  We present a globally convergent method to accelerate density-based topology
optimization using projection-based reduced-order models (ROMs) and
trust-region methods. To accelerate topology optimization, we replace the
large-scale finite element simulation, which dominates the computational cost,
with ROMs that reduce the cost of objective function and gradient evaluations
by orders of magnitude. To guarantee convergence, we first introduce a
trust-region method that employs generalized trust-region constraints and prove
it is globally convergent. We then devise a class of globally convergent
ROM-accelerated topology optimization methods informed by two theories: the
aforementioned trust-region theory, which identifies the ROM accuracy
conditions required to guarantee the method converges to a critical point of
the original topology optimization problem; a posteriori error estimation
theory for projection-based ROMs, which informs ROM construction procedure to
meet the accuracy conditions. This leads to trust-region methods that construct
and update the ROM on-the-fly during optimization; the methods are guaranteed
to converge to a critical point of the original, unreduced topology
optimization problem, regardless of starting point. Numerical experiments on
three different structural topology optimization problems demonstrate the
proposed reduced topology optimization methods accelerate convergence to the
optimal design by up to an order of magnitude.
