Research On Element-free Galerkin Method Of The Structure Under Uncertain Parameters

With the developments of the computer technique and the numerical computing methods, the effects of numerical simulation in the science and engineering areas are enhanced gradually, the utility of which has been shifted from an assistant tool to an essential means of the scientific research, and plays an important role both in the theoretical and the experimental areas. Generally, we establish a engineering structural analysis model and the design model based on deterministic physical significance. That is, we deal with the factors as deterministic ones during analysis process. However, errors and uncertainties that caused by loads, the initial conditions, the boundary constraints, manufacture and assembling widely exist in the real engineering world. If these uncertain factors are roughly concerned as deterministic parameters, we may receive conflict or unreasonable results on occasion. Some numerical solution, such as the stochastic Finite Element (FE), the interval FE and the fuzzy FE, have been widely used in the uncertain analysis of the engineering structures. However, they are all evolved from the FEM, and they encounter difficulties in dealing with some problems caused by their own drawbacks, such as crashes, shaping, crack propagation, phase change in materials, and super-large deformation. Element-Free Method (EFM) is a promising method, which can overcome the dependency on meshes, and eliminate the mesh partition thoroughly or partly. It possesses an apparent advantage rather than FEM over the above problems, and attracts more and more scientists'interests.The EFM has developed so fast in the past decades, many contributions have been made both in basic theoretical research and engineering application, which focused on algorithm proof, efficiency of calculation, boundary constraints application, quick and stable domain integration technique, generalization of FEM and EFM, and improvement and application that overcome the challenge of FEM. But all these results were based on deterministic models; they did not consider the uncertainties which exited comprehensively in engineering structures. Hereby, how to extend the EFM to uncertain problems becomes a significant task.The traditional Element Free Galerkin method (EFGM) based on moving least square approximate may lead to ill conditioned system of equations, and will limit its application as a result. Based on the improved local weighted orthogonal basis function, an improved element-free Galerkin method was proposed in this paper, and named as the Local Orthogonal Element Free Galerkin method (LOEFGM). Perturbation theorem, interval mathematics and fuzzy set theorem provide powerful mathematical tools for analysis of uncertain problems. EFGM takes apparent advantage than FEM when dealing with some special problems. It will be a valuable orientation if combine the EFGM with perturbation theorem, interval mathematical, and fuzzy set theorem to solve uncertain problems. We hereby proposed the Perturbation Stochastic Local Orthogonal Element Free Galerkin method (PSLOEFGM), the Interval Local Orthogonal Element Free Galerkin method (ILOEFGM), and the Fuzzy Local Orthogonal Element Free Galerkin method (FLOEFGM). And we also applied the methods mentioned above to solve uncertain problems successfully. This will enrich and develop the EFM theory and method in dealing with structral mechanic problems with uncertainties.The main achievements of this dissertation are:Based on local weighted orthogonal basis functions, the improved Element Free Galerkin method so called the Local Orthogonal Element Free Galerkin method (LOEFGM) was established. LOEFGM can also preserve the character of the orthogonal basis function, but it makes the formulation simplify, and what's more, it is easy to calculate the inverse matrix and can avoid the singularity of matrix, thus, it takes advantage of programming and promotes the computation efficiency. The numeral example was taken to compare LOEFGM with the traditional EFGM, the results showed that LOEFGM was more efficient than the traditional EFGM.By studying the improved EFGM and the perturbation method, the perturbation stochastic local orthogonal EFGM was proposed, and the random variational principle and the discrete scheme of the EFM were deduced in detail. The natural boundary conditions were added by using the penalty function. The numerical examples of bar, the beam and the square plate with circular hole, which include random parameters were studied, respectively. The results showed that the proposed method is correct, feasible and efficient.By studying the interval mathematics, combined with the inner product space and the improved EFGM, the interval local orthogonal EFGM was proposed. The singular function of the linear elastic fracture mechanics was added as an enhance function into the local weighted orthogonal basis function of the moving least square approximation. The interval number decomposition method was used to solve the interval equilibrium equations. The interval J-integral formula was deduced in detail. The uncertain problem of the cracked structure was analyzed. From the results of the example, it can be seen that the interval local orthogonal EFGM is correct and effective in solving the uncertain problems.The fuzzy local orthogonal EFGM was proposed and the fuzzy minimum potential energy principle was given. Then, the fuzzy local orthogonal element-free equilibrium equations were deduced. According to the membership function of the given fuzzy numbers, the possibility distributions of the structural response quantities can be obtained; the fuzzy local orthogonal EFGM was used to obtain the fuzzy displacements of the nodes, and then the direct displacement method was used to compute the fuzzy stress intensity factor. The method discussed here provides a new pathway of computing the stress intensity factor in the crack tip of the cracked structures which including uncertain parameters.The research was carried out which focused on the interface edge singular stress field of the structure with interface crack under uncertainties. The Interval Local Orthogonal Element Free Galerkin analysis model was established on the structure with interface crack. The singularity of stress field of the crack tip was simulated with enriched basis functions, and the discontinuousness of crack was dealed with diffraction method. This provided an EFM for the calculation of stress intensity factor of structure with interface crack, and also provided a new application category of EFGM. The proposed method can present the response interval of structures effectively, even if the probability statistics is unknown whereas the range of the interval is given. This datum provides a valuable reference for uncertain structure analysis and design.