There has been continuing effort in developing analytical, numerical, and empirical models of laser-based additive manufacturing (AM) processes in the literature. However, advanced physics-based models that can be directly used for feedback control design, i.e., control-oriented models, are severely lacking. In this paper, we develop a physics-based multivariable model for directed energy deposition. One important difference between our model from the existing work lies in a novel parameterization of the material transfer rate in the deposition as a function of the process operating parameters. Such parameterization allows an improved characterization of the steady-state melt-pool geometry compared to the existing lumped-parameter models. Predictions of melt-pool geometry and temperature from our model are validated using experimental data obtained from deposition of Ti-6AL-4V and deposition of Inconel® 718 on a laser engineering net shaping (LENS) AM process and finite-element analysis. Then based on this multivariable model, we design a nonlinear multi-input multi-output (MIMO) control, specifically a feedback linearization (FL) control, to track both melt-pool height and temperature reference trajectories using laser power and laser scan speed.

## Introduction

Adoption of the techniques of direct melting of metal powder can be traced back to late 1990s [1,2]. Various additive manufacturing (AM) processes for metals have been developed in the last decade, including selective laser melting [3,4], direct metal deposition [5], laser engineered net shaping (LENS) [6,7], and electron beam melting [8]. Parts can be built from metal wire or powderized feedstock, either in a powder-feed process or a powder-bed process. All metal-based AM processes are similar in that a three-dimensional part represented by a computer-aided design file is sliced into layers, then a high power energy source (laser or electron beam) is used to heat and melt metal powder (or wire), which solidifies to form a fully dense layer. Complex shapes can then be realized by repeating the process for a number of layers. Applications range from customized medical implants to aerospace fuel nozzles and turbine blade repair [911]. Despite its huge potential, AM has not yet achieved widespread adoption in industry due to multiple challenges in the process, including accuracy of part geometry, mechanical and material properties of the processed part, surface roughness, etc. To resolve these issues, advanced technologies for sensing, modeling, and control are in demand to improve the accuracy and process stability of AM.

There has been continuing effort in developing analytical, numerical, and empirical models of AM processes [7,12,13], and quite some success has been achieved in finite-element (FE) modeling of AM processes recently [1424]. However, partial differential (difference) equations (PDEs) commonly used in FE models are difficult to be used for control systems design [25]. Physics-based ordinary differential equation models, which can be directly used for feedback control designs, are severely lacking in the literature. A black-box empirical dynamic model was developed in Refs. [26,27] for a cladding process, where the transient response relating laser power to melt-pool temperature was modeled by a linear time-invariant (LTI) system and the model matrices were obtained by applying a subspace system identification to randomly modulated laser powers and the resulting melt-pool temperatures. A first-order transfer function was identified empirically and used for temperature or height control in Refs. [2830]. A knowledge-based (semi-empirical) model was proposed in Refs. [31,32] for the height control in laser solid freeform fabrication, where the model consists of a first-order linear time invariant description of height connected with a static nonlinear output map. One important lumped-parameter model was derived by Doumanidis and Kwak, which characterized the dynamics of melt-pool geometry and temperature as a function of the laser power, material feed rate, and laser scan speed based on the principles of mass, momentum, and energy conservation in a molten puddle [33]. This work was further extended to layer-dependent process models in Refs. [34,35]. Along a similar line, a set of simplified first-order dynamic models for melt-pool height and temperature were derived in Ref. [36].

There are limited studies on the design and implementation of closed-loop controls for laser-based AM processes. Most of the existing controllers are restricted to classical controllers, e.g., on/off bang–bang controller [37], Proportional integral derivative (PID) controllers [28,29,38,39], or PID combined with a feedforward controller [31], to name a few. The PID controllers were designed either based on pure sensing information without a model [38,39], or based on relatively simplistic single-input single-output (SISO) empirical models, e.g., a first-order transfer function [29], or a knowledge-based empirical model [28,31]. A fuzzy logic controller was designed to follow the desired clad height in a cladding process by controlling the laser power [40]. A sliding mode controller, together with a PID controller, was developed in Ref. [32] to control clad height based on an empirical model. A generalized predictive controller was designed based on a second-order empirical model [26,27]. For laser metal deposition, an iterative learning control (ILC) was developed for height control by varying powder flow rate in Refs. [34,41,42], and for melt-pool temperature control by varying laser power in Refs. [30,43]. An ILC was also used to control wire feed rate for laser wire deposition in Ref. [44].

In this paper, we focus on the modeling and control of directed energy deposition processes. We follow the same line of the lumped-parameter modeling work conducted by Doumanidis and Kwak [33] to characterize the dynamics of molten-pool geometry and temperature with respect to process input parameters such as laser power and laser scan speed, from the principles of mass balance and energy balances of the molten puddle. However, a simple derivation can show that the steady-state melt-pool height derived from the lumped-parameter model by Doumanidis and Kwak [33] is not explicitly related to the laser power (see the corresponding mathematical expression in Remark 1 in Sec. 2), which does not agree with the measurements from our experiments. One important difference between our model from the existing work such as Ref. [33] lies in a novel parameterization of the material transfer rate in the deposition as a function of the process operating parameters, which renders an improved characterization of the dependence of melt-pool height with respect to laser power, in contrast to the derivation that the steady-state melt-pool height is not explicitly related to laser power from Ref. [33]. Predictions of melt-pool geometry from our model are validated using experimental data collected from deposition of two different types of materials on a LENS machine, namely, Ti-6AL-4V and Inconel® 718. Model predictions of average melt-pool temperature are validated using FE computation.

