  We investigate critical points of a Landau-de Gennes (LdG) free energy in
three-dimensional (3D) cuboids, that model nematic equilibria. We develop a
hybrid saddle dynamics-based algorithm to efficiently compute solution
landscapes of these 3D systems. Our main results concern (a) the construction
of 3D LdG critical points from a database of 2D LdG critical points and (b)
studies of the effects of cross-section size and cuboid height on solution
landscapes. In doing so, we discover multiple-layer 3D LdG critical points
constructed by stacking 3D critical points on top of each other, novel pathways
between distinct energy minima mediated by 3D LdG critical points and novel
metastable escaped solutions, all of which can be tuned for tailor-made static
and dynamic properties of confined nematic liquid crystal systems in 3D.
