, Volume 54, Issue 1–2, pp 239–260 | Cite as

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

  • T. V. LisbôaEmail author
  • Ch. Zhang
  • R. J. Marczak


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.


Adomian decomposition method Elliptic differential operators Convergence analysis Anisotropic stationary problems 



The authors would like to acknowledge CAPES (Coordination for the Improvement of Higher Educational Personnel—Brazil), CNPq (National Council for Scientific and Technological Development—Brazil) and DAAD (German Academic Exchange Service—Germany) for the funding the research project.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringFederal University of Rio Grande do SulPorto AlegreBrazil
  2. 2.Faculty of Science and Technology, Chair of Structural MechanicsUniversity of SiegenSiegenGermany

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