Identifying uncertainty in laser powder bed fusion additive manufacturing models

Abstract

As additive manufacturing (AM) matures, models are beginning to take a more prominent stage in design and process planning for AM. A limitation frequently encountered in AM models is a lack of indication about their precision and accuracy. Often overlooked, information on model uncertainty is required for validation of AM models, qualification of AM-produced parts, and uncertainty management. This paper presents a discussion on the origin and propagation of uncertainty in laser powder bed fusion (L-PBF) models. Four sources of uncertainty are identified: modeling assumptions, unknown simulation parameters, numerical approximations, and measurement error in calibration data. Techniques to quantify uncertainty in each source are presented briefly, along with estimation algorithms to diminish prediction uncertainty with the incorporation of online measurements. The methods are illustrated with a case study based on a transient, stochastic thermal model designed for melt pool width predictions. Model uncertainty is quantified for single track experiments and the effect of online estimation in overhanging structures is studied via simulation. The application of these concepts to estimation and control of the L-PBF process is suggested.

1. Introduction

Additive manufacturing (AM) is the use of layer-based processes for producing parts directly from computer (CAD) models, without part-specific tooling [1]. Since its introduction in the mid-1980s [2,3], AM has become popular because of its ability to produce complex geometries that were impossible with traditional manufacturing techniques. After decades of being primarily led by polymer prototypes, these technologies are now employed in the production of functional parts made of polymers, ceramics, and metals [4–6].

AM technologies still present some unresolved challenges that hinder their widespread adoption. Among these challenges are high process variability, unsatisfactory part quality, and lack of process standards; all of which originate from the limited knowledge of this relatively new set of processes. Numerous models have been developed to improve the understanding of AM processes and to aid design and process engineers in predicting the quality of AM-produced components. Such quality predictions are expected to aid qualification and certification of AM designs and manufacturing plans, which currently rely only on extensive experimentation. Although most models published in the literature have been compared with experimental measurements, they often ignore process variability and lack measures of the precision and accuracy of their deterministic predictions.

Knowledge of uncertainty in AM models is required for applications such as:

1. Model validation, which may use comparisons between simulation results and experimental data, accounting for uncertainty in both sources. Comparison of simulation results obtained with different models is expected to require information on their uncertainty as well.

2. Decision making, where model predictions and their probabilities may be used to make informed decisions. In the case of AM, for example, one would expect to use models to accelerate the qualification and certification of designs and manufacturing plans for AM-produced parts. Model-enabled certification will depend on the probability of predicted key performance indicators being within admissible bounds, which are still unknown for most AM processes.

3. Uncertainty management, to identify the sources with the largest relative contributions to overall prediction error, and determine effective strategies to more accurate predictions.

This paper presents a general discussion on uncertainty in computational models of metal-based AM, and in particular, of laser powder bed fusion (L-PBF). The discussion begins with how key performance indicators (KPIs) drive the development of models, seeking simulation as a method to be used for qualification of L-PBF produced parts. We present a general description of the modeling process, and the generation and propagation of errors. We then conduct a deep dive into how errors are commonly introduced into AM models, and the contribution of each individual error source in such predictions. Uncertainty quantification (UQ) methods suitable for the L-PBF process are discussed, and Bayesian estimation is presented as a method to extend UQ by including online measurements to reduce overall prediction uncertainty. These concepts are illustrated on a low-order stochastic model.

The contribution in this paper is three-fold. It provides: 1) A discussion on methods for uncertainty quantification in L-PBF models; 2) An example of uncertainty quantification for computational models in which all sources of error are considered; and 3) A method for quantifying unmodeled process perturbations with potential applications in detection of anomalies and feedback control.

