This dissertation concerns theoretical and computational methodologies in topology optimization in continuum mechanics.

A general procedure for treating problems of this kind starts with the establishment of a well-posed continuum problem formulation, in the sense that one can prove existence of at least one solution to the problem. When straightforward mathematical modelling of a physical situation results in an ill-posed formulation, i.e., no solutions exist, the problem is regularized by means of restriction methods in order to remedy this. These are general and appealing regularization approaches that apply to a large class of problems and have increased in popularity of later years. In this dissertation, some new restriction methods are introduced, and an overview and characterization of such methods are made that indicate theoretical and numerical advantages and drawbacks.

Topology optimization of elastic solids and structures is a well established discipline  but little efforts have previously been made to apply the methodology in fluid mechanics. In this dissertation, topology optimization of viscous fluids in Stokes flow is successfully performed, yielding many potential applications. -while regularization often is needed for the problems encountered in topology optimization of continua, this problem is a rare exception. The underlying physics allows for a mathematical problem formulation that is well-posed without any regularization. The presented methodology can also be seen as a guideline for how to proceed with research in topology optimization in fluid mechanics.

A well-posed problem formulation serves as a basis for a finite element discretization procedure performed such that finite element convergence can be proved. This asserts an a priori reliability in the sense that discrete solutions will not suffer from typical artifacts known of in the field. The discretized problem is solved numerically by mathematical programming algorithms, a general and powerful approach that is applicable to many problems of varying degree of complexity. In addition to using powerful solution algorithms, object-oriented implementation techniques and high performance computers and computing are incorporated as means to facilitate encounter of new applications and to improve computational efficiency. The advantages of these approaches are exemplified by solving problems from several disciplines of physics within the same object-oriented framework and by solving large scale "real-life" problems in an industrial environment using parallel computing.