Premium
Efficient linear and nonlinear heat conduction with a quadrilateral element
Author(s) -
Liu Wing Kam,
Belytschko Ted
Publication year - 1984
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620200510
Subject(s) - quadrilateral , finite element method , mathematics , eigenvalues and eigenvectors , mathematical analysis , nonlinear system , mixed finite element method , quadrature (astronomy) , norm (philosophy) , physics , quantum mechanics , political science , law , optics , thermodynamics
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 L 2 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.