Translate_Topology_optimization

毕业设计

Topology optimization is the major numerical method to synthesize the concept design of the ACG in this study.

The detailed formulation of the proposed soft-add topology optimization method is firstly introduced in this section.
Then the optimal design of the ACG based on the proposed soft-add topology optimization method is discussed.

The comparison of the optimal results from the proposed soft-add method and traditional soft-kill method are provided to demonstrate the computational efficiency of the proposed method.

2.1 The Soft-add Topology Optimization Method
The proposed soft-add topology optimization method can be used to synthesize the optimal layout of the analyzed com- pliant mechanism or structure. The flowchart of the soft-add topology optimization method is given in Fig. 1. The first step of the topology optimization method is to discretize the analysis domain based on the finite element formulation [48, 49] with pre-specified boundary conditions; then finite ele- ment method is used to solve the force equlibrium problem. The sensitivity number for each element can be computed by calculating the gradient of objective function with respect to design variable. A filter scheme is used to avoid the pos- sible mesh-dependency and checkerboard pattern problems, then an averaging scheme is used to increase the numerical

Fig. 1
Flowchart of the soft-add topology optimization method
stability in the topology optimization process. A new soft- add scheme is developed to update sensitivity number and pseudo density for each element. Unlike traditional hard-kill and soft-kill methods, the special characteristic of the pro- posed soft-add scheme is that the elements are equivalent to be numerically added into the analysis domain. Finally, a convergence criterion is used to evaluate the variation of objective function values until convergence.
The objective function for traditional topology optimiza- tion of continuum structure is to minimize compliance (maximize stiffness) of the design, which can be formulated as:
Minimize : Subject to :
C=DTKD KD = F

where C is compliance; D is the nodal displacement vector corresponding to the input force vector F; K is global stiff- ness matrix; V is calculated volume; Ne is total number of elements in the finite element model; xi is the pseudo den- sity of the ith element; Ve is volume of an element; vf is volume fraction constraint; V is initial volume of analysis domain; xmin is a small positive value. The pseudo density (which is the design variable in the topology optimization problem) of each element can be varied from xmin to 1 with x as the increment. The calculated volume for each itera- tion is equal to the summation of the pseudo density times the volume for each element. The converged volume is equal to the volume fraction times initial volume of the analysis domain.
For design of compliant mechanisms, the objective func- tion in this study is to maximize the output displacement. The topology optimization problem for compliant mecha- nism can be defined as:
Maximize : Subject to :
out = DT1 KD2/P2 KD1 = F1
KD2 = F2

where out is the output displacement at output port of the analyzed compliant mechanism [6, 11, 19, 50]; P2 is an unit dummy force (P2 = 1); D1 is the nodal displacement vector corresponding to the input force vector F1; D2 is the nodal displacement vector corresponding to the dummy force vector F2 .
The global stiffness matrix K can be represented as:

where Ks is the spring matrix (which is a void matrix for compliance minimization problem); p is penalty factor; Ke is the elemental stiffness matrix. The formulation for the stiffness matrix of a two-dimensional, linear square element used in this study is given in Appendix.
The sensitivity number for each element is defined by differentiation of objective function with respect to design variable. Based on the formulations given in Eqs. 1 and 3, the sensitivity number for the compliance minimization problem of continuum structures can be formulated as:

where αe,i is the elemental sensitivity number of the ith ele- ment; Di is the nodal displacement vector of the ith element. Similarly, based on the formulations given in Eqs. 2 and 3, the sensitivity number for the output displacement maximization problem of compliant mechanisms can be represented as:


where D1,i is the nodal displacement vector corresponding to the input force vector of the ith element; D2,i is the nodal displacement vector corresponding to the dummy force vector of the ith element.
After the calculation of the sensitivity number for each element, the BESO filter scheme and averaging scheme [3] are used in this study. The soft-add scheme summarized in Fig. 2 is proposed to update the pseudo densities, sensitiv- ity numbers, and the volume constraint. Unlike traditional topology optimization methods which define density value for each element as 1 at the first iteration (xi = 1 when iter = 1), the pseudo density of each element in the soft-add scheme is initially specified with a very small positive value (xi = xmin when iter = 1). The calculated volume for each iteration in the soft-add scheme is defined as: