Abstract

Melt pool modeling is critical for model-based uncertainty quantification (UQ) and quality control in metallic additive manufacturing (AM). Finite element (FE) simulation for thermal modeling in metal AM, however, is tedious and time-consuming. This paper presents a multifidelity point-cloud neural network method (MF-PointNN) for surrogate modeling of melt pool based on FE simulation data. It merges the feature representations of the low-fidelity (LF) analytical model and high-fidelity (HF) FE simulation data through the theory of transfer learning (TL). A basic PointNN is first trained using LF data to construct a correlation between the inputs and thermal field of analytical models. Then, the basic PointNN is updated and fine-tuned using the small size of HF data to build the MF-PointNN. The trained MF-PointNN allows for efficient mapping from input variables and spatial positions to thermal histories, and thereby efficiently predicts the three-dimensional melt pool. Results of melt pool modeling of electron beam additive manufacturing (EBAM) of Ti-6Al-4V under uncertainty demonstrate the efficacy of the proposed approach.

1 Introduction

Metallic additive manufacturing (AM) is a disruptive manufacturing technology, which makes three-dimensional (3D) metal components layer upon layer based on computer-aided design (CAD) models [13]. Compared to traditional manufacturing processes, AM has significant advantages in fabricating intricate and customized parts, including high-efficiency and cost-saving [4,5]. Nevertheless, the inferior process consistency and components quality hinder the wide application of metallic AM techniques, which are chiefly caused by propagation and aggregation of various uncertainty sources (e.g., microstructural heterogeneity, variation in properties of the powder, and fluctuation in temperature boundary) in AM process [6]. Therefore, it is necessary to develop effective uncertainty quantification (UQ) methods for the quality control of the metallic AM process under uncertainty.

Uncertainty quantification is generally adopted by constructing a correlation between quality and uncertainty sources on the quantities of interest (QoI), followed by process optimization [7]. Although UQ techniques have been widely applied for traditional manufacturing processes, their application in the metallic AM process is still at its early stage. To date, the available studies of UQ in AM are limited [812]. Currently reported UQ approaches in metallic AM can be categorized into experiment-based UQ and model-based UQ [13]. The experiment-based UQ relies heavily on repetitive experiments, which are not only tedious and time-consuming but also demanding of costly material consumption. Powered by advanced simulation techniques, the model-based UQ used a large volume of computational data to achieve quality control in a cheap yet effective manner. Among various models (e.g., energy consumption model, melting pool model, solidification model, and so on) adopted in the model-based UQ, the melt pool model is one of the decisive models to investigate the effects of the metallic AM process on the microstructure and mechanical properties of the as-fabricated AM parts [14]. The thermal phenomena in melt pools are changeable and complicated, which contains convective, conductive, and radiative heat transfer interactions between the part, material, powder, and energy source. Actually, these thermal aspects in metallic AM, which govern the thermal field (i.e., temperature distribution) in the components, are in turn a function of the material properties, component design, and the process parameters [15]. Hence, geometrical and thermal modeling of melt pools is essential for comprehensive UQ.

Recently, finite element (FE) models have been widely adopted for studying thermal characteristics of AM process at the part-level [1618]. Though FE-based models provide high-fidelity (HF) simulations of the AM process, the high computational cost is still the major disadvantage. To substitute the FE-based models and reduce the computational cost, surrogate models, including the Kriging model, support vector machine (SVM), and neural networks (NN), are successfully adopted in UQ [19]. Due to the high dimensionality of the thermal field, only a few examples of surrogates for thermal modeling in AM have been reported in the literature [2023]. Nath et al. [20] developed a thermal surrogate model using singular value decomposition (SVD) and the Kriging model, which achieve the superior prediction of the original 3D thermal field. Wang et al. [21] further adapted the SVD-Kriging to deal with surrogate modeling of 3D steady temperature field and microstructure statistical moments under uncertainty in electron beam additive manufacturing (EBAM). Without uncertainty, Ren et al. [22] designed a two-dimensional (2D) thermal field model in laser aided additive manufacturing, which uses recurrent neural networks (RNN) and deep neural networks. Zhu et al. [23] proposed a physics-informed neural network (PINN) framework for 3D AM processes modeling, which fed physical knowledge into the PINN to improve the prediction accuracy. Unfortunately, in the presence of uncertainties, since the Kriging model and NN-based model need abundant HF data to achieve desirable accuracy, these methods based solely on FE simulation data are still computationally tormenting.

