Abstract: A new numerical algorithm is presented to solve viscoelastic problems by combining a temporally piecewise adaptive technique with the isogeometric analysis. By expanding variables at a discretized time interval, a series of recursive scaled boundary finite element method equations in spatial domain is established with non-uniform rational B-spline discretization in the circumferential direction. The proposed algorithm not only takes advantages of conventional scaled boundary finite element method in dealing with singularity and unbounded domains, but also provides a more accurate geometric description of the boundary, resulting in more accurate special solutions. A piecewise adaptive process is utilized to fully maintain a steady accuracy in the time domain for different sizes of time step. Numerical examples are presented to demonstrate the effectiveness of the proposed model in term of computational accuracy and convergence.