Improved accuracy of continuum surface flux models for metal additive manufacturing melt pool simulations

Abstract

Computational modeling of the melt pool dynamics in laser-based powder bed fusion metal additive manufacturing (PBF-LB/M) promises to shed light on fundamental mechanisms of defect generation. These processes are accompanied by rapid evaporation so that the evaporation-induced recoil pressure and cooling arise as major driving forces for fluid dynamics and temperature evolution. The magnitude of these interface fluxes depends exponentially on the melt pool surface temperature, which, therefore, has to be predicted with high accuracy. The present work utilizes a diffuse interface finite element model based on a continuum surface flux (CSF) description of interface fluxes to study dimensionally reduced thermal two-phase problems representative for PBF-LB/M in a finite element framework. It is demonstrated that the extreme temperature gradients combined with the high ratios of material properties between metal and ambient gas lead to significant errors in the interface temperatures and fluxes when classical CSF approaches, along with typical interface thicknesses and discretizations, are applied. It is expected that this finding is also relevant for other types of diffuse interface PBF-LB/M melt pool models. A novel parameter-scaled CSF approach is proposed, which is constructed to yield a smoother temperature field in the diffuse interface region, significantly increasing the solution accuracy. The interface thickness required to predict the temperature field with a given level of accuracy is less restrictive by at least one order of magnitude for the proposed parameter-scaled approach compared to classical CSF, drastically reducing computational costs. Finally, we showcase the general applicability of the parameter-scaled CSF to a 3D simulation of stationary laser melting of PBF-LB/M considering the fully coupled thermo-hydrodynamic multi-phase problem, including phase change.

Figures (12)

• Figure 1: Diffuse interface two-phase domain: (left) The domain $\Omega$ consists of two phases: a gas domain ${\Omega}_{\textnormal{g}}$ and a liquid domain ${\Omega}_{\ell}$, that are separated by an interface $\Gamma^{(\ell\textnormal{g})}$ characterized as the zero contour of the signed distance function $d_{\Gamma}$. A narrow band with thickness $w_{\Gamma}$ around $\Gamma^{(\ell\textnormal{g})}$ comprises the diffuse interface region $\Omega_{\Gamma}$. (center) The indicator $\varphi(d_{\Gamma})$ specifies the phases, being $\varphi = 0$ in ${\Omega}_{\textnormal{g}}$ and $\varphi = 1$ in ${\Omega}_{\ell}$ with the transition function according to \ref{['eq:indicator']}. In $\Omega_{\Gamma}$, $\varphi(d_{\Gamma})$ transitions smoothly between the two phases, attaining $\varphi = 0.5$ at $\Gamma^{(\ell\textnormal{g})}$. (right) Shape of the symmetric delta function $\delta_{\epsilon}(\varphi)$ of the classical CSF model, i.e., the norm of the indicator gradient \ref{['eq:norm_of_indicator_gradient']}.
• Figure 2: Classical CSF modeling of an interface heat flux in the 1D heat equation for different ratios in the volume-specific heat capacity: (upper right) effective volume-specific heat capacity $c_{\textnormal{v,eff}} = c_{\textnormal{v,a}}$\ref{['eq:parameter_transition_arithmetic']} for the ratios $\frac{c_{\textnormal{v,}\ell}}{c_{\textnormal{v,g}}}\in\{e2, e3, e4\}$ with the fixed value $c_{\textnormal{v,}\ell} = 1J\per m^3K$; (lower left) continuum surface heat flux $\tilde{q}_{\Gamma} = q_{\Gamma}\,\delta_{\epsilon}$ with $q_{\Gamma} = 1W\per m^2$ and the smoothed approximation of the Dirac delta function $\delta_{\epsilon}$ defined by \ref{['eq:norm_of_indicator_gradient']}; (lower right) temperature rate $\dot{T}_{\textnormal{CSF}}$ for the different ratios of the volume-specific heat capacities as the result of the continuum surface heat flux divided by the effective volume-specific heat capacity.
• Figure 3: Sketch of the 1D example.
• Figure 4: Temperature profile resulting from interface heating with an interface heat source of $q_{\Gamma} = e10W\per m^2$ using the classical CSF model. The instationary reference temperature profile $T_{\textnormal{ref}}$ is determined using a sharp interface approach, and the steady state temperature profile is the analytical solution $T_{\textnormal{ref}}$ according to \ref{['eq:onedim_stat_analytical_max_temperature']}.
• Figure 5: Parameter-scaled CSF modeling of an interface heat flux in the 1D heat transfer equation \ref{['eq:onedim_heat_equation']} for different interpolations of the volume-specific heat capacity and corresponding delta functions $\delta_{\epsilon,i}$ for $i\in\{\text{a}; \text{h}; \text{a,a};\text{h,a}\}$ according to Table \ref{['tab:parameter_scaled_CSF_cases']}. For all cases, the densities of the phases are ${{\rho}_{\ell} = 1kg\per m^3}$ and ${{\rho}_{\textnormal{g}} = e-2kg\per m^3}$ and the specific heat capacities are ${c_{\textnormal{p,}\ell} = 1J\per(kg\,K)}$ and ${c_{\textnormal{p,g}} = e-1J\per(kg\,K)}$ which results in the volume-specific heat capacities of ${c_{\textnormal{v,}\ell} = 1J\per(m^3 kg)}$ and ${c_{\textnormal{v,g}} = e-3J\per(m^3 kg)}$: (upper right) effective volume-specific heat capacity $c_{\textnormal{v}}$; (lower left) continuum surface heat flux $\tilde{q}_{\Gamma} = q_{\Gamma}\,\delta_{\epsilon,i}$ with $q_{\Gamma} = 1W\per m^2$ and a smoothed Dirac delta function $\delta_{\epsilon,i}$; (lower right) temperature rate $\dot{T}_{\textnormal{CSF}}$ due to the continuum surface heat flux as the result of the continuum surface heat flux divided by the effective volume-specific heat capacity. Due to the significant scale difference, V1 and V3 use the scale on the left $y$-axis, and V2 and V4 use the scale on the right $y$-axis to improve readability.