Instead, multifidelity (MF) metamodels [24,25] provides a promising solution to address the above drawbacks, which can be trained with low-fidelity (LF) data and HF data simultaneously. During the MF modeling process, lots of LF samples are first employed to provide the initial trend of the response, then few HF samples are used to calibrate the LF model [26]. The popular MF approaches, including Monte Carlo (MLMC), co-Kriging, and Scaling-function-based MF, are already well-established in the engineering domain [2628]. However, these methods may fail to deal with high-dimensional problems when the LF and HF data feature stochastic independence and nonlinear correlation [29]. Lately, the deep learning method combined with the MF scheme has been a hot topic. Liu and Wang [30] first introduced the theory of MF into a physics-constrained neural network (MF-PCNN) to solve partial differential equations (PDEs) and perform materials modeling. The MF-PCNN blends two PCNN, including LF-PCNN and discrepancy artificial neural networks, to perform the final prediction. Conversely, De et al. [31] presented deep MF multilayer perceptron (MLP) for uncertainty propagation, which used transfer learning (TL) and the bi-fidelity weighted learning in one MLP to balance the accuracy and the computational cost. Meng and Karniadakis [32] developed an MF-PINN for inverse PDEs problems, which could effectively construct the nonlinear and linear correlations between the LF and the HF data. Inspired by Refs. [30] and [31], a TL-based MF-PINN for PDEs problems was proposed, which adapted to construct the mapping between the known LF data and unknown HF data [33].

Based on the discussion above, existing NN-based surrogate methods (i.e., deep neural networks, PCNN, PINN) with the MF scheme for application in 3D AM thermal modeling still has some critical limitations: (1) The governing PDEs used in PINN or PCNN are mainly built on the uncertainty-free assumptions, and will usually fail when it comes to uncertainties; (2) Most LF data used in MF approaches are derived from FE-based numerical models with coarser mesh than HF data. Nevertheless, the required computational cost is also relatively large; (3) The PINN adopted for 3D AM processes modeling in Ref. [23] is constructed by several over-parameterized fully-connected neural networks (FNN, i.e., MLP), which is notoriously memory-intensive with large inputs. These methods may achieve inferior performance if the input data are unstructured from irregular grids.

To overcome the above limitations, this paper proposes a multifidelity point-cloud neural network (MF-PointNN) for 3D thermal modeling in EBAM, which is based on one-dimensional convolutional neural networks (1D CNN) and FNN. Different from some of the previous researches, a 3D AM thermal analytical model is adopted as an efficient LF data generator under uncertainty for the first time. In MF-PointNN, the spatial coordinates of the mesh grid or nodes from HF and LF models are used as one of the inputs, while the corresponding temperatures are the outputs. Another input is the uncertainty parameters, including design variables and uncertainty sources parameters. In that case, the MF-PointNN can learn an end-to-end mapping between uncertainties, spatial positions, and temperature field to perform 3D thermal modeling. Meanwhile, the fine-tune method of TL is employed for the MF problem, allowing for a significant reduction of computational costs. The results of a numerical example show that the proposed surrogate method achieved the highest accuracy compared with other surrogate methods with a similar computational cost.

The rest of the paper is structured as follows: The background of thermal models for EBAM is introduced in Sec. 2. Following that, Sec. 3 briefly reviews current Kriging-based melt pool surrogate modeling methods. Next, Sec. 4 presents the proposed method. The experimental results are presented and discussed in detail in Sec. 5. Finally, the conclusions are given in Sec. 6 along with future works.

2 Thermal Models for Electron Beam Additive Manufacturing

This section briefly reviews the thermal models, including the analytical model (i.e., Rosenthal's model) and numerical model (i.e., finite element-based model), for modeling and simulation of the temperature field during the EBAM process.

2.1 Rosenthal's Model.

Rosenthal's model was first developed for analytical thermal modeling in fusion welding [34]. Then, Rosenthal [35] further adopted the Rosenthal equation to predict the quasi-steady-state temperature distribution. Benefiting from various similarities between welding processes and AM, Rosenthal equation and its variants have been widely extended to AM [35]. The Rosenthal equation can be expressed as follow
$Tr(x,y,z)=Tpre+η·P·12πkRexp(−v(x+R)2α)R=(x2+y2+z2)0.5α=kρ·Cp$
(1)

where $Tr(x,y,z)$ is the temperature at coordinate $(x,y,z)$, $Tpre$ is the preheating temperature (unit: K), $η$ is the absorption efficiency of beam power, $P$ is the power of electron beam, $k$ is the thermal conductivity (unit: W·m−1·K−1), $R$ is the distance from the of electron beam, $v$ is the beam velocity (unit: m·s−1), $α$ is the thermal diffusivity (unit: m2·s−1), $ρ$ the density of the powder materials (unit: kg·m−3), and $Cp$ is the specific heat capacity (unit: J·kg−1·K−1).

