In this dissertation, a new meshless semi-analytical method for the solution of fluid flow equations with moving boundaries and interaction with rigid structures (weak interaction) is presented. This method is based on the solution of incompressible inviscid fluid flow equations, using a Lagrangian formulation. Hence, in each time step, the pressure Laplace equation is solved in a meshless style, similar to the method of fundamental solutions (MFS) and other fluid variables such as acceleration, velocity and displacement are calculated accordingly. Problem geometry is then updated based on the Lagrangian formulation of motion. In this method, the approximate solution of the Laplace equation is expressed by a linear combination of exponential basis functions. These functions are obtained by satisfying the Laplace equation. Constant coefficients of the series are evaluated through a point collocation on domain boundaries via an especial discrete transformation; Fluid-structure interfaces are supposed as slip impermeable boundaries. The developed Lagrangian algorithm is used to test variety of linear and nonlinear benchmark examples and good results are obtained as comparing with solutions of analytical and other numerical methods. Considering the computational cost and time, changing geometries and moving boundaries can be analyzed by such a meshless method much easier compared with the conventional Finite Element Methods. In this investigation, the solution of elasticity and steady Navier-Stokes equations for fully incompressible materials has also been presented in a similar manner. For these equations, polynomial functions are added to exponential basis functions. Contrary to standard displacement formulation of elastic problems (that fails when Poisson’s ratio becomes 0.5 or when the material becomes incompressible), with these basis functions formulation, the standard equations can be solved with Poisson’s ratio 0.5. It is not required to use mixed methods, iterative solutions and so on. Pressure basis functions can be evaluated by calculating the limit of bulk modulus multiplied by volumetric strain when Poisson’s ratio becomes 0.5.
