\documentclass[10pt]{article}
\usepackage{fullpage}
\usepackage{setspace}
\usepackage{parskip}
\usepackage{titlesec}
\usepackage[section]{placeins}
\usepackage{xcolor}
\usepackage{breakcites}
\usepackage{lineno}
\usepackage{hyphenat}
\PassOptionsToPackage{hyphens}{url}
\usepackage[colorlinks = true,
linkcolor = blue,
urlcolor = blue,
citecolor = blue,
anchorcolor = blue]{hyperref}
\usepackage{etoolbox}
\makeatletter
\patchcmd\@combinedblfloats{\box\@outputbox}{\unvbox\@outputbox}{}{%
\errmessage{\noexpand\@combinedblfloats could not be patched}%
}%
\makeatother
\usepackage{natbib}
\renewenvironment{abstract}
{{\bfseries\noindent{\abstractname}\par\nobreak}\footnotesize}
{\bigskip}
\titlespacing{\section}{0pt}{*3}{*1}
\titlespacing{\subsection}{0pt}{*2}{*0.5}
\titlespacing{\subsubsection}{0pt}{*1.5}{0pt}
\usepackage{authblk}
\usepackage{graphicx}
\usepackage[space]{grffile}
\usepackage{latexsym}
\usepackage{textcomp}
\usepackage{longtable}
\usepackage{tabulary}
\usepackage{booktabs,array,multirow}
\usepackage{amsfonts,amsmath,amssymb}
\providecommand\citet{\cite}
\providecommand\citep{\cite}
\providecommand\citealt{\cite}
% You can conditionalize code for latexml or normal latex using this.
\newif\iflatexml\latexmlfalse
\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}%
\AtBeginDocument{\DeclareGraphicsExtensions{.pdf,.PDF,.eps,.EPS,.png,.PNG,.tif,.TIF,.jpg,.JPG,.jpeg,.JPEG}}
\usepackage[utf8]{inputenc}
\usepackage[ngerman,english]{babel}
\begin{document}
\title{Finite element model of heating in AM}
\author[1]{Toni Ivas}%
\author[2]{VIktor LindstrÃ¶m}%
\author[2]{eric.boillat}%
\affil[1]{EPFL}%
\affil[2]{Affiliation not available}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\sloppy
\subsection*{Numerical methods}
{\label{768731}}
We used the Abaqus for modeling the heat transfer in the~Additive
Manufacturing process. The heat transfer in the SLM process is modeled
using the heat equation with Fourier's law and laser heat source:
\par\null
\begin{equation}\label{eq:heat}
\frac{\partial (\rho c_p T)}{\partial t} = \nabla (k \nabla T) + S
\end{equation}
in which $\rho$ is density, $c_p$ is specific heat capacity, $\kappa$ is thermal conductivity and $T$ is the temperature of system. All material parameters are temperature dependent and were obtained from literature data (refs). The source term $S$ in Eq (\ref{eq:heat}) can be modeled using different source terms as explained in recent study . The Goldak source term \hyperref[csl:1]{(\textit{1})} defined by:
\begin{equation}\label{eq:goldak}
S=\frac{6\sqrt{3}P\eta f}{abc \pi \sqrt{\pi}}exp[-(3(x+u*t)^2/a^2 +3(y+v*t)^2/b^2+3(z+w*t)^2/c^2)]
\end{equation}
is the one that is mostly used by community because it captures essential
characteristics of laser heat source.
Equation (\ref{eq:goldak}) represents the volumetric source were
$P$ is power of the laser, $\eta$ is process efficiency, and $x,y,z$ are coordinates of the double ellipsoid model; $a$,$b$,and $c$ are the ellipsoid axes
which represent width, depth and tail of the heat source. In most cases is $a$ axis set to deposition half width and $b$ to the melt pool depth. The velocities of the laser beam are denoted by $u,v,w$.
The initial condition of 298 K were assumed at time $t = 0$. The bottom and sides of the system shown in Fig.(\ref{}) were isolated and top surface we implemented the boundary conditions as:
\begin{equation}\label{eq:top}
(-\kappa \nabla T)*\bar{n} = h(T-T_e)+\epsilon\sigma(T^4-T^4_e)
\end{equation}
where $h$ represents the heat convection coefficient, $\epsilon$ is thermal radiation coefficient, and $\sigma$ is Stefan-Boltzmann constant.
The terms on the right side of Eq.(\ref{eq:top}) are heat convection loss due to flowing of the gas, and radiation loss due to Stefan-Boltzman law.