To use the Rosenthal equation in EBAM, three assumptions are made: (1) the thermal properties are independent of temperature; (2) temperature predicted at the origin of the beam is infinite; (3) the maximum temperature does not exceed the liquidus temperatures [36]. In this study, Rosenthal's model is employed to obtain LF data [37,38].

2.2 Finite Element-Based Model.

In this study, an FE-based heat transfer model incorporating a moving heat source is utilized to predict the in-process temperature distribution [21], which are then extracted as HF data. Specifically, the tempo-spatial evolution of temperature field with external heat input is governed by
$∇·(k∇T)+ρQe=∂ (ρcpT)∂ t$
(2)
where $T(x,y,z,t)$ is the tempo-spatial temperature field, $k$ is thermal conductivity, $ρ$ is density and $cp$ is specific heat. Of special importance is the external heat input from the moving electron beam, $Qe$, which is described using the Gaussian distribution equation as the following [39]
$Te(x,y,z,t)=ηP4ln(0.1)πd2zeexp (4ln(0.1)((x−v⋅t)2+y2)d2){−3(zze)2−2(zze)+1}ze=2.1×10−5×Ve2ρ$
(3)

where $η$ is the power absorption efficiency of the powder bed, $P$ is the nominal power of electron beam, $d$ is the diameter of the electron beam, $v$ is the beam velocity and $ze$ is the absolute penetration depth of electron beam associated with the acceleration voltage, $Ve$.

The boundary conditions of the FE-based thermal model, as graphically illustrated in Fig. 1, are summarized as follows: (1) the initial boundary condition of substrate and deposits is $T0=Tpre$, where $Tpre$ is the preheating temperature; (2) the types of heat transfers at the surface are convective and radiative; (3) in view of the limited thermal conductivity of loose powders, adiabatic conditions are imposed on the sides of the printing part, thus with raw powders physically ignored in the simulation; and (4) evaporation is not incorporated in the FE-based thermal model, as normally did in AM thermal models. The above FE-based thermal model is realized in ABAQUS 6.10 using a custom DFLUX user subroutine [40]. The extraction of temperature field (upon reaching steady-state) as HF data is automated by Python script.

3 Current Surrogate Modeling Methods of Melt Pool Based on Kriging

In this section, we briefly review two existing methods based on Kriging for melt pool surrogate modeling. These two methods will be compared with the proposed methods in the result section.

3.1 Melt Pool Surrogate Modeling Using Kriging and Singular Value Decomposition.

The basic principles of Kriging and SVD for melt pool surrogate modeling are provided in this section. More details are available in Refs. [20,21]. Figure 2 presents the overall flowchart of the thermal surrogate modeling using the Kriging and SVD methods.

For given the controllable variables $d$ (e.g., preheating temperature, power of electron beam, and beam velocity) and the uncontrollable uncertain parameters $θ$ (e.g., power absorption efficiency of powder bed, thermal conductivity, specific heat capacity, and density of the powder materials), the steady temperature field can be expressed as $T(d, θ,s)$, where $s$ is the $(x, y,z)$ coordinates of all nodes. Then temperature field under different conditions $T(d(i),θ(i),s),i=1,2,…,Nsample$ are generated through HF FE-based model, where $(d(i),θ(i)),i=1,2,…,Nsample$ are the $Nsample$ training points using the sampling approach. For dimensionality reduction, the SVD is first employed to approximate the raw thermal field data matrix. The data matrix $T(d, θ,s)$ is decomposed using SVD as $T(d, θ,s)=UMVT$, where $U$ is a $Nsample×Nsample$ orthogonal matrix, $V$ is a $np×np$ orthogonal matrix, $np$ is the number of data points, $M$ is a $Nsample×np$ diagonal matrix with singular values. For given matrix as $γ=UM$, the raw thermal field data matrix can be expressed as follow:
$T(d(i), θ(i),s)≈γ(i)VT≈∑j=1m γj(i)τj(s),∀i=1,2,…,Nsample$
(4)

where $γj(i), j=1,2,…,m$ (the element of $γ$ at $i$-th row and $j$-th column) are the responses in the latent space, $m$ is the number of significant features used in SVD; $τj(s), j=1,2,…,m$ ($j$-th row of $VT$) are the significant features.

After that, the Kriging models [41] are built based on $γj(i), j=1,2,…,m$, which as expressed by:
$γj(i)≈ĝj(d(i),θ(i)),∀i=1,2,…,Nsample$
(5)

where $ĝ(·)$ is the Kriging model.