The second contribution of this paper lies in a multi-input multi-output (MIMO) nonlinear control. Based on our new models on melt-pool geometry and temperature, we have designed a feedback linearization control that uses laser power and laser scan speed to track both melt-pool height and temperature reference trajectories at the same time.

Part of the preliminary results on validation using measurements from deposition of a Ti-6AL-4V track was submitted to an invited session of the 2016 American Control Conference [45]. In this paper, we have added model validation on deposition of an Inconel® 718 track. We also explore model variation to account for changes in melt-pool dynamics for different materials. In the end, we have added model validation for deposition on a preheated substrate.

The paper is organized as follows: Section 2 presents our physics-based model for melt-pool geometry and temperature. In addition, an FE model for computing melt-pool temperature is introduced. The FE computed melt-pool temperature is used to validate the temperature prediction from the model in Sec. 3. In addition, Sec. 3 shows the model validation using experimental measurements from deposition of a Ti-6AL-4V track and deposition of an Inconel® 718 track. Section 4 presents a feedback linearization controller, which can track both melt-pool height and temperature reference trajectories using laser power and laser scan speed at the same time.

## Modeling

In Sec. 2.1, we present our lumped-parameter model for melt-pool geometry and temperature. Since it is commonly known that accurate measurement of melt-pool temperature is difficult to obtain, in this paper, we compare the lumped model-predicted melt-pool average temperature against the melt-pool temperature computed using an FE model, which is introduced in Sec. 2.2.

### Analytical Lumped-Parameter Model for Melt-Pool Geometry and Temperature.

Mass conservation of the molten puddle requires that its mass change rate equals the difference of the material transfer rate by deposition minus material loss rate due to solidification. Let ρ denote the melt density and assume it to be constant, and let V(t) denote the volume of the molten-puddle, then the mass change rate of the molten-puddle can be described by $ρV˙(t)$. Let v denote the laser scan speed, and let A(t) denote the maximum cross-sectional area of the molten puddle, then the material loss rate due to solidification can be represented by $ρA(t)v(t)$. Let μ denote mass transfer efficiency (or powder catchment efficiency) of the deposited material, and let f denote the powder flow/feeding rate, then the material transfer rate can be computed as $μf(t)$. As a result, the mass conservation of the molten puddle can be written as follows:
$ρddt(V(t))+ρA(t)v(t)=μf(t)$
(1)

Assuming ellipsoidal geometry for the molten puddle, the volume V(t) and maximum cross-sectional area A(t) of the half-ellipsoid molten puddle can be calculated as $V=(π/6)w(t)h(t)l(t), A=(π/4)w(t)h(t)$, with $w(t),h(t)$, and l(t) denoting the puddle width, height, and length, respectively.

Remark 1. The powder catchment efficiency μ in general is a time-varying parameter. If the laser power is not high enough to melt the powder, the catchment efficiency equals zero irrespective of the powder feeding rate. Under the same powder feeding rate, the higher the melt-pool temperature, the larger the volume of the melt pool and the more powder the melt pool will catch. Hence, not accounting for the dependence of powder catchment efficiency with respect to laser power or melt-pool temperature could lead to an erroneous conclusion, e.g., the steady-state melt-pool height in Ref. [33] by Doumanidis and Kwak can be shown not explicitly related to the laser power, since the steady-state melt-pool width and height satisfy $w¯=μfv/(1−cos θ)·(γGL−γSL)$ and $h¯=−4(1−cos θ)·(γGL−γSL)/ρπv2$, respectively, with γGL and γSL denoting the gas–liquid and solid–liquid surface tension coefficient, respectively, and θ denoting the wetting angle. In Ref. [36], the powder catchment efficiency was assumed to take an exponential decay function with respect to the melt-pool temperature. Though it provided a link between the melt-pool geometry and temperature, the exponential function for the powder catchment efficiency was not backed up by experiment data. Furthermore, how the powder catchment efficiency varies with respect to temperature is highly dependent on individual AM process and AM machine. Experimental estimation of the powder catchment efficiency will have to rely on the measurement of melt-pool geometry or surface temperature and reverse calculation of the amount of powder being absorbed by the melt pool [46].

