The method of fundamental solutions (MFS) is a boundary-type and truly meshfree method, which is recognized as an efficient numerical tool for solving boundary value problems. The geometrical shape, boundary conditions, and applied loads can be easily modeled in the MFS. This capability makes the MFS particularly suitable for shape optimization, moving load, and inverse problems. However, it is observed that the standard MFS lead to inaccurate solutions for some elastostatic problems with stress concentration and/or highly anisotropic materials. In this work, by a numerical study, the important parameters, which have significant influence on the accuracy of the MFS for the analysis of two-dimensional anisotropic elastostatic problems, are investigated. The studied parameters are the degree of anisotropy of the problem, the ratio of the number of collocation points to the number of source points, and the distance between main and pseudo boundaries. It is observed that as the anisotropy of the material increases, there will be more errors in the results. It is also observed that for simple problems, increasing the distance between main and pseudo boundaries enhances the accuracy of the results; however, it is not the case for complicated problems. Moreover, it is concluded that more collocation points than source points can significantly improve the accuracy of the results.

The MFS is an integration-free meshfree method, which has found a wide application because of its accuracy and simplicity. In the MFS, the solution is expressed in terms of known fundamental solutions, which exactly satisfy the governing equations of the problem. This semi-analytic nature of the MFS makes it suitable for obtaining accurate solutions [

The literature review shows a few studies on the MFS for the analysis of elastostatic problems in anisotropic media. Raamachandran et al. [

Other variants of the MFS have also been used for anisotropic elasticity. Liu et al. [

Base on the above literature review, it can be seen that the MFS and its variants have been employed for solving anisotropic elastostatic problems; however, the parameters, which have significant influence on the accuray of the MFS for solving these problems, have not been studied yet. In this work, influences of three important parameters, i.e., the degree of anisotropy of the problem, the ratio of the number of collocation points to the number of source points, and the distance between main and pseudo boundarires, on the accuracy of the MFS results are numerically investigated. Determining a suitable configuration of source points is a major issue in the MFS. There are many studies on the location of source points and collocation points in the MFS. Among them, one can refere to the works on Laplace and Helmholtz equations [

Consider a plane stress/strain problem in the anisotropic domain

Representing the displacement components in the _{1} and _{2}) directions by _{1} and _{2}), respectively, the strain components can be expressed as follows [

The relationship between strain and stress components can be written as follows:

where _{ij}

for plane stress problems, and

for plane strain problems. The constants _{ij}

_{i}_{i}_{i}x_{j}_{ij}

In plane stress/strain problems, two boundary conditions are considered for a boundary point that can be expressed as follows:

where _{1} and _{2} are given values for each boundary point. _{1} and _{2} represent linear combinations of

The MFS formulation of the problem starts with considering

where

The components of the strain tensor can be found as follows:

where

The components of the stress tensor are computed as follows:

where

There are 2

If the number of collocation points is greater than the number of source points, i.e.,

A point worth emphasizing here is that equations corresponding to the boundary conditions, i.e., _{0} and _{0} and

We represent the displacement fundamental solution with _{0}, _{0}) and

which is common in the BEM [_{01}, _{02}, _{1}, and _{2} in

where

Two complex roots of _{ik}

The real and imaginary parts of _{ji}

where

where _{11} = _{12} = 1,

In this section, by presenting several numerical examples, the important parameters, which have significant influence on the accuracy of the MFS results for 2D anisotropic elastostatic problems, are investigated. These parameters are:

1-The degree of the anisotropy of materials

2-The ratio of the number of collocation points to the number of source points, i.e.,

3-The distance between main and pseudo boundaries, as denoted by

Anisotropic materials show various degrees of anisotropy. The largeness of the ratios _{1}/ _{2} and

_{1}/ _{2} |
||
---|---|---|

Silicon carbide/ceramic [ |
1.09 | 2.75 |

Oseton (bone) [ |
1.88 | 1.96 |

E-glass/epoxy [ |
4.0 | 9.5 |

Pine (Wood) [ |
12.82 | 12.34 |

Carbon/epoxy [ |
14.2 | 21.3 |

To investigate the effect of the distance between main and pseudo boundaries, i.e.,

where

Two types of anisotropic elastostatic problems can be analyzed by the MFS. In the first type, the boundary condition is of Dirichlet type and is prescribed by a function which satisfies the governing equation of the problem. In the second type, which is more practical, the boundary conditions are of mixed type and are prescribed by functions which do not satisfy the governing equations. As mentioned in reference [

A circular domain (disk) of unit radius centered at the origin of the coordinate system under the plane stress conditions is considered. The geometry of the disk is suitable for the MFS because it is simple and smooth. Simple boundary conditions of Dirichlet type are also considered for the problem. It is assumed that the radial and circumferential displacements of the boundary are _{r}

