热传导差分边界元技术的双方程方法

THE DOUBLE EQUATIONS METHOD OF DIFFERENCE-BOUNDARY ELEMENT TECHNIQUE FORHEAT PROBLEM

由热传导方程经过对时间差分离散得到的椭圆型方程-Δu+k^2u=f,其边界积分方程的类型,关于未知热势导数是第一类积分方程,关于未知热势,第二类积分方程。本文以守恒积分为工具,推导出新型边界积分方程,其类型与经典方程相反,关于未知热势是第一类积分方程,关于未知热势导数是第二类积分方程。此外,对Direchlet问题与混合边值问题作边界元离散时,采用双方程方法,即:在不同类型的边界段上采用不同的边界积分方程,算例表明该算法比经典边元界法具有更高的精度。

The elliptic type equation-Δu+k2u=f (1)is derived from heat conduction equation by finit difference discreting. The ordinary boundary integral equation of ( 1) is of the second kind for the unknown potential, but is the first kind for the unknown exterior normal derivative of potential. In this paper, a new type of boundary integral equation has been derived by conservation integral method, which is of the second kind for the unknown exterior normal derivative of potential and is the frist kind for the unknown poential, and the two equations method for the mixed boundary value prblem of ( 1 ) is presented, which use the ordinary boundary integral equation at the collocation points in the boundary segment in which the potential is unknown, and use the new type of boundary integral equation at the collocation points in the boundary segment in which the exterior normal derivative is unknown. Some numerical examples indicate this method has higer accuracy than ordinary boundary element method.