2. Background

A common aim among design and process engineers is model-based qualification (MBQ), which is expected to guide users on the manipulation of design and process parameters and their effect in the final part. MBQ in L-PBF will require control of several qualities, such as: identification and reduction of over- and under-melting defects, microstructure and mechanical properties, and reduction of residual stresses. Given the general variability in AM processes [7], accounting for uncertainty is critical if simulation is to be used as an effective tool in design and process planning. In this article, it is suggested that melt pool dimensions be chosen as KPIs due to their direct relationship with the thermal processes that define these qualities, and due to previous studies that reported improved quality when controlling melt pool geometry [8].

Under- and over-melting are known to be caused by insufficient or excessive heat deposition on the melt pool, respectively. The former may result in porosity, which is known to have an adverse influence on tensile and fatigue strength [9]; while the latter may compromise dimensional accuracy and surface roughness [10]. The origin of both kinds of defects may be traced back to the dimensions of the melt pool [11]. Conversely, grain size and morphology, microstructural parameters responsible for yield strength and other mechanical properties [12]; and residual stress, known to have a strong influence on fatigue crack growth [13], are dependent on the thermal history during solidification, which is in turn related to melt pool dimensions as well [14–16].

Out of the set of melt pool dimensions, melt pool width is chosen as the primary KPI in this first study because it can be traced both during and after the build. Other approaches, intended only for pre-process optimization, may focus in different KPIs (e.g. cross-sectional area, length-to-depth ratio) [12,15].

Until recently, the issues of model validation and verification (V&V) and uncertainty quantification (UQ) had for the most part eluded the AM community. Some of the few examples of UQ in AM models can be found in the papers by Moser [17] and Ma [18], both of which studied the sensitivity of their models to uncertainty in input parameters; and Kamath [11], who discussed uncertainty in data-driven surrogate models.

In engineering, computational models are designed as approximations of physical reality and, as such, are subjected to a cascade of errors and uncertainties. Fig. 1 illustrates recognized sources of modeling errors [19], that we have adapted for an additive manufacturing application. Introduction of errors begin as early as the selection of the physical process to model (e.g. heat diffusion, melting/solidification, free-surface flows, etc.), which is approximated with an imperfect model of reality given by constitutive equations. All physical information that cannot be represented by the adopted mathematical model is considered to be modeling error. The mathematical model is calibrated with incomplete information of model parameters, based on incomplete calibration data gathered with imperfect sensors. Error in the determination of input parameters is propagated through the simulation. The mathematical model, often unsolvable with analytical methods, is approximated with numerical methods that inject numerical errors in the model predictions. Additionally, in the case of model validation, measurement error must be kept in mind for comparisons between simulation results and test data.

Fig. 1:

Cascade of sources of error in computer models of additive manufacturing.

3. Identifying uncertainty in L-PBF models

All the aforementioned sources of error are present in L-PBF process models. L-PBF involves multiple physical phenomena occurring at different length scales. Due to the complexity of the process, most computational models limit their scope to a subset of physical phenomena at a given scale, neglecting dynamics not captured by them. Modeling assumptions that neglect certain dynamics are the origin of modeling uncertainty. It should be noted that only relationships of the causal type contribute to modeling uncertainty. For instance, if temperature distribution or a similar variable is chosen as the KPI, the lack of a structural model would not result in a relevant contribution to modeling uncertainty because the systems are only weakly coupled and the effect of stresses on the thermal history is insignificant.

Some common examples of modeling uncertainty in L-PBF can be found in: a) surface tension and particle-level dynamics neglected in continuum models, b) the choice of inaccurate distributions for laser power acting on the powder bed, or c) an inappropriate choice of boundary conditions that neglects track-to-track and layer-to-layer interactions. In the case of particles in the powder bed, studies have shown that the size and distribution of powder particles influence the formation mechanisms for pores, spatter, and denudation zones [20, 21]. The exact distribution of particles in the powder bed, however, is never known precisely and should be accounted for as a source of uncertainty, even if the computational model is able to account for powder particles. Regarding the effect of track-to-track and layer-to-layer interactions, simple continuum models often assume isotropic thermophysical properties [18,22,23] which do not take into account how thermal conductivity changes when in contact with loose powder or consolidated material. Error in determining the effect of variable thermophysical properties is expected to increase the uncertainty in predicted melt pool geometries [8].

