Nonlinear thermal simulation of laser metal deposition

ABSTRACT

Simulation of Laser Metal Deposition (LMD) is central to the planning of Additive Manufacturing processes. This manuscript presents the computational implementation of a 2D-plus-thickness nonlinear thermal simulation of LMD, which considers: (i) temperature-dependent material properties, (ii) heat losses due to convection and radiation, (iii) geometrical update during material deposition, (iv) phase change and (v) the interaction between the laser and the substrate. The implementation computes the history of the temperature field at a cross-cut normal to the laser trajectory and the history of the bead accumulation. The material deposition model is based on the spatial distribution of the delivered powder. This manuscript presents the mathematical and numerical foundations to execute an efficient local re-meshing of the growing bead. The numerical estimation of the bead geometry is compared with experimental results found in the existing literature. The present model shows reasonable accuracy to predict the bead width ($15\mathrm{\%}$ error) and bead height ($22\mathrm{\%}$ error). This implementation is an in-house one, which allows for the inclusion of additional physical effects. Additional work is needed to account for the particle (thermo) dynamics over the substrate, responsible for a significant material and energy waste, which in turn leads to the actual temperature and molten depth being over-estimated in the executed simulations.

KEYWORDS:

• Laser metal deposition
• additive manufacturing
• nonlinear finite element method

