## 5 Lifetime and Weight Co-Optimization  
### Multi-Objective Optimization Function  
The lifetime of an engineering structure is related to geometry dimension, material properties, and bearing load. However, there is permissible deviation for the dimensions of structure, and the manufacturing process would also result in the uncertainty of geometry. The material properties could change because of different forming conditions and artificial operating. The dynamic loads are bound to keep varying in service time. Therefore, those uncertainty variables of geometry dimensions, material properties, and bearing loads were all taken into account in lifetime and weight co-optimization in this paper. The thickness of front and back plates in this frame would vary in ranges, as shown in Figure 19, as well as the cyclic material parameters \(\sigma^{\prime}_{f}\), \(e^{\prime}_{f}\), \(b\) and \(c\). The four main bearing loads at four connection sites (marked as A1, A2, A3, and A4 in Figure 19) were also defined to be variables. Then, the multi-objectives optimization function was described as:  
\[\begin{array}{c}\min\quad Mass(x0,x1,x2)\\ \max\quad Life(x0,x1,x2,x3,x4,x5,x6,x7,x8,x9,x10)\\ s.t.\quad Life\geq 2.0e5;mass\leq 4.0e3 kg;0\leq x0\leq 100;15\leq x1\leq 25;2 4\leq x2\leq 40;\\ 872.6\leq x3\leq 1454.3;-0.1554\leq x4\leq-0.0932;0.119\leq x5\leq 0.198;\\ -0.3269\leq x6\leq-0.1961;-2.50e5\leq x7\leq-1.50e5;7.88e4\leq x8\leq 1.31e5;\\ 7.95e4\leq x9\leq 1.32e5;-5.01e5\leq x10\leq-3.01e5;\end{array}\]  
where _Mass_ function stood for the optimization objective of weight and \(Life\) function stood for the optimization objective of lifetime. The distance of weld metal away from the back plates was defined as \(x0\), marked as \(d\) in Figure 19. The weighting coefficient of two optimization objectives was assigned to be 0.3 and 0.7 in the light of engineering experience. The geometry variables \(x1\) and \(x2\) were thickness of front and back plates. The cyclic material parameters \(x3\), \(x5\) were the fatigue strength coefficient and the fatigue ductility coefficient, and the parameters \(x4\), \(x6\) the fatigue strength exponent and the fatigue ductility exponent, respectively. The bearing loads \(x7\), \(x8\), \(x9\), and \(x10\) were the peak vertical force along Y direction at connection site A1, the peak lateral force along X direction at connection site A2-A4, respectively.  
In order to build the relationship between objectives and variables, the Latin hypercube sampling method was first used to get the 50 sample points, listed in Table 5.  
Figure 23: Fatigue life contour of welded A-type frame (cycles).  
When the optimization design variables were determined, the corresponding response values could be obtained from the finite element analysis. Then, their approximation relationship could be constructed to start the multi-objectives optimization. In this paper, the Kriging approximation model was chosen to describe their mathematical links, while the NSGA-II was used to search for the optimal solution in this optimization problem. These two methods are described in my previous works [22; 23]. The whole optimization process is shown in Figure 24.