Meshfree methods made big progress in the past decades which have been applied to many areas. Roughly, these methods can be divided into two categories: Galerkin-type meshfree methods based on weak form and collocation-type meshfree methods based on strong form. On one hand, direct collocation method (DCM) suffers low accuracy and instability compared with Galerkin-based meshfree methods. On the other hand, since most approximation functions in meshfree methods are rational functions, it’s quite hard to achieve high-accuracy integration with high efficiency in Galerkin-type meshfree methods. Therefore, to combine the advantages of Galerkin-type and collocation-type meshfree methods, we propose a new meshfree method - stabilized collocation method (SCM). The functions which can satisfy the high order consistency conditions such as reproducing kernel (RK) function and Lagrange interpolation function can be utilized as the approximation function. The presented method can satisfy the high order integration constraints which can conserves the high order consistency conditions in the integration form. This property leads to the high accuracy and optimal convergence for the proposed method. When RK approximation is introduced, gradient reproducing kernel (GRK) approximations can be utilized to promote the efficiency, especially for the problems governed by high order partial differential equations. When Lagrange interpolation function is introduced as the approximation function, since it has Kronecker delta property, the essential boundary conditions can be simply and exactly imposed like finite element method (FEM), which further improves the accuracy of this method. Numerical examples validate the high accuracy and convergence as well as good stability of the presented method, which can outperform DCM and Galerkin based meshfree method utilizing the same approximation function.

Based on the SCM, a Lagrangian-Eulerian stabilized collocation method (LESCM) is further proposed for the fluid-structure interaction problems involving free surface flow, in which the structure is modelled by a rigid body. This method is an evolution of the material point method and particle-in-cell methods which are based on the hybrid Lagrangian-Eulerian description. The problem domain of the fluid and structures is discretized into the Lagrangian particles which carry the information, and the problem domain covering the entire movement space is discretized into the uniformly distributed Eulerian background nodes. The coupling governing equations of the fluid, structures and interfaces are solved by the meshfree SCM employing the RK approximation on the Eulerian nodes. The solution is very efficient since the Eulerian nodes are set to be the initial positions in each time step and it's no need to reconstruct the shape function. The information mappings between the Lagrangian particles and the Eulerian nodes are also conducted by the RK approximation which can keep the mass and momentum conservation of the solution. The cell-cut algorithm is introduced to couple the fluid and the structures which can solve the fluid pressure and the fluid-structure interactional force simultaneously and avoid the complicated iterations of the traditional interaction algorithms. Several numerical examples including the collapse of water column with a rigid barrier, water entry of a half-buoyant circular cylinder and a rigid box rotating and sinking in water are simulated, which demonstrate the high accuracy, high efficiency and good stability of the proposed method. This method can be extensively applied to the engineering applications of fluid-rigid body interactions.