  We numerically investigate magnetization, pinning, ordering, and dynamics of
vortices interacting with pinning arrangements which have a density gradient.
We focus on conformal crystal structures obtained by conformally transforming a
spatially uniform periodic array, as well as non-conformal gradient structures
and structures with quasiperiodic order. The conformal structures feature a
density gradient and local ordering. Using magnetization simulations we find
that conformal pinning arrays exhibit enhanced pinning compared to
non-conformal gradient arrays as well as compared to random, periodic, and
quasiperiodic arrays, for a broad range of fields. The effectiveness of
conformal arrays arises from the continuum of length scales introduced into the
arrays by the conformal transformation, allowing for a broad range of local
commensuration effects. At higher vortex fillings above the range of conformal
effectiveness, we show that a non-conformal rectangular gradient array exhibits
strong pinning due to a novel commensuration effect and vortex ordering. Using
transport simulations where vortices are driven along the gradient and at an
angle to the gradient, we confirm the effectiveness of conformal pinning at
increasing the critical current. For a rotated drive, the gradient arrays
produce a strong vortex guidance effect in the direction perpendicular to the
gradient.
