水坝瞬态热传导分析中的拟初始条件边界元法

Application of quasi-initial condition boundary element method in transient heat conduction problem on gravity dams

使用一种时域边界元方法对混凝土水坝进行瞬态热传导分析。在对时间积分进行离散计算时,采用一种拟初始条件法,即在时间步迭代计算的过程中,将之前计算结果对当前时间步的影响都视作当前时间步的初始条件。在所取时间步长较小的情况下,这种处理方法容易导致数值结果不稳定,即每一步的计算误差会累计放大,最终导致计算崩溃。本文提出一种虚拟时刻方法以缓解这类数值不稳定现象,在该方法中,时间步长首先放大至合适尺度,计算某个虚拟时刻(往往在真实计算时刻之后)的温度和流量分布,再通过插值方法换算出真实时刻的温度和流量分布。在虚拟时刻点上的温度和流量计算过程中,边界已知温度或流量由真实时刻的温度或流量进行外插得到。本文简单证明了该方法在温度和流量随时间呈线性变化情况下的正确性,最后给出了两个分析实例,验证了该方法的准确性和稳定性。

This paper applies a quasi-initial condition boundary element method （BEM） to solve the transient heat conduction problems on gravity dam. In this application,a quasi-initial condition method is utilized to calculate the integral over time. In the time-stepping scheme,the effects of the results compu- ted forward on the current step are treated as the initial condition of the current step computation. In the quasi-initial condition method, however, numerical stability problem appears when the utilized time step is very small. The computational error increases gradually with the time stepping. To solve this numerically unstable problem,a virtual time point method is presented in this paper. In this method,the computational time step is amplified to some virtual time step which is usually with larger scale. The physical variables at the virtual time point are evaluated at first. In the evaluation of the physical variables at virtual time point, the boundary condition at that time is assumed and computed through a linear extra-interpolation scheme. The variables at the computational time point are then evaluated through an interpolation scheme. Validity of this method in the case that the physical variables vary linearly with time is proved. Two numerical examples are presented finally to show the accuracy and the stability of the present method.