A new functional perturbation method for linear non-homogeneous materials

A functional perturbation method (FPM), for solving boundary value problems of linear materials with non-homogeneous properties is introduced. The FPM is based on considering the unknown field such as displacements or temperatures, as a functional of the non-uniform property, i.e., elastic modulus or thermal conductivity. The governing differential equations are expanded functionally by Fréchet series, leading to a set of differential equations with constant coefficients, from which the unknown field is found successively to any desirable degree of accuracy. A unique property of the FPM is that once the Fréchet functions are found, the solution for any morphology is obtained by direct integration, without re-solving the differential equation for each case. The FPM procedure is outlined first for general linear differential equations with non-uniform coefficients. Then, four examples are solved and discussed: a 1D tensile loading of a rod with continuously varying and discontinuous moduli, beam bending, beam deflection on non-uniform elastic foundation and a unidirectional heat conduction problem. FPM results are compared with the exact (if exists) or numerical solution. The FPM accuracy for the bending problem is also compared to the common Rayleigh-Ritz and Galerkin methods. It is shown that the FPM is inherently more accurate, since the convergence rate of the other methods depends on the arbitrarily chosen shape functions, while in the FPM, these functions are obtained as generic results of each order of the solution. The FPM solution is analytical, and is shown to be suitable for large variations in material properties. Thus, a direct insight of each functional perturbation order is possible. Advantages and limitations of the FPM as compared to other existing methods are discussed in detail.