In this way, the SVD-based Kriging model is built, for given test point $(dt, θt)$, the predicted thermal field response of the SVD-based Kriging model can be given by:
$T̂(dt, θt,s)≈∑j=1m μĝj(dt, θt)τj(s),∀j=1,2,…,m$
(6)

where $μĝj(dt, θt)$ is the mean prediction of the $j$-th latent space Kriging model, and $T̂(dt, θt,s)$ is the predicted thermal field response on test point $(dt, θt)$.

3.2 Multifidelity Surrogate Modeling Using Kriging and Singular Value Decomposition.

A standard MF SVD-Kriging using additive scaling function [26,42] will be illustrated in this section. The scaled LF SVD-Kriging model is first built based on inputs and response of LF data. Then, the discrepancies between the HF model and the scaled LF SVD-based Kriging model are calculated, using inputs and response of few HF samples. In this context, the scaling function is constructed using the discrepancies. Finally, the MF SVD-Kriging is proposed to predict the thermal field, which is combined with the scaled LF SVD-Kriging model and the scaling function. The details of the standard MF SVD-Kriging for melt pool surrogate modeling can be summarized as follows,

Firstly, LF training points $(d(li),θ(li)),li=1,2,…,NHFtrain$ and HF training points $(d(hi),θ(hi)),hi=1,2,…,NLFtrain$ are collected, where $NHFtrain$ and $NLFtrain$ are the number of training points in HF dataset and LF dataset, respectively. Then, HF thermal responses $THF(d(hi),θ(hi),s),hi=1,2,…,NHFtrain$ and LF thermal responses $TLF(d(li),θ(li),s),li=1,2,…,NLFtrain$ are developed by HF and LF model, respectively.

Secondly, the thermal field of training LF data and HF data matrix using SVD are approximated as follows:
$THF(d(hi),θ(hi),s)≈∑j=1m γHFj(hi)τHFj(s),∀hi=1,2,…,NHFtrain$
(7)
$TLF(d(li),θ(li),s)≈∑j=1m γLFj(li)τLFj(s),∀li=1,2,…,NLFtrain$
(8)

where $γHFj(hi), j=1,2,…,m$ and $γLFj(li), j=1,2,…,m$ are respectively the responses in the latent space of HF data and LF data, $m$ is the number of significant features used in SVD; $τHFj(s), j=1,2,…,m$ and $τLFj(s), j=1,2,…,m$ are the significant features of HF data and LF data, respectively.

Thirdly, the LF SVD-Kriging model is built using $γLFj(li), j=1,2,…,m$, as expressed by
$γLFj(li)≈ĝLFj(d(li),θ(li)),∀li=1,2,…,NLFtrain$
(9)
$T̂LF(dt, θt,s)≈∑j=1m μĝLFj(dt, θt)τLFj(s),∀j=1,2,…,m$
(10)

where $ĝLF(·)$ is the Kriging model of LF data; $μĝLFj$ is the mean prediction of the $j$th latent space LF SVD-Kriging model, and $T̂LF(dt, θt,s)$ is the predicted thermal field response of LF SVD-Kriging model on test point $(dt, θt)$.

Based on that, the discrepancies between the HF model and the scaled LF SVD-based Kriging model are calculated by,
$TD(d(hi),θ(hi),s)≈THF(d(hi),θ(hi),s)−ω·T̂LF(d(hi),θ(hi),s),≈∑j=1m γDj(hi)τDj(s),∀hi=1,2,…,NHFtrain$
(11)

where $TD(d(hi),θ(hi),s)$ is the discrepancies between the HF model and LF model at HF data $hi$, $ω$ is the LF scale factor; $γDj(hi), j=1,2,…,m$ is the responses in the latent space of discrepancies, $τDj(s), j=1,2,…,m$ is the significant features of the discrepancies. For original MF metamodel, the LF scale factor is set as 1.

Then, the scaling function is constructed using the $γDj(hi), j=1,2,…,m$, which shown as
$γDj(hi)≈ĝDj(d(hi),θ(hi)),∀hi=1,2,…,NLFtrain$
(12)
$T̂D(dt, θt,s)≈∑j=1m μĝDj(dt, θt)τDj(s),∀j=1,2,…,m$
(13)

where $ĝD(·)$ is the Kriging model of discrepancies; $μĝDj$ is the mean prediction of the $j$-th latent space of the scaling function, and $T̂D(dt, θt,s)$ is the predicted thermal field response of the scaling function on test point $(dt, θt)$.

