Structural Optimization 13, 258-266 ~) Springer-Verlag 1997 S-relaxed approach in structural topology optimization G.D. Cheng and X. Guo Dalian University of Technology, Dalian 116023, PR China Abstract This paper presents a so-called $-relaxed approach for structural topology optimization problems of discrete struc- tures. The distinctive feature of this new approach is that unlike the typical treatment of topology optimization problems based on the ground structure approach, we eliminate the singular optima from the problem formulation and thus unify the sizing and topol- ogy optimization within the same framework. As a result, numer- ical methods developed for sizing optimization problems can be applied directly to the solution of topology optimization problems without any further treatment. The application of the proposed approach and its effectiveness are illustrated with several numer- ical examples. 1 Introduction Structural optimization has seen remarkable progress in re- cent years and it is now recognized as a practical design tool. There are three major structural optimization problems: (1) sizing, (2) shape and (3) topology optimization problems. A considerable amount of work has been done in the last three decades on optimal structural design. Most of the work is re- lated to the optimization of the sizing and shape of the struc- ture, and fewer contributions have been devoted to optimal topology design because of its complexity. It is recognized, however, that optimization of the structural topology results in large savings of material and greatly improves the design, so more and more research activities have concentrated on this field recently. Topology optimization of discrete structures was first ad- dressed by Dorn et al. (1964), who applied a linear pro- gramming method to optimize the topology of trusses, and presented the so-called ground structure approach. The au- thor defined the ground structure as a grid which included all the structural joints, supports, load positions and was connected by all the potential members. Since then, the ground structure approach in which mathematical program- ming techniques are employed to remove redundant members during the process of optimization has been frequently used to find the optimal topologies of discrete structures. Based on this approach, Dobbs and Felton (1968) used member cross- sectional properties as topology design variables and applied a "steepest descent-alternate mode" algorithm to find the op- timal topologies of trusses subjected to stress constraints and multiple loading cases. In their approach, once the cross- sectional area of a member was reduced to zero value, it was removed from the ground structure and not allowed to re-enter the design problem. Sheu and Schmit (1972) ap- plied a branch and bound method to optimize the topology of trusses under multiple loading conditions in order to ob- taln the global optimum of the problem. In their study, stress and displacement constraints have all been taken into consid- eration. Ringertz (1986), in addition to the member areas, considered joint displacements simultaneously as design vari- ables to the solution of truss topology optimization problems. A branch and bound search technique was also employed in this work. Kitsch (1989), and Kirsch and Topping (1992) suggested a heuristic but very attractive two-stage design approach for topology optimization of discrete structures. A typical such procedure is to evaluate an approximate solution at the first stage and modify it at the second stage to achieve the final optimum. For topology optimization of large scale ground structures subject to stress, local buckling, as well as displacement constraints, the DCOC method was developed by Zhou and l~ozvany (1991). With the use of this approach, the capability of traditional optimization methods can be in- creased by several orders of magnitude. Singular optima in structural topology optimization were first treated by Sved and Ginos (1968). They investigated a three-bar truss subjected to three loading cases and pointed out that a global optimum could be obtained only by remov- ing one of the members from the ground structure and, in effect, violating the stress constraint for that member. The type of problem where optimal topology is singular has been studied by Kitsch (1986, 1987, 1990). Kirsch suggested that "in case of singular solutions, it might be difficult or even impossible to arrive at a true optima by numerical search algorithms". Cheng and Jiang (1992) studied the singular optima from a different point of view. Illustrating by truss topology opti- mization problems subject to stress constraints, they demon- strated that the discontinuity of the stress constraint function of the bar when its cross-sectional area takes zero value is the essential cause of the existence of singular optima. They also have shown that a singular optimum is not an isolated point but just end point of line segments attached to the feasible domain. Cheng's work indicated that there is an essential difference between topology and sizing optimization and it is very important to establish a rational formulation which can unify the sizing and topology optimization within the same framework if one wants to apply the sizing optimization tech- niques to solve the topology optimization problems. Cheng (1995) discussed the shape of the feasible domain for truss topology design when buckling and compliance constraints are imposed. For a wide range of structural topology opti- mization problems, Cheng and Guo (1995) introduced some basic definitions such as the topology design variable and its critical value, the topology-dependent behaviour constraints