The interaction of Nd-YAG laser with wavelength ($\lambda$=1064 nm) used in our system with the material is modeled using Eq.(\ref{eq:goldak}). The power of the laser is $P$ = 200 W and laser beam diameter is 100 $\mu m$.
The build-up process of during the additive manufacturing process is modeled using so called "birth" element method as described in \hyperref[csl:2]{(\textit{2})}. In this model the elements are inactive at start of the AM process and are subsequently activated as material is deposited to
the substrate. We use Abaqus user subroutine OUTVOLACTIVATED to activate the elements. The activation in SLM process is done after deposition of the powder layer.
\subsection*{Test of the FEM~}
{\label{191341}}
In this section we are comparing results of Abaqus with the well known Rosenthal solution for moving heat source. The solution obtained by Rosenthal is given by equation:
\begin{equation}\label{eq:rosen}
\theta = \frac{P_L}{4\pi k R(T_m-T_0)} exp(-V(R+x)/2\kappa)
\end{equation}
where the $\theta = (T-T_0)/(T_m-T_0)$ is dimensionless temperature, $\kappa$ is the thermal diffusivity, $P_L$ the applied laser power and $R$ the distance from the heat source location. This equation is not good for accurately predict the temperature field in the neighbourhood of the finite heat source. However further away from the source the isotherms should be correctly represented with the Rosenthal's solution.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/SolutionRosenthal/SolutionRosenthal}
\caption{{Showing the solution of the Rosenthal equation for the material data of
Ti6Al4V material.
{\label{776822}}%
}}
\end{center}
\end{figure}
The uniform heat source is more appropriate to model the finite heat source as modeled by M. Van Elsen et. al. \hyperref[csl:3]{(\textit{3})}. For heat flux at point (x,y,z) within a uniform source is given by:
\begin{equation}
\dot{q}(x,y,z) = \frac {P_L} {4 a_h b_h c_h} = \begin{cases}
-c_h < x < c_h \\
-b_h < y < b_h \\
0 < z < b_h
\end{cases}
\end{equation}
The solution is based on the Rosenthal solution:
\begin{equation}
dT_{t^{'}}=\frac{\dot{q}dt}{\rho c\left[4 \pi \kappa(t-t^{'})\right]^{3/2}} exp\left(-\frac{(x-x^{'})^2+(y-y^{'})^2+(z-z^{'})^2}{4\kappa(t-t^{'})}\right)
\end{equation}
Integration over time finally gives:
\begin{equation}\label{eq:fin_eq}
T-T_0 = -\frac{P_L}{2^5 \rho c a_h b_h c_h}\int_{0}^{t} Erfh(x+V(t-t^{'}),c_h,t^{'}) Erfh(y, a_h, t^{'}) Erfh(z, b_h, t^{'})
dt^{'}
\end{equation}
This equation can be evaluated using the numerical integration and compared with finite element solution. For most practical cases is steady state reached already after 1s.
The numerical evaluation for the material parameters of Ti-6Al-4V summarized in Table\ref{tab:prop} is done.\selectlanguage{english}
\begin{table}
\begin{tabular}{ c| c| c| c| }
Symbol & Description & Value & Units \\
$C_p$ & Specific Heat & 564 & J/kg K \\
$h $ & Convective coefficient & 15 & $W /(m^2 K)$ \\
$k $ & Thermal conductivity & 6 & W/mK \\
$P_L$ & Source power & 25 & W \\
$T_m$ & Melting temperature & 1933 & K \\
$T_0$ & Room temperature & 293 & K \\
$\rho$ & Density & 4450 & $kg/m^3$ \\
\end{tabular}
\caption{{Material data for Ti-6Al-4V \label{tab:prop}}}
\end{table}
using the previous data we can calculate the solution of
Equation~{\ref{eq:fin_eq}} and give~the solution~in
Figure :
~\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/SolutionFiniteRosenthal/SolutionFiniteRosenthal}
\caption{{{\ref{190418}}Solution to
Eq.~{\ref{eq:fin_eq}} using the material parameters in
Table~~{\ref{tab:prop}}.
{\label{593041}}%
}}
\end{center}
\end{figure}
Using the previous solutions we can compare it with the simulation results from the FEM software. In this case we are using Abaqus ver. 2018. Two different meshes are used for simulations one contained 80000 and second contained 160000 elements. We modeled movement of the laser in x-direction with velocity $v=0.05 m/s$ and dimension of the uniform heat source with
$a_h$=0.15 mm $b_h$=0.05 mm and $c_h$=0.15 mm.
