This section aims to build the topology and size cooperative optimization design models with the force-performance-structure, to realize the light-weight design of a complex part structure and size which can meet the multi-performance requirements. It is noteworthy that the design models include both a mathematical optimization model and a physical optimization model. Taking the static and dynamic performance as objective function, the light weight as constraint and the material density or the feature size as optimization variables, the mathematical optimization model is established. Moreover, the physical optimization model with loads, constraints, optimal design domains, non-optimal design domains is developed.

### 2.1 Mathematical Optimization Model

#### 2.1.1 Mathematical Optimization Model of Structural Configuration

Topology optimization is the process of determining the connectivity, shape, and location of voids in given design domains [20]. For a complex part design, the goal of the structure optimization design is to achieve a light weight structure configuration with optimal static and dynamic performance [21]. As the variations of working conditions of the part directly affect the optimization design results, the static and dynamic combined strain energy under multiple conditions should be considered for the structural optimization design, sequentially, the optimal mathematical model is expressed as below:

$$\left\{ \begin{aligned} & \hbox{min} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} S(\varvec{x}) = \sum {\omega_{i} }\varvec{\mu}_{i} (\varvec{x})^{\text{T}} \varvec{K\mu }_{i} (\varvec{x}) + NORM\frac{{\sum {\omega_{j} /\lambda_{j} (\varvec{x})} }}{{\sum {\omega_{j} } }}, \hfill \\ & {\text{s}} . {\text{t}} . ,\left\{ \begin{array}{l} V_{i} (\varvec{x})/V_{ 0} \le\varvec{\varDelta},\hfill \\ 0 \le x_{k} \le 1,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k = 1,2, \ldots ,N, \hfill \\ \end{array} \right. \hfill \\ \end{aligned} \right.$$

(1)

where *S*(*x*) is the weighted strain energy, *ω*_{
i
} and *ω*_{
j
} are the weighted coefficient of the *i*th operating condition of strain energy and dynamic characteristic, and *μ*_{
i
}(*x*) is the node displacement vector of the *i*th operating condition, respectively. Moreover, *K* is the system stiffness matrix, *λ*_{
j
}(*x*) is the *j*th order eigenvalue, *NOMR* is the correction coefficient to correct the strain energy and eigenvalue contribution degree, *V*_{
i
}(*x*) is the total volume after optimization, *V*_{0} is the initial volume, *Δ* is the optimization volume ratio constraint, generally taken 0 − 1, *x*_{
k
} is the design variable of material density, varying between 0 and 1.

#### 2.1.2 Mathematical Optimization Model of Structural Feature Sizes

The optimal structural feature sizes of a part can be determined after its structural configuration design, to meet the final performance requirements, which are related to the stiffness, strength and weight [22]. Considering the stiffness, strength and weight of the part, the mathematical model of multi-objective size optimization is formulated as

$$\left\{ \begin{aligned} & \mathop {\hbox{min} }\limits_{\varvec{X}} (D_{1} (\varvec{X}), \ldots ,D_{i} (\varvec{X}),D_{p} (\varvec{X})), \, i = 1,2,3, \ldots ,p, \hfill \\ & {\text{s}} . {\text{t}} . , { }f_{1} (\varvec{X}) \ge f, \hfill \\ & \varvec{X} = (X_{1} ,X_{2} ,X_{i} , \ldots ,X_{n} ), \, i = 1,2,3, \ldots ,n, \hfill \\ \end{aligned} \right.$$

(2)

where *D*_{
i
}(*X*) is the optimization objective, *f*_{1}(*X*) is the constraint function including the maximum stress and strain of a part, *f* is the constraint boundary of the constraint function, *X*_{
i
} is the optimization variables of structural feature size, respectively.

In this paper, the second order response surface method [23] is used to construct the part approximation model to solve the objective function and the constraint function. The established structural size optimization model by using the second order response surface method is given by

$$D_{k} (\varvec{X}) = a_{0} + \sum\limits_{i = 1}^{n} {b_{i} \varvec{X}_{i} + } \sum\limits_{i = 1}^{n} {c_{ii} \varvec{X}_{i}^{2} + } \sum\limits_{ij(i \prec j)} {c_{ij} \varvec{X}_{i} \varvec{X}_{j} } ,$$

(3)

where *X*_{
i
} and *X*_{
j
} are the input feature size optimization variables, *a*_{0}, *b*_{
i
}, *c*_{
ii
} and *c*_{
ij
} are the polynomial coefficients. In order to reduce the model error, the least squares method is used to regress the coefficients.

### 2.2 Physical Optimization Model

Considering the functions, connections, geometry, overall size characteristics, loads, constraints and optimization design interval of a part, the physical optimization model is built, including geometric model, design domains, loads and constraints equivalence.

The geometric model is constructed, which follows the functional rule, geometry rule and size rule in this paper. Functional rule is to determine the basic structure based on support, installation, auxiliary operation and other functional constraints of the part. The geometry rule is that the part should be made up of basic geometry structures or their combination, such as revolving body and non-swivel body of rectangular parallelepiped. The size rule is to determine the geometric sizes according to the machine-related parameters, the movement space of the adjacent parts and the positions of the loads. The design domains mean the variable areas of model during the topology optimization process. The non-design domains mean the non-variable areas of model to satisfy some installation requirements, such as moving contact surfaces, connection structures of the part, etc.

A complex part often works under multi-condition, of which the types, directions, magnitude, locations and numbers of working loads will vary correspondingly during operation. Theoretically, the working loads should be calculated using the load spectrum, however, the actual load spectrum is often unknown. Therefore, the working loads of multi-condition is weighted equivalent performed based on dangerous conditions, typical conditions and the working frequency. The constraints on the connection surfaces constrain the motion and deformation of a part called degree of freedom constraint and stiffness constraint respectively. According to the types of connection, the mobility constraints are classified into movable connection and stable connection constraints [24]. Furthermore, the stiffness characteristics of a part are difficult to accurately solve due to plenty of affecting factors. Therefore, it is important that the constrained degree of freedom and stiffness equivalent rules are simplified to establish constraint equivalent models for simulating the effect of the motion and deformation. The degree of freedom equivalent rule is to determine the number and direction of the constrained motion based on the type of connection. The stiffness equivalent rule is to equal the stiffness characteristic of actual joint surface by using spring equivalent and contact equivalent method.

Taking a gantry-type machining center bed as an example, two dangerous conditions and a typical working condition are considered. The loads are applied to the joint of the guide rail and the bearing seat. Considering the constrained bottom area of the bed, the physical optimization model of the bolt connection structure is shown in Figure 1.