No Essential Boundary Conditions For The Implementation Of The Grid Method

In recent years, meshless methods have been proposed and achieved remarkable progress in numerical computations, the main feature of which is that they only need information of nodal parameters instead of elements information and the mesh can be eliminated wholly or partly. So the difficulty in mesh generation of the structure is reduced. Furthermore, they also have other merits, including the high accuracy, the convenience of the postprocessor and so on. A lot of sorts of meshless methods were presented by native or foreign scholars and applied to the engineering practice in the past decades. Many theoretic and applied achievements were acquired.So far, theoretical researches as well as relative applications about meshless methods are still in the stage of infancy. Considering the existing key problems of meshless methods, improved techniques are discussed in detail.On the basis of reading a lot of literatures, the advantages and disadvantages of EFGM and basis theory of meshless method are introduced. The physical meaning of MLS, reversibility of matrix A, selection of weight function, and the Penalty factor, the proper radius of support domain and the characteristic of approximate function are investigated in detail.A special treatment is needed to realize the essential boundary conditions for mesh-free method while moving least squared method is used to approximate the unknown field functions. This is caused by the fact that moving least square approximation is just a kind of fitting method without interpolation characteristics. In this paper, after introducing the meshless method simply, we discuss every kind of methods on implementation of essential boundary conditions in detail and compares advantages and shortcomings of several kinds of methods. Besides the above, a new method that the Modified—Penalty is proposed, essential boundary condition was forced by introducing Lagrange multipliers firstly. To enrich the positive definite of stiffness matrix, the Lagrange multipliers is identified as boundary force, then a modified variational principle was obtained. To improve the computing accuracy, Penalty factor is induced in the obtained principle. Comparing with the Penalty method, the Modified—Penalty method has the following several advantage: smaller condition number, less sensitivity to Penalty factor, higher computing accuracy. Comparing with Lagrange multipliers method, the present method has less unknown...