The results obtained from Abaqus are shown in Figure \ref{190418}:\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.98\columnwidth]{figures/Solution-Abaq-Neumann2xM/Solution-Abaq-Neumann2xM}
\caption{{The solution obtained from Abaqus with 160000 elements at time 0.03 s
using the Neumann boundary conditions.
{\label{190418}}%
}}
\end{center}
\end{figure}
The results along the direction of the scanning of the laser between the
analytical solution and FEM results are shown in Figure
{\ref{842728}}:
\par\null\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/best-fit-fem-analy/best-fit-fem-analy}
\caption{{The dimensional temperature comparison between the analytical solution
and FEM results for~test case. The best fit is obtained with the Neumann
boundary conditions with finer mesh.~
{\label{842728}}%
}}
\end{center}
\end{figure}
As shown in Figure~{\ref{842728}} we can observe the
largest error at the center of the uniform heat source, the relative
error of the complete results is approx. R = 11 \%. This value is
reasonable~concerning~the approximation used in~these simulations.~ The
maximal error is R = 21 \% which around the maximal temperature of
profile.~ ~
~
\subsection*{Examples of heat transfer simulations in
AM}
{\label{541381}}\par\null
In this section, we describe several examples of heat transfer
simulations in AM modeling.
Example 1 is a comparison between the FEM model implemented in Abaqus
and in-house heat transfer program (name). The main difference is that
an in-house program has sophisticated ``chimera'' mesh capabilities and
handle the adsorption of the laser beam in powder and bulk with
sophisticated models.~
The first example of the heat transfer in AM is given by single track
modeling. In Figure~{\ref{940315}} we present the mesh
used to model the heat transfer for the single track experiments.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/mesh-export-fine/mesh-export-fine}
\caption{{Mesh used to model heat transfer in single track simulations. The light
color represents the powder region while green color region is the bulk
material.
{\label{940315}}%
}}
\end{center}
\end{figure}
The thermophysical properties used for simulation are given in Table
{\ref{730072}}\selectlanguage{english}
\begin{table}[h!]
\centering
\normalsize\begin{tabulary}{1.0\textwidth}{CCC}
& & \\
& Temperature & Specific Heat \\
& & \\
& 273 & 380000000 \\
& 1153 & 512690000 \\
& 1297 & 484520000 \\
& 1573 & 439078000 \\
& 2573 & 2714730000 \\
& 3573 & 2007240000 \\
\end{tabulary}
\caption{{The specific heat of Bronze sample used in our simulations.
{\label{730072}}%
}}
\end{table}
The density and conductivity of the bronze bulk are set to $\rho=8.8*10^{-9}\, t/mm^3$ and $k=62\, mW/mmK$. For the powder we used density $\rho=6.2*10^{-9}\, t/mm^3
$ and conductivity $k = 6.2\, mW/mmK$.
The laser heating of the powder and substrate in AM technique is implement as user subroutine DFLUX in Abaqus FEM software. The source term is modeled by Equation (\ref{eq:source}):
\begin{equation}\label{eq:source}
Q=\frac{2*P}{\pi*\omega^2}exp(-2 \frac{r^2}{\omega^2})*exp(-z/\delta)
\end{equation}
The results of the simulations are shown in Figure
{\ref{922822}}:
\par\null\par\null\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/TemperatureBronzeNewDens-5e-3s1/TemperatureBronzeNewDens-5e-3s1}
\caption{{The temperature distribution after 0.005 s of the laser illumination.
The maximum temperature is around 2122 K.
{\label{922822}}%
}}
\end{center}
\end{figure}
The temperature profile in direction of the laser scanning at 0.005 s is
displayed in Figure~{\ref{242185}} :
\par\null\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Graph2/TemperatureProfileTestBronzeNewDens}
\caption{{Temperature profile in the direction of the laser scanning for time
t=0.005s.
{\label{242185}}%
}}
\end{center}
\end{figure}
\selectlanguage{english}
\FloatBarrier
\section*{References}\sloppy
\phantomsection
\label{csl:1}1. J. Goldak, A. Chakravarti, M. Bibby, \textit{Metallurgical Transactions B}, in press, doi:10.1007/bf02667333.
\phantomsection
\label{csl:2}2. P. Michaleris, \textit{Finite Elements in Analysis and Design}, in press, doi:10.1016/j.finel.2014.04.003.
\phantomsection
\label{csl:3}3. M. V. Elsen, M. Baelmans, P. Mercelis, J.-P. Kruth, \textit{International Journal of Heat and Mass Transfer}, in press, doi:10.1016/j.ijheatmasstransfer.2007.02.044.
\end{document}