The finite element method (FEM) is a numerical method that divides a domain into small connected sub-do- mains (elements), such that differential equations can be solved by solving a set of polynomial equations. With the development of computer science, more and more complicated mechanic problems in different fields are now solvable with FEM. One feature of FEM we need here is the definition of parameters for each element individually. This is important for the implementation of topology optimization, since the design variables should be set for every single element locally.

Topology optimization is a mathematical approach that optimizes the material layout within a given design space, for a given set of load and boundary conditions. It is widely used in the field of shape or structure design, especially for the lightweight design without losing much stiffness. Different kinds of topology optimization algorithms have already been carried out. Xie and Steven introduced the Evolutionary Structural Optimization (ESO) to eliminate the less effecting elements of a structure [1] [2] . Querin et al. proposed Bi-directional Evolutionary Structural Optimization (BESO) as an improvement of ESO that can also add elements in a design domain to compensate the structure defect caused by a single directional elimination of elements [3] . Bendsøe presented the Solid Isotropic Material with Penalization (SIMP) method to update the element-density-depen- dent material properties in order to reach the final optimum [4] . The penalization factor is applied to reduce the amount of intermediate densities. However, the introduction of penalization usually has a negative effect on both computational time and final results. In order to reduce such influences, Dadalau [5] introduced SIMP with an adaptive penalization scheme. In this paper the SIMP-model is used. As shown in Equation (1), it regards every element density as a single optimization variable. All the variables are updated with a specific scheme which is derived by Lagrange multipliers [6] .

Here denotes the density at k-th step. Variables and control the possible change in one iteration with the typical values of 0.5 and 0.2. This scheme adds material to the structure if and removes it if without violating the bound of. is given by

where is the displacement at k-th step, which is calculated from the equilibrium function of the structure, is a Lagrange multiplier and is the penalization factor.

Classical methods of parameter optimization, which are based on sensitivity calculation and the gradient descent method like the Newton method, are already well developed with high accuracy. However, the drawbacks are also obvious. It is expensive to calculate the needed sensitivities, if the variables are not explicit in the objective function. Moreover, in many discretized situations, the loss of continuity is usually problematic. Furthermore, the global optimum is difficult to reach, since it can be easily trapped by a local optimum. The Particle Swarm Optimization (PSO) method, known as a purpose optimization method, was originally proposed by Kennedy and Eberhart [7] . It optimizes a problem by looking for the best candidate solution in a search domain with a set of particles. The general equations for moving particles can be written as follows,

where is the search velocity of i-th particle for k-th step, is the position of i-th particle for k-th step, is the inertia of particle, is the best point i-th particle ever met, is the global best point among personal bests, and are weight parameters. Particles are randomly generated at first and they find the next step with the influence of personal and global best points. After several iterations, all the particles could gradually converge to the optimal design point. Various researches about the behavior due to parameter selection in PSO have been presented, such as Shi and Eberhart [8] , Van Den Bergh [9] and Trelea [10] . As the choice of position for the next iteration is relatively random, this method can be easily applied in both the continuous and the discretized problem.

The task of moving the hole position during the optimization process must be simple to handle. Normally, the empty area is not included in mesh. When the hole position is changed, meshing and assembling of the stiffness matrix need to be repeated, which decreases the efficiency greatly. This problem can be solved by the passive element method, with which the hole in the design domain can be defined as “zero” density element (in the program) instead of a description of the geometric information. The mechanism of the passive element is to set the density in the hole area from the last iteration to one and set those in a new hole area to zero, when the position is changed during the searching procedure. The nodes in the lower part of the hole can also be easily selected to add the load. With the passive element method, only the material density parameter is applied to evaluate the global stiffness matrix, the remeshing process is no longer needed.

The penalization factor plays an important role in topology optimization. By, which means the density and the Young’s modulus of an element are exactly linearly related, the optimality of the topology optimization is guaranteed. However, the optimized structure will be filled with large numbers of intermediate densities, which are not realistic to be manufactured. With a large penalization factor, such as, the structure will be more realistic and closer to a 0-1 structure. Nevertheless, high penalization factors also have some disadvantages. Table 1 shows that the compliance by is worse, and it converges more slowly than that by.

Due to the desired efficiency, the topology optimization during the PSO should be fast, thus the penalization factor within the fast topology optimization (Step 2 in Figure 3) is fixed to one.

The convergence condition also has a great effect on the convergence speed. Normally, the topology optimization is terminated when the largest change of the element density is smaller than a preset value [6] [12] . Under this condition, a very fine result can be achieved. Nevertheless, the change of density can fluctuate on some positions of the domain, which makes the process difficult or sometimes impossible to converge, although the change of compliance is already very small. The simplified convergence condition terminates the optimization loop, if the change of compliance is smaller than a preset value. According to Table 1, it decreases the convergence loops greatly without losing much accuracy. Through the simplified convergence condition and unity penalization, the fast topology optimization will converge in around 10 loops instead of almost 100 loops each time. Along with the flexibility of the program, the strict condition with a high penalization value can still be used to get the accurate final structure (Step 5 in Figure 3).

The optimized hole position found by the combination method is converged on (93, 40) with the compliance value of 63.64, which is just one element width beside the best position of the reference (Figure 2(b)). Actually, due to the property of the PSO method, the converged result is not always the best but very close to it. Figure 4 describes the two structures after two separate steps in a sequential method. The final structure resulting from the combined method is shown in Figure 5.