選擇性雷射熔融製程是一項快速發展的技術，但在廣泛的工業用途上，我們仍對其缺乏相關知識與管理方法。由於在此製程中，會發生許多複雜的物理現象以及諸多參數的調控，而造成模擬的成本過高且費時。因此為了精確有效地模擬316L不鏽鋼金屬粉末在選擇性雷射熔融製程下的熔池尺寸和溫度分布，本文引入了三維架構下之徑向基有限差分無網格法的模型。
徑向基有限差分是一種新穎的無網格法，並且具有高度可編程性和程式平行化的優點。因此本文透過自撰之程式，開發了應用在選擇性雷射熔融製程上之高效率計算程序。此外，為了驗證該數值方法，本文利用了一些柏松方程式來做數值試驗，包括狄利克雷邊界條件下的柏松方程、諾伊曼邊界條件下的柏松方程和圓形邊界下的柏松方程。
本文採用了兩種不同的體積移動熱源模型進行比較，分別為射線追蹤法熱源模型和雙橢球熱源模型。除此之外，基於實驗數據以及等效的材料特性，本文也考慮了隨溫度變化的材料性質以及相變化過程。在結論的部分，本文中所提出之三維徑向基有限差分無網格法模型已通過文獻比對，驗證了熔池尺寸大小的結果。至此，我們利用此無網格法模型去預測不同雷射功率與不同雷射速度下對於熔池最高溫度、深度、寬度和長度，以及溫度梯度沿深度方向變化的影響。最後，本文也對此無網格模型進行了程式平行化之性能分析，比較了前處理與主程式在多進程情況下之運行速度的提升。
本研究得出的結論是，當雷射功率從35 W增加到55 W時，射線追踪法熱源模型的熔池最高溫度、深度、寬度和長度分別增加了9.23％，6.33％，0.98％和3.06％，而雙橢球熱源模型則分別增加了8.61％，3.38％，1.42％和4.79％。至於當雷射速度從80 mm / s增加到160 mm / s時，射線追踪法熱源模型的熔池最高溫度、深度、寬度和長度分別降低了2.14％，3.37％，0.97％和增加了26.65％，而雙橢球熱源模型則分別降低了1.67％，2.15％，1.49％和增加了12.03％。

Selective laser melting (SLM) is a fast-growing technology which still lacks knowledge and management for wider industrial use. Due to the complexity of phenomena and the various selection of parameters that take place in the SLM process, the simulation is costly and time-consuming. Thus, a three-dimensional radial basis function-finite difference (RBF-FD) meshless model is introduced in this paper in order to accurately and efficiently simulate the molten pool size and temperature distribution during the SLM process of material stainless steel 316L.
RBF-FD is a novel meshless method, which has the advantages of high degree of programmability and parallelization. We have developed the computationally efficient procedures on SLM process which is implemented by self-written program. To verify the numerical method, this paper studied several Poisson equations for the numerical tests, including Dirichlet boundary conditions, Neumann boundary conditions and the problems with circular boundary.
In this paper, two different volumetric moving heat source models are presented, namely ray-tracing method heat source model and double-ellipsoidal shape heat source model. Besides, the temperature-dependent material properties and phase change process are also considered based on the experiments and effective models. The results of models for the molten pool size are validated with the literature. The models are then used to predict the effect of different laser power and scan speed on maximum temperature, molten pool depth, width and length, and temperature gradient along depth direction. Finally, the parallelization performance of the RBF-FD model is introduced in the pre-processing part and main program part as well.
The present study conclude that as the laser power increases from 35 W to 55 W, the maximum temperature, the depth, the width and the length of molten pool increase by 9.23%, 6.33%, 0.98% and 3.06% respectively for ray-tracing method heat source model while these values increase by 8.61%, 3.38%, 1.42% and 4.79%, respectively for double-ellipsoidal shape heat source model. However, as the scan speed increases from 80 mm/s to 160 mm/s, the maximum temperature, the depth, the width and the length of molten pool decrease by 2.14%, 3.37%, 0.97%, and increase by 26.85%, respectively for ray-tracing method heat source model while these values decrease by 1.67%, 2.15%, 1.49%, and increase by 12.03%, respectively for double-ellipsoidal shape heat source model.

