# A Fast Approach to Optimize Trad Pattern Shape for Tire Noise Reduction
## 3 Basis Vector Method for Shape Generation  
During the early stage of low-noise tire pattern design, tire designers often resort to trial and error to refine the design schemes, which is very time-consuming. Optimization is a highly effective approach for mitigating tire noise and can be considered a viable method. Various structural optimization methods, such as size, shape, and topology optimization, are available for consideration. Because of a tire's complex geometric structure, which includes ribs, lugs, sipes, and other elements, it is difficult to represent the tread pattern shape mathematically. This study aims to propose a valuable means of designing a tread pattern. The basis vector method [32; 37] may be suitable because the optimized shape could be determined by finding the best possible combination of basis vectors. When using the basis vector method, it is necessary to use several slightly different shapes, called basis vectors. The synthetic shape is generated in terms of a linear combination of vectors as follows [33; 37]:  
\[G=G_{0}+\sum\limits_{i=1}^{n}w_{i}(G_{i}-G_{0}) \tag{10}\]  
where \(G\) and \(G_{0}\) are the synthetic and original shapes, respectively, \(G_{i}\) is the i-th basis vector, and \(w_{i}\) is the weighting factor. Different shapes and their weighting factors are illustrated in Figure 2b. Figure 2: An example of shape generation process by basis vector method. (**a**) Original shape and basis vectors, (**b**) synthetic shape at different weighting factors. I, II, III are three shape change vectors. As illustrated in Figure 2, the principle of the basis vector method is elucidated. Figure 2a shows the original shape and the basis vector. As the initial shape is a rectangle, base shape I is generated by offsetting the upper edge by \(1/2\) in the lateral direction, and base shape II is generated by offsetting the left node of the upper boundary along the lateral direction. By reducing the upper-end edge of the shape by one-half along the symmetry axis, base shape III is generated. Subsequently, the coordinates of the four shapes are arranged in the same order, represented by vectors \(G_{0}\), \(G_{1}\), \(G_{2}\), and \(G_{3}\). \(G_{0}\) is the original vector, while \(G_{1}\), \(G_{2}\), and \(G_{3}\) are the three basis vectors. To generate a new shape, each coordinate of the shape change vector, \(G_{1}-G_{0}\), \(G_{2}-G_{0}\), and \(G_{3}-G_{0}\), is multiplied by a weighting factor: \(w_{1}\), \(w_{2}\), and \(w_{3}\). The new node coordinate vector is then determined as follows:  
\[G=G_{0}+w_{1}(G_{1}-G_{0})+w_{2}(G_{2}-G_{0})+w_{3}(G_{3}-G_{0}) \tag{11}\]  
By changing the design variables \(w_{1}\), \(w_{2}\), and \(w_{3}\), the corresponding node coordinates of the new shape can be calculated directly using Equation (11). Considering that the design variables \(w_{1}\), \(w_{2}\), and \(w_{3}\) are continuous, the number of shapes can be imagined to be infinite. In general, the process of basis vector generation involves the following three steps:  
1. Create a basis vector \(G_{i}\) based on the original design shape \(G_{0}\). 2. Evaluate the correlation between the two shape change vectors \(v_{i}\) and \(v_{j}\). Once the base vector is determined, the MAC (Modal Assurance Criterion) value should be calculated [37]: \[MAC(v_{i},v_{j})=\frac{(v_{i}^{T}v_{j})^{2}}{(v_{i}^{T}v_{j})(v_{i}^{T}v_{j})}\] (12) where \(v_{i}\) is the shape change vectors of two shapes: \(v_{1}=G_{1}-G_{0}\). The MAC represents a normalized vector dot product with values between 0 and 1. If the MAC = 1, this indicates no correlation between the two vectors. If the MAC = 0, this means that there is a proportional relation, \(v_{1}=Kv_{j}\), between the shape change vectors \(v_{i}\where \(F_{i}(X)\) and \(g_{i}(X)\) are the objective and constraint functions. \(X_{i}^{L}\) and \(X_{i}^{U}\) are the lower and upper bound, respectively, of each design variable.