Computational Mechanics

, Volume 60, Issue 6, pp 943–967 | Cite as

A scaled boundary finite element formulation with bubble functions for elasto-static analyses of functionally graded materials

  • E. T. OoiEmail author
  • C. Song
  • S. Natarajan
Original Paper


This manuscript presents an extension of the recently-developed high order complete scaled boundary shape functions to model elasto-static problems in functionally graded materials. Both isotropic and orthotropic functionally graded materials are modelled. The high order complete properties of the shape functions are realized through the introduction of bubble-like functions derived from the equilibrium condition of a polygon subjected to body loads. The bubble functions preserve the displacement compatibility between the elements in the mesh. The heterogeneity resulting from the material gradient introduces additional terms in the polygon stiffness matrix that are integrated analytically. Few numerical benchmarks were used to validate the developed formulation. The high order completeness property of the bubble functions result in superior accuracy and convergence rates for generic elasto-static and fracture problems involving functionally graded materials.


Functionally graded materials Scaled boundary finite element method Shape functions Bubble functions Polygon elements Heterogeneous 


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.School of Engineering and Information TechnologyFederation University of AustraliaBallaratAustralia
  2. 2.School of Civil and Environmental EngineeringThe University of New South WalesSydneyAustralia
  3. 3.Department of Mechanical EngineeringIndian Institute of Technology, MadrasChennaiIndia

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