The disk is assumed to be made of the unidirectional S-glass/epoxy composite material. The material is orthotropic and its elastic constants in its principal material directions are

We use the standard MFS with

It is observed that by increasing the value of _{1} and

_{1}/ _{2} |
||
---|---|---|

Material 1 | 1.09 | 2.75 |

Material 2 | 4.0 | 9.5 |

Material 3 | 14.2 | 21.3 |

For each material, the problem is solved using the standard MFS (

where

Material 1 | 1.20^{−5} |
1.15^{−5} |
9.86^{−6} |
8.40^{−8} |
7.36^{−8} |
6.52^{−8} |

Material 2 | 5.12^{−3} |
1.47^{−3} |
2.84^{−3} |
7.79^{−5} |
3.48^{−5} |
4.15^{−5} |

Material 3 | 5.98^{−3} |
2.43^{−3} |
3.50^{−3} |
8.22^{−5} |
3.36^{−5} |
4.83^{−5} |

In the previous section, a disk with a simple boundary condition was considered. In this section, a complicated geometry with more challenging boundary conditions is considered. As shown in

There is no exact solution for this problem. An accurate reference solution is obtained by the finite element method (FEM) with a fine mesh using ANSYS software. Maximum absolute values of stresses occur on the circle and the error of solution is defined in terms of these stresses as follows:

Importantly, a large number of nodes and elements should be considered for finite element (FE) analysis of this problem to obtain relatively accurate solutions. In

Number of nodes | |||
---|---|---|---|

27307 | 2.0786 | 2.8727 | 2.4730 |

107995 | 2.1224 | 2.8773 | 2.5208 |

427599 | 2.1393 | 2.9125 | 2.5189 |

In the first case, the problem with Material 1 is solved using the MFS. Since the degree of the anisotropy of Material 1 is relatively small, by considering only 164 source points (100 and 64 source points for outer and inner boundaries, respectively), sufficiently accurate solutions can be obtained. Moreover, we consider different values of the location parameter of source points, i.e.,

1 | 0.003620 | 0.002849 |

2 | 0.003036 | 0.0002779 |

3 | 0.003497 | 0.0002805 |

4 | 0.003653 | 0.0002856 |

In the second case, the problem with Material 2, which has a higher degree of anisotropy, is solved. In this case, 100+128 = 228 source points, i.e., 100 source points for the outer boundary and 128 source points for the inner boundary, are considered instead of 164. The corresponding errors are listed in

0.8682 | 0.01120 | |

0.002727 | 0.005994 | |

0.002731 | 0.006368 | |

0.002720 | 0.006428 |

Solving the problem with Material 3 is far more difficult because the degree of anisotropy of Material 3 is larger than the previous cases. In

Material 1 | 0.001214 | 0.0001916 | 0.0001641 |

Material 2 | 0.001809 | 0.001009 | 0.001597 |

Material 3 | 0.03615 | 0.02874 | 0.07752 |

Material 1 | 0.001006 | 0.0001864 | 0.0001666 |

Material 2 | 0.001380 | 0.0008963 | 0.0009256 |

Material 3 | 0.1223 | 0.002202 | 0.005767 |

Material 1 | 0.0009517 | 0.0001799 | 0.0001666 |

Material 2 | 0.001261 | 0.0008971 | 0.0009115 |

Material 3 | 0.05698 | 0.001640 | 0.002011 |

For better clarification, the contours of the stress components in the rectangular plate obtained by the FEM with 427599 nodes, and by the MFS with 912 source points and

The important parameters, which have significant influence on the accuracy of the MFS for the analysis of two-dimensional anisotropic elastostatic problems, were numerically studied in this work. Although the findings reported in this research is not based on mathematical analysis, some suggestions regarding the MFS for the elastostatic analysis of anisotropic bodies are offered that can help solve practical problems. According to the numerical studies conducted in the work, the following conclusions are drawn:

Three materials with low, moderate, and high degrees of anisotropy were examined in simple and complicated elastostatic problems. As observed, in the same conditions, the accuracy of the results corresponding to the problem with highly anisotropic material was far lower than the same problem with lower degree of anisotropy.

Anisotropic elastostatic problems with simple geometry and simple boundary conditions can be efficiently solved using the standard MFS with equal numbers of source and collocation points, i.e.,

For anisotropic elastostatic problems with complicated geometry and/or complicated boundary conditions, more collocation points than source points render far more accurate results. Through the examples presented in this work, it was observed that

In simple problems, raising the value of

Unlike the element-based methods, meshfree methods do not require any re-meshing process [

Assume that the first principal material direction makes an angle of _{ij}

where