This paper considers the problem of piecewise affine abstraction with polytopic partitions of nonlinear systems, i.e., the over-approximation of nonlinear dynamics by a pair of piecewise affine functions over polytopic subdo-mains/partitions in the sense of the inclusion of all possible trajectories. Specifically, to tackle the "boundary effect" that may make the over-approximation incorrect for polytopic partitions, we propose two mesh-based affine abstraction approaches based on expanding the partitions to simultaneously find the polytopic partitions and the pair of piecewise functions over the partitions. The effectiveness of the proposed approaches are compared with existing methods using hyperrectangular partitions, and demonstrated by computing abstractions of swarm dynamics and applying them for swarm intent identification.