# Shape Optimum Design by Basis Vector Method Considering Partial Shape Dependence
## 1 Introduction  
To improve the quality of machine products, applying topological optimization technology to actual product structure design has received much attention. Since the 1980s, in the industrial world such as aerospace and automobiles, the structural analysis technique using the finite element method and a mathematical programming method had been the norm [1]. Research on structural optimization for obtaining the optimum structural composition has been actively conducted after satisfying restrictive conditions such as strength and rigidity [2]. Since the beginning of the 2000s, researchers have improved the analysis technology using supercomputers and other hardware, collision analysis and noise analysis, along with the approximate analytical model, have introduced the experimental design method and the neural network, and the range of study of optimization and the scale of analysis model are further expanded. Recently, with the progress of analytical technology, the need for structural topological optimization has been increasing more and more from the field of design development. Regarding conventionalstructural shape optimization, depending on the type of design variable, it can be classified into size optimization, phase optimization, and shape optimization. During size optimization, the most general setting should consist of a determination for every point in space, whether there is material at that point or not, such [PERSON] proposed as a solution to shape optimization problems of domain boundaries in which boundary value problems of partial differential equations are defined [3]. Alternatively, for a FEM (Finite element _method_) discretization, every element is a potential void or structural member. Since the design variables of dimension optimization are simple, it is currently the situation that there are relatively few cases that can be applied to complicated structural topological optimization problems. During topological optimization, the element types and element densities of the analysis model are used as design variables, and the number of elements increases or decreases during the optimization analysis [4]. [PERSON], [PERSON], et al., mentions only the critical substructure to which the internal force of the structure is transmitted remains, when the so-called optimum phase structure can be obtained. However, the shape of the optimum structure obtained by phase optimization is difficult to predict in advance, it is difficult to use at the stage of detailed design, and there are features to be applied to the initial design [5,6]. During topological optimization, by changing the boundary shape of the structure using a parameter of a combination of appropriate interpolation functions, and by directly changing the design variable via the design variable, the design variable is changed in the middle of the optimization analysis, therefore, it is possible to control the shape change [7]. [PERSON], [PERSON], et al., find it is common to use the definition and numerical formulation of CAD (computer-aided design) shape data to parameterize the structural shape change [8]. However, similar to dimension optimization, there is a remaking problem that appropriately adjusts the analysis mesh inside the structure under the shape change of the structure boundary. Conversely, the researchers proposed a topological optimization method based on the reverse variation principle on the edge, since it is a method for finding the optimum boundary shape mathematically to realize the design target under the given load condition and constraint condition [9,10]. Although it is applied to structural design or independent part design, it is easy to see that it is difficult to apply to complicated topological optimization problems often found in the design site [11].