一种无条件稳定的时域推进法与有限元混合分析瞬态热传导问题

AN UNCONDITIONALLY STABLE TIME-STEP-TIME ALGORITHM COUPLED WITH FINITE ELEMENT METHOD FOR TRANSIENT HEAT CONDUCTION ANALYSIS

本文将一种新的无条件稳定的时域推进法与有限元结合,用于分析瞬态热传导问题。根据微-积分型热传导控制方程,对时间变量在小区闻内插值积分,变初-边值问题为一系列离散时刻的边界值问题,再应用有限元法求解之。由于推进中的每一时刻的解,都严格满足原给定的初始条件,就消除了累积误差的影响。此外,这种时域推进法对时间变量的数值积分也优于数值微分。因而,与现行的基于微分型控制方程,采用时-空有限元模型同时离散,再逐步求解的直接积分算法相比,计算精度可大大提高。不难预料,当求解较长时间后的瞬态值时,本算法的优越性会更加明显。

This paper proposes a new and unconditionally stable time-step-time algorithm coupled with the finite element method for analysing transient heat conduction problems. The essential feature of the present method hings on the discretization to the equivalent governing equation of the integral type not the originally differential type. The algorithm adopts a finite-order interpolating polynomial in a partitioned small interval of the time domain with a special function as the interpolation coefficient. After integrating the governing equation with respect to the time variable, a series of elliptic equations for the coefficients are derived. The finite element method is then applied to the solutions to these equations. Since the exact initial condition is directly related to every coefficient and there is no other approximate initial value from the previous solution to be required at each step solution, the accumulative error from a direct integration method is then avoided. Another advantage of the present algorithm comes from the fact that the accuracy of numerical integration is generally higher than that of numerical differentiation. These two substantial accuracy improvements make the present method much more efficient than a direct integration algorithm.