This thesis examines efficient solution procedures for the structural analysis problem within topology optimization. The research is motivated by the observation that when the nested approach to structural optimization is applied, most of the computational effort is invested in repeated solutions of the analysis equations. For demonstrative purposes, the discussion is limited to topology optimization problems within the field of structural mechanics. Nevertheless, the results can be relevant for a wide range of problems in structural and topology optimization. The main focus of the thesis is on the utilization of various approximations to the solution of the analysis problem, where the underlying model corresponds to linear elasticity. For computational environments that enable the direct solution of large linear equation systems using matrix factorization, we propose efficient procedures based on approximate reanalysis. For cases where memory limitations require the utilization of iterative equation solvers, we suggest efficient procedures based on alternative termination criteria for such solvers. These approaches are tested on two- and three-dimensional topology optimization problems including minimum compliance design and compliant mechanism design. The topologies generated by the approximate procedures are practically identical to those obtained by the standard approach. At the same time, it is shown that the computational cost can be reduced by up to one order of magnitude. The main observation in the context of optimal design of linear structures is that relatively rough approximations are acceptable, in particular in early stages of the optimization process. The thesis also addresses topology optimization of structures exhibiting nonlinear response. In such cases, the computational effort invested in the solution of the nested problem is even more dominant since nonlinear equation systems are to be solved repeatedly. Efficient procedures for nonlinear structural analysis are proposed, based on transferring solutions and factorized tangent stiffnesses from one design cycle to the following one. This approach is demonstrated on several design problems involving either geometric or material nonlinearities. The suggested procedures are shown to be effective mainly for problems that do not involve path-dependent solutions.