Rather than estimate [46] or iteratively learn the powder catchment efficiency [43], we propose to approximate and parameterize the entire material transfer rate $μf$ altogether. We follow the modeling approach by Eagar and Tsai [47] on dimensionless analysis of the size and shape of arc weld pools with respect to welding process variables and material parameters. Specifically, starting from the dimensionless form of a Gaussian heat distribution, the dimensionless weld shape was modeled as a function of the dimensionless operating parameter
$n=qv4πa2ρcl(Tm−T0)$
(2)

where q denotes the power of the heat input, v denotes the travel speed, a denotes the thermal diffusivity, cl denotes the melt specific heat, Tm is the melting temperature, and T0 is the initial temperature. Note that n can be rewritten as $n=v⋅λ/4πa⋅q/k(Tm−T0)λ$, where k denotes the thermal conductivity (the thermal conductivity k and thermal diffusivity a satisfy $a=k/(ρcl)$), λ denotes some characteristic length, e.g., the thickness of the workplace. It is also noted that $q/(k(Tm−T0)λ)$ corresponds to the dimensionless net heat input to the workpiece. Hence the parameter n essentially captures the travel speed, the thermal diffusivity, and the net heat input to the workpiece. Such operating parameter n was first used by Christensen et al. [48] in the derivation of the dimensionless form of the Rosenthal's solution of a traveling point source of heat [49], and then used by Eagar and Tsai in characterizing the weld shape under a Gaussian heat distribution [47].

Inspired by this modeling work by Eagar and Tsai [47], we propose that the steady-state dimensionless cross-sectional area of the melt pool ($A¯=Av2/4a2$) can be parameterized in terms of a dimensionless operating parameter $n¯$, i.e.,
$A¯=Γ(n¯)$
(3)
with the function $Γ(·)$ to be determined and $n¯$ defined as a modification of n in Eq. (2)
$n¯=η(Q−Qc)v4πa2ρcl(Tm−T0)$
(4)

where η denotes the laser absorption efficiency, Q denotes the laser power, and Qc denotes a critical value of the laser power. Compared to the operating parameter n in Eq. (2) used by Christensen et al. [48] as well as by Eagar and Tsai [47], the modified $n¯$ captures the net heat input being absorbed by metal powder when it is melted by laser, noting that metal powder will not melt if the laser power Q is less than a critical value Qc.

Consequently, the steady-state cross section area satisfies
$Av2/4a2=Γ(η(Q−Qc)v4πa2ρcl(Tm−T0)),Q>Qc$
(5)
Note that the mass balance equation (1) at steady state implies $μf=ρAv$. If we further approximate the function $Γ(·)$ to be a linear function with coefficient β, i.e., $Γ(n¯)=βn¯$, the steady-state cross section area follows:
$A=βη(Q−Qc)πρcl(Tm−T0)v,Q>Qc$
(6)
It agrees with our observation that the melt-pool cross section area increases with the increase of the laser power and decreases with the increase of the laser scan speed. Then the material transfer rate can be approximated as follows:
$μf(t)≈βη(Q−Qc)πcl(Tm−T0), Q>Qc$
(7)

and $μf=0$ if $Q≤Qc$, where the initial temperature T0 corresponds to the ambient temperature $T∞$ if the substrate is not heated, and equals to the temperature of the substrate if it is preheated. Note that for a given powder feeding rate f(t), Eq. (7) essentially approximates the material transfer rate by considering that the powder catchment efficiency has reached steady state under a given laser power input.

This updated model of material transfer rate (7) provides a physics-based parameterization with respect to AM process input variables and material parameters, and the model parameters β and Qc can be calibrated using experimental data. As a result, the improved mass conservation equation takes the following form:
$ρddt(V(t))+ρA(t)v(t)=βη(Q(t)−Qc)πcl(Tm−T0), Q>Qc$
(8)

Remark 2. The function $Γ(·)$ could be different for different types of material. Consider a linear approximation, $Γ(n¯)=βn¯$, the value of the coefficient β could be a function of the dimensionless Gaussian distribution parameter $σ¯g=vσg/2a$ in terms of the laser scan speed v, spread of the Gaussian heat source σg, and the thermal diffusivity a.

Remark 3. Compared to Eq. (1), the updated mass conservation equation (8) has removed the direct dependency on powder feeding rate, which could seemingly be more restrictive since it reduces the number of control knobs for the AM process. However, we would like to argue that the powder feeding rate is not reliable to be used as a control knob to control melt-pool geometry. Our experiments often show that under certain operating conditions, there exists a plateau region where increasing powder feeding rate will not increase the material being melted (the extra powder brought in by the higher feeding rate could just be blown away without being melted) and thus will not increase the melt-pool size. Therefore for control purpose, we suggest that the AM process should run at a constant powder feeding rate and only control laser power and laser scan speed.

It is not uncommon to assume that the molten puddle has a fixed aspect ratio, e.g., to assume w = l [43] and to assume a fixed ratio r of the melt-pool width over height, i.e., $w=r⋅h$ [36]. In this case, the mass balance equation (8) can be further simplified to lead to the following dynamic equation for the melt-pool height:
$ρπ2r2h2(t)dh(t)dt+ρπ4rh2(t)v(t)=βη(Q(t)−Qc)πcl(Tm−T0)$
(9)

The dynamics of melt-pool temperature can be derived from energy balance of the puddle. For the completeness of the paper, we include below the derivation by Doumanidis and Kwak [33] on energy balance. In this paper, we compute a certain model parameter in the energy conservation, e.g., the convection coefficient from molten puddle to the substrate, using FE-based heat flux computation (see Sec. 2.2).

