# The Influence of the Load Term of the Strain Energy Sensitivity in the Shell Shape Optimization
## 1 Introduction  
Shape optimization is an approach for finding the optimal geometry of a structure to capture a desired structural performance, subject to some given constraints [1]. The shape optimization of structures, especially the free-form surface structure, has gained significant attention in the last few decades, because shape optimization is an effective tool to generate a free-form shell structure, and the architecture form and structure form of the free-form surface structure are very close and architecturally pristine [2; 3; 4; 5; 6; 7]. Many shape optimization methods have been developed by researchers, such as the sensitivity method [8; 9; 10; 11; 12; 13], firefly algorithm [4], genetic algorithm [14], and so on. The objective functions are usually one or several among the structural behaviors, such as the compliance [11; 12; 13], the buckling load [15; 16], the plastic collapse load [17], the Eigen frequency, or the dynamic response [18; 19], etc. Shape optimization based on the strain energy sensitivity method [8; 9; 10] has been applied in engineering projects. [PERSON] [10] chose strain energy as an objective function and used the sensitivity method to design the free-form surface reinforced concrete shell structure. These projects have attracted scholars' attention, and strengthen the research on the free-form shell structures. Minimizing the strain energy is equivalent to maximizing the stiffness. The strain energy is considered to be a very promising objective for shells because it minimizes bending, yielding a membrane-orienteddesign [20]. Evolutionary speed based on the strain energy sensitivity method is faster than the optimization methods which are not used the gradient information, for instance, the genetic algorithm, and the optimization result can be controlled by the initial structural shape, as it is a kind of local optimization method. Such methods based on strain energy sensitivity [7; 8; 9; 10; 11; 12; 13] assume that the load does not change with the design variables when deducing the sensitivity information of the structure. In the traditional optimization method, the structural weight is only considered as the external load, and its influence on strain energy sensitivity is not considered. However, the structural self-weight and surface load are a kind of design dependent load, which will change with the structural shape [21]. Sensitivity analysis and evolutionary procedure need to accommodate the load variation condition. [PERSON] et al. [22] concluded that the influence of the load term on the sensitivity of optimization should be taken into account for the engineering structures with self-weight and inertia as the main load, and took the displacement sensitivity of the truss structure as an example to provide an explanation. Moreover, some scholars, [PERSON] and [PERSON] [23], and [PERSON] etc. [24], considered that the load is a function of the design variables, but they did not show the difference between these two situations. In this paper, we focus on the method based on strain energy sensitivity to optimize the shape of a free-form shell, which was named the height adjusting method [8] of curved surfaces. In this method, the height of the curved surface is adjusted gradually according to the differential of the strain energy around the height of the curved surface, calculated by the finite element approach. It is appropriate to approximate the load, as it is evenly distributed, and approximate the loading on each node. This loading will be proportional to the horizontal projection of the surface area carried by the node and is therefore a function of the geometrical positions of the nodes, which are variable. The structural self-weight is also a function of the geometrical positions of the nodes. The paper is organized as follows. In Section 2, the formula of the surface shape optimization is shown. In Section 3, we show the derivation process of the strain energy sensitivity with and without the load term.