This project concerns method development for numerical optimization. Here, we are interested in design optimization problems covering a range of different disciplines, such as linear and nonlinear elasticity, transportation, acoustics, and electromagnetics. Primarily, we work with topology optimization algorithms. These algorithms simultaneously search for the optimal placement and shape of an unknown number of components constituting the total design and are typically used in the initial stages of the design process to find a conceptual design. In the material distribution method, the presence of material is modeled by using a binary valued material indicator function. In the discrete case, the optimization problem is to determine which type of material that occupies each pixel. More precisely, the computational problem is cast a large-scale non-linear programming problem over the coefficients in a partial differential equation. The solution of such optimization problems relies on repeated numerical solutions, for example, of the 3D Maxwell equations in time domain, where coefficients in the equations are successively updated using a gradient-based optimization algorithm, and where the gradients are supplied through solutions of associated adjoint equations. In this project, HPC resources will be used both to solve the underlying partial differential equations as well as the large-scale optimization problems.