Energy balance of the puddle requires that the internal energy rate of accumulation in the puddle is equal to the difference of power inflow of material addition minus the power outflow loss to solidified material, plus total external heat transfer rate from the puddle surface. If it is assumed that the powder added to the puddle does not go through preheating, the specific internal energy of cold material being added to melt pool equals zero, and thus the power inflow of material addition equals zero. Then the energy balance of the puddle leads to
$ddt(ρV(t)e(t))=−ρA(t)v(t)eb+Ps$
(10)
where e denotes the specific internal energy of the melt pool
$e=cs(Tm−T0)+L+cl(T−Tm)$
(11)
where cs denotes the solid material specific heat and L denotes the specific latent heat of solidification. The specific energy of the solidified bead material eb can be calculated as $eb=cs(Tm−T0)$. The total thermal transfer at the puddle surface Ps is
$Ps=ηQ−Asαs(T−Tm)−AGαG(T−T0)−AGεσ(T4−T04)$
(12)

which consists of the power influx from the laser power Q with a laser absorption efficiency η to the puddle, heat outflux convected to the substrate boundary and conducted into the substrate by convection coefficient αs, heat loss at the free surface via gaseous convection, with a heat transfer efficiency αG, and radiation with a total hemispherical emissivity of the melt surface ε. The Stefan–Boltzmann constant is denoted by σ. Corresponding to an ellipsoidal puddle geometry, the areas of the puddle interface to the substrate, denoted by As, and to the free surface, denoted by AG, are $As=(π/4)wl, AG=π/23(whl)2/3$.

### Finite-Element Model for Melt-Pool Temperature.

Finite-element modeling is performed here to compute the 3D-transient distribution of temperature within the melt pool. Then the FE model-computed average temperature of the melt pool is used as reference in comparison to the predicted temperature by Eqs. (10)(12) of the lumped-parameter model.

The melt-pool temperature is obtained by determining the thermal equilibrium of a body subject to a volumetric heat source within a Lagrangian reference frame, described as follows:
$ρCpdTdt=−∇⋅q(x,t)+Qs(x,t)$
(13)
where Cp is specific heat capacity, q is the heat flux vector, x is the position vector, and Qs is the body heat source. The heat flux vector satisfies
$q(x,t))=−k(T)∇T$
(14)

where k(T) is the temperature-dependent thermal conductivity, which is assumed to be isotropic. Assume at $t=0s, T=T∞$, where $T∞$ denotes the temperature of the ambient environment.

The boundary heat losses are due to free convection, forced convection from the nozzle gas flow near the heat source [23], and radiation. Heat losses are modeled via Newton's law of cooling as follows:
$qs=h(Ts−T∞)$
(15)
where qs denotes the surface heat flux, h is the heat transfer coefficient, and $Ts$ is surface temperature. The heat transfer coefficient h needs to account for all three cooling effects, i.e.,
$h=hfree+hforced+hrad$
(16)
where $hfree$ is the free convection coefficient, $hforced$ is the forced convection coefficient, and $hradiation$ is the effective convective coefficient for thermal radiation, which is determined by linearization of the following Stephan–Boltzmann equation for heat loss due to radiation:
$qrad=εσ(Ts4−T∞4)$
(17)
The linearization gives $qrad=hrad(Ts−T∞)$, where hrad satisfies
$hrad=εσ(Ts+T∞)(Ts2+T∞2)$
(18)

#### Numeric Implementation of the FE Model.

A nonlinear Newton–Raphson-based FE code, CUBES® by Pan Computing LLC [50], is used to solve the discretized transient thermal equilibrium of a component subject to a body heat source. This FE software has been previously validated for the LENS process [23]. Additional modeling details, including the activation strategy and boundary condition application method, can be found in Ref. [20].

As the FE method is a conduction-based model, it does not attempt to capture any of the complex melt-pool fluid behavior. It has been well established that the flow within the melt pool (Marangoni Flow) and the resulting heat transfer (Marangoni Convection) deepen and widen the melt pool while also reducing both the peak and average melt-pool temperatures [51]. A common technique to approximate the Marangoni effects in FE modeling is to use an artificially high thermal conductivity above the solidus temperature (Tsolid) [52]. For the present work, the thermal conductivity at the liquidus temperature is 150% of the reported highest temperature value, following the recommendation of Goldak et al. in Ref. [52].

The FE mesh is produced through a two-step process. First, a coarse mesh is generated within Patran (by MSC Software). Then, during simulation, the mesh is refined by CUBES® according to user inputs. Past simulations have shown that thermal behavior converges with a mesh having one element per laser radius. However, to improve the resolution in the melt-pool region, the adaptive subroutine controls are set to yield a mesh with four elements per laser radius. A sample refined mesh for deposition of Inconel® 718 under varying laser power and a fixed laser scan speed (see Sec. 3.3) is displayed in Fig. 1.