[1] ISO/ASTM 52900:2015, Additive Manufacturing – General Principles – Terminology, International Organization for Standardization, Geneva, Switzerland
[2] Jesse Maida (2020), Metal Additive Manufacturing Market by Application and Geography - Forecast and Analysis 2020-2024, Technavio Research
[3] Additive Manufacturing Research Group, The 7 Categories of Additive Manufacturing, Mechanical and Manufacturing Engineering, Loughborough University, UK. Retrieved from https://www.lboro.ac.uk/research/amrg/about/
[4] Di Wang, Shibiao Wu, Fan Fu, Shuzhen Mai, Yongqiang Yang, Yang Liu and Changhui Song (2017), “Mechanisms and characteristics of spatter generation in SLM processing and its effect on the properties”, Materials and Design, 117(5), pp.121-130
[5] J.-P. Kruth, G. Levy, F. Klocke and T. H. C. Childs (2007), “Consolidation phenomena in laser and powder-bed based layered manufacturing”, CIRP Annals, 56(2), pp.730-759
[6] F. Verhaeghe, T. Craeghs, J. Heulens and L. Pandelaers (2009), “A pragmatic model for selective laser melting with evaporation”, Acta Materialia, 57(20), pp.6006-6012
[7] C. D. Boley, S. A. Khairallah and A. M. Rubenchik (2015), “Calculation of laser absorption by metal powders in additive manufacturing”, Applied Optics, 54(9), pp.2477-2482
[8] M. Badrossamay and T. H. C. Childs (2007), “Further studies in selective laser melting of stainless and tool steel powders”, International Journal of Machine Tools and Manufacture, 47(5), pp.779-784
[9] Gibson, I., Rosen, D.W., Stucker, B., (2010), Additive Manufacturing Technologies, Springer.
[10] M. Matsumoto, M. Shiomi, K. Osakada and F. Abe (2002), “Finite element analysis of single layer forming on metallic powder bed in rapid prototyping by selective laser processing”, International Journal of Machine Tools and Manufacture, 42(1), pp.61-67
[11] A. V. Gusarov, I. Yadroitsev, Ph. Bertrand and I. Smurov (2009), “Model of Radiation and Heat Transfer in Laser-Powder Interaction Zone at Selective Laser Melting”, Journal of Heat Transfer, 131(7):072101
[12] I. A. Roberts, C. J. Wang, R. Esterlein, M. Stanford and D. J. Mynors (2009), “A three-dimensional finite element analysis of the temperature field during laser melting of metal powders in additive layer manufacturing”, International Journal of Machine Tools and Manufacture, 49(12-13), pp.916-923
[13] Ahmed Hussein, Liang Hao, Chunze Yan and Richard Everson (2013), “Finite element simulation of the temperature and stress fields in single layers built without-support in selective laser melting”, Materials & Design (1980-2015), 52, pp.638-647
[14] Donghua Dai and Dongdong Gu (2015), “Tailoring surface quality through mass and momentum transfer modeling using a volume of fluid method in selective laser melting of TiC/AlSi10Mg powder”, International Journal of Machine Tools and Manufacture, 88(1), pp.95-107
[15] Saad A. Khairallah, Andrew T. Anderson, Alexander Rubenchik and Wayne E. King (2016), “Laser powder-bed fusion additive manufacturing: Physics of complex melt flow and formation mechanisms of pores, spatter, and denudation zones”, Acta Materialia, 108(15), pp.36-45
[16] Ali Foroozmeha, Mohsen Badrossamay, Ehsan Foroozmehr and Sa’id Golabi (2016), “Finite Element Simulation of Selective Laser Melting process considering Optical Penetration Depth of laser in powder bed”, Materials and Design, 89(5), pp.255-263
[17] Hong-Chuong Tran and Yu-Lung Loa (2018), “Heat transfer simulations of selective laser melting process based on volumetric heat source with powder size consideration”, Journal of Materials Processing Technology, 255(1), pp.411-425
[18] C. Teng, H. Gong, A. Szabo, J.J.S. Dilip, K. Ashby, S. Zhang, N. Patil, D. Pal and B. Stucker (2017), “Simulating melt pool shape and lack of fusion porosity for selective laser melting of cobalt chromium components”, Journal of Manufacturing Science and Engineering, 139(1):11009
