Element-freed Numerical Manifold Method Based On MLS

The paper summarizes the advantages and disadvantages of numerical manifold method(NMM) and meshless method (EFM) after studying their theory. NMM can uniformly deal with calculating problems of continuous and discontinuous objects, but NMM has some difficulties in the preprocess and the simulation of open crack propagation because it is based on the grid. EFM has advantages of simple preprocess, high computation precision and low computation difficulty, but it also has some disadvantages,such as high computation cost. It is found that NMM and EFM supplements each other very well. If the advantages of both methods can be combine,or if the kernel thought and numerical technique can be amalgamated, the problems in preprocess and the simulation of open crack propagation will be resolve well, also the calculating problems of continuous and discontinuous objects can uniformly deal with.Based on the theory study of NMM and EFM especially element free Galarkin method (EFGM), a element-freed numerical manifold method based on MLS (EFNMM) is introduced according to their advantages and disadvantages in this paper. The proposed method can avoid the element meshing and simplify the construction of approximate functions in the discrete nodes inside object field by using the finite circle cover theory and shepard function. The spline curve is choosed to be the weight function, and essential boundary conditions is implemented with punishment function method. Then Gauss quadrature is carried out with the background grid. It can also be regarded as a special form of NMM. The proposed method keeps the advantages of numerical manifold method and meshless method, and gets off their disadvantages. It largely simplifies the preprocess and can uniformly deal with the calculating problems of continuous and discontinuous objects without meshes.It is a more usefull numerical method. At last, EFNMM is applied to calculate displacement of the cantilever which bears the concentrated loading on its end, and compared with NMM and MM. Results demonstrates that EFNMM has advantages of simple preprocess, high accuracy, low computation cost, and easy programming.