A Barycentric Lagrange Interpolation Collocation Method For Steady Heat Conduction In Functionally Graded Materials

In recent years,functionally gradient materials(FGMs)have attracted considerable attention in many engineering fields due to its specific characteristic.FGMs are composite materials,which usually merge two or more materials by varying gradually in specific properties(such as thermal conductivity,specific heat,density,etc.)according to a specific rule.Because of the excellent properties of many materials,such as heat resistance and high hardness,FGMs are usually used in extremely high temperature environments,such as aerospace structures and thermal barrier coatings for nuclear fusion reactors.Therefore,accurate heat transfer analysis is very important for the design,optimization and engineering application of FGMs.Compared with homogeneous materials,the physical parameters of FGMs are usually functions of coordinate variables.Therefore,it is more difficult to solve the heat conduction problem of FGMs.In general,the analytical method has great limitations in solving the heat conduction problem of FGMs except for some simple cases.In recent years,finite element method and other numerical methods have been widely used to solve FGMs heat conduction problems,and many research results have been achieved.However,the inherent problems such as low computational accuracy and difficulty in meshing,also have emerged.In this paper,a new meshless method,named collocation method with Baiycentric Lagrange Interpolation(BLIC),is presented for analyzing the steady,state heat conduction in FGMs.Firstly,one and two-variables Barycentric Lagrange interpolation function at Chebyshev nodes are introduced.Secondly,the two-variables Barycentric Lagrange interpolation and its derivatives are introduced into the controlling equations and boundary conditions of FGMs directly,the linear discretized equation is obtained.Then,the least squares method is applied to obtained the whole nodal temperatures.Lastly,the temperature fields of the exponential,quadratic,trigonometrical and anisotropic FGMs models are simulated by the BLIC method.The numerical results show that the present method that the present BLIC method takes advantages of both barycentric Lagrange interpolation and collocation method such as low computational cost,stable,accurate and easy to numerical implementation.The BLIC method can be extended to solve the transient heat conduction and thermal stress analysis of FGMs.