[19] Wayne E. King, Holly D. Barth, Victor M. Castillo, Gilbert F. Gallegos, John W. Gibbs, Douglas E. Hahn, Chandrika Kamath and Alexander M. Rubenchik (2014), “Observation of keyhole-mode laser melting in laser powder-bed fusion additive manufacturing”, Journal of Materials Processing Technology, 214(12), pp.2915-2925
[20] Ruidi Li, Jinhui Liu, Yusheng Shi, Li Wang and Wei Jiang (2012), “Balling behavior of stainless steel and nickel powder during selective laser melting process”, The International Journal of Advanced Manufacturing Technology, 59(9-12), pp.1025-1035
[21] Damien Buchbinder, Wilhelm Meiners, Norbert Pirch, and Konrad Wissenbach (2014), “Investigation on reducing distortion by preheating during manufacture of aluminum components using selective laser melting”, Journal of Laser Application, 26(1):012004
[22] K. Dai and L. Shaw (2005), “Finite element analysis of the effect of volume shrinkage during laser densification”, Acta Materialia, 53(18), pp.4743-4754
[23] L.B. Lucy (1977), “A numerical approach to the testing of the fission hypothesis”, Astronomical Journal, 82, pp.1013-1024
[24] R.A. Gingold and J.J. Monaghan (1977), “Smoothed particle hydrodynamics: theory and application to non-spherical stars”, Monthly Notices of the Royal Astronomical Society, 181(3), pp.375-389
[25] B. Nayroles, G. Touzot and Pierre Villon (1992), “Generalizing the finite element method: diffuse approximation and diffuse elements”, Computational Mechanics, 10, pp.307-318
[26] T. Belytschko, Y. Y. Lu and L. Gu (1994), “Element‐free Galerkin methods”, International Journal for Numerical Methods in Engineering, 37(2), pp.229–256
[27] Wing Kam Liu, Sukky Jun and Yi Fei Zhang (1995), “Reproducing kernel particle methods”, International Journal for Numerical Methods in Fluids, 20(8-9), pp.1081-1106
[28] Armando Duarte, C. and Tinsley Oden, J. (1996), “H-p Clouds - An h-p Meshless Method”, Numerical Methods for Partial Differential Equations, 12(6), pp.673-705
[29] S. N. Atluri and T. Zhu (1998), “A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics”, Computational Mechanics, 22, pp.117-127
[30] G.R. Liu and Y. T. Gu (2001), “A point interpolation method for two‐dimensional solids”, International Journal for Numerical Methods in Engineering, 50(4), pp.937–951
[31] G.R. Liu, L. Yan, J.G. Wang and Y.T. Gu (2002), “Point interpolation method based on local residual formulation using radial basis functions”, Structural Engineering and Mechanics, 14(6), pp.713-732
[32] E.J. Kansa (1990), “Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates”, Computers and Mathematics with Applications, 19(8-9), pp.127-145
[33] E.J. Kansa (1990), “Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations”, Computers and Mathematics with Applications, 19(8-9), pp.147-161
[34] Rolland L. Hardy (1971), “Multiquadric equations of topography and other irregular surfaces”, Journal of Geophysical Research, 76(9), pp.1905-1915
[35] Richard Franke (1982), “Scattered data interpolation: tests of some methods”, Mathematics of Computation, 38, pp.181-200
[36] Y.C. Hon and X.Z. Mao (1998), “An efficient numerical scheme for Burgers' equation”, Applied Mathematics and Computation, 95(1), pp.37-50
[37] C. Franke and R. Schaback (1998), “Solving partial differential equations by collocation using radial basis functions”, Applied Mathematics and Computation, 93(1), pp.73-82
[38] E. Larsson and B. Fornberg (2003), “A numerical study of some radial basis function based solution methods for elliptic PDEs”, Computers & Mathematics with Applications, 46(5-6), pp.891-902
[39] A.I. Tolstykh and D. A. Shirobokov (2003), “On using radial basis functions in a “finite difference mode” with applications to elasticity problems”, Computational Mechanics, 33(1), pp.68-79