Fig. 1
Fig. 1
Close modal

#### Melt Pool Convection.

In the lumped model, the heat conduction into the substrate at the liquid–solid boundary of the melt pool is modeled as a convection with a coefficient αs in Eq. (12). The finite-element model is used to compute the convection αs by equating the conductive heat loss from the FE model to the convective heat loss of the lumped model
$∫Asq⋅nda=Asαs(Tpool−Tm)$
(19)
where As denotes the surface of the liquid–solid boundary, n denotes the normal direction vector to As, and Tpool is the mean temperature of the melt pool. Values of Tpool and q are computed from the FE thermal model discussed earlier in this section. This allows the calculation of αs as follows:
$αs=1As(Tpool−Tm)∫Asq⋅nda$
(20)

## Model Validation

In this section, we validate the lumped-parameter model prediction of melt-pool geometry using experimental measurements, and validate the lumped model prediction of melt-pool average temperature using the FE computation.

### Experiment Setup.

Figure 2 shows a schematic plot of the LENS system used for the experimental data collection. The LENS machine was operated in a manually programmed mode, which allowed communication with external data acquisition systems and dynamic change of process input parameters. Deposition of a single track (with a single layer) using one of the two different types of materials, Ti-6AL-4V and Inconel® 718, was conducted in the experiments. In addition, we examined the following two scenarios on varying process parameters in deposition:

Fig. 2
Fig. 2
Close modal
• Scenario I: Vary laser power to follow a prescribed multistep trajectory (shown in the bottom subfigure of Fig. 3
Fig. 3

Ti-6AL-4V: Our model prediction versus experiment measurements of melt-pool geometry and FE prediction of melt-pool temperature, subjected to a multistep trajectory of laser power Q and a fixed scan speed $v=10.58 mm/s$

Fig. 3

Ti-6AL-4V: Our model prediction versus experiment measurements of melt-pool geometry and FE prediction of melt-pool temperature, subjected to a multistep trajectory of laser power Q and a fixed scan speed $v=10.58 mm/s$

Close modal
) but fix the laser scan speed to $v=10.58 mm/s$.
• Scenario II: Vary laser scan speed to follow a prescribed multistep trajectory (shown in the bottom subfigure of Fig. 4
Fig. 4

Ti-6AL-4V: Our model prediction versus experiment measurements of melt-pool geometry and FE prediction of melt-pool temperature, validating against a new set of process inputs: a multistep trajectory of laser scan speed v and a fixed laser power $Q=450 W$

Fig. 4

Ti-6AL-4V: Our model prediction versus experiment measurements of melt-pool geometry and FE prediction of melt-pool temperature, validating against a new set of process inputs: a multistep trajectory of laser scan speed v and a fixed laser power $Q=450 W$

Close modal
) but fix the laser power to be $Q=450 W$.

Work distance was maintained at 9.27 mm in both cases. The experimental measurements on bead height and width were obtained from processing the 3D spatial optical profilometry data.

### Ti-6AL-4V.

Note that the updated mass balance equation (8) has two model parameters, β and Qc, which need to be determined. We calibrate these two model parameters using experimental data. Equation (8) indicates that the steady-state cross–sectional area $A=β(η(Q−Qc)/πρcl(Tm−T0)v)$. The parameters β and Qc can be identified by conducting the following least-squares (LS) optimization in matching the predicted cross-sectional areas with the measurements:
$minβ,Qc∑i(Aim−βη(Qi−Qc)πρcl(Tm−T0)vi)2$
(21)

where Qi, vi, and $Aim$ denote a sequence of values for laser power, scan speed, and the resulting measured steady-state melt-pool cross-sectional area, respectively. We use the experimental data collected in the aforementioned Scenario I, where $Qis$ correspond to the seven step changes in laser power shown in Fig. 3, $vi=10.58mm/s$, and $Aim$ corresponds to the steady-state mean cross-sectional area resulting from Qi and vi. The calibrated model parameters are $β=0.3026$ and $Qc=111.72W$, which are listed in Table 1.

