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.
This is a preview of subscription content, log in to check access.
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.
Karimpour A, Ganji DD (2008) An exact solution for the differential equation governing the lateral motion of thin plates subject to lateral and in-plane loadings. Appl Math Sci 2(34):1665–1678MathSciNetzbMATHGoogle Scholar
Almazmumy M, Hendi FA, Bakodah HO, Almuzi H (2012) Recent modifications of adomian decomposition method for initial value problem in ordinary differential equations. Am J Comput Math 2(3):228–234Google Scholar
Biazar J, Babolian E, Islam R (2004) Solution of the system of ordinary differential equations by adomian decomposition method. Appl Math Comput 147(3):713–719MathSciNetzbMATHGoogle Scholar
Wazwaz Abdul-Majid (2000) A new algorithm for calculating adomian polymials for nonlinear operators. Appl Math Comput 111(1):33–51MathSciNetGoogle Scholar
Garcia-Olivares A (2003) Analytical solution of nonlinear partial differential equations of physics. Kybernetes 32(4):548–560MathSciNetzbMATHGoogle Scholar
Tu YO (1968) The decomposition of an anisotropic elastic tensor. Acta Crystallogr Sect A 24(2):273–282ADSGoogle Scholar
Browaeys JT, Chevrot S (2004) Decomposition fo the elastic tensor and geophysical applications. Geophys J Int 159(2):667–678ADSGoogle Scholar
Norris AN (2006) The isotropic material closest to a given anisotropic material. J Mech Mater Struct 1(2):223–238Google Scholar
Lisbôa TV, Marczak RJ (2017) A recursive methodology for semi-analytical rectangular anisotropic thin plates in linear bending. Appl Math Model 48:711–730MathSciNetGoogle Scholar
Lisbôa TV, Marczak RJ (2018) Adomian decomposition method applied to moderately anisotropic thick plates in linear bending. Eur J Mech A Solids 70:95–114MathSciNetzbMATHGoogle Scholar
Cowin SC, Mehrabadi MM (1995) Anisotropic symmetries of linear elasticity. Appl Mech Rev 48(5):247–285ADSzbMATHGoogle Scholar
Chadwick P, Vianello M, Cowin SC (2001) A new proof that the number of linear elastic symmetries is eight. J Mech Phys Solids 49(11):2471–2492ADSMathSciNetzbMATHGoogle Scholar
Ting TCT (2003) Generalized Cowin–Mehrabadi theorems and a direct proof that the number of linear elastic symmetries is eight. Int J Solids Struct 40(25):7129–7142MathSciNetzbMATHGoogle Scholar
Lisbôa TV, Geiger FP, Marczak RJ (2017) A recursive methodology to determine the mechanical response of thin laminated plates in bending. J Aerosp Technol Manag 9(4):409–422Google Scholar
Šolin P (2005) Partial differential equations and the finite element method. Wiley, New JerseyzbMATHGoogle Scholar
Marczak RJ, Denda M (2011) New derivations of the fundamental solution for heat conduction problems in three-dimensional general anisotropic media. Int J Heat Mass Transf 54(15–16):3605–3612zbMATHGoogle Scholar
Hahn DW, Necati Özişik M (2012) Heat conduction. Wiley, New JerseyGoogle Scholar
Reddy JN (2007) Theory and analysis of elastic plates and shells. CRC Press, Boca RatonGoogle Scholar
Ventsel E, Krauthammer T (2001) Thin plates and shells: theory, analysis and applications. Marcel Dekker Inc, New YorkGoogle Scholar
Reddy JN (2004) Mechanics of laminated composite plates and shells. CRC Press, Boca RatonzbMATHGoogle Scholar
Alternbach H (1998) Theories for laminated and sandwich plates. Mech Compos Mater 34(3):243–252ADSGoogle Scholar