Topology Optimization Problems of Density Variation Type

From this chapter, we finally examine shape optimization problems in continua. Firstly, let us think about a problem seeking the appropriate arrangement of holes in a domain where the boundary value problem of a partial differential equation is defined. Such a problem is known as the topology optimization problem. Here, the term topology refers to the study of geometrical properties and spatial relation of objects unaffected by the continuous change of their shape or size. In mathematics, two mathematical objects are said to belong to the same topology if they are images of two homotopic maps; that is, if one can be continuously deformed into the other. Therefore, letting n be a natural number, a set of n-connected domains are regarded as belonging to the same homotopy groups. Here, the term “topology optimization” in the topology optimization problem refers to the determination of the connectivity of the design domain that optimizes an object’s material distribution through insertion and arrangement of holes in its structure. However, as will be explained in detail later, the shape of the holes actually becomes the target. Therefore, the problems dealt with in this chapter also become included in shape optimization problems in a wider sense. In this book, it will be referred to as the topology optimization problem in the sense that topology is also in the scope of the design.