Table 1

Material properties and model parameters

Parameter (unit)SymbolTi-6AL-4VInconel®
Density (kg/m3)ρ44308145
Solid material specific heat (J/(kg K))cs610652
Molten material specific heat (J/(kg K))cl700778
Melting temperature (K)Tm19231570
Ambient temperature (K)T0292292
Specific latent heat of solidification (J/kg)L3 × 1052.5 × 105
Heat transfer coefficient (W/(m2 K))αG7575
Convection coefficient in varying power (W/(m2 K))αs5.8 × 105–9.7 × 1051.09 × 106–1.37 × 106
Convection coefficient in varying speed (W/(m2 K))αs4.1 × 105–6.1 × 1051.13 × 106–1.18 × 106
Surface emissivityε0.40.53
Stefan–Boltzmann constant (W/(m2 K4))σ5.67 × 10−85.67 × 10−8
Thermal diffusivity (m2/s)a2.48 × 10−65.89 × 10−6
Laser transfer efficiencyη0.40.4
Melt-pool height–width ratio1/r0.130.2039
Critical laser power (W)Qc111.72132.63
Coefficient in Γβ0.30260.3322
Parameter (unit)SymbolTi-6AL-4VInconel®
Density (kg/m3)ρ44308145
Solid material specific heat (J/(kg K))cs610652
Molten material specific heat (J/(kg K))cl700778
Melting temperature (K)Tm19231570
Ambient temperature (K)T0292292
Specific latent heat of solidification (J/kg)L3 × 1052.5 × 105
Heat transfer coefficient (W/(m2 K))αG7575
Convection coefficient in varying power (W/(m2 K))αs5.8 × 105–9.7 × 1051.09 × 106–1.37 × 106
Convection coefficient in varying speed (W/(m2 K))αs4.1 × 105–6.1 × 1051.13 × 106–1.18 × 106
Surface emissivityε0.40.53
Stefan–Boltzmann constant (W/(m2 K4))σ5.67 × 10−85.67 × 10−8
Thermal diffusivity (m2/s)a2.48 × 10−65.89 × 10−6
Laser transfer efficiencyη0.40.4
Melt-pool height–width ratio1/r0.130.2039
Critical laser power (W)Qc111.72132.63
Coefficient in Γβ0.30260.3322

Remark 4. In this paper, we directly calibrated β using experimental data. As mentioned in Sec. 2.1, β is considered to be a function of a dimensionless Gaussian distribution parameter $σ¯g=vσg/2a$. So alternatively, it is possible, in the future work, to model explicitly the function β in terms of $σ¯g$, and then use experimental data to help to choose the appropriate value for the spread of the Gaussian heat source σg.

By setting $1/r=0.13$, which is an average ratio of height over width from experimental measurements, we then simulate the melt-pool height dynamics (9) and temperature dynamics (10)(12), as shown in Fig. 3. The value of the heat transfer efficiency αG in Eq. (12) is obtained from an experimental analysis [23]. The convection coefficient αs is evaluated through the FE heat flux computation described in Sec. 2.2. As the reference for comparison, the melt-pool average temperature is also computed using the FE code CUBES® as described in Sec. 2.2. Thermal property parameters used in the FE computation are given in Table 2, and some other material property parameters such as density, latent heat of fusion, and emissivity used in the FE computation share the same values as those used in the lumped-parameter model prediction and they are given in Table 1. The values of αG and αs as well as other material property parameters used in the lumped model predictions are given in Table 1.

Table 2

Thermal properties used in FE computation

Ti-6AL-4V [53]
T (°C)k (W/m °C)Cp (J/kg °C)
206.6565
937.3565
2049.1574
31510.6603
42612.6649
53714.6699
64917.5770
76017.5858
87117.5959
Ti-6AL-4V [53]
T (°C)k (W/m °C)Cp (J/kg °C)
206.6565
937.3565
2049.1574
31510.6603
42612.6649
53714.6699
64917.5770
76017.5858
87117.5959
Inconel® 718 [54]
2011.4427
10012.5441
30014.0481
50015.5521
70021.5561
Inconel® 718 [54]
2011.4427
10012.5441
30014.0481
50015.5521
70021.5561

Table 3 lists the root mean square error (RMSE) between the model-predicted melt-pool height (or width) and the respective measurements, as well as the RMSE between the model-predicted melt-pool average temperature and the FEA computed temperature.

Table 3

Root mean square error (RMSE) between model prediction and measurement (or FE calculation)

Inconel® 718

Ti-6AL-4V

Varying power

Varying speed
Varying powerVarying speed
RMSERoom-tempPreheatRoom-tempPreheat
w (μm)107.8111.6135.1166.673.7175.0
h (μm)14.921.833.035.425.569.8
T (K)175.962.2235.784.7203.5219.9

Inconel® 718

Ti-6AL-4V

Varying power

Varying speed
Varying powerVarying speed
RMSERoom-tempPreheatRoom-tempPreheat
w (μm)107.8111.6135.1166.673.7175.0
h (μm)14.921.833.035.425.569.8
T (K)175.962.2235.784.7203.5219.9

Figure 3 shows that our model predictions for the melt-pool width follow the experimental measurements very well, and both the predictions and measurements show a clear pattern of step changes, correlating to the step changes of the laser power. However, it is noted that the measured melt-pool height trajectory does not show a clear stepwise pattern, especially a “balling” effect is observed during the distance intervals of $2×104−4×104μm$ and $8×104−10×104μm$. Nevertheless, our predictions capture, to a certain extent, the varying trend of melt-pool height following the change of laser power, and thus we consider them to have better reflected the height measurements than the prediction by Doumanidis and Kwak [33], where the mathematical expression for the steady-state melt-pool height in Remark 1 indicates no direct relation to the laser power.

