A theoretical method base on the zero-slip boundary and a finite element model with thermoviscous membrane-acoustic interaction are proposed for investigating a viscously damped micro-perforated rectangular membrane. Both the theoretical and finite element methods well predicted the sound absorption performance of a dielectric elastomer micro-perforated membrane, and the results agree with those from experiments. The thermoviscous membrane-acoustic interaction is further explored. The properties which lead to better acoustic performance are studied using the developed numerical model. With the optimized parameters, a duct silencer with high transmission loss (>10 dB) and broad noise attenuation band (from 300 Hz to 1700 Hz) is designed, fabricated, and tested. The potential engineering application and the background physical mechanism are also addressed.