The r-adaptive mesh method in isogeometric analysis is a technique for redistributing an initial mesh based on specific criteria, such as high-gradient regions or areas identified by error estimators, while preserving the tensor product structure of B-splines and maintaining the original data structure. Traditional approaches often involve reparameterizing the initial mapping by adjusting control points, which introduces challenges with boundary constraints and geometric complexities when solving additional problems. Instead, we adopt the approach proposed in \cite{bahari}, which preserves the initial mapping and composes it with a new mapping that transforms the parametric domain (square) into itself. This method avoids geometric complexities, since the Laplace equation is solved exclusively on the square, leveraging a fast diagonalization algorithm. In this presentation, we propose an enhanced algorithm based on this method that ensures injectivity without requiring a mixed variational formulation, allowing for acceleration by means of fast diagonalization. Additionally, we present benchmarks comparing r-adaptive, h-adaptive \cite{Carlotta}, and combined adaptive mesh strategies, along with various applications.