  We present a detailed description of a quantum scalar field theory within a
flat spacetime confined to a cavity with perfectly reflecting moving
boundaries. Moreover, we establish an equivalence between this time-dependent
setting and a field theory on an acoustic metric with static Dirichlet boundary
conditions. We discuss the classical and quantum aspects of the theory from the
latter perspective, accompanied by the introduction of novel numerical
techniques designed for the (nonperturbative) computation of particle
production attributed to the Dynamical Casimir effect, applicable to arbitrary
boundary trajectories. As an illustrative example of these methodologies, we
compute the particle production for a massless field in 1+1 dimensions.
Notably, our approaches readily extend to encompass scenarios involving massive
fields and higher dimensions
