  We present an improved method for topology optimization with both adaptive
mesh refinement and derefinement. Since the total volume fraction in topology
optimization is usually modest, after a few initial iterations the domain of
computation is largely void. Hence, it is inefficient to have many small
elements, in such regions, that contribute significantly to the overall
computational cost but contribute little to the accuracy of computation and
design. At the same time, we want high spatial resolution for accurate
three-dimensional designs to avoid postprocessing or interpretation as much as
possible. Dynamic adaptive mesh refinement (AMR) offers the possibility to
balance these two requirements. We discuss requirements on AMR for topology
optimization and the algorithmic features to implement them. The numerical
design problems demonstrate (1) that our AMR strategy for topology optimization
leads to designs that are equivalent to optimal designs on uniform meshes, (2)
how AMR strategies that do not satisfy the postulated requirements may lead to
suboptimal designs, and (3) that our AMR strategy significantly reduces the
time to compute optimal designs.
