# Topology Optimization Considering Porosity Defects in Metal Additive Manufacturing
## 1 Introduction  
Topology optimization is a calculation method that achieves the optimal material configuration in the design domain according to the given boundary conditions and the load conditions. In the past decades, many effective topology optimization methods have been studied and applied, e.g., the homogenization method [1; 2; 3; 4; 5], the Solid Isotropic Material with Penalization (SIMP) method [6; 7; 8; 9], the Evolutionary Structural Optimization (ESO) method [10; 11; 12; 13; 14], the Moving Morphable Component (MMC) method [15; 16; 17; 18], and the level set method [19; 20; 21; 22; 23; 24; 25]. In particular, the level set method is introduced as an alternative new method in the structural optimization field. This method uses a higher-dimensional level set function to implicitly represent the structural boundary (zero level set) and then obtains the updated structural boundary via the evolution of the level set function during the optimization process. This implicit expression can avoid the relaxation of design variables and numerical instability. As a result, the level set topology optimization is recognized by many researchers. The studies and applications of numerous optimization methods have made the field of structural optimization to develop rapidly. For example, [PERSON] et al. [26] established a method of designing and manufacturing micro-structured ultralight components to achieve the goal of lightweight. Topology optimization achieves lightweight structure but results in a more complex structure. Complex structures cannot be manufactured easily by traditional manufacturing processes (turning, milling, casting, etc.). However, the rapidly developing Additive Manufacturing (AM) technology is a forming method in which materials are overlapped layer by layer, which fully eliminates the limitation ofgeometric complexity. AM makes it possible to manufacture complex structures derived from topology optimization and provides technical support for the application of topology optimization. On the other hand, affected by the cost of AM, how to reduce the consumption of AM has become an urgent problem to be solved. The topology optimization can effectively reduce material consumption under the condition of satisfying structural performance, which provides a solution for reducing the manufacturing cost of AM. In theory, the characteristics and advantages of both topology optimization and AM enable them to complement each other. AM has unlimited potential, but it does not have unlimited capabilities. Laser AM is based on the technical principles of point-by-point melting, line-by-line scanning, and layer-by-layer accumulating. The metal material undergoes complex thermodynamic behavior under the action of the laser, and the parts are fabricated after the cyclic process of heating, melting, cooling, and solidifying. When affected by many conditions such as laser power, scanning speed, powder materials, and temperature, defects will inevitably occur in the fabricated parts. In recent years, these defects of laser AM have been studied widely [27; 28; 29; 30; 31; 32; 33; 34; 35]. Porosity is one of the most common defects in laser metal AM. The shape of pores is relatively regular and mostly spherical, and the size is generally small (within 100 \(\upmu\)m) [36; 37; 38; 39]. Most of these defects are generated by gas remaining in the molten pool due to excessive or unstable energy input. In the process of material melting and solidifying, the gas in the molten pool does not have sufficient time to escape. Furthermore, the higher the temperature of the molten pool during the melting process, the higher the solubility of the gas in the molten pool. As the molten pool cools, the temperature and the solubility decrease, and the residual gas increases [39]. In the AM process, eliminating or suppressing porosity defects is a challenging problem. [PERSON] et al. [40] presented Epitaxial laser metal forming (E-LMF) to effectively reduce porosity defects in AM process. [PERSON] et al.