Nonlinear transient heat transfer is a common and important phenomenon in practical engineering and attracts many research attentions during past years. In this paper, Precis Time Integration (PTI) is proposed for the numerical analysis of nonlinear heat conduction. Precise Time Integration method is a very special scheme possessing unconditionally stable and very high precision for linear transient system. When it is extended to nonlinear system, the method is still stable, and unlike the conventional methods, independent to the numerical precision. So it is easy to construct a computational algorithm has both stability and high precision. In the paper, the Precise Time Integration method is introduced firstly, and then, its extension to transient heat conduction problem is presented. The symmetry of matrix exponential of PTI in heat conduction is proved and used in sub-domain integration to reduce computational expense in great deal. As for nonlinear heat conduction, the Predictor-Corrector algorithm is employed to solve the nonlinear equations. Numerical examples validate the method. It is concluded that Precis Time Integration method is an accurate and stable numerical scheme for the linear and nonlinear transient heat conduction problems.

The precise time integration method proposed for linear time-invariant dynamic system can give precise numerical results approaching the exact solution at the integration points. However, difficulties arise when the algorithm is used for the non-homogenous dynamic systems due to the inverse matrix calculation required. In this paper, a new algorithm is proposed to convert non-homogenous dynamic equations into homogenous ones by means of the dimensional expanding method. With this conversion, the inverse matrix calculation is not required in the precise time integration method. The new algorithm has enhanced the precise time integration method by benefiting the programming implementation and the numerical stability; it has improved the computational efficiency as well. Numerical examples are given to demonstrate the validity and efficiency of the algorithm.

In this article, the precise time integration (PTI) method in conjunction with the finiteelement method is applied to conduct the heat transfer analysis of functionally graded materials. Two techniques are introduced to improve the computational efficiency and accuracy. The first one is an adaptive algorithm of PTI, which can determine the algorithm parameters according to the problem characteristics and the prescribed computational error tolerance. The second one is the isoparametric graded finite-elements method, which can be compatible with the problem of spatially varying material properties. Numerical examples are presented to demonstrate the advantages of the present method.