To further validate our model, the same model parameter values in Table 1 are used to simulate Scenario II, where the laser scan speed follows a multistep trajectory but with a fixed laser power $Q=450W$. Figure 4 shows that the predictions for both melt-pool width and height match the measurements reasonably well in the distance interval of $2×104−11×104μm$. In addition, the model-predicted temperature is close to the FE-computed melt-pool average temperature.

### Inconel® 718.

This section considers deposition of Inconel® 718. For parameterization of the steady-state dimensionless cross-sectional area of the melt-pool, a linear function and a square-root function for $Γ(·)$ in Eq. (3) are both examined. We observe that though for Ti-6AL-4V, a linear function approximation for $Γ(·)$ is sufficient to provide enough modeling accuracy, approximating $Γ(·)$ as a square-root function has a much better goodness of fit for deposition of Inconel® 718 as described below.

Considering a square root function for $Γ(·)$, i.e., $Γ(n¯)=βn¯$, the steady-state cross-sectional area is approximated by
$Av2/4a2=βη(Q−Qc)v4πa2ρcl(Tm−T0), Q>Qc$
(22)
Consequently, following Eq. (21), the following least-squares optimization needs to be solved:
$minβ,Qc∑i{Aim−βη(Qi−Qc)(4a2)πρcl(Tm−T0)vi3}2$
(23)

We use the measurement data from Scenario I for Inconel® 718 to solve the above least-squares optimization to identify the parameters β and Qc, i.e., $Qi′s$ correspond to the seven-step changes in laser power shown in Fig. 5, $vi=10.58mm/s$, and $Aim$ corresponds to the steady-state mean cross-sectional area resulting from Qi and vi. The least-squares optimization leads to $β=0.3322$ and $Qc=132.63W$. Model validation was then conducted using the measurement data collected from Scenario II for Inconel® 718 (varying laser speed with a fixed laser power). Figure 6 shows the comparison of using the linear function versus the square-root function for Γ to parameterize the dimensionless cross-sectional area. It can be observed that though the two different function forms have similar goodness of fit for the changing power scenario (see Fig. 6(a), where the coefficient of determination R2 for using linear function $Rlin2=0.713$ and the R2 for using square-root function $Rsqr2=0.717$), the prediction given by the square-root function has a higher goodness of fit than the linear function under the changing speed scenario (see Fig. 6(b), where $Rlin2=0.827$ and $Rsqr2=0.943$).

Fig. 5
Fig. 5
Close modal
Fig. 6
Fig. 6
Close modal

Remark 5. One possible reason why Ti-6AL-4V and Inconel® 718 take different functions for dimensionless melt-pool cross-sectional area (linear versus square root in terms of the dimensionless parameter $n¯$) could be due to that compared to Inconel® 718, Ti-6AL-4V has lower thermal conductivity (see the values of thermal conductivity k(T) of Ti-6AL-4V and Inconel® 718 in Table 2), and thus stays at molten state longer, which leads to a larger cross-sectional area.

For $Γ(n¯)=βn¯$, in terms of Eq. (22), approximation of the material transfer rate ($μf≈ρAv$) in Eq. (7) is updated as follows:
$μf(t)≈βη(Q−Qc)ρ(4a2)πcl(Tm−T0)v, Q>Qc$
(24)
which leads to the following updated dynamic equation for the melt-pool height:
$ρπ2r2h2(t)dh(t)dt+ρπ4rh2(t)v(t)=βη(Q(t)−Qc)ρ(4a2)πcl(Tm−T0)v$
(25)

For $β=0.3322, Qc=132.63W$, and a melt-pool width and height ratio of $w/h=1/0.2039$, Figs. 5 and 7 show the prediction from the melt-pool height dynamics (25) and temperature dynamics (10), under varying laser power and varying laser speed, respectively. The corresponding RMSEs between the model predictions and measurements (or FE computation) are given in Table 3. The width and height ratio $w/h=1/0.2039$ is an average value obtained from experimental measurements and it is noted that such ratio did not hold constant during the deposition. Assuming a fixed ratio of width over height does contribute to misfit of the model-predicted melt-pool height with respect to measurements (see Fig. 7), despite that the model-predicted steady-state cross-sectional areas match the measurement data very well in Fig. 6(b).

Fig. 7
Fig. 7
Close modal

### Deposition With Preheated Substrate.

Experimental measurements and simulations in Secs. 3.2 and 3.3 are for deposition on a substrate under room temperature $T∞$. For Inconel® 718, experiments were also conducted for deposition on a preheated substrate, where the substrate was heated to $317 °C(590.15K)$ before deposition. Experimental data show that for deposition on a preheated substrate, melt-pool height increases by 10–20% and melt-pool width increases by 10–15%, compared to deposition on a room-temperature substrate.