Glossary

AM	=	Additive manufacturing.
FEA	=	Finite element analysis.
FEM	=	Finite element method.
LMD	=	Laser metal deposition.
PL	=	Piecewise linear.
$\mathrm{\Omega }\subset {\mathbb{R}}^2$	=	2D-plus-thickness FEA domain.
$\mathrm{\partial }\mathrm{\Omega }$	=	1D border of domain $\mathrm{\Omega }$.
$M=(V,T)$	=	FEA triangular mesh defined by the set of nodes $V$ and the set of triangles $T$.
$\mathrm{\Delta }z\in \mathbb{R}$	=	Thickness of the domain $\mathrm{\Omega }$ [mm]
$\mathrm{\Delta }t\in \mathbb{R}$	=	Time increment for the FEM simulation [s]
$T(\mathbf{x},t):{\mathbb{R}}^2\times \mathbb{R}\to \mathbb{R}$	=	Temperature at $\mathbf{x}\in \mathrm{\Omega }$ in the instant $t$ [K]
$T^h(\mathbf{x},t):{\mathbb{R}}^2\times \mathbb{R}\to \mathbb{R}$	=	Approximated temperature function at $\mathbf{x}\in \mathrm{\Omega }$ in the instant $t$ [K]. It is the result given by the finite element method.
$\theta (t):\mathbb{R}\to {\mathbb{R}}^{N_{nodes}}$	=	Global vector of nodal temperatures at time $t$ [K].
${\theta }^s\in {\mathbb{R}}^{N_{nodes}}$	=	Global vector of nodal temperatures at time $t^s$ [K].
${\theta }^{s,k}\in {\mathbb{R}}^{N_{nodes}}$	=	Global vector of nodal temperatures at time $t^s$ and iteration $k$ in the Newton-Raphson scheme [K].
$\mathbb{M}(\mathbf{x},t):{\mathbb{R}}^2\times \mathbb{R}\to {\mathbb{R}}^{N_{nodes}}\times {\mathbb{R}}^{N_{nodes}}$	=	Global mass matrix in the FEM formulation [J/K].
${\mathbb{M}}^s(\mathbf{x},t):{\mathbb{R}}^2\times \mathbb{R}\to {\mathbb{R}}^{N_{nodes}}\times {\mathbb{R}}^{N_{nodes}}$	=	Global mass matrix at time $t^s$ [J/K].
$\mathbb{K}(\mathbf{x},t):{\mathbb{R}}^2\times \mathbb{R}\to {\mathbb{R}}^{N_{nodes}}\times {\mathbb{R}}^{N_{nodes}}$	=	Global conductivity matrix in the FEM formulation [W/K].
${\mathbb{K}}^s(\mathbf{x},t):{\mathbb{R}}^2\times \mathbb{R}\to {\mathbb{R}}^{N_{nodes}}\times {\mathbb{R}}^{N_{nodes}}$	=	Global conductivity matrix at time $t^s$ [J/K].
$\mathbf{f}(\mathbf{x},t):{\mathbb{R}}^2\times \mathbb{R}\to {\mathbb{R}}^{N_{nodes}}$	=	Global force vector in the FEM formulation [W].
${\mathbf{f}}^s(\mathbf{x},t):{\mathbb{R}}^2\times \mathbb{R}\to {\mathbb{R}}^{N_{nodes}}$	=	Global force vector at time $t^s$ [W].
$T^0(\mathbf{x}):{\mathbb{R}}^2\to \mathbb{R}$	=	Initial (at $t=0$) temperature at $\mathbf{x}$ [K]
$\bar{T}(\mathbf{x},t):{\mathbb{R}}^2\times \mathbb{R}\to \mathbb{R}$	=	Temperature function at the region with Dirichlet boundary conditions [K]
$\bar{q}(\mathbf{x},t):{\mathbb{R}}^2\times \mathbb{R}\to \mathbb{R}$	=	Heat flux function at the region with Neumann boundary conditions [W/m2]
$\mathbf{R}$	=	System of nonlinear equations associated to the semi-discrete FEM formulation.
$\mathbf{q}(\mathbf{x},t):{\mathbb{R}}^2\times \mathbb{R}\to {\mathbb{R}}^2$	=	Heat flux into or out of the medium at $\mathbf{x}\in \mathrm{\Omega }$ at time $t$ [W/m2]
$s(\mathbf{x},t):{\mathbb{R}}^2\times \mathbb{R}\to \mathbb{R}$	=	Volumetric heat sources at $\mathbf{x}\in \mathrm{\Omega }$ in the instant $t$ [W/m3]
$\mathbf{n}(\mathbf{x}):{\mathbb{R}}^2\to {\mathbb{R}}^2$	=	Outward unit normal to the boundary at $\mathbf{x}\in \mathrm{\Omega }$
$\rho (T):\mathbb{R}\to \mathbb{R}$	=	Density of the material [kg/m3] as a function of the temperature.
$\kappa (T):\mathbb{R}\to \mathbb{R}$	=	Thermal conductivity of the material [W/(m K)] as a function of the temperature.
$C(T):\mathbb{R}\to \mathbb{R}$	=	Specific heat capacity of the material [J/(kg K)] as function of the temperature.
$C_{\mathrm{e}\mathrm{q}}(T):\mathbb{R}\to \mathbb{R}$	=	Equivalent specific heat capacity of the material [J/(kg K)], used to incorporate phase change into the simulation.
$N_1(\xi ,\eta ),N_2(\xi ,\eta ),N_3(\xi ,\eta ):{\mathbb{R}}^2\to \mathbb{R}$	=	Shape functions in the FEM formulation for 3-node triangular elements. Interpolation functions inside the triangular elements.
$N_1^q(\xi ),N_2^q(\xi ):\mathbb{R}\to \mathbb{R}$	=	Shape functions in the FEM formulation for the edges of 3-node triangular elements. Interpolation functions along the edges with Neumann boundary conditions.
$I(x,z,t):{\mathbb{R}}^2\times \mathbb{R}\to \mathbb{R}$	=	Laser energy intensity distribution [W/mm2]
$P_L\in \mathbb{R}$	=	Laser nominal power [W]
$R_L\in \mathbb{R}$	=	Laser beam radius [W]
$[T_s,T_l]\subset \mathbb{R}$	=	Melting range of the material [K].
$\mu \in \mathbb{R}$	=	Material flow rate [kg/s]
$f(x,z,t):{\mathbb{R}}^2\times \mathbb{R}\to \mathbb{R}$	=	Powder particle distribution projected by the nozzle onto the substrate at time $t$ [kg/(mm2 s)].
$H(x,z,t):{\mathbb{R}}^2\times \mathbb{R}\to \mathbb{R}$	=	Height of the bead at time $t$ [mm].
$L\in \mathbb{R}$	=	Latent heat of fusion [J/kg].
$h_c\in \mathbb{R}$	=	Convection coefficient [W/(m2 K)].
$\epsilon \in \mathbb{R}$	=	Material thermal emissivity.
$\sigma \in \mathbb{R}$	=	Stefan-Boltzmann constant [W/(m2 K4)].
$T_{\mathrm{\infty }}\in \mathbb{R}$	=	Ambient temperature [K].

