基于双层位势的非定常扩散方程的虚边界元解法

On virtual boundary element method based on double layer potential for unsteady diffusion equation

对于二维非定常扩散方程边值问题,采用与时间有关的基本解,基于双层位势的延拓,建立虚边界积分方程,然后用虚边界元法求解.通常的虚边界积分公式是利用单层位势的延拓来建立虚边界元积分方程,但对带时间变量的单层位势,要涉及到指数积分函数的计算.提出了基于双层位势的方法,计算时没有涉及到对基本解的时间积分,避免了用直接边界元方法求解时遇到的指数积分函数.最后,通过数值算例验证了该方法的有效性和可行性.

Initial boundary value problem of two dimensional unsteady diffusion equations is solved in this paper.By using time-dependent fundamental solution of two dimensional diffusion equations and the extension of double layer potential,virtual boundary integral expression of diffusion equations are established.Then virtual boundary element method is used to implement the numerical computation.The traditional virtual boundary integral expression is based on the extension of single layer potential,for the integral formulas related to single layer potential for parabolic problem,the numerical computation of the exponential integral function is unavoidable.In this thesis,the virtual boundary integral equation is based on double layer potential and the exponential integral function is not involved in it,so numerical computation for the exponential integral function is avoid.Finally,numerical examples illustrate the feasibility and the efficiency of the proposed method.