Finally, the standard MF SVD-Kriging with the scaling function and the scaled LF SVD-Kriging model is constructed as follow:
$T̂MF(dt, θt,s)≈1·T̂LF(dt, θt,s)+T̂D(dt, θt,s)$
(14)

where $T̂MF(dt, θt,s)$ is the predicted thermal field response of the standard MF SVD-Kriging model on test point $(dt, θt)$.

4 Proposed Method

This section describes the proposed method for constructing multifidelity surrogates for predicting the thermal field in AM. The dataset generation under uncertainty is discussed in Sec. 4.1, Sec. 4.2, and Sec. 4.3 present the structure of the PointNN and the details of the proposed MF-PointNN, respectively.

4.1 Dataset Generation Under Uncertainty.

In this study, the controllable variables $d$ and the uncontrollable uncertain parameters $θ$ of the melt pool is similar to that in Ref. [21]. The controllable variables $d$ include the preheating temperature $Tpre$, the power of electron beam $P$, and the beam velocity $v$. The uncontrollable uncertain parameters $θ$ include the absorption efficiency of beam power $η$, the thermal conductivity $k$, the specific heat capacity $Cp$, and the density of the Ti-6Al-4V powders $ρ$. Table 1 shows the distribution of controllable variables and uncontrollable uncertain parameters.

For given $d$ and $θ$ (i.e., inputs), the steady thermal field (i.e., outputs or response) is defined as $T(d, θ,s)$, where $s∈Ωxyz$ represents the all spatial coordinates of the nodes. In this research, the Latin hypercube sampling approach [43] is used to generate 280 sample points for $d$ and $θ$. After that, the HF thermal simulations using the FE-based model (i.e., Sec. 2.2) are performed for all sample points and we obtain the steady thermal fields denoted as $THF(d(hi), θ(hi),s),hi=1, 2,…, 280$. Similarly, the LF dataset $TLF(d(li), θ(li),s),li=1, 2,…, 280$ can be generated using Eq. (1). The generation of all the LF dataset only requires several seconds (based on Intel Core i7-9700K), which shows the high efficiency of analytical model. Notably, the EBAM of cuboid-shaped test part is simulated on top of a build plate with the dimension of 1.5 $×$ 1.5 $×$ 12 mm and layers along the z-axis. The whole part is transformed into a set of 18,746 (13 $×$ 14 $×$ 103) discrete nodes, then, the spatial coordinates of these nodes are saved as $s∈Ωxyz$. Figures 3 and 4 show comparisons of steady thermal field predictions from LF and HF models for two samples. Following that, Fig. 5 presents the absolute differences between HF and LF predictions for the two samples depicted in Figs. 3 and 4.

For the present problem, the thermal field around the melting pool and in the active region of the laser is critical, while the far-field data are relatively unimportant. In this study, a set of 4,096 discrete nodes around QoI are randomly sampled according to the temperature gradient. For the sampled data, the range of grid point distribution is that $x∈[0, 0.62 mm]$, $y∈[0.983 mm, 1.5 mm]$, $z∈[0.824 mm, 10.2 mm]$. Therefore, the LF dataset ($TLF(d(li), θ(li),sL(li)),li=1, 2,…, 280,sL(li)∈s$) and HF dataset ($THF(d(hi), θ(hi),sH(hi)),hi=1, 2,…, 280,sH(hi)∈s$) are successfully built. Figures 6 and 7 show comparisons of critical steady thermal fields from HF dataset and LF dataset for two samples. Figure 8 depicts the absolute differences of critical thermal fields between the HF and LF predictions for the two samples.

4.2 Point-Cloud Neural Network.

As illustrated in Fig. 9, a PointNN is developed to perform the prediction of the 3D thermal field under uncertainty. The inputs of the model are unstructured grid points $Npoints×3$ including the geometry information and uncertainty parameters $1×Npara$. The output of the model is the computed thermal field $Npoints×4$, where each unstructured grid point is concatenated with the corresponding temperature.

Indeed, the PointNN is a modified architecture that uses multiple transformation networks (T-Net) and Shared MLP, which are derived from the PointNet [44]. PointNet provides an end-to-end solution for classification and scene segmentation of point clouds, which has the advantages in efficiency and easy implementation compared to 3D CNN or other multiview-based methods [45]. Lately, Kashefi et al. [46] first introduced the PointNet into the surrogate model of computational fluid dynamics (CFD), which achieves efficient prediction of 2D fluid flow fields on irregular geometries. To our best knowledge, none of the PointNet-based methods has been used in AM domain.

