Meshfree Simulations for Additive Manufacturing Process of Metals

The motion, deformation, and temperature distribution of a material system can be determined by solving the conservation laws, including the mass conservation:the linear momentum conservation:the angular momentum conservation:and the energy conservation:where ρ and ρ₀ is the current and reference density, respectively. φ : Ω₀ × [t₀, t] → ℝ³ is a time-dependent deformation mapping, X is the coordinate, F is the deformation gradient, J = det F is the Jacobian of the deformation, P^e is the equilibrium part of the first Piola Kirchhoff stress tensor and P^v is the viscous part, B is the body force per unit mass, T is temperature, N is the entropy per unit undeformed volume, q is the heat flux applied on the surface, Q is the distributed heat source per unit mass, and Y is the driving forces for the internal variables Z.

In order to develop the numerical solutions for the conservation laws, the weak form of the governing equations can be formulated by using the variational framework proposed in Yang et al. [10]. In the HOTM method, the variational structure is extended by taking into account the inertia term. This framework describes the corresponding action of a thermomechanical system with general dissipative mechanisms, which follows:where \( K=\frac{1}{2}{\rho}_0{\left|\dot{\varphi}\right|}^2 \) is the kinetic energy, A is the Helmholtz free energy, Δ is the dissipation potential, \( \widehat{N} \) is the normal direction. The state function \( \varTheta \left(F,T,Z\right)=\frac{\partial U}{\partial N} \) is derived from the internal energy, which can be treated as internal temperature, while T is the external temperature field. At the thermal equilibrium condition, the equation Θ = T is satisfied.

The Helmholtz free energy A includes the energy stored in the material due to heat capacity, W^h, which can be described as:where C(T) is the temperature-dependent specific heat. To take into account the effect of latent heat during phase change, the apparent heat capacity method is employed, such that C(T) can be described as:where α(T) is a phase transition function introduced to describe the smooth transition between two phases. L denotes the latent heat during the phase transition, C_phase1 and C_phase2 are the heat capacity coefficient of phases 1 and 2, respectively.

The dissipation potential Δ of Eq. (5) can be decomposed into three terms:where \( G=-\frac{1}{T}{\nabla}_0T, \)ϕ^∗ is the internal dissipation potential, ψ^∗ is the viscous dissipation, and χ^∗ is the heat conduction dissipation potential.

This variational formulation works for materials with arbitrary constitutive relations including finite elastic and plastic deformation, rate-dependency, thermal softening, and strain hardening rules. Thus, the thermal and mechanical balance equations, the constitutive relations, and the equilibrium between the external temperature and the internal temperature can be obtained as the Euler-Lagrange equations by taking variations of the potential Φ and enforcing stationarity, i.e.,

Li et al [11] propose the optimal transportation meshfree (OTM) method to discretize the variational framework for adiabatic systems. The OTM method is an incremental updated Lagrangian meshfree scheme capable of solving general fluid and solid flows, possibly involving multiple phases, viscosity, and general inelastic and rate-dependent constitutive relations, arbitrary variable domains with discontinuity and boundary conditions. The OTM method is constructed through integration of optimal transportation theory with local maximum entropy (LME) meshfree approximation and material point sampling [12].

The optimal transportation variational framework results in geometrically exact updates of the local volumes and mass densities, thus bypassing the need for solving a costly Poisson’s equation for the pressure and eliminating the mass conservation errors that afflict Eulerian formulations. Furthermore, by adopting a discrete Hamilton principle based on a time-discrete action furnished by optimal transportation theory, the discrete trajectories have exact conservation properties including symplecticity, and linear and angular momentum. Material point sampling is introduced into the framework for spatial discretization. All the field information is carried by two sets of points, nodes, and material points, as shown in Fig. 1. In specific, the node x_a,k carries the kinematic information, such as displacement, velocity, and acceleration, while the local states of materials, such as strain, stress, material properties, and internal variables, are evaluated at the material point x_p,k. The material point of the OTM method is very similar to the quadrature points in the FEM method, which provides an efficient way for numerical integration without a background mesh. The nodes and material points can be initialized by taking the nodes and barycenters of a conforming finite element mesh of the computational domain. The connection between material points and nodes is initialized by the connectivity table of the finite element mesh. Later in the calculations, the connection is dynamically reconstructed by using a search algorithm based on the deformation-dependent geometrical information.

We follow the same procedure as described in [11] for the discretization of the variational structure in Eq. (5). In particular, we consider the standard Ritz-Galerkin approach to approximate the displacement, temperature, deformation gradient, and thermodynamic force field using the local maximum entropy (LME) meshfree shape functions. The LME shape functions are affine on the boundary, which enables the direct coupling of solid/liquid/gas phases. In addition, the LME shape functions have the key property of possessing a Kronecker delta property at the boundary, which overcomes a common difficulty in meshfree approximation schemes and enables direct imposition of temperature boundary conditions on the domain without special treatment. To this end, the displacement and temperature field are interpolated by:where φ_a is the nodal displacement, T_a is the nodal temperature, and N_a is the shape function of node x_a.

Finally, variations of the semi-discrete incremental potential δΦ_n are taken. The stationarity conditions yield the fully discrete mechanical and thermal balance equations as:where m_a,n + 1 denotes the lumped mass of the node x_a at t_n + 1.is a central difference approximation of the nodal acceleration.

The internal nodal force \( {f}_{a,n+1}^{int} \) and external nodal force \( {f}_{a,n+1}^{ext} \) can be obtained as:

where v_p is the volume of material point x_p.

