瞬态热传导问题的精细积分-双重互易边界元法

Precise integration DRBEM for solving transient heat conduction problems

采用双重互易边界元法结合精细积分法求解二维含热源的瞬态热传导问题。针对边界积分方程中热源项和温度关于时间导数项引起的域积分,采用双重互易法处理,将域积分转换为边界积分。采用边界元法将边界积分方程离散后,得到关于时间的微分方程组,并利用精细积分法处理其中的指数型矩阵;对于微分方程组中由边界条件和热源项引起的非齐次项,采用解析的方法计算。为了比较精细积分-双重互易边界元法的计算效果,同时使用有限差分法计算温度对时间的导数项。通过数值算例验证了本文方法的有效性和精确性。计算结果表明:时间步长对于精细积分-双重互易边界元法的结果影响较小,而有限差分法对时间步长比较敏感且只在时间步长选取较小时有效;当选取较大时间步长时,精细积分-双重互易边界元法依然具有良好的计算精度。

The dual reciprocity boundary element method(DRBEM) and the precise integration method(PIM) are combined to analyze transient heat conduction problems with heat sources. The boundary integral equation contains two domain integrals corresponding to the heat source term and the time derivative of temperature, respectively. The dual reciprocity method(DRM) is applied to transform domain integrals into boundary integrals. After the boundary integral equation is discretized, an ordinary differential equation system(ODE) is obtained. The precise integration method is adopted to solve the exponential matrix. The analytical method is used to calculate the inhomogeneous terms in the ODE. Meanwhile, the finite difference method(FDM) is also used to treat the time derivative of temperature and compared with the precise integration method. Numerical examples are presented to validate the efficiency and accuracy of this method. The results show that the time steps have no effect on PI-DRBEM, whereas only in the case of a small time step the FD-DRBEM can obtain accurate results. Even for a very large time step, satisfactory results can still be obtained by the present approach.