Optimum shape design of thin-walled cross sections using a parameter-free optimization method

Numerous industrial processes and products employ the use of thin-walled structures, mostly press formed or welded, as load-carrying components. Regarding current environmental protection regulations and the urgent need to reduce the use of fossil fuels, the current and future design of thin-walled structures has largely focused on weight reductions. This has initiated the need for maximizing the desirable sectional properties of the structures through optimization. To this end, various shape and size optimization methods have been developed. These methods, however, require a strong understanding of shape parametrization, which is underexplored in the literature. In an attempt to resolve this problem, researchers have explored the possibility of developing effective free-form optimization methods for efficiently determining the shapes of thin-walled structures.

In a recent research published in the Thin-Walled Structures Journal, Professor Masatoshi Shimoda and Kousaku Ishikawa (Ph.D. student) from Toyota Technological Institute, together with Professor Yang Liu from Sojo University, developed a parameter-free method to optimize the cross-sectional shapes of thin-walled structures. They specifically minimized the circumference length of a cross-section, taking into consideration the constraints of the sectional properties. Parameter-free method involves theoretical derivation, numerical computation and determination of the optimal shape gradient. The key constraints considered included centroid, moment of inertia, and the shear center of the cross-sectional area. By applying a distributed shape optimization approach, it was possible to derive the shape gradient function using material derivative and Lagrange multiplier techniques.

Several design examples were used to validate the feasibility of the presented parameter-free method. In these examples, the determined optimal shapes were satisfied by the linearized constraint equation. These examples also used some parameters: thickness, the cross-section area of truss member, Young’s modulus and Poisson’s ratio as 1mm, 1mm², 200GPa, 0.3 respectively. The first design case involved a closed circular thin-walled section, and going by the results, the proposed method was effective in obtaining desirable shape conditions. In the design of a closed rectangular thin-walled section, the circumferences of the initial and optimized shapes were observed to depend on both the given constraints as well as the initial shapes. For design example three, an open thin-walled section was designed. Two main constraints considered in this case were torsion constant and shear center of the cross-section. The last example involved designing a closed side sill section, and just like in other examples, the minimized circumference length satisfied the sectional constraints.

In summary, the study presented a new parameter-free method for the optimization of the cross-sections of thin-walled structures. The authors proved, using four different design examples, that the given constraints of sectional properties can be satisfied in obtaining a minimized circumference. Also, being a gradient-based method, the circumference length depends on the initial shapes and the given constraints. The influence of the mesh size was further investigated, and both the coarse and fine mesh produced almost similar shapes. It was noted that reducing the mesh sizes could result in a smoother curvature distribution. In a statement to Advances in Engineering, authors recognized their efforts in developing new methods of designing thin-walled cross-section structures aiming at combating current global problems like reducing energy consumption by reducing the mass of the structures.