Research On Element-Free Method And Application In Concrete Crack Analysis

The element-free Galerkin method (EFGM), as a new numerical method developed recently, is similar to finite element method. In EFGM, in order to get a numerical solution for a partial differential equation, the shape function is constructed by moving least squares (MLS), the control equation is derived from the weak form of variational equation and essential boundary conditions are imposed by penalty function method. The advantages of EFGM are: (1) Node information only, no need to join nodes into elements and mesh elements; (2) The function and its gradient for the whole region are continuous;(3) High accuracy can be achieved and post process is easy, etc.The mathematical foundation of element-free Galerkin method is moving least squares method. When the shape function is constructed, only a series of nodes are laid in the region with moving least squares method, so the EFGM need no elements. While the approximation function of MLS method is not necessary to pass through all the calculations, unless the weight functions are singular. Therefore, the imposing of essential boundary conditions and the dealing with concentrated loads has become complex, but the disadvantages are negligible compared with the advantages.The development and the basic theory of element-free Galerkin method (EFGM) are introduced in this paper. The basic equation of EFGM based on the moving least-squares method, the dealing with of continuity, the imposing of essential boundary conditions, and discrete programs are described emphatically. The three main factors (weight function, nodes on the field and the density of nodes) which can affect accuracy of element-free Galerkin method are analyzed separately and then the guiding significance conclusions are obtained.At present, the main research of EFGM is focused on the flexible material. In this paper, the finite element method is introduced to the EFGM, and OTTOSEN constitutive model is adopted, then the max tensile strain principle is selected as destructive principle. And then a concrete beam is analyzed by nonlinear EFGM. Based on the proposed method, the spreading of crack is estimated and then the cracking track is obtained, according the situation that the crack is analyzed utilizing FEM is avoided. In the paper, we select concrete model and reinforced concrete model to carry on the simulation, and the results prove that the EFGM is feasible to solve the problem of concrete crack.