Input uncertainty is the result of inaccurate simulation parameters, adopted in lieu of more precise knowledge or as result of uncertainty in the training data. In the case of L-PBF models, common sources can be found in: a) absorption coefficient, which quantifies the amount of irradiated laser power that heats up the powder bed [24]; b) thermal conductivity in loose powder, which depends on the distribution of powder particles [18]; c) thermo-physical parameters at high temperatures [25]; d) convection and radiation coefficients [25]; and e) enhancing coefficients occasionally used to account for the effect of advection in the liquid phase [26, 27]. It is difficult to determine the precise values of these parameters for use in L-PBF models, and it is common to observe some of them used only as adjusting coefficients.

Commercial packages, based on finite element methods, are often preferred as tools to solve AM mathematical models, but other methods (e.g. discrete element methods, arbitrary Lagrangian Eulerian, lattice Boltzmann methods) are rapidly capturing the attention of the AM community. The choice of the numerical method depends on the physical processes included in the mathematical model (some methods are tailored for free-surface and particle-to-particle interactions), and their suitability for parallelization and implementation in high performance computing (HPC) systems. Commercial packages used for L-PBF models include convergence studies to ensure that the error is small, but its magnitude is seldom reported.

Measurement uncertainty is independent of the choice of numerical model, and depends solely on the methods and instruments used to gather test data. The choice of appropriate measurement techniques for L-PBF is an unsolved issue, and it depends on the KPI of interest [28]. In the case of thermal variables, non-intrusive thermographic techniques hold the promise of providing online temperature measurements in the powder bed, but difficulties determining the correct emissivity of the material makes these predictions partially unreliable [29].

It is apparent that the different sources of uncertainty can be conceptualized but not completely defined until the mathematical model and measurement system are selected. This discussion will be expanded in the section dedicated to our case study, where each source will be identified and quantified.

4. Uncertainty quantification

Quantification of uncertainty in computational models is based on the comparison of simulation solutions with experimental data, to identify and track every source of error [19]. A comprehensive discussion on uncertainty sources in heat transfer and fluid mechanics models can be found in the standard ASME V&V 20, along with methods to quantify them [30]. The fact that AM involves thermally-activated consolidation processes makes this standard suitable for this application. Fig. 2 illustrates the process of prediction and validation followed in UQ with ASME V&V 20.

Fig. 2:

V&V and UQ in computational models as suggested in ASME V&V 20

In this example, a known set of processing parameters and material properties are fed to the model to obtain a simulation result S. Information about the grid is used to estimate the numerical error δ_num that results from the numerical method using methods based on Richardson extrapolation [31] or Roache’s grid convergence index (GCI) [32]. Meanwhile, an assumed probability distribution in simulation parameters is propagated through the model into the predicted quantity of interest, resulting in an estimate of the error due to inaccurate inputs δ_input. Finally, simulation results S are confronted with measurements D and measurement error δ_D. The difference between model prediction and measurement determines the bias E, which acts as an estimate of modeling error δ_model. All sources of error are merged in the calculation of prediction uncertainty, which is reported along the bias as described in the first section of ASME V&V 20.

At this stage, simulation predictions and estimates of their uncertainty could potentially be incorporated in decision making systems for pre-process adjustments to designs or manufacturing plans. The probability of desired or undesired events could be used to guide such model-based adjustments. Additionally, prediction uncertainty may be used to reconcile potentially conflicting models by assigning higher importance to more precise predictions.

Process variability in AM models is accounted for as uncertainty due to unknown inputs, if it can be traced back to simulation parameters, or as modeling uncertainty otherwise. Herein, we describe an application of Bayesian estimation [33] to reduce modeling error by mapping sources of variability to random simulation parameters that are identified in real time. In the case of L-PBF, the set of identified parameters may include random variables that attempt to model the variable thermal characteristics of the material that surrounds the melt pool.