The PointNN is actually an encoder-decoder structure, the encoder branch consists of two T-Net, five shared MLP and a max-pooling layer. The shared MLP is applied with T-Net interchangeably, which is employed for local features aggregation with transformation-invariant characteristics. After a max-pooling layer, the global features of geometry at the QoI are obtained. To merge the local and global knowledge, the per point local features $Npoints×64$ after the second T-Net are concatenating with the global features, which is employed for the generation of combined point features $Npoints×1088$. In addition, to obtain new point features $Npoints×(1088+Npara)$ under uncertainty, the parameters $1×Npara$ of each point are concatenated with combined point features. In the decoder branch, the features $Npoints×(1088+Npara)$ are fed into three Shared MLP with output sizes $(512, 256, 128)$. Then, the extracted features are fed into one Conv1D layer and concatenated with input points $Npoints×3$ to produce the predicted thermal field. Herein, the Sigmoid activation function is used in the final Conv1D layer, which limits the outputs in the range of $[0,1]$.

The details of the shared MLP and T-Net are shown in Fig. 10. The Shared MLP contains only one one-dimensional convolution (Conv1D) layer with Batch Normalization [47] and Rectified Linear Unit (ReLU) [48] activation function, Conv1D layer has $dout$ filters with a kernel of $din×1$ and a stride of $1×1$, where $din$ and $dout$ are respectively the dimensionalities of input features and output features. For up-sampling from inputs features $Npoints×din$ to output features $Npoints×dout (dout>din)$, the trainable parameters of the Conv1D layer are $din×dout$, while that of the FNN are $din×Npoints×dout×Npoints$. Hence, the shared MLP shared the feature and parameter of each point, which has the advantage of reducing the computational costs.

Furthermore, the T-Net is composed of three Shared MLP modules, one max pooling, and three FNN, which is served as a mini PointNet to perform the prediction of the transformation matrix. Then the transformation matrix is applied to the input features for joint alignment. In this way, the learned representation by point clouds is invariant to geometric transformations, which is beneficial for predicting the 3D thermal field of the melt pool.

4.3 Multifidelity-Point-Cloud Neural Network With Transfer Learning.

After the dataset is successfully built, the problem of the 3D thermal field approximating is solved by constructing surrogates using training data of both the LF dataset and HF dataset. On the concept of TL [49], an LF PointNN is first optimized and built based on inputs and responses of LF training samples. Then, the parameters of LF PointNN are partly transferred to a new PointNN. After that. the new PointNN is updated and fine-tuned using the smaller HF data to obtain the optimal MF-PointNN. During the training process of the new PointNN, the parameters transferred from LF PointNN are freeze. Thus, the features learned from the LF model are reserved in the new PointNN, which is beneficial for the promotion of the convergence from the beginning and reduction of the computation costs. Finally, the optimized MF-PointNN is proposed to predict the 3D thermal field. The general procedures of the proposed method can be summarized as follows:

• Step 1. The MF dataset is built under uncertainty, which includes the LF dataset and HF dataset (i.e., Sec. 4.1).

• Step 2. Randomly split the MF dataset into two subsets of training and testing.

$THF(d(hi), θ(hi),sH(hi)),hi=1,2,…,NHFtrain$ and $TLF(d(li), θ(li),sL(li)),li=1,2,…,NLFtrain$ are firstly confirmed, where $NHFtrain$ and $NLFtrain$ are the sample size of train subset in HF dataset and LF dataset, respectively. Then, $THF(d(hi),θ(hi),sH(hi)),hi=1,2,…,Ntest$ is selected, where $Ntest$ is the sample size of test subset.

• Step 3. Optimize the LF PointNN based on $TLF(d(li), θ(li),sL(li)),li=1,2,…,NLFtrain$.

The optimization problem can be formulated as Eq. (15), which can be solved by a gradient descent-based algorithm.
$find {ωL,bL}min JLJL=∑i=1n Lloss(T̂LF(d(li), θ(li),sL(li)),TLF(d(li), θ(li),sL(li))),∀li=1,2,…,NLFtrain$
(15)
where ${ωL,bL}$ is the optimal parameters of PointNN, $Lloss(·)$ is the loss function, the $JL$ is the total loss to be minimized, and $T̂LF(d(li), θ(li),sL(li))$ is the predicted thermal field response of the LF PointNN model at the training data $li$.
The LF PointNN with the optimal parameters ${ωL,bL}$ is constructed as follow,
$T̂LF≈MLF̂((d, θ,s);{ωL,bL}),$
(16)
where $MLF̂(·)$ is the optimal LF PointNN, $T̂LF(d, θ,s)$ is the predicted thermal field response of the LF PointNN model.
• Step 4. A new PointNN $MNeŵ((d, θ,s);{ωLpart+ωnew,bLpart+bnew})$ is constructed, where $MNeŵ(·)$ is the new PointNN, ${ωLpart,bLpart}$ is the parameters transferred from the LF PointNN, and ${ωnew,bnew}$ is the parameters need to be optimized.

