Part I. Topology optimization is the most general form of design optimization in which the optimal layout of material within a given region of space is to be determined. Filters are essential components of many successful density based topology optimization approaches. The generalized fW-mean filter framework developed in this thesis provides a unified platform for construction, analysis, and implementation of filters. Extending existing algorithms, we demonstrate that under special albeit relevant conditions, the computational complexity of evaluating generalized fW-mean filters and their derivatives is linear in the number of design degrees of freedom. We prove that generalized fW-mean filters guarantee existence of solutions to the penalized minimum compliance problem, the archetypical problem in density based topology optimization. In this problem, the layout of linearly elastic material that minimizes the compliance given static supports and loads is to be determined. We formalize the connection between mathematical morphology and the notion of minimum length scale of a layout of material and thereby provide a theoretical foundation for imposing and assessing minimum length scales in density based topology optimization. Elaborating on the fact that some sequences of generalized fW-mean filters provide differentiable approximations of morphological operators, we devise a method capable of imposing different minimum length scales on the two material phases in minimum compliance problems.
Part II. The notion of Friedrichs systems, also known as symmetric positive systems, encompasses many linear models of physical phenomena. The prototype model is Maxwell's equations, which describe the evolution of the electromagnetic field in the presence of electrical charges and currents. In this thesis, we develop well-posed variational formulations of boundary and initial–boundary value problems of Friedrichs systems on bounded domains. In particular, we consider an inhomogeneous initial–boundary value problem that models lossless propagation of acoustic disturbances in a stagnant fluid. Galbrun's equation is a linear second order vector differential equation in the so-called Lagrangian displacement, which was derived to model lossless propagation of acoustic disturbances in the presence of a background flow. Our analysis of Galbrun's equation is centered on the observation that solutions to Galbrun's equation may be formally constructed from solutions to linearized Euler's equations. More precisely, the Lagrangian displacement is constructed as the solution to a transport-type equation driven by the Eulerian velocity perturbation. We present partial results on the well-posedness of Galbrun's equation in the particular case that the background flow is everywhere tangential to the domain boundary by demonstrating mild well-posedness of an initial–boundary value problem for linearized Euler's equations and that our construction of the Lagrangian displacement is well-defined. Moreover, we demonstrate that sufficiently regular solutions to Galbrun's equation satisfy an energy estimate.