The concept behind the moving mesh (r-refinement) method is to concentrate the mesh in specific areas of interest while maintaining the initial number of degrees of freedom. This strategy holds appeal due to its potential to enhance computational efficiency for more accuracy. In the realm of isogeometric analysis, researchers have introduced the r-adaptive parameterization technique. While optimization techniques based on a posteriori error estimation and the utilization of Winslow's mapping, employing such technologies in isogeometric analysis poses challenges. Any modification of control points at the boundary leads to a deviation from the exact parameterization of the geometry, a critical property to preserve in IGA, aiming to eliminate boundary-induced errors. In this particular scenario, an alternative path is taken, tackling these challenges through the lens of the optimal transport problem \cite{bahari}. This fresh methodology avoids the previously mentioned issues. We introduce a new modified and adapted algorithm as a novel technique for dynamic mesh generation that represents a significant stride in addressing partial differential equations across multi-dimensional spaces. The approach involves deriving a mapping from a scalar mesh potential function satisfying a non-linear parabolic Monge-Ampere equation \cite{Budd}. The method is distinguished by its simplicity, requiring the solution of one time-dependent scalar equation using a fast diagonalization method adaptable to any dimension.