The weak patch test for nonhomogeneous materials modeled with graded finite elements

Glaucio H. Paulino, Jeong Ho Kim

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


Functionally graded materials have an additional length scale associated to the spatial variation of the material property field which competes with the usual geometrical length scale of the boundary value problem. By considering the length scale of nonhomogeneity, this paper presents the weak patch test (rather than the standard one) of the graded element for nonhomogeneous materials to assess convergence of the finite element method (FEM). Both consistency (as the size of elements approach zero, the FEM approximation represents the exact solution) and stability (spurious mechanisms are avoided) conditions are addressed. The specific graded elements considered here are isoparametric quadrilaterals (e.g. 4, 8 and 9-node) considering two dimensional plane and axisymmetric problems. The finite element approximate solutions are compared with exact solutions for nonhomogeneous materials.

Original languageEnglish (US)
Pages (from-to)63-81
Number of pages19
JournalJournal of the Brazilian Society of Mechanical Sciences and Engineering
Issue number1
StatePublished - 2007
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Automotive Engineering
  • Aerospace Engineering
  • General Engineering
  • Mechanical Engineering
  • Industrial and Manufacturing Engineering
  • Applied Mathematics


  • Finite element method (FEM)
  • Functionally graded material (FGM)
  • Graded element
  • Patch test
  • Weak patch test


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