The process for online estimation is illustrated in Fig. 3. In this case, a simulation is performed for a given set of simulation parameters and an initial state with its associated uncertainty, which originate from a previous step of the simulation. Uncertainty in the initial state is propagated in time using the model and process uncertainty, which is expected to capture the effect of modeling error, numerical error δ_num and error due to unknown inputs δ_input. The propagated state and uncertainty are then compared with the measurement and its error δ_D to result in estimates of the state at the next time step, its uncertainty, and an updated estimate of process uncertainty in the case of adaptive filtering.

Fig. 3:

Online estimation in predictive models

In online operation, estimates and their uncertainty can be used to determine the probability of process anomalies and to make in-situ adjustments to processing conditions, which can be added to the adjustments already made in the design phase.

5. Case study: Uncertainty in a stochastic model for L-PBF AM

In this case study, an Isotherm Migration Method (IMM) model, developed for laser cladding [34], is adjusted for use in L-PBF. The model provides a set of ordinary differential equations (ODEs) that describe the motion of isotherms on the surface of the powder bed. If one of these isotherms is assigned to the melting temperature (T_m), the model can be used to dynamically track the location of the solidification front and predict melt pool width. The method is similar to Rosenthal’s solution for temperature distribution due to a moving point source [22], but it allows the use of temperature-dependent material properties. Also, instead of solving for the distribution of temperature T(x,y,z,t), the system is solved for the half-widths y(T,t) of isotherms on the bed surface. The array of half-widths corresponds to a user-defined, uniformly-spaced temperature grid T = [T₁ T₂ … T_m … T_N], for ΔT < 0.

$$\frac{d{y}_1}{dt}=-\frac{2\mu {\alpha }_1}{y_1}\left(\frac{y_2}{y_2-y_1}\right)-\frac{A\times P}{\pi \rho C_p\Delta T{y}_1^2}\left(1+\frac{v}{2\mu {\alpha }_1}{y}_1\right)\text{exp}\left(-\frac{v}{2\mu {\alpha }_1}{y}_1\right)+\frac{v^2}{4\mu {\alpha }_1}\frac{T_1-A_l}{\Delta T}\left(y_2-y_1\right)$$ (1)

$$\frac{d{y}_i}{dt}=-\frac{\mu {\alpha }_i}{y_i}\left(\frac{y_{i+1}}{y_{i+1}-y_i}-\frac{y_{i-1}}{y_i-y_{i-1}}\right)+\frac{v^2}{8\mu {\alpha }_i}\frac{T_i-A_l}{\Delta T}\left(y_{i+1}-y_{i-1}\right),\text{ for }i=2,3,\dots ,m-1$$ (2)

$$\frac{d{y}_m}{dt}=-\frac{\mu {\alpha }_m}{\left(1+St{e}^{-1}\right)y_m}\left(\frac{y_{m+1}}{y_{m+1}-y_m}-\frac{y_{m-1}}{y_m-y_{m-1}}\right)-\frac{v^2}{4\mu {\alpha }_m\left(1+St{e}^{-1}\right)}\frac{A_l-2T_m+T_0}{\Delta T}(\frac{1}{y_{m+1}-y_m}+{\frac{1}{y_m-y_{m-1}})}^{-1}\left\{1+2St{e}^{-1}{\left[1+{\left(\frac{A_l-2T_m+T_0}{\Delta T}\frac{v}{2\mu {\alpha }_m}\right)}^2{\left(\frac{1}{y_{m+1}-y_m}+\frac{1}{y_m-y_{m-1}}\right)}^{-2}\right]}^{-1}\right\}$$ (3)

$$\frac{d{y}_i}{dt}=-\frac{\mu {\alpha }_i}{y_i}\left(\frac{y_{i+1}}{y_{i+1}-y_i}-\frac{y_{i-1}}{y_i-y_{i-1}}\right)+\frac{v^2}{8\mu {\alpha }_i}\frac{T_i-T_0}{\Delta T}\left(y_{i+1}-y_{i-1}\right),\text{ for }i=m+1,\dots ,N-1$$ (4)

$$\frac{d{y}_N}{dt}=-\frac{\mu {\alpha }_N}{y_N}\left(\frac{y_N-2y_{N-1}}{y_N-y_{N-1}}\right)-\frac{v^2}{4\mu {\alpha }_N}\left(y_N-y_{N-1}\right).$$ (5)

