A popular class of methods for solving Fluid-Structure Interactions (FSI) problems computationally are the moving mesh deformation methods. These methods generally involve deforming the fluid domain so that it conforms to the structure as it undergoes changes. An efficient and robust moving mesh technique involves interpolation of displacements using Radial Basis Functions (RBFs). But it is noticed that large deformations in the computational domain restrict the applicability of RBF methods due to resulting poor mesh quality. This further worsens the accuracy of the simulation by introducing additional numerical errors. The standard practice to overcome this deficiency has been to introduce the computationally expensive step of re-meshing the entire domain. The current thesis aims to tackle this issue of poor mesh quality caused due to large deformations of the structure by introducing localized corrections in regions of poor quality. But deforming the mesh, then calculating the quality and then applying corrections would become an expensive operation of its own. Therefore, firstly a methodology is devised to predict the state of the mesh after the deformation apriori to performing the deformation step. It is found that the gradient information from the deformation functions can be utilized to accurately predict the various algebraic mesh quality metrics. The algorithm is tested on 1D and 2D meshes and grid convergence studies are performed to validate the predicted quality with the actual quality post-deformation. The theoretical framework for the 3D prediction algorithm is also laid out. Once the mesh quality is predicted, based on a user-defined threshold, if the quality is deemed to be poor in a certain region then localized corrections are implemented to improve the mesh quality. These corrections are introduced in the form of enrichment functions on additional control points within the domain. The gradient of the deformation function is constrained using these functions such that a good quality and smooth mesh deformation is obtained post-deformation. Multiple enrichment functions are initially tested in 1D in a global sense. The findings from this analysis are then carried forward to analyze where to introduce these additional control points for local corrections and the nature of the correction itself. Finally, the methodology is tested on 2D unstructured grids. It is found that the imposition of gradients is more complicated in the 2D sense as global information can not be used directly as in the 1D case. Therefore, a parametric study is performed to analyze if a logical conclusion can be made with regard to the imposition of the gradients. It is found that such a logical trend is not clear, therefore the strategy is modified such that the interpolation coefficients are imposed explicitly instead based on data available from eigen decomposition of the deformation gradients. It is found that this strategy performs admirably when compared to the results obtained from the standard RBF model. In conclusion, the method significantly improves the robustness of the mesh deformation process by ensuring that large deformations can be accommodated in the domain without huge computational costs.