Streamwise streaks are ubiquitous in transitional boundary layers, particularly when subjected to high levels of free-stream turbulence. Using the steady boundary-layer approximation, the upstream disturbances experiencing maximum spatial energy growth are numerically calculated. The calculations use techniques commonly employed when solving optimal-control problems for distributed parameter systems. The calculated optimal disturbances consist of streamwise vortices developing into streamwise streaks. The maximum spatial energy growth was found to scale linearly with the distance from the leading edge. Based on these results, a simple model for prediction of transition location is proposed. Available experiments have been used to correlate the single constant appearing in the model.

Standard weak solutions to the Poisson problem on a bounded domain have square-integrable derivatives, which limits the admissible regularity of inhomogeneous data. The concept of solution may be further weakened in order to define solutions when data is rough, such as for inhomogeneous Dirichlet data that is only square-integrable over the boundary. Such very weak solutions satisfy a nonstandard variational form (u,v) = G(v). A Galerkin approximation combined with an approximation of the right-hand side G defines a finite-element approximation of the very weak solution. Applying conforming linear elements leads to a discrete solution equivalent to the text-book finite-element solutionto the Poisson problem in which the boundary data is approximated by L ² -projections. The L ² convergence rate of the discrete solution is O(h ^s ) for some $s\in(0,1/2)$ that depends on the shape of the domain, assuming a polygonal (two-dimensional) or polyhedral (three-dimensional) domain without slits and (only) square-integrable boundary data.

We show how the computational power and programmability of modern graphics processing units (GPUs) can be used to efficiently solve large-scale pixel-based material distribution problems using a gradient-based optimality criterion method. To illustrate the principle, a so-called topology optimization problem that results in a constrained nonlinear programming problem with over 4 million decision variables is solved on a commodity GPU.