# Topology Optimization Considering Porosity Defects in Metal Additive Manufacturing
## 1 Introduction
[41] designed a real-time molten pool monitoring system which consists of an optical sensor, data processing, reference database, and quality evaluation. In this system, the sensor collects the melt pool information in the form of an optical signal, and then transmits it to the data processing module to establish the melt pool image. Then, the location and size of the defects are determined by comparing with the reference database. Finally, the process parameters are optimized by the feedback system to reduce the porosity. Although the proposed methods can effectively reduce the porosity defects, they cannot completely remove the porosity defects. It is well known that stress concentration occurs near the defects of the fabricated part. Under the external load, the stress near the defects will increase significantly. When the ultimate strength of the material is exceeded, cracks are generated and gradually expand until fracture failure. Thus, the existence of porosity has a significant impact on the mechanical properties of the fabricated parts. Moreover, the pores not only are the source of fatigue cracks but also accelerate the expansion of fatigue cracks under the stress concentration, greatly reducing the fatigue life of the fabricated part. The more pores in the fabricated parts, the more cracks may occur. It is difficult to satisfy the requirements with the existence of porosity, which restricts the usability of the fabricated parts. According to our investigation, few researchers consider the porosity defects of the AM in the topology optimization problem. In this paper, the concept of topological sensitivity is employed to construct a porosity constraint function in the form of domain integral and a level set topology optimization model considering the effects of porosity is proposed. To improve the robustness of the algorithm, the topology optimization model is solved by two steps. At first, the classical level set method is used to initially optimize the structure. Secondly, the porosity is taken into topology optimization of the structure after the initial optimization, and a robust structure that minimizes the impact of porosity is obtained. Since the reinitialization retains the properties of the signed distance function, the holes cannot be automatically nucleated in the solid design domain, i.e., the topology change can only rely on the existing initial structural boundary. In the past few decades,some solutions have been proposed for hole nucleation. [PERSON] et al. [42] proposed a level set topology optimization method that inserts holes according to the value of topological sensitivity. [PERSON] et al. [43] added topological sensitivity as a diffusive term of the Hamilton-Jacobi equation, so that new holes were automatically nucleated during the optimization process. [PERSON] et al. [44] proposed a hole nucleation method in combination with bi-directional evolutionary structural optimization (BESO) based on the material removal scheme. [PERSON] et al. [45] proposed a filtering level set method to nucleate holes and avoid re-initialization of level set function. Although the above hole nucleation method makes the level set method unnecessary to guess the initial structure, the effect of hole nucleation is not ideal for various reasons. Therefore, it is necessary to propose a new hole nucleation method. In the present work, a new level set hole nucleation method is proposed by combining the BESO method with topological sensitivity. This method can automatically find and nucleate holes within the design domain. The rest of this paper is organized as follows: Section 2 reviews the level set topology optimization methods. Section 3 describes the hole nucleation method combining BESO and topological sensitivity. In Section 4, a level set topology optimization model considering porosity defects is formulated, and the shape sensitivity of the constraint function is calculated. Section 5 lists the optimization procedure. Section 6 gives numerical examples and discussions.