Recall that in Eqs. (7), (9), and (25), the initial temperature T0 equals to the temperature of the substrate if it is preheated. We set $T0=590.15K$ in Eq. (25) as well as in Eqs. (11) and (12), and then simulate the melt-pool height dynamics (25) and temperature dynamics (10)(12). Figure 8 shows the model-predicted melt-pool geometry and temperature under varying laser power (Scenario I) and Fig. 9 shows the predictions under varying laser speed (Scenario II). We can see that our model prediction, though performed worse than the case of deposition on the substrate with room temperature, still matches the measurement reasonably well under the varying laser power scenario, with the RMSEs for melt-pool width, height, and average temperature being $111.6μm, 21.8μm$, and $62.2K$, respectively, (see Table 3). There is increased discrepancy between the model prediction and measurement under the varying laser speed scenario, where the RMSEs for melt-pool width, height, and average temperature are $166.6μm, 35.4μm$, and $84.7K$, respectively.

Fig. 8
Fig. 8
Close modal
Fig. 9
Fig. 9
Close modal

It should be noted that the same parameter values for β and Qc, which are solved from Eq. (23) using the room temperature T0, are used here to predict the melt-pool geometry under $T0=590.15K$. This might be the main source causing the degraded performance of model prediction for melt-pool geometry, suggesting that the parameters β and Qc might need to be recalibrated in order to improve the prediction accuracy for deposition on a preheated substrate.

## Nonlinear MIMO Control Design

Based on the dynamic model developed in Sec. 2, a feedback linearization (FL) control [55] can be designed to track both melt-pool height and temperature reference trajectories using laser power and laser scan speed at the same time. In this section, we design and simulate an FL process controller for the deposition of Ti-6AL-4V. An FL process controller for the deposition of Inconel® 718 can be designed following a similar procedure.

The melt-pool height dynamics (9) and the energy balance equation (10) can be rewritten into the following compact form:
$[h˙T˙]=f+G⋅[vQ̃]$
(26)
where $Q̃=Q−Qc$. The nonlinear vector field $f=[f1;f2]$ takes the following form:
$f1=0$
(27)
$f2=1ρclV[ηQc−Asαs(T−Tm)−AGαG(T−T0)−AGεσ(T4−T04)]$
(28)
and the matrix $G=[Gij],i,j∈{1,2}$ is defined as follows:
$G11=−1/(2r)$
(29)
$G12=2βηr2ρπ2clh2(Tm−T0)$
(30)
$G21=1ρclV[ρAe−ρAcs(Tm−T0)]$
(31)
$G22=1ρclV[−βηeπcl(Tm−T0)+η]$
(32)

where in calculating V and A, it is assumed that w = l and $w=r⋅h$.

Then a feedback linearization control is derived as follows:
$[vQ̃]=G−1(−f+u)$
(33)
with $u=[u1;u2]$ as the new control input vector. As a result,
$[h˙T˙]=[u1u2]$
(34)
Then any linear control methodologies can be applied to design u, and afterward the original control inputs v and Q can be derived based on u using Eq. (33). One simple linear control for u can be chosen as follows:
$u1=−τh(h−hr)$
(35)
$u2=−τT(T−Tr)$
(36)

where hr and Tr denote reference trajectory for the melt-pool height and average temperature, respectively; τh and τT are control design parameters that determine the tracking speed of the resulting close-loop system.

For $τh=0.5$ and $τT=0.6$, as well as a constant $αs=9.3×105W/m2K$, Fig. 10 shows the simulation results on applying the FL controller (Eqs. (33), (35), and (36)) to the deposition process of Ti-6AL-4V to track the reference melt-pool height and temperature, where the dash lines represent the reference trajectories. It should be noted that if there is no constraint on the melt-pool temperature, either scan speed v or laser power Q can be used to achieve the target melt-pool height (single-input single-output (SISO) control). For example, to increase melt-pool height, we can either increase laser power or decrease scan speed. When both height reference and temperature reference need to be tracked, Fig. 10 shows that when Q increases to achieve the $200μm$ reference height, v also increases, rather than decreases, as in the SISO control. This can be explained by the energy balance equation (10) at steady state, which requires v to increase so that $ρA(t)v(t)eb$ can cancel the growing Ps due to the increase of Q.

Fig. 10
Fig. 10
Close modal

Due to uncertainties associated with the AM process, robustness is a major concern for any control algorithms to be implemented in the AM process. Nevertheless, the proof-of-concept study in this section suggests the possibility of designing a nonlinear controller using the proposed AM process model.

## Conclusions

In this paper, we developed a physics-based model, in terms of ordinary differential equations, for the melt-pool geometry and temperature dynamics in a directed energy deposition AM process. For two different types of material, Ti-6AL-4V and Inconel® 718, our model-predicted melt-pool geometry matched well with the experimental data collected from deposition of a single track on a LENS system. The model-predicted average melt-pool temperatures are also close to the FE model predictions. We also evaluated the model-predicted melt-pool geometry and temperature in deposition of Inconel® 718 on a preheated substrate. In the end, based on the developed model, we designed a nonlinear MIMO controller, a feedback linearization controller specifically, which can track both melt-pool height and temperature trajectories simultaneously using both laser power and scan speed. In future work, we will evaluate the proposed feedback linearization controller using experiments.

## Acknowledgment

The experimental work of this paper was supported by Office of Naval Research, under Contract No. N00014-11-1-0668.

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