The Improvement And Application Of Element-free Galerkin Method

Element-free Galerkin method is a new computational method arising recently which discretizes the whole solution domain into independent nodes and there is no need to connect these nodes into element. In this way, there is no need to create and rearrange the meshes. The approximation of the displacement field adopts the function fittings based on nodes to make sure that the basic field variables are continuous in the entire solution domain. Because of the high precision, good stability and fast constringency, element-free Galerkin method based on MLS (Moving Least Square) approximation has the most developing prospect among the many element-free methods.Nowadays, the finite element method has become a general computational method to solve engineering problems in the aspect of computational mechanics and has many remarkable achievements, however, along with the development of computational mechanics, lots of researchers still try to study other fields to compensate the deficiencies of the finite element method theory. Element-free Galerkin method has now been being researched comprehensively all over the world, and its range of application is expanding day by day. However, it is not perfect in the aspects of strict mathematical reasoning, the treatment of boundary conditions, the treatment of discontinuities, the efficiency and effectiveness of calculation and a great deal of engineering applications. Also, there is no full-blown general software for it. However its characteristics, which is better than finite element method, has become a hot spot in the research of the engineering field.The research of this article about Element-free Galerkin method is organized as follows:Chapter one: The background and significance of why this subject being chosen and the retrospection of the history and the current status of Element-free Galerkin method are presented. The general comments about Element-free Galerkin method are presented, and the main contents of this dissertation are determined.Chapter two: The basic theory of Element-free Galerkin method is summarized. Application of the basis function and the patulous basis function, selection of the weight function, the characteristics of the shape function, discrete strategy, the adding of natural boundary condition, the treatment of discontinuity and the solution of the integral are illustrated and two examples are given.Chapter three: The orthogonal basis function and its principle and method of construction are presented, deducing the shape functions and its derivatives which have used the orthogonal basis function and the shape functions and its derivatives which have used the polynomial basis function, getting the conclusion that the results of them are the same. Two examples are given to verify this conclusion.Chapter four: The element-free Galerkin method is applied to solve the problem of the 2-dimensional solid linearly elastic static mechanics,the problem of the 2-dimensional solid linearly elastic dynamics and the problem of stress intensity factor of the crack tip. Examples are used to verify the correctness and the validity of the method.Chapter five: The main conclusions of this article are given, and further development of the element-free Galerkin method is predicted.The improvement and significance of Element-free Galerkin method is accounted as follows:1. The orthogonal basis function are presented, deducing the shape functions and its derivatives which have used the orthogonal basis function and the shape functions and its derivatives which have used the polynomial basis function, getting the conclusion that the results of them are the same. Two examples are given to verify this conclusion. Using the orthogonal basis function to change matrix A into diagonal matrix, and it can eliminate the possibility of the morbidity of matrix A and make it easy to get the inverse matrix of A. It does not only keep the calculation accuracy but also reduce the calculation time, which makes the improved method become more practical2. The element-free Galerkin method is applied to solve the problem of the 2-dimensional solid linearly elastic static mechanics,the problem of the 2-dimensional solid linearly elastic dynamics and the problem of stress intensity factor of the crack tip. When resolving the 2-dimensional solid linearly elastic dynamics, the structural dynamical basic equations are discretized without meshes, and the element-free dynamical equations are achieved. Using penalty function to satisfy the natural boundary conditions, and achieve the natural frequency and vibration mode, being compared with the finite element method, it proves the correctness and validity of the 2-dimensional dynamical element-free program designed in this dissertation. When solving the stress intensity factor of the crack tip, the diffraction principle is used to deal with the discontinuous problem, the program of solving the plane problem of the fracture mechanics is made, the displacement field and the stress field of the uniaxial tensile plate with center crack are solved, and the results are coincidence with the results of the ABAQUS software which is authorized in the field of the international structural analysis. In the mean time, the element-free J integral method is used to deduce the stress intensity factor of the crack tip, compared to the boundary collocation method, the error is only 3.33%. The correctness and the validity of the element-free Galerkin method are proved. It provides theoretical foundations for solving some complex practical engineering problems.