NUMERICAL MANIFOLD METHOD FROM MATHEMATICAL THEORY AND ITS APPLICATION

The solving equations of numerical manifold method (NMM) used to be formulated by minimum potential energy principle. For many practical problems, it is difficult to find the functional of the governing equations, hence the minimum potential energy principle or the variational principle cannot be used to obtain the solving equations of the numerical manifold method. A general method to obtain the solving equations of NMM must be identified. Hence, in the paper, the solving equations of the numerical manifold method are derived by the method of weighted residuals (MWR). For elastic problems, the same solving equations of the numerical manifold method by the minimum potential energy principle or the variational principle can be obtained by the method of weighted residuals by choosing suitable weight functions. It is shown that WMR is the mathematical foundation of NMM. The proposed method is more general than the minimum potential energy principle in obtaining the solving equations of the numerical manifold method. At last, the validity and accuracy of the proposed NMM method are illustrated by a numerical example, the numerical results of which agree with the analytical ones. At the same time, variational rule of the stress and displacement fields at the tip of crack under tension is given.