The internal heat \( {Q}_{a,n+1}^{\mathrm{int}} \) and external heat \( {Q}_{a,n+1}^{ext} \) are defined as:where H is the outward heat flux.

A fully explicit solution for Eqs. (12) and (13) faces stability issues, and the well-known Courant-Friedrichs-Lewy (CFL) conditions restrict the time step size [13]. Alternatively, a fully implicit method involves a very expensive calculation of the tangent of the mechanical forces, which is further exacerbated in large scale three-dimensional simulations. However, the time scale for heat transfer is much larger than the one required for the wave propagation in the material. In this paper, an operator splitting approach is used to construct a recursive staggering scheme for the solution of the strong thermomechanical coupling Eqs. (12) and (13). In specific, the mechanical balance equations are solved explicitly first based on the current temperature distribution. Assuming a constant motion, the thermal balance equations are solved by an implicit method based on the Newton-Raphson solver to recompute the temperatures. The material properties are then updated according to the new resultant temperature field. The iterations increment until reaching the total simulation time.

To model the experiments conducted by the AM benchmark tests, i.e., the process of a laser scanning over an Inconel 625 bare plate, various heat flux models are introduced as the boundary conditions. First of all, the laser beam is modeled as a heat flux applied to the surface of the domain. The most widely used heat flux model in the field of AM is a Gaussian distributed heat flux with scanning velocity [14], which is given by:where A is the absorptivity of the heat flux by the AM material, P and r are the power and radius of the laser beam, respectively, and x_c(t) and y_c(t) is the current location of the center of the laser spot. The profile of the Gaussian heat flux is shown in Fig. 2. x_c(t) and y_c(t) in Eq. (19) can be described by any time-dependent functions. Therefore, it furnishes an effective means of controlling the movement and scanning strategies of the laser beam in the simulations, such as the stripe hatch, the meander hatch, and the island scan strategy. The influence of various scanning speed on the deformation of the melt pool is studied in this paper by adopting different x_c(t) and y_c(t) in the heat flux model. In addition, the shape, power, and other important processing parameters of the power beam can also be easily manipulated to study their relations to the melt pool dimensions.

The energy loss due to convective heat transfer between the plate and the atmosphere is not negligible. In our simulations, a convective heat flux is applied on the plate. As observed in experiments, this flux can cool down the melt pool. The convective heat flux takes the simple form as:where C is the convective coefficient and T₀ is the temperature of the environment.

Radiation heat flux is another boundary condition applied on the surface nodes of the domain. The Stefan Boltzmann law,is employed. Here, the Stephan’s constant σ = 5.669 × 10⁻⁸W/m²K⁴ is employed. The emissivity ε varies with temperature and surface chemistry in physical experiments. For simplicity, the average value of the emissivity is adopted for the solid state and the liquid state of the material, respectively.

To predict the stress at material points, a phase-aware thermomechanical constitutive model is developed, since the AM process involves multiphase transitions and interactions. The temperature T directly determines the local state of the material point. That being said, a material point is in solid phase with a local temperature lower than the melting temperature. As the temperature increases beyond the melting or boiling temperature, it automatically transfers to the liquid phase or gas phase, which is accomplished by using temperature-dependent material coefficients in the multiphase constitutive relations. For instance, the material model of the solid state is assumed of the thermoelastic form:where K(T) is the temperature-dependent bulk modulus, J is the Jacobian, α is the linear thermal expansion rate, ΔT is the temperature difference between the material point temperature T_p and the reference temperature, δ_ij is the Kronecker delta, μ(T) is the temperature-dependent shear modulus, and \( {\varepsilon}_{ij}^{dev} \) is the deviatoric strain. As the material point transfers to the liquid state, the viscoelastic model with Murnaghan-Tait equation of state is employed [15], i.e.,where C is a constant related the bulk modulus of the liquid, which is usually taken between 7 × 10⁶ and 7 × 10⁷ Pa to allow for compressibility, γ is a material parameter and equals to 7 for water, P₀ is the ambient pressure, η(T) is the temperature-dependent viscosity, and \( {\dot{\varepsilon}}_{ij} \) is the strain rate. The last term in Eq. (23) corresponds to the viscous stress. For liquid phase, shear modulus automatically goes to zero.

The model geometry is shown in Fig. 3a. The dimensions of the domain are 10 mm × 10 mm × 3.2 mm. The geometry is modeled by an adaptive mesh with mesh size ranged from 25 to 1000 μm, as shown in Fig. 3b, which leads to 244,624 elements. Since the HOTM method is a meshfree method, the mesh is only used to determine the location of the material points and nodes as described in the “Optimal Transportation Meshfree Implementation” section. After the initialization, the connectivity between material points and nodes will be updated automatically throughout the simulations.

The AM benchmark tests are conducted using two different machines, and the laser parameters used by each machine are different. Our simulation results are compared against experimental data measured from the AMMT machine. Three AM benchmark experiments are simulated using the HOTM method. For each case, a laser with different laser power and scanning speed is adopted. The parameters of these three types of lasers are shown in Table 1. Only one single laser scan track is simulated in each case. Particularly, the scan track at the center of the plate is 700-μm long and performs from left to right, as shown in Fig. 3a.

Due to the lack of material property data for Inconel 625 in liquid and gas state, constant heat conductivity and heat capacity are employed for all the material states of Inconel 625. The heat conductivity is set as 20 W/m K, and the heat capacity 600 J/kg K. The laser absorptivity of the material is set to be 20%, as listed in [14]. The emissivity of the material is set as 0.8, as the measure in experiments.