  Many biological microswimmers locomote by periodically beating the
densely-packed cilia on their cell surface in a wave-like fashion. While the
swimming mechanisms of ciliated microswimmers have been extensively studied
both from the analytical and the numerical point of view, the optimization of
the ciliary motion of microswimmers has received limited attention, especially
for non-spherical shapes. In this paper, using an envelope model for the
microswimmer, we numerically optimize the ciliary motion of a ciliate with an
arbitrary axisymmetric shape. The forward solutions are found using a fast
boundary integral method, and the efficiency sensitivities are derived using an
adjoint-based method. Our results show that a prolate microswimmer with a 2:1
aspect ratio shares similar optimal ciliary motion as the spherical
microswimmer, yet the swimming efficiency can increase two-fold. More
interestingly, the optimal ciliary motion of a concave microswimmer can be
qualitatively different from that of the spherical microswimmer, and adding a
constraint to the ciliary length is found to improve, on average, the
efficiency for such swimmers.
