势问题的数值流形方法

以往的数值流形方法都是以最小势能原理或变分原理为基础来建立求解方程的。但在实际工程中有些实际问题,无法应用变分方法来建立数值流形方法的求解方程,必须寻找较一般的方法来推导数值流形方法的求解方程。本文研究了如何从加权残数法出发建立拉普拉斯方程数值流形方法的求解方程。通过建立拉普拉斯方程的数值流形方法,充实了数值流形方法的数学基础,并拓宽了其应用领域。最后以热传导和渗流为例,验证了本文方法的正确性。

Numerical manifold method for the potential problem

Usually,the governing equations of the numerical manifold method(NMM) are derived from the minimum potential energy principle.For many practical problems,it is very difficult to derive the governing equations of the numerical manifold method with the variational principle.So we should implement the method of weighted residuals to derive the governing equations of the NMM.The numerical manifold method of Laplace equation was presented,it was also more general than the minimum potential energy principle to obtain the governing equations of the NMM.At the same time,the method enriched the mathematical foundation of NMM and extended fields of application.At last,the validity of the method was illustrated by use of two numerical examples.