An 81.25 MHz continuous wave (CW) radio frequency quadrupole (RFQ) accelerator has been designed for the Low Energy Accelerator Facility (LEAF) at the Institute of Modern Physics (IMP) of the Chinese Academy of Science (CAS). In the CW operating mode, the proposed RFQ design adopted the conventional four-vane structure. The main design goals are providing high shunt impendence with low power losses. In the electromagnetic (EM) design, the pi-mode stabilizing loops (PISLs) were optimized to produce a good mode separation. The tuners were also designed and optimized to tune the frequency and field flatness of the operating mode. The vane undercuts were optimized to provide a flat field along the RFQ cavity. Additionally, a full length model with modulations was set up for the final EM simulations. Following the EM design, thermal analysis of the structure was carried out. In this paper, detailed EM design and thermal simulations of the LEAF-RFQ will be presented and discussed. Structure error analysis was also studied.