1. First, let us assume that our goal is to obtain a shapedesign based exclusively onamacroscopic distribution of solid, homogeneous material and void (we refer to this objective as the macroscopic topology design problem). This situation typically arises when considerations of cost preclude the use of porous or composite microstructures, despite their potential for improving performance. Clearly then, the necessary inclusion in the feasible design space of porous microstructures makes relaxed formulations unsuitable formacroscopicdesign - if we insist on a design tool that can directly synthesize a feasible solution. However, we hasten to add that relaxed formulations can nonetheless be very useful in macroscopic design problems. For example, solutions to relaxed problems provide bounds on the performance that can be achieved with conventional designs, and thereby provide an effective benchmark for evaluating macroscopic designs. Furthermore, the pattern of microstructural orientation and density distribution in a relaxed solution can often be manually interpreted by a knowledgeable engineer to obtain a manufacturable, macroscopic design. In addition, relaxed solutions provide efficient global distributions of material that can be used very effectively to initiate a design algorithm (based on another problem formulation) that eventually returns asolid-void macroscopicdesign (Bends 0e and Rodrigues, 1991, Bremicker et al, 1992, Olhoff et al, 1992, Maute and Ramm, 1995, Allaire et al, 1997). Finally, solutions to the relaxed problem can often be interpreted as the limiting cases of non-convergent minimizing design sequences forsomecorresponding ill-posed basic topology design problem. Therefore, the study of relaxed solutions is extremely important for advancing our understanding of what "optimal" means.