# Multi-Objective Optimization of Spatially Truss Structures Based on Node Movement
## 1 Introduction  
Optimizing a discrete structure is a multi-objective optimization problem with some constraining conditions that include not only strength issues but also weight and rigidity factors. As early as 1904, [PERSON] [1] researched the structural design of stress constraint under a single load using an analytic method and proposed a theory of truss optimization design that was called Michell theory. In a subsequent study, [PERSON] [2] found that the Michell structure was a more flexible structure than others. In the single loading case, the bar direction is distributed along the main stress. In 1964, [PERSON] et al. [3] proposed a \"Ground Structure Approach\", which introduced the numerical method into the optimization region for the first time. The \"Ground Structure\" was connected by many components, which included all loading points and supporting points. The principle of this method was that: on the basis of the ground structure model, some optimization algorithm (mathematical programming approach or criterion method) can be used to omit needless bars from the ground structure according to regulations and constraints. For example, the sectional area of a bar attaches to the zero or floor level, which is unnecessary, and ultimately, the remaining bars form the optimization topology of the structure. Then, there are the other numerical algorithms, such as the weight method, constraint method [4], simplex method [5] and gradient method [6]. In recent years, the artificial intelligence algorithm has been widely used for solving the truss optimization problem. [PERSON] et al. [7] listed all connection types of truss structure first and then seta bound of threshold value for a section surface on the truss member. Subsequently, the section surface of the truss member was used as a design variable, which was optimized by the GA (Genetic Algorithm), and in this method, some trusses with section area that was less than the floor threshold level were omitted to achieve topological optimization. [PERSON] et al. [8] used the PSO (Particle Swarm Optimization) algorithm for topological optimization of a truss structure, and this theory translated the structure optimization problem constraint with stress and displacement into an unconstrained optimization issue by the penalty function. Under this condition, the optimization problem of truss structure constraint with stress and displacement was finally solved. In recent years, Free Gradient Methods for short (FGMs ) have been applied to many industrial applications in aerospace and civil engineering [9]. However, regardless of using the traditional numerical optimization method or a new artificial intelligence optimization method, there is no fundamental change in the optimization variables. Thus, most of the topological optimization of the truss structure is based on the ground structure, and the result of the optimal solution is actually a subset of given layouts. Thus, the range of the ground structure adoption is defined from the beginning of the optimization, which leads to an approximate optimal solution and loss of the optimal solution because the true value is not included. Therefore, the real topological optimization is supposed to search optimization in any given region, which means that researchers should consider the effect of the ultimate topological form on node movement of the structure. For optimization goals, one of the biggest features of the evolutionary algorithm search solution is that it is a group search algorithm that can deal with the parallel solving of the problem of multi-objective optimization with two or more goals. In 1967, [PERSON] proposed a solution to a multi-objective optimization problem using a genetic search algorithm in his doctoral thesis, while VEGA (Vector Evaluated Genetic Algorithm) did not appear until 1985, which was the first multi-objective evolutionary algorithm (MOEA) [10]. Then, many researchers improved this method later.