## 2 Theoretical Foundation
### Topology Optimization  
This study employed a topology optimization method using the density function approach, wherein material density is treated as a design variable in structural design. A relationship between the material density and Young's modulus is established for each finite element. The density function approach, based on material interpolation, involves penalty factors and can be further categorized as solid isotropic material with penalization (SIMP) and rational approximation of material properties (RAMP). Therefore, this method is also referred to as the penalty function-based density function approach. The ANSYS  
Figure 2: Schematic diagram for ultrasonic-assisted cutting. Workbench R18.1 software can automatically determine which interpolation method to use, with SIMP as shown in Equation (1) and RAMP as shown in Equation (2). \[E=\rho^{P}E_{0} \tag{1}\]  
\[E=E_{0}\frac{\rho}{1+P(1-\rho)} \tag{2}\]  
where \(\rho\) is the specific density value of the element ranges from 0 to 1, and 0 indicates the absence of the element while 1 indicates its presence. \(E_{0}\) is the real Young's modulus for the materials. \(E\) is used to determine the Young's modulus during the topology optimization process. \(P\) as a penalty factor, it typically ranges from 2 to 4, with a default value of 3 in the software. Based on the above information, it can be observed that when the penalty coefficient is greater than one, elements with lower specific density undergo an exponential operation, causing their E-values to decrease, facilitating their removal from the system. Ultimately, this allows for the optimization of the topological shape (model) based on different density distributions. SIMP and RAMP methods primarily involve the steps of setting design variables and defining objective functions in the context of topology optimization. Specifically: 1. design variable setting: in the SIMP method, design variables are typically associated with the material density of each element. The material density of each element serves as a design variable that can be adjusted during the optimization process. Similar to the SIMP method, in the RAMP method, design variables are also related to the material density of each element. The material density of each element is used to describe the topological distribution. 2. Objective function definition: in the SIMP method, the objective function usually involves minimizing the deformation or volume of the structure. This is achieved by adjusting the material density of each element. The definition of the objective function often includes a relationship between the density and material properties, introducing a penalization factor to control the material distribution. The objective function is related to the relationship between the density and material properties. However, as compared to the SIMP method, the RAMP method introduces rational approximation to more accurately describe the material properties. The objective function involves minimizing the deformation or volume of the structure and utilizes rational approximation of the material properties. Therefore, both methods play a crucial role in the design variable and objective function definition stages of topology optimization. The choice between these methods often depends on the specific engineering requirements and the need for accurate representation of the material properties. Structural optimization is a process of iteratively solving finite element models by continuously updating design variables and modifying the finite element model to meet the design requirements based on the constraint conditions. Topology optimization, during the analysis process, continuously modifies the material density within the optimization domain, effectively removing elements from the analysis model to achieve the optimization objectives. Optimization analysis must be conducted with well-defined objective function and constraints in place, requiring a standardized procedure for the smooth execution of the optimization analysis.