[40] Shu, C. (2003), “Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations”, Computer Methods in Applied Mechanics and Engineering, 192(7-8), pp.941-954
[41] Grady B. Wright and Bengt Fornberg (2006), “Scattered Node Compact Finite Difference-Type Formulas Generated from Radial Basis Functions”, Journal of Computational Physics, 212(1), pp.99-123
[42] G. Chandhini and Y. V. S. S. Sanyasiraju (2007), “Local RBF‐FD solutions for steady convection–diffusion problems”, International Journal for Numerical Methods in Engineering, 72(3), pp.352-378
[43] David Stevens, Henry Power, Michael Lees and Herve Morvan (2009), “The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems”, Journal of Computational Physics, 228(12), pp.4606-4624
[44] Varun Shankar, Grady B. Wright, Robert M. Kirby and Aaron L. Fogelson (2015), “A Radial Basis Function (RBF)-Finite Difference (FD) Method for Diffusion and Reaction–Diffusion Equations on Surfaces”, Journal of Scientific Computing, 63(3), pp.745-768
[45] Varun Shankar (2017), “The overlapped radial basis function-finite difference (RBF-FD) method: A generalization of RBF-FD”, Journal of Computational Physics, 342, pp.211-228
[46] Eduardo Divo and Alain J. Kassab (2007), “Localized Meshless Modeling of Natural-Convective Viscous Flows”, Numerical Heat Transfer, Part B: Fundamentals, 53(6), pp.487-509
[47] Eduardo Divo and Alain J. Kassab (2007), “An Efficient Localized Radial Basis Function Meshless Method for Fluid Flow and Conjugate Heat Transfer”, Journal of Heat Transfer, 129(2), pp.124-136
[48] Gregor Kosec and Bozidar Sarler (2013), “Solution of a low Prandtl number natural convection benchmark by a local meshless method”, International Journal of Numerical Methods for Heat and Fluid Flow, 23(1), pp.189-204
[49] P. P. Chinchapatnam (2009), “A compact RBF-FD based meshless method for the incompressible Navier—Stokes equations”, Proceedings of the Institution of Mechanical Engineers, Part M: Journal of Engineering for the Maritime Environment, 223(3), pp.275-290
[50] Zhi Heng Wang, Zhu Huang, Wei Zhang and Guang Xi (2014), “A Meshless Local Radial Basis Function Method for Two-Dimensional Incompressible Navier-Stokes Equations”, Numerical Heat Transfer, Part B: Fundamentals, 67(4), pp.320-337
[51] Flyer, N., Lehto, E., Blaise, S., Wright, G. B., and St-Cyr, A. (2012), “A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere”, Journal of Computational Physics, 231(11), pp.4078-4095
[52] B. Fornberg and G. Wright (2004), “Stable computation of multiquadric interpolants for all values of the shape parameter”, Computers and Mathematics with Applications, 48(5-6), pp.853-867
[53] Bengt Fornberg and Cécile Piret (2007), “A Stable Algorithm for Flat Radial Basis Functions on a Sphere”, SIAM Journal on Scientific Computing, 30(1), pp.60-80
[54] Bengt Fornberg, Elisabeth Larsson and Natasha Flyer (2011), “Stable Computations with Gaussian Radial Basis Functions”, SIAM Journal on Scientific Computing, 33(2), pp.869-892
[55] Bengt Fornberg, Erik Lehto and Collin Powell (2013), “Stable calculation of Gaussian-based RBF-FD stencils”, Computers & Mathematics with Applications, 65(4), pp.627-637
[56] Goldak John, Chakravarti Aditya and Bibby Malcolm (1984), “A new finite element model for welding heat sources”, Metallurgical Transactions B, 15(2), pp.299-305
[57] A.V. Gusarova, T. Laouib, L. Froyenc and V.I. Titova (2003), “Contact thermal conductivity of a powder bed in selective laser sintering”, International Journal of Heat and Mass Transfer, 46(6), pp.1103-1109
[58] Hodge, N. E., Ferencz, R. M. and Solberg, J. M. (2014), “Implementation of a thermomechanical model for the simulation of selective laser melting”, Computational Mechanics, 54(1), pp.33-51