Data Availability Statement

The authors declare that the data supporting the findings of this study are available within the article. Additional data is not provided due to industrial confidentiality restrictions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work has received funding from the Eusko Jaurlaritza/Basque Government under the grants KK-2018/00115(ADDISEND) and KK-2018/00071 (LANGILEOK).

Notes on contributors

Diego Montoya-Zapata

Diego Montoya-Zapata obtained a B.Sc. degree in Mathematical Engineering (2016) and a M.Sc. degree in Engineering (2018) from Universidad EAFIT, Colombia. In 2016, Diego Montoya-Zapata joined the Laboratory of CAD CAM CAE at Universidad EAFIT under the supervision of Prof. Oscar Ruiz-Salguero. In 2018, Diego Montoya-Zapata started a research internship in Vicomtech, Spain. In 2019, he enrolled in the Ph.D. program in Engineering at Universidad EAFIT under the direction of Prof. Ruiz-Salguero (Universidad EAFIT) and Dr. Ing. Jorge Posada (Vicomtech Spain) on the topic Computational Geometry Applied to Additive Manufacturing. His main research interests are applied computational geometry and computational mechanics.

Juan M. Rodríguez

Juan M. Rodríguez is full time Professor in the Mechanical Engineering Department at EAFIT University. He received a B.Sc in Mechanical Enginering at Los Andes University; and M.Sc. in Engineering at Los Andes University and a Ph.D in Computational Mechanics at Universidad Politécnica de Cataluña. His current research interest includes computational solid mechanics, computational contact mechanics and particle-based methods.

Aitor Moreno

Aitor Moreno received a Ph.D degree in Computer Science in 2013 from the University of the Basque Country. His thesis titled “Urban and forest fire propagation and extinguishment in Virtual Reality training scenarios” received the “Cum Laude” designation. In 2002, he received a degree in Computer Science from the University of the Basque Country. In 2002, Aitor Moreno joined Vicomtech (http://www.vicomtech.org) as a full-time researcher. From 2002 to 2013, he collaborated in the Intelligent Transport Systems and Engineering department. After 2013, he is a senior researcher in the Industry and Advanced Manufacturing department. He is involved in several regional, national and international projects focusing mainly on Computer Graphics, VR and interactive simulations. He has published several papers in international conferences and journals. His main interests are Computer Graphics, Simulation, Virtual Reality and advanced interaction techniques (Human-Computer Interaction).

Oscar Ruiz-Salguero

Prof. Oscar Ruiz - Salguero (born Tunja, Colombia) earned B.Sc. degrees in Mechanical Engineering and Computer Science (1983, 1987) at Los Andes University, Colombia, a M.Sc. degree (1991) and a Ph.D. (1995) from the Mechanical Eng. Dept. of University of Illinois at Urbana- Champaign, USA. He has been Visiting Researcher at Ford Motor Co. (USA), Fraunhofer Inst. for Computer Graphics (Germany), University of Vigo (Spain), Max Planck Inst. for Informatik (Germany) and Purdue University (USA). Prof. Ruiz-Salguero's CV is ranked in the Official top quartile (Senior Investigator rank) in Colombia. He coordinates the CAD CAM CAE Laboratory at U. EAFIT. His research and teaching interests are Computational Geometry applied on Design, Manufacturing, Medicine, etc. A summary of Prof. Ruiz - Salguero research group highlights and publications may be found in www1.eafit.edu.co/cadcamcae

Jorge Posada

Jorge Posada is Scientific and Associate Director of Vicomtech. He holds a Ph.D. in Engineering and Computer Science from the Technische Universität Darmstadt (Germany), and an Executive MBA from IE Business School. His research lines include Visual Computing, Digital Manufacturing, Industry 4.0, Knowledge Engineering and other related fields.