With the assistance of the moving least-squares (MLS) interpolation functions, a two-dimensional finite element code is developed to consider the effects of a stationary or moving solid body in a flow domain. At the same time, the mesh or grid is independent of the shape of the solid body. We achieve this goal in two steps. In the first step, we use MLS interpolants to enhance the pressure (P) and velocity (V) shape functions. By this means, we capture different discontinuities in a flow domain. In our previous publications, we have named this technique the PVMLS method (pressure and velocity shape functions enhanced by the MLS interpolants) and described it thoroughly. In the second step, we modify the PVMLS method (the M-PVMLS method) to consider the effect of a solid part(s) in a flow domain. To evaluate the new method's performance, we compare the results of the M-PVMLS method with a finite element code that uses boundary-fitted meshes.