Intro to IMLS Part 2
...oint Method describe the necessity to incorporate a stabilization procedure. The stabilization procedure is included because the point collocation procedure has been considered to have the incapacity to satisfy the equilibrium equations over a region of points of finite size by sampling the equations only at the collocation points within the region. It has been reported also that these deficiencies are more pronounced in the regions near the boundary due to the usual lack of symmetry of the regions in the boundary (Onate et al., 2001). These deficiencies typically lead to an ill-conditioned system of equations and the solution of the system of equations produces unstable and inaccurate results. These stabilization procedures, however, require additional computational cost since stabilizing terms are added to the set of governing differential equations. The additional computational cost is considerably large because typically the computation of the third derivatives of the shape functions is required to form the stabilizing terms. All published works concerning the Finite Point Method employ the Weighted Least-Squares (WLS) approximation instead of the MLS approximation. This leads to another difficulty since in the WLS approximation the interpolation is defined in each interpolation domain and in general the interpolation will vary from one domain to another. Different interpolation domains can yield different shape functions. As a consequence, a point belonging to one or more overlapping interpolation domains will have different shape function values. The interpolation will be multivalued within the problem domain and so a condition must be applied to reduce the interpolation to a single value. The governing partial differential equations in the Finite Point Method are enforced at each nodal point in such a way that results in a familiar global ¡°stiffness matrix¡± and global ¡°force vector¡± that may be used to solve for the unknowns. For large problems, the global stiffness matrix and global force vector require a large amount of computer memory storage (especially considering that the matrix, in most meshless methods, is unbanded or has a relatively high bandwidth). The computational effort required to solve the resulting large set of algebraic equations may also be significantly larger than in the finite element method, again d...