For instance, Xie and Steven [29] presented a simple meth- od for structural optimization with frequency constraints. The structure is modeled by finite elements. At the end of each analysis, part of the material is removed from the structure so that the frequencies of the resulting structure shifted to- wards a desired direction. Wang [30] presented a numerical method for structural shape and topology optimization. The method relies on a novel approach for the representation of the design boundaries with level set models. A structural optimi- zation is formulated as a mathematical programming problem with a design objective and a set of constraints, utilizing the level set models for the incremental shape changes. Huang and Xie [31] demonstrated the effectiveness and efficiency of the BESO method on the minimization compliance prob- lem with fixed external loads. They considered the minimiza- tion of mean compliance for continuum structure subjected to design-dependent self-weight loads. Tcherniak [32] studied on the layout optimization of resonating actuators using the SIMP topology optimization method. The goal of the optimi- zation is maximization of the magnitude of steady-state vibra- tions for a given excitation frequency. Yildiz and Saitou [20] developed a novel approach for multi-component topology optimization of continuum structures using a multi-objective genetic algorithm to obtain Pareto optimal solutions that ex- hibits trade-offs among stiffness, weight, manufacturability, and assembly ability. Fourie and Groenwold [21] applied PSO algorithm to shape optimization of a torque arm and to size optimization of truss structures. In their PSO algorithm, the concept of craziness is redefined, and elitism operator borrowed by GA was used. Their results showed that PSO algorithms were better than GA and the gradient-based recur- sive quadratic programming algorithm. Perez and Behdinan [22] proposed a particle swarm approach for structural design optimization. The effectiveness of the improved PSO algo- rithm on structural optimization is shown through the use of four classical truss optimization examples. Results from the three tested cases using an improved PSO illustrate the ability of the algorithm to find optimal results which are better than, or at the same level of, other structural optimization methods. Yildiz [25] developed a novel hybrid optimization method (HRABC) based on the artificial bee colony algorithm and Taguchi method. The proposed approach is applied to struc- tural design optimization of a vehicle component and a multi- tool milling optimization problem. Mahdavi [26] developed an improved harmony search (IHS) algorithm for solving op- timization problems. IHS employs a novel method for gener- ating new solution vectors that enhances accuracy and con- vergence rate of harmony search algorithm.

Most of the existing research work on sheet metal forming is concentrated on the numerical simulation of different kinds of forming processes to improve the precision of produced parts [33]. For example, Wang et al. [34] accomplished a series of numerical simulations concerning the influence of shape error and non-uniformity in thickness distribution of sheet metal parts. Farsi and Arezoo [35] developed a system for operation sequencing sheet metal part that includes bend- ing and stamping operation. They used classification and fuzzy rules for determination of the sequence of the bending operations. Yan and Klappka [36] studied the spring-back be- havior of panel forming using multi-point stretch forming technique. Fazli and Arezoo [37] presented an analytical method for estimating the limiting drawing ratio (LDR) of the redrawing stages in deep drawing process of axisymmetric components.

As mentioned, many research works were carried out to develop the forming conditions of sheet metal forming oper- ations. However, there has been a small number of research works in die structural optimization, from which a few are the optimization of the structure of stamping and stretch forming dies. These are presented as follows: