# Lightweight Research in Engineering: A Review
## 3 Lightweight Mathematical Model and Solution  
The problem of lightweight ultimately comes down to the maximum and minimum problem of solving the objective function under certain constraints. The difficulty is that the constraints are not easy to establish in different engineering problems, or different working conditions, differentbackgrounds, and different hard requirements. The current solution steps can be roughly simplified, as shown in Figure 1.  
There are generally two modeling ideas of structural optimization methods [1; 5]: One is to seek structural stiffness maximization (minimum compliance) under volume or mass constraints, and the other is to seek structural minimum volume or mass under stiffness constraints. Mathematical modeling is the first step in lightweight. Regardless of whether the optimization is a static problem, a dynamic problem, or a shape, size, and topology problem, it can generally be expressed in the form of nonlinear programming. The standard nonlinear programming model is as follows:  
\[Minf(X) \tag{1}\]  
\[S.t.\quad\mathbf{g}_{j}(X)=0,j=1,2,\cdots,p\] \[\mathbf{g}_{j}(X)\leq 0,j=p+1,p+2,\cdots,m \tag{2}\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad X^{L} \leq X\leq X^{U}\]  
where, \(f(X)\) is the objective function, generally taking the structural weight; \(g_{j}(X)\) is the constraint function, which may include the physical equation and the coordination equations, static or dynamic strength, stiffness limit, etc. ; \(X=\left(X^{1},X^{2},\cdots,X^{N}\right)^{T}\) is the design variable; and \(X^{U}\) and \(X^{L}\) are the upper and lower limits of \(X\), respectively. It is necessary to make the following explanation about the model:  
* Design variables can be either continuous or discrete. For engineering structure design, the variables are usually a lot. * The objective function and the constraint function are continuously differentiable in most cases and may also be noncontinuous and nondifferentiable. * The constraint function is usually implicit and has a nonlinear nature. The degree of nonlinearity is different for different problems or different design points of the same problem. Because the constraint conditions in complex engineering may be very diverse, the degree of nonlinearity and linearity are also different for the structure's design. At the same time, our solution was obtained through iterative optimization. For the optimization result, different variables are grouped with different requirements. Due to the fact that each set of variables must be fully analyzed, one by one, the amount of calculations is usually very large. Therefore, the number of times structural analysis is usually an important indicator of the efficiency of an optimization method [1]. After the lightweight approach and the mathematical model are determined, we need to solve the model. In recent years, there have been many methods for solving lightweight optimization problems, such as mathematical programming, optimization criterion method, emerging meta-heuristic bionic optimization algorithm, etc., which have attracted many experts and scholars, and have been widely used in engineering field. From the point of view of engineering and mechanics, the criterion method [6] offers some criteria that should be met when the structure reaches the optimal design (such as synchronous failure  
Figure 1: The current solution steps. criterion, full stress criterion, energy criterion, etc.). Then the solution satisfying these criteria is obtained by the iterative method. The method is characterized by fast convergence, no direct relation between the number of times of reanalysis and the number of design variables, and a small amount of calculation. However, it is limited in its application, which is mainly applicable to the case where the structural layout and geometric shape have been determined. Although the standard method has its shortcomings, from the perspective of engineering application, it is more convenient.