In this paper, a new hypervolume-based evolutionary multi-objective optimization algorithm (EMOA), namely R2HCA-EMOA (R2-based Hypervolume Contribution Approximation EMOA), is proposed for many-objective optimization. The core idea of the algorithm is to use an R2 indicator variant to approximate the hypervolume contribution. The basic framework of the proposed algorithm is the same as SMS- EMOA. In order to make the algorithm computationally efficient, a utility tensor structure is introduced for the calculation of the R2 indicator variant. Moreover, a normalization mechanism is incorporated into R2HCA-EMOA to enhance the performance of the algorithm. Through experimental studies, R2HCA-EMOA is compared with three hypervolume-based EMOAs and several other state-of-the-art EMOAs on 5-, 10- and 15-objective DTLZ, WFG problems and their minus versions. Our results show that R2HCA-EMOA is more efficient than the other hypervolume-based EMOAs, and is superior to all the compared state-of-the-art EMOAs.