## 1 Introduction
According to Archimedes' principle, the relative density can be calculated as follows [23]:  
\[\rho=m_{in\ air}\frac{\rho_{liquid}}{m_{in\ air}-m_{in\ liquid}}, \tag{2}\]  
where \(\rho_{liquid}\) is the density of the liquid generating buoyancy, \(m_{in\ air}\) is the mass of the sample in the air, \(m_{in\ liquid}\) is the mass of the sample in liquid. ASTM Standard B311-13 uses Equation (2), which does not consider the effect of open cavities. In order to consider open cavities in the sample, Equation (2) needs to be re-written as Equation (3) [24],  
\[\rho=m_{in\ air}\frac{\rho_{liquid}}{m_{wet}-m_{in\ liquid}}, \tag{3}\]  
where \(m_{wet}\) is the mass of the liquid-impregnated sample in air, \(m_{in\ liquid}\) is the mass of the liquid-impregnated sample in liquid. Here, the liquid impregnating of the sample can be the same or different from the liquid generating buoyancy. According to the ASTM Standard B962-13 [24], the liquid for liquid-impregnation should be the oil with a viscosity of 20 to 60 cSt. Finally, the relative density \(\rho_{relative}\) in the current work was calculated as follows:  
\[\rho_{relative}\left(\%\right)=\frac{\rho}{\rho_{theoretical}}\times 100, \tag{4}\]  
where \(\rho\) is the actual density of the sample; for AlSi10 Mg, theoretical density \(\rho_{theoretical}\) is 2.68 g/cm\({}^{3}\). Some papers reported the optimization of SLM parameters in terms of relative density using the Archimedes method [7; 25], X-ray scanning [13], or image analysis (micrograph of a cross-section) [3; 16]. A comparison of the Archimedes method and image analysis showed that for high part densities (\(\geq\)98%), the significant differences are within about 1%. [13] X-ray scanning can deliver 3D information of the internal imperfections of a sample, such as distributions of porosities, but the accuracy of this method depends on the resolutions of scanning. High resolutions result in long computing time. Compared with image analysis, the Archimedes method would deliver more reliable results, as the whole sample volume is taken into account instead of some single cross-sections. However, image analysis can reflect the effect of unmelted powder on the results of relative density, as the cavities containing unmelted powder can be detected by macrograph of a cross-section. These cavities have a significant influence on the results of relative density, especially for the sample with a low density. Design of experiment (DOE) reduces the number of experiments to be conducted, and hence is labor and cost effective and, at the same time, provides statistically reliable results [26; 27; 28; 29]. Many researchers use DOE to optimize powder melting processes in SLM technology [2; 25; 27; 30]. DOE can be categorized three classes of full factorials, fractional factorials, and response surface methodology (RSM). RSM is suitable for experiments of three or less than three factors. Doehlert design presents advantages over other DOE for response surface methodology, such as central composite designs and Box-Behnken designs [31], as it requires fewer experiments, which are easier and more efficient. Doehlert matrixes are easily expanded in both the variables space and the experimental space [26]. The combined effects were represented by means of a polynomial model [27] for each parameter. The surface methodology was then applied to represent these effects. The Doehlert design was applied to optimize the SLM processing parameters of Ni-based superalloy [22], however, it was not applied to the optimization of Al-Si-Mg alloy in past reports. Some papers reported the optimization of SLM processing parameters in terms of relative density measured by the Archimedes method [7; 25], however, detailed measuring procedures were not provided considering the difference of relative density. In the most recent studies focusing on optimizing the SLM process parameters [17; 18], the laser power reached less than 400 W because of the limitation of equipment.