  Design studies, for accelerator modules based on an injector cyclotron and a
superconducting ring cyclotron able to accelerate H$_2^+$ molecules, are
presented. H$_2^+$ molecules are stripped by a foil creating a proton beam,
with a maximum energy of 800 MeV and a beam power of 8 MW (CW). This beam would
be sent to a beam dump where neutrinos would be produced from pion and muon
decays at rest for the Decay At rest Experiment for $\delta_{CP}$ At the
Laboratory for Underground Science - DAE$\delta$ALUS. We are discussing the
advantage of H$_2^+$ molecules for acceleration and present precise beam
dynamics simulations w.r.t. extraction and beam losses. In general, beam losses
are one of the most challenging parts in such a high power cyclotron design and
must be addressed very early on in the design. We are also addressing H$_2^+$
dissociation and the stripping process, two other characteristic challenges in
the DAE$\delta$ALUS design.
