Research On Meshless Methods And Its Application In Helmholtz Problems

Recently, the meshless method is an active topic in the areas of computational science and approximation theory. Based on the approximate nodes, this method can eliminate meshes completely or partly, and need not initiatory plot along with rebuild of meshes, also can handle the disadvantage of the approximate function of the finite method. Over past decades, meshless methods have applied in many different areas ranging from artificial intelligence, computer graphics, image processing and optimization to the numerical solution of all kinds of partial differential equations.Helmholtz problems have been used widely in many fields, such as physics, mechanics, engineering, and so on. Hence, the research on its numerical solutions not only has theoretical values, but also has practical significances.The opening chapter introduces the basic principle and limitation of traditional numerical methods such as the finite element method, delimits the background, developing history and research status of the meshless methods, and summarizes the excellence and limitation of the meshless methods. In the second chapter of this paper, the theory about solving processes for the meshless methods is discussed. Several nodes generating algorithms for discretion the domain, several methods for discretion the differential equations with their respective characteristics and the GMRES algorithm for solving algebraic equations are introduced primarily. In the third chapter, several popular methods for creating shape functions are discussed. With the uniqueness of the Helmholtz equations, an improving point interpolation method using trigonometric functions as the base functions is advanced. Images about the shape functions and its derivative are given in the one dimensional. Through the specific example of functions approximation, the point interpolation is compared with the moving least square method and its superiority in the collocation method is obvious. In the fourth chapter, the collocation method based on trigonometric interpolation is used to solving Helmholtz problems. Through the study of different types of Helmholtz problems, the feasibility and adaptability of the method is certified. Against with the unstable of the collocation method when differential equations have derivative boundary conditions, various technologies are summed up and new processing technology is improved in the one-dimensional in the fifth chapter. Through various numerical examples, every technology's advantages are obvious.