# Shape Optimum Design by Basis Vector Method Considering Partial Shape Dependence
## 1 Introduction
Observations from earlier studies [12] during the actual machine part topological optimization problem find, as the external boundary shape of the structure is changed, the internal shape of the hole or the uneven partial shape is always adjusted according to requirements such as the assembly relationship and the processing condition between the part and the part. It is common to have a certain partial shape dependency. Alternatively, it is required that a partial shape dependency relationship always exists between the boundary shape outside the structure and the internal partial shape in the course of shape optimization [13,14]. To solve this partial shape subordination problem, some researchers have applied the basis vector method based on the basic shape change pattern [15,16], like [PERSON], [PERSON], and [PERSON], et al., who proposed a COBYLA optimization method, and [PERSON], [PERSON], et al., who proposed evaluation of manufacturing constraints from the early stages of the conceptual design to perform an optimization coherent with the manufacturing technology chosen. Actually, when optimizing the structure using the basis vector method, one first considers the structural processing condition of the design object and the assembling relation with other parts, from the designer [17,18], then some basic shape change patterns and their change range. Then, each new basic shape change pattern is multiplied by a weighting coefficient and added to form a new structure shape. Each weighting coefficient is used as a design variable as it is, thereby obtaining an optimal weighting factor [19,20]. Here, since each basic shape change pattern is expressed in the form of a node coordinate vector having a similar phase configuration, this method is called a basis vector method. To the authors' knowledge, there are no studies of shape optimization considering the partial shape dependence problem. However, when the basic vector method is applied to complex structural, topological optimization problems, there is no theoretical research on the partial shape dependency problem, and the basic vector problems which depend on the relationship of multiple basic shape change patterns are not solved [21]. This has a significant impact on the solution of complex structure topological optimization problems. While these simple approaches are adequate for many applications, it is nevertheless clear for practitioners that they can become insufficient to address the partial shape dependency optimization method [22; 23]. Conversely, many researchers have been making efforts to add optimum design functions in general-purpose analysis codes such as ANSYS and HyperWorks but, due to the complexity and difficulty of partial shape subordination problems, many cases have stopped halfway. Until now, the partial shape dependency problem in complicated structural shape optimization has not been solved very well, and there is no theoretical research on the partial shape dependency problem. Here, to solve the complicated structural shape optimization problems, some important issues are handled and applied in detail when applying the basis vector method. First, considering the partial shape dependence problem and the basis vector method, the subordinate relationship of the partial shape inside the structure is divided into three basic cases with the change of the external shape of the structure. The correctness of the basic vector method is verified using mathematical theory, which provides a theoretical basis for the basic vector method used in complex shape optimization design. Additionally, the basic vector generation and mesh regeneration problems in the practical application of the basic vector method are discussed, and the corresponding algorithms are given. The basic vector method proposed in this paper is used to analyze the complex shape optimization design of a two-dimensional turbine blade and a three-dimensional engine base, and the optimization design results verify the local shape dependence problem.