A local iteration scheme for nonlinear two-dimensional steady-state heat conduction: a BEM approach

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13 Scopus citations


An efficient algorithm is proposed to solve the steady-state nonlinear heat conduction equation using the boundary element method (BEM). Nonlinearity of the heat conduction equation arises from nonlinear boundary conditions and temperature dependence of thermal conductivity. Using Kirchhoff's transformation, the case of temperature dependence of thermal conductivity can be transformed to the nonlinear boundary conditions case. Applying the BEM technique, the resulting matrix equation becomes nonlinear. The nonlinearity, however, only involves the boundary nodes that have nonlinearboundary conditions. The proposed local iterative scheme reduces the entire BEM matrix equation to a smaller matrix equation whose rank is the same as the number of boundary nodes with nonlinear boundary conditions. The Newton-Raphson iteration scheme is used to solve the reduced nonlinear matrix equation. The local iterative scheme is first applied to two one-dimensional problems (analytical solutions are possible) with different nonlinear boundary conditions. It is then applied to a two-region problem. Finally, the local iterative scheme is applied to two cavity problems in which radiation plays a role in the heat transfer.

Original languageEnglish (US)
Pages (from-to)650-657
Number of pages8
JournalApplied Mathematical Modelling
Issue number12
StatePublished - Dec 1993


  • boundary element method
  • nonlinear heat conduction
  • sintering

ASJC Scopus subject areas

  • Modeling and Simulation
  • Applied Mathematics


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