A modified decomposition method to solve linear elliptic partial differential equations in anisotropic domains by enhancing isotropic responses

Abstract

A modified version of Adomian decomposition method is presented and applied to solve linear elliptic differential equations in anisotropic domains in a recursive manner. A complementary constitutive decomposition, guided by a constitutive hierarchy, governs the superposition of the operator—a step of the Adomian’s method—and is defined as the original constitutive tensor being constructed by an isotropic tensor added to an anisotropic one. The recursive system obtained by the application of Adomian decomposition method is related to an enhancement of the problem’s isotropic solution by the domain’s anisotropy. Alternative solution procedures as Rayleigh–Ritz and and finite element methods are considered. Requirements for absolute convergence are presented and are related to the decomposition as well as to the material’s anisotropy. The rate of convergence is close related to the eigenvalues of the decomposed constitutive terms. The methodology is demonstrated for two- and three-dimensional of heat conduction, two- and three-dimensional elasticity problems and for homogeneous and heterogeneous thin and thick plates. Numeric and semi-analytic results are presented generalised plane stress elasticity as well as for anisotropic thin and laminated thick plates.