  We introduce a computational framework for the topology optimization of
cellular structures with spatially varying architecture, which is applied to
functionally graded truss lattices under quasistatic loading. We make use of a
first-order homogenization approach, which replaces the discrete truss by an
effective continuum description to be treated by finite elements in a
macroscale boundary value problem. By defining the local truss architecture
through a set of Bravais vectors, we formulate the optimization problem with
regards to the spatially varying basis vectors and demonstrate its feasibility
and performance through a series of benchmark problems in 2D (though the method
is sufficiently general to also apply in 3D, as discussed). Both the
displacement field and the topology are continuously varying unknown fields on
the macroscale, and a regularization is included for well-posedness. We argue
that prior solutions obtained from aligning trusses along the directions of
principal stresses are included as a special case. The outlined approach
results in heterogeneous truss architectures with a smoothly varying unit cell,
enabling easy fabrication with a tunable length scale (the latter avoiding the
ill-posedness stemming from classical nonconvex methods without an intrinsic
length scale).
