Isogeometric analysis (IGA) was introduced to integrate methods for analysis and computer-aided design (CAD) into a unified process. High-quality parameterization of a physical domain plays a crucial role in IGA. However, obtaining high-quality parameterizations for complex geometries retains a challenge. Triangle configuration based bivariate simplex splines (TCB-splines) and their rational extensions provide an attractive alternative to classical nonuniform rational B-splines (NURBS) in the context of IGA, as they not only share many desired properties with NURBS but also can be defined over general polygonal domains. In this work, using TCB-splines in IGA is suggested to overcome the limits posed by the tensor-product structure of NURBS. We present the methodology for IGA to use rational TCB-splines over general polygonal domains. Then a method to generate the parametric domain according to given physical domain boundaries is proposed. This allows us to obtain a high-quality parameterization easily without resorting to the optimization method. Several numerical examples with complex physical domains demonstrate the flexibility of our TCB-spline-based IGA method, the high quality of the parameterization, and the accuracy of the numerical solutions. (C) 2022 Elsevier B.V. All rights reserved.