Efficient linear and nonlinear heat conduction with a quadrilateral element

Wing Kam Liu*, Ted Belytschko

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


A method is presented for performing efficient and stable finite element calculations of heat conduction with quadrilaterals using one‐point quadrature. The stability in space is obtained by using a stabilization matrix which is orthogonal to all linear fields and its magnitude is determined by a stabilization parameter. It is shown that the accuracy is almost independent of the value of the stabilization parameter over a wide range of values; in fact, the values 3, 2 and 1 for the normalized stabilization parameter lead to the 5‐point finite difference, 9‐point finite difference and fully integrated finite element operators, respectively, for rectangular meshes; numerical experiments reported here show that the three have identical rates of convergence in the L2 norm. Eigenvalues of the element matrices, which are needed for stability limits, are also given. Numerical applications are used to show that the method yields accurate solutions with large increases in efficiency, particularly in nonlinear problems.

Original languageEnglish (US)
Pages (from-to)931-948
Number of pages18
JournalInternational Journal for Numerical Methods in Engineering
Issue number5
StatePublished - May 1984

ASJC Scopus subject areas

  • Numerical Analysis
  • Engineering(all)
  • Applied Mathematics


Dive into the research topics of 'Efficient linear and nonlinear heat conduction with a quadrilateral element'. Together they form a unique fingerprint.

Cite this