瞬态热传导问题的精细积分边界元法

Precise Integration Boundary Element Method for Transient Heat Conduction Problems

热传导为最普遍的工程问题之一,为解决含内部热源的三维瞬态热传导问题,通过双重互易边界元法(dual reciprocal boundary element method, DRBEM)与精细积分法(precise integration method, PIM)耦合进行求解。该方法根据含有内部热源的各向同性介质瞬态常系数热传导问题的控制方程,通过加权余量法推导出相应的边界积分方程,然后用双互易法(dual reciprocity method, DRM)处理得到的边界积分方程,将热源项和温度关于时间导数项引起的域积分通过径向基函数(radial basis function, RBF)逼近后转化为边界积分。之后将边界积分方程离散,得到与时间相关的一阶常系数微分方程组,最后,在获得解析解的过程中,通过PIM处理其中的矩阵指数函数(matrix exponential function, MEF)。最后通过3个数值算例进行验证。结果表明:在3种边界条件下,数值求解结果都具有很好的准确性。可见该方法的可行性和稳定性。

Heat conduction is one of the most common engineering problems.In order to solve the three-dimensional transient heat conduction problem with internal heat source,it was solved by coupling the dual reciprocal boundary element method(DRBEM)and the precise integration method(PIM).According to the governing equation of transient constant coefficient heat conduction problem in isotropic medium with internal heat source,the corresponding boundary integral equation was derived by the weighted residual method.The dual reciprocity method(DRM)was applied to deal with the obtained boundary integral equation.The domain integral caused by the heat source term and the time derivative term of temperature was transformed into the boundary integral after being approximated by radial basis function(RBF).Then,the boundary integral equation was discretized to obtain the first-order constant coefficient differential equations related to time.Finally,in the process of obtaining the analytical solution,the matrix exponential function(MEF)was processed by PIM.Finally,three numerical examples were used to verify the results.The results show that the numerical results have good accuracy under the three boundary conditions.It can be seen that the method is feasible and stable.