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The paper analyzed the thermal problem of the 2D FGM beam using meshless weighted least-square (MWLS) method. The MWLS as a meshless method is fully independent of mesh, and an approximate function was used to construct a series of linear equations to solve the unknown field variable, which avoided the troublesome task of numerical integration. The effectiveness and accuracy of the approach were illustrated by a clamped-clamped FGM beam which was subjected with interior heat source. The volume fraction of FGM beam was assumed to be given by a simple power law distribution. The effective material properties of the FGM beam were assumed to be temperature independent and calculated by Mori-Tanaka method. The results showed that a good agreement was achieved between the proposed meshless method and commercial COMSOL Multiphysics.

FGMs can resist high temperatures and are proficient in reducing the thermal stress and have received more attention from the researchers [

Liu and Gu [

Zhou et al. presented steady-state [

The solution of MWLS analysis to the thermal problem is described in this section. The shape functions in MWLS analysis is a moving least-squares approximation scheme which is originally developed for the smooth interpolation of irregularly distributed data.

Construct the local approximate function

To obtain

Solving

The steady-state heat conduction equation and the thermal boundary conditions of FGM objects are as follows, respectively,

Substituting the unknown temperature field variable

We use an alternative discrete equation to avoid integration and (

The system equations of the MWLS method for solving steady-state heat conduction equations are written in a matrix form_{1},_{2,} and_{3} is the number of interior nodes in the boundary

Consider the 2D FGM anisotropic linear elastic body defined in the domain

Substituting

Substituting the unknown field variable

The thermal stresses is written in the matrix form

Two homogenization methods are often used to evaluate the effective material properties for FGMs. One is the rule of mixtures, and the other is the micromechanical model. The former is simply a linear rule of mixtures and the effective value can be determined by

The most widely used micromechanical model is Mori-Tanaka model [

In order to demonstrate the efficiency and accuracy of the presented method, firstly, we choose a isotropic square region (Case 1) with defined boundary conditions; through heat conduction analysis the results are compared with the analytical solutions and FDM. Then a clamped-clamped FGM beam (Case 2) which was subjected with interior heat source is analyzed using MWLS method.

A 100×100m isotropic square region is shown in Figure

Geometry of square region and boundary conditions.

An analytical solution is given as

The design region is discretized as 20×20,

Maximum of relative error.

Cartesian grid | MWLS | FDM |
---|---|---|

10×10 | 0.15% | 0.35% |

20×20 | 0.05% | 0.077% |

30×30 | 0.024% | 0.033% |

Temperature field distribution (20×20).

2D display

Top boundary

A clamped-clamped FGM beam is shown in Figure

Material property.

Material property | Al | SiC |
---|---|---|

k(W/mK) | 233 | 65 |

E(Pa) | 7e10 | 4.27e11 |

| 0.3 | 0.17 |

2.34e-5 | 4.3e-6 |

Initial condition.

The beam is assumed to be in a state of plane strain normal to the xy plane, and the design region is discretized as 31×15 and 61×30. The effective material properties are determined by the Mori-Tanaka model (Eq. (

Comparison of MWLS method and COMSOL Multiphysics.

method | Temperature/K | X displacement /mm | Y displacement/mm | |||
---|---|---|---|---|---|---|

maximum | minimum | maximum | minimum | maximum | minimum | |

MWLS (31×15) | 361.8 | 300 | .596 | -.873 | 2.95 | -2.8 |

MWLS (61×30) | 361.3 | 300 | .71 | -.72 | 3.88 | -3.84 |

COMSOL (948 triangle mesh) | 361.1 | 300 | .639 | -.639 | 3.81 | -3.81 |

Comparison of our method with COMOSOL of a=0.

Temperature field of our method

Temperature field of COMSOL

x-displacement of our method

x-displacement of COMSOL

y-displacement of our method

y-displacement of COMSOL

For a=2, we analyzed the heat conduction and thermoelastic problem using MWLS method. Temperature field distribution, x-displacement, and y-displacement are plotted in Figures

Comparison of different material distribution in different heat source.

MWLS (31×15) | Temperature/K | Temperature/K | ||
---|---|---|---|---|

Q=5e5W/m^{3} | Q=5e6W/m^{3} | |||

maximum | minimum | maximum | minimum | |

a=0 | 361.8 | 300 | 910.8 | 300 |

a=2 | 425.6 | 300 | 1558.2 | 300 |

a=3 | 436.1 | 300 | 1660.9 | 300 |

Distribution of temperature field.

x-displacement/m.

y-displacement/m.

Stress results comparison of thermoelastic analysis among a=0, a=2, and a=3.

In this paper, a novel thermoelastic analysis of FGM beam based on MWLS method was presented. We do thermoelastic and heat conduction analysis aimed at a clamped-clamped thick beam which is subjected with interior heat source. The FGM beam is assumed to be given by a simple power law distribution. Material properties of the FGM beam are obtained by Mori-Tanaka method. Through being compared with analytical solution and the commercial software of COMSOL Multiphysics, the effectiveness and accuracy are verified. We also listed the comparison of thermal stresses with the variation of power law index. The present method of analysis will be also useful in the design and optimization of FGM objects.

Coefficient

Matrices of computation

Radius of the circular support domain

The nodal parameter of the field variable at node

Node indices

Thermal conductivity

Number of neighbor points

Number of nodes

Outward surface normal

Shape functions

Number of interior nodes in the boundary

Basis function

Heat source

Components of the Cauchy stress tensor

Complementary parts of the boundary

Stiffness matrix

Young’s modulus

Coefficient of thermal expansion

Volume fraction

Temperature

Ambient temperature

The heat transfer coefficient

The moving least-square approximation function

The positions of the nodes

Fixed domain

Closed boundary of

Dirichlet boundary

Neumann boundary

Mixed boundary

Weight functions

Normal heat flux

Given traction on

Displacement components

Displacement of x, y

Infinitesimal strain vector

Poisson’s ratio

Shear modulus

Bulk modulus.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

The work described in this paper was supported by a grant from the National Natural Science Foundation of China (Projects No. 51505131, U1504106) and Program for Innovative Research Team (No. T2017-3) of Henan Polytechnic University. The correlative members of the projects are hereby acknowledged.