  We present a new scheme for constructing initial data for the Einstein field
equations using the conformal thin-sandwich formulation that does not assume
conformal flatness or approximate Killing vectors. This includes a method for
determining free data based on superposition, as well as a way to handle black
hole singularities without excision. We numerically solve the constraint
equations using a multigrid algorithm with mesh refinement. We demonstrate the
efficacy of the method with initial data solutions for several applications: a
quasicircular binary black hole merger, a dynamical capture black hole-neutron
star merger, and an ultrarelativistic collision.
