Reduced numerical integration in thermal transient finite element analysis

Abstract

The stability and accuracy of the so-called “ga-family”, two level time integration algorithms for thermal transient problems are considered. In particular, the influence on the solution characteristics of the order of numerical integration rule employed in the evaluation of the element matrices is examined. Consideration is restricted to the four node bilinear and eight node biquadratic isoparametric elements. The time integration schemes considered in detail are the Crank Nicolson, Galerkin, Modified Galerkin and Euler Backward algorithms. The stability of each process is related to both the eigenproperties of the conductivity and heat capacity matrices and to the eigenvalue spectrum of the thermal equation system. It is shown that inaccuracies and instabilities can arise in solution which are associated with the poor prediction of the element properties by reduced numerical integration rules. For eight node biquadratic elements, the source of error is the inadequate modelling of the eigenvalue spectrum of the heat capacity matrix by reduced quadrature; with both full and reduced integration rules providing very similar values for the conductivity (or stiffness matrix). However, for four node bilinear elements, reduced quadrature can result in an inadequate representation of both the conductivity and heat capacity matrices. It is shown that these inaccuracies are confined to the higher thermal frequencies which can be dominant in the initial stages of a transient response and cause deterioration of the numerical solution. Therefore an “adaptive” scheme is proposed in which a full integration order is used for the first few timesteps, after which a reduced integration rule is employed. Recommendations are made for the selection of time integration algorithms, quadrature rules and element dimensioning which are substantiated by the solution of several numerical examples.