The computational model shown in equations (1) to (5) describes the evolution of the half-widths y=[y₁ y₂ … y_m … y_N], where each half-width y_i corresponds to a temperature T_i. In these equations, α_i denotes the thermal diffusivity evaluated at temperature T_i, A is the absorption coefficient, P is laser power, ρ is density, C_p is specific heat, v is scan speed $A_l=T_0-h_l/\left(C_p+2\mu {\alpha }_i{C}_p/v/y_m+2{\mu }^2{\alpha }_i^2{C}_p/y_m^2/v^2\right)$ is the apparent ambient temperature for the liquid phase, and Ste = −C_pΔT/h_l is the Stefan number.

The set of ODEs can be expressed in compact form as $\dot{x}=f(x,u)$, where x = [y μ] denotes the vector of state variables and u = [P v] the vector of control inputs. In this case, μ is the diffusion efficiency, a random variable used to correct for variable sideways thermal diffusion due to unmodeled process perturbations, allowing modeling error to be mapped to a simulation variable. Its value is set to 1 in nominal cases, when the melt pool is surrounded by fully-dense material. In the case of overhanging structures, for example, a decrease of thermal diffusivity toward the bottom improves heat transfer to the sides, increasing the value of μ. The proposed model is fast, returning a melt pool width prediction in 0.1 s when solved with MATLAB R2014b’s ode23 function running on an Intel Core i7-3770 CPU.

5.1. Uncertainty quantification

Uncertainty is quantified by comparing simulation results for fully-dense material with melt pool width measurements gathered using an EOSINT M270 system on an alloy 625 plate, as described by Montgomery [35]. Single bead tests were performed using different combinations of laser power and scan speed, both on a bare plate (no added powder) and one with a 20 μm layer of powder added. To ensure that diffusion efficiency does not introduce extra uncertainty, only scans on bare plate are compared with model predictions.

Error is approximated within the interval δ_model ∈ [E − u_val,E + u_val] centered around E = S–D, the bias between the simulation result S and experimental measurements D. Validation uncertainty u_val, which accounts for uncertainty from all sources, can be computed following $u_{val}=\sqrt{u_{num}^2+u_{\mathit{\text{input}}}^2+u_D^2}$ under the assumption that all error sources are independent.

The first steps toward quantification of modeling error are code and solution verification; in other words, assessing that the code is correct (free of bugs) and estimating the error in the numerical approximation. Code was verified with a manufactured solution and it was observed that the method converges to the analytical solution given by Rosenthal for constant material properties as ΔT → 0 and ΔT_max → ∞. Similar convergence studies were performed for predictions of melt pool width w (in μm) using temperature-dependent material properties. Results are presented in Fig. 4, where successive grid refinement was used to identify an order of accuracy of p = 1.9. The formal order of accuracy of the method was found at p = 3, but nonlinearities in the temperature-dependent properties insert iteration error reducing the effective value of p.

Fig. 4:

Numerical error as a function of mesh size.

Numerical uncertainty was quantified using Roache’s Grid Convergence Index (GCI) [30] for the prediction obtained with T₁ = 2560 °C and ΔT = −248 °C, corresponding to a grid of 10 isotherms. The GCI is an estimated 95 % uncertainty obtained by multiplying the absolute value of a Richardson extrapolation error by an empirically determined factor of safety. In this case, the numerical prediction for melt pool width for L-PBF of alloy 625 with 195 W and 800 mm/s is (127.3±2.7) μm (±2.12 %).