Based on above remark, the new PointNN is updated and fine-tuned using the smaller HF data to obtain the optimal MF-PointNN, which can be expressed as
$find {ωnew,bnew}min JHJH=∑hi=1NHFtrain Lloss(T̂HF(d(hi), θ(hi),sH(hi)),THF(d(hi),θ(hi),sH(hi))),∀hi=1,2,…,NHFtrain$
(17)
Then, the MF-PointNN with the optimal parameters ${ωM,bM}$ is constructed as follow,
$T̂MF≈MMF̂((d, θ,s);{ωM,bM})ωM={ωLpart+ωnew}bM={bLpart+bnew}$
(18)
where $MMF̂(·)$ is the optimal MF-PointNN, $T̂MF(d, θ,s)$ is the predicted thermal field response of the MF-PointNN model.
• Step 5. At the testing stage, the testing samples $THF(d(hi), θ(hi),sH(hi)),i=1,2,…,Ntest$ are input into the MF-PointNN to get the final predicted 3D thermal field.

Figure 11 summarizes the overall procedure for the training of MF-PointNN with transfer learning.

5 Results and Discussion

In this section, the steady temperature field developed during electron beam scanning of a long track (i.e., Sec. 4.1) is used as a case study, to validate the efficiency and effectiveness of the proposed method. All experiments were performed on Windows 10 with 32GB RAM, Intel Core i7-9700K processor, and an NVIDIA GeForce RTX 2080 GPU. The proposed method is implemented in Python 3.8.5 with PyTorch 1.7.1.

5.1 Training Details.

The main goal of the proposed method is to predict temperature in each $(x,y,z)$ coordinate. Hence, the mean square error is defined as the loss function $Lmse$ to be optimized, which can be formulated as:
$Lmse=1Ns×Np∑i=1Ns∑j=1Np(Tj,i−T̂j,i)2$
(19)

where $Ns$ is the number of training samples in a mini-batch, $Np$ is the number of $(x,y,z)$ coordinates, $Tj,i$ and $T̂j,i$ are respectively the true and predicted thermal history.

During the training process, Rectified Adam (RAdam) optimizer [50] combined with LookAhead schemes [51] is used to adjust and optimize the loss function $Lmse$, which has better learning stability compared to Adam optimizer [52]. The default parameters of RAdam and LookAhead are selected, with the initial learning rate (LR) of $3×10−2$ and the batch size of 32. To improve the performance of PointNN, A LR scheduler with cosine annealing [53] is utilized to decay the LR for each batch (shown in Fig. 12), as follows:
$ηt=ηmin+12(ηinitial−ηmin)(1+cos (TcurTmaxπ))$
(20)

where $ηt$ is the current LR, $ηinitial$ is the initial LR, $ηmin$ is the minimum of the LR, $Tmax$ is number of epochs in each restart, and $Tcur$ is the number of current epochs. In this research, $Tmax$ and $ηmin$ is fixed as 100 and $3×10−6$, respectively.

Besides, the temperature of each point is normalized and scaled in a range of $[0, 1]$, which is given by,
$Tnormal=T−min(T)max(T)−min(T),$
(21)

where $Tnormal$ is the normalized temperature histories, $T$ is the original temperature histories.

In this case, the sample size of test subset $Ntest$ is defined as 28. Moreover, the mini-batch size of training and testing are fixed at 32 samples and 1 sample, respectively. The total number of training epoch is set as 5000 in our experiments. To avoid contingency in the testing process, all experiments are conducted 10 times and the average values are reported as the final results for analysis.

5.1.1 Evaluation Metrics.

As mention in Sec. 5.1, the proposed method is aims to predict the thermal field in the component. Therefore, the max absolute error (MAE) and root mean square error (RMSE) are defined as effectiveness metrics, which are given by,
$MAE=max|Ti−T̂i|,i=1,…,Ntest$
(22)
$RMSE=1Ntest∑i=1Ntest(Ti−T̂i)2$
(23)

where $Ntest$ is the number of testing samples, $Ti$ and $T̂i$ are respectively the true thermal history and predicted thermal history of test sample $i$.

