Meshless And Hybrid Trefftz Finite Element Methods Using Radial Basis Function

As alternatives to the general finite element method (FEM) and boundary element method (BEM), the meshless numerical methods have recently become popular in computational mechanics. Since the meshless methods discretize the domain by means of separate points, instead of elements, they have such properties as simple data preparation, avoiding mesh generation or remeshing, and avoidance of mesh locking, distortion in large-deformation problems, for example, metal forming, high-speed crash, and so on. The paper presents a new meshless algorithm by combining the classic method of fundamental solutions (MFS), radial basis function (RBF) approximation and the analog equation method (AEM). According to the definition of the fundamental solutions for the differential operator of the problem, the linear combination of the fundamental solutions completely satisfying the governing equation inside the domain just needs to satisfy the specified boundary conditions at boundary nodes. This is the main idea of the MFS, which can be viewed as one of boundary-only methods. However, for the inhomogeneous problems involving internal distributed sources or body forces and problems with unavailable fundamental solutions, there are some inconveniences in the implementation of the classic MFS. In order to overcome these drawbacks, the proposed method employs the analog equation method (AEM) to convert the original governing partial differential equation (PDE) to an equivalent one, and then, the MFS and RBF interpolation are used to construct the homogeneous part and particular part, respectively. Finally, making the approximated solution of unknown field variable satisfies the original governing PDE at the interpolation points and boundary conditions at boundary nodes to give all unknowns. Much numerical practices show that the presented meshless method possesses such advantages as simple theory requirement, ease of programming, good accuracy and convergence, and has a potential in solving other linear or nonlinear problems in computational mechanics.Moreover, based on the similar idea, considering the shortcomings of classic hybrid Trefftz FEM (HTFEM), that is, it is difficult to treat nonhomogeneous problems and strongly depends on T-complete functions of the problems under consideration, the paper presents an improved method combining the HTFEM, RBF approximation and AEM. The analog equation method firstly is used to obtain the equivalent equation to the original governing PDE, and then the RBFs interpolation and HTFEM are used to produce the related particular and homogeneous parts, respectively. Finally making the approximated formula of field variable satisfies the original PDE to obtain all unknowns. Results for solving the nonlinear minimal surface problems show that the presented method has good numerical accuracy and convergence, and inherits the advantage of full boundary integral of classic HTFEM. In addition, the process demonstrates that the proposed method can be easily used to solve other problems.Besides above works, the paper also studies the construction and smoothing strategies of RBFs, which can be regarded as different extensions of fundamental solutions after removing singularity. Numerical computations tell us that elaborated RBFs based on characteristics of problems considered can displace the commonly used RBFs and produce good results. In addition, different smoothing strategies evidently affect the accuracy and stability, and should be treated carefully.