The second source of uncertainty comes from imperfect knowledge of input parameters, and the effect they may have on predictions. Six factors were selected for a Monte Carlo study to determine the propagation of uncertainty in inputs: laser power (P), scan speed (v), absorption coefficient (A), latent heat (h_l), melting temperature (T_m), and thermal diffusivity (α_i(T_i)). All factors are assumed to be normally distributed and are described in Table 1. Nominal values for temperature-dependent material properties were obtained with the TCN16 thermodynamic database within the Thermo-Calc software [36], while nominal absorptivity was identified experimentally. The variability assumed for most input parameters is based on the work of Ma [18], except for that of absorptivity which was enhanced to replicate the expected variability of this tuning coefficient.

Table 1:

Assumed distributions for normally distributed input parameters.

Input	Nominal	Std. dev. (% nominal)
P	195 W	2.5%
v	0.800 m/s	1.5%
A	0.6	25%
h l	2.97 × 105 J/kg	5.0%
T m	1320 °C	5.0%
α i	αi (Ti)	10.0%

A Monte Carlo approximation of the probability distribution of melt pool width is obtained following for 4000 samples. The resulting distribution, which resembles a normal distribution, is shown in Fig. 5, where the 95 % confidence interval is found in (131.6 ± 37.3) μm.

Fig. 5:

Normalized histogram of predicted melt pool widths.

The last source of uncertainty comes from the experiments used for validation. This study uses measurements that were taken in the middle of single bead tracks obtained for different combinations of power and speed, which were imaged using a Zeiss AxioVision AX10 optical microscope. The image was measured 15 times along the width at approximately equal spacing, as shown in Fig. 6. These 15 measurements were then averaged for each melt pool. In this study, measurements in the steady-state region have standard deviations of close to 5.2 μm, suggesting a ±10.4 μm confidence interval.

Fig. 6:

Sample image with measurement points marked (No powder added, 125 W and 600 mm/s) [35]. A similar process was repeated for various combinations of power and speed.

Measured melt pool widths are shown in Fig. 7 and compared to predictions obtained from the IMM model, only for data points close to nominal operating conditions (195 W and 800 mm/s). The region of calibration for this model was delimited between 150 W and 195 W, and 600 mm/s and 1000 mm/s. Assuming that all error sources are independent, validation uncertainty is estimated at (127.3 ± 38.8) μm (±30.5 %) for nominal operating conditions. Some observations can be made from these results:

1. Modeling uncertainty is relatively large, as expected due to the simplification of the thermal problem by assuming a point source instead of a distributed one. The absence of other physical phenomena considered important for melt pool dynamics, such as surface tension, also contributes to modeling error.

2. Numerical uncertainty is negligible compared to input or experimental error, even for coarse grids.

3. Input parameters, the error source most widely studied, have a significant contribution to model uncertainty (±29.3 %). This is partly due to the large uncertainty assumed for the absorption coefficient A, used as a tuning coefficient in this example.

4. Uncertainty due to unknown inputs depends on the confidence in the chosen set of input parameters. Different users may choose different input uncertainties, resulting in different prediction uncertainties.

5. The relatively large prediction uncertainty (±30.5 %) is compensated by the speed of the model.

6. Extrapolation of the modeling error to the other points in the region of calibration matches the obtained measurements, as observed in Fig. 8, where power and scan speed were kept constant. It is important not to extrapolate predictions outside the region of calibration of each predictive model, which is often not reported.

7. Model was validated by comparison with scans on bulk material, ignoring the addition of a powder layer. Montgomery reported that the effect of adding a layer of powder does not have a significant effect on melt pool width [35], but its effect was not quantified in this study.

8. The model in its current form can be used as a first step towards process planning by providing users with computationally-inexpensive predictions to explore the effects of laser power and speed in melt pool geometry.

Fig. 7:

Comparison of model predictions with experiments.

