We report on our efforts to optimize the geometry of neutron moderators and converters for the TRIUMF UltraCold Advanced Neutron (TUCAN) source using MCNP simulations. It will use an existing spallation neutron source driven by a 19.3 kW proton beam delivered by TRIUMFs 520 MeV cyclotron. Spallation neutrons will be moderated in heavy water at room temperature and in liquid deuterium at 20 K, and then superthermally converted to ultracold neutrons in superfluid, isotopically purified $^4$He. The helium will be cooled by a $^3$He fridge through a $^3$He-$^4$He heat exchanger. The optimization took into account a range of engineering and safety requirements and guided the detailed design of the source. The predicted ultracold-neutron density delivered to a typical experiment is maximized for a production volume of 27 L, achieving a production rate of $1.4 cdot 10^7$ s$^{-1}$ to $1.6 cdot 10^7$ s$^{-1}$ with a heat load of 8.1 W. At that heat load, the fridge can cool the superfluid helium to 1.1 K, resulting in a storage lifetime for ultracold neutrons in the source of about 30 s. The most critical performance parameters are the choice of cold moderator and the volume, thickness, and material of the vessel containing the superfluid helium. The source is scheduled to be installed in 2021 and will enable the TUCAN collaboration to measure the electric dipole moment of the neutron with a sensitivity of $10^{-27}$ e cm.
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