Fig. 8:

Melt pool width predictions (continuous line) and measurements (points) for single track scans with alloy 625

5.2. Bayesian estimation

Diffusion efficiency may be allowed to vary in time to account for unmodeled track-to-track and layer-to-layer interactions, which were ignored in the previous example. In this section, we present an example that illustrates how online thermographic monitoring could potentially be used to identify unmodeled dynamics and decrease uncertainty in melt pool width predictions.

The presented case, illustrated in Fig. 9, is designed to represent a horizontal overhanging plane which is scanned in a direction perpendicular to the solid-to-powder transitions. This case study, designed and published by Kruth et al. [8], showed that melt pool area increases threefold when going through this kind of overhangs. Synthetic data was generated to mimic this event by artificially perturbing μ and assuming that it varies instantaneously from 1 to 2.2 when melting on top of loose powder. The value of the assumed perturbed diffusion efficiency was chosen to increase steady-state melt pool width approximately by $\sqrt{3}$, assuming that melt pool length changes by the same ratio.

Fig. 9:

Case simulated in “perturbed” scenario.

In this study, it has been assumed that the isotherms corresponding to T_i = {576,824,1072}° C can be detected with thermographic sensors. To simulate measurement uncertainty, noise was added to the measured isotherm widths following a standard deviation of 26 μm, which corresponds to half the pixel width in a similar thermographic setting.

Process estimation, using a linear stochastic version of the IMM model and a Kalman filter [37], results in the estimates shown in Fig. 11, where the null hypothesis of normal operation (no overhang, H₀ : μ = 1) is rejected in favor of the alternative hypothesis of an anomaly (H_A : μ ≠ 1) in the shaded region. The perturbation in heat diffusion is detected between 0.47 ms and 1.35 ms, lagging from the 0.37 ms and 1.10 ms in which they occur in the simulation. Response speed, and accuracy and uncertainty in the estimates are expected to be dependent on the process and measurement uncertainty used for estimation, which were assumed in this example and will have to be adjusted in an experimental study to close the loop shown in Fig. 3.

Fig. 11:

Estimated diffusion efficiency and melt pool width. Predictions are plotted as continuous lines and 95% confidence intervals are given in dashed lines.

An important point to be observed is the low uncertainty in the melt pool width prediction even in the region of anomalous operation. Without online measurements, models would have to account for the potential variation in diffusion efficiency using large uncertainties for μ, increasing uncertainty in melt pool width predictions. For example, if the study to determine sensitivity to input parameters is repeated letting μ vary following μ ~ Unif[1.0,2.5], the obtained prediction is (193.6 ± 106.6) μm (±55.1 %), which is much larger than the confidence intervals reported in Fig. 11 (±4.0 μm).

6. Conclusions

As metal-based AM gains popularity, closer attention has been paid to the computational models developed to predict quality in manufactured components. Such predictions could be used to aid design and process planning by allowing engineers to make adjustments for improved quality. One aspect that has been traditionally ignored in these models is that, if they are to be used in model validation or for certification of parts, one must know how accurate and precise these models are. Uncertainty quantification presents a set of challenges that have often been ignored both by manufacturing and modeling engineers.

The series of steps that go from a physical process to a numerical representation involve successive assumptions in the mathematical models, model parameters, numerical scheme, and calibration data. It is important to quantify the relative effect of each error source to identify the ones that will result in the most significant reductions in prediction uncertainty.

A method to decrease modeling error, by mapping it to random simulation inputs that are identified in real time, is illustrated. Inclusion of random inputs requires that the assumed randomness is validated and adjusted, which can be done with adaptive filtering. The proposed estimation method could potentially be used for real-time control to maintain desired melt pool geometries in L-PBF even when process perturbations are detected.

Even though the case study presented in this paper is based on a low-order model, the same ideas can be extended to high-order models. The algorithms used for uncertainty quantification, however, are different. For instance, the high computational cost of Monte Carlo methods prevents their application in the propagation of uncertainty in input parameters. Methods based on the Karhunen-Loève expansion (e.g. polynomial chaos [38]) are often preferred in such scenarios.

Fig. 10:

Simulation of melt pool width through overhang assuming μ = 2.2.