The MAE reflects the local accuracy of the model, and the RMSE evaluates the global accuracy. The lower the values of RMSE and MAE are, the more accurate the metamodel is. In addition, the efficiency is measured by computational budget and computation (i.e., testing) time. The computational budget is defined as the number of HF samples used during the training process [27]. Computation time is defined as the average CPU time per sample during the test process.

5.2 Comparison of Different Methods.

In this section, the proposed method is compared with FE analysis, SVD-Kriging [20] with HF data (i.e., Sec. 3.1), and standard MF SVD-Kriging (i.e., LF scale factor fixed as 1 [26], reviewed in Sec. 3.2). These methods are implemented in MATLAB R2020b [54] with the Statistics & Machine Learning Toolbox. In this case, 252 LF data and HF data randomly chosen to train surrogates, the number of significant features used in SVD-Kriging is defined as 10. Notably, randomly sampling without replacement is used, which makes testing data are unseen in the training data.

As discussed in Sec. 4.1, the temperatures in data are randomly sampled according to the gradient, which is randomly scatter and unsorted. The Kriging-based method may fail to construct using this kind of data. For a fair comparison, the sorted data is also used in the experiment. The comparison results of different methods for 3D thermal modeling on the test part are listed in Table 2. As shown in Table 2, although the SVD-Kriging-based methods achieve superior performance with sorted data, these methods fail to predict the thermal field with unsorted data. However, the FE data is usually in form of random scattered or complex grids in practical application, which is a challenge for conventional surrogate models.

Conversely, considering unsorted data, the proposed PointNN has the best performance in both global accuracy and local accuracy with an MAE of 129.47, and an RMSE of 13.01 using 252 HF data. The proposed MF-PointNN also achieves the superior performance with an MAE of 167.23, and an RMSE of 14.38 using 140 HF data and 252 LF data, while the PointNN only has the performance with a MAE of 287.96, and a RMSE of 43.55 using 140 HF data.

Additionally, as shown in Table 2, the computation time of proposed the MF-PointNN is only 3.6 ms, while those of SVD-Kriging and standard MF SVD-Kriging are respectively 0.51 ms and 1.06 ms. Thus, the proposed method shows equal advantages to the conventional surrogate model in computation time. Furthermore, the computation time for 3D thermal field of a surrogate model is within 4 milliseconds, as opposed to over 3,000,000 ms with FE analysis.

A testing example for comparison between surrogate model prediction and FE thermal model is shown in Fig. 13. The input parameters of this example are {$Tpre=993.357,P=438.756,v=0.484,η=0.720,k=4.789,Cp=530.050,ρ=4299.37$}. As evident from Fig. 13, the proposed method has superior performance in prediction of temperature field. The temperature concentration and distribution around the melt pool are clearly demonstrated. Figure 14 shows the absolute errors of surrogate model predictions compared with FE-based thermal model. Obviously, from Fig. 14, the PointNN and MF-PointNN achieve lower absolute errors even near the electron beam focus, while the Kriging-based method obtains relatively large prediction errors. Figure 15 shows the probability density function (PDF) of absolute prediction errors of different methods. It indicates the absolute prediction errors of PointNN are much smaller than its counterpart from the SVD-Kriging based methods. It should be noted that the accuracy of PointNN with 252 HF data is similar to that of MF-PointNN. But MF-PointNN only uses 140 HF data samples. This demonstrates the effectiveness of the proposed MF-PointNN method in reducing the number of required MF data samples in surrogate modeling. Consequently, the proposed method can be an effective and efficiency substitute for FE-based thermal model.

6 Conclusions

This paper proposes a novel MF-PointNN method combining PointNN and TL-based MF schemes for efficient and accurate surrogate modeling of 3D thermal field in metallic AM. PointNN has shown a powerful high-dimensionality modeling ability, while a TL-based MF scheme has been proposed to reduce computational cost. Firstly, the Rosenthal's analytical model is adopted to generate cheap LF dataset under uncertainty, and the FE-based numerical model is used for the HF dataset generator. After dataset generation, a LF-PointNN is adjusted and optimized by LF dataset, then, the LF-PointNN is fine-tuned and updated using the smaller HF data by TF-based MF scheme. In this way, MF-PointNN is construct and employed as an automatic feature extractor to improve capability of the information from both LF data and HF data. Finally, the proposed method has been validated to predict the 3D thermal field around melt pool on EBAM of Ti-6Al-4V in cuboid geometry. Numerical experimental results suggest that the proposed method can not only improve the prediction performance of 3D thermal field but also effectively reduce the computational costs.

In our future work, we will focus on investigating different analytical models [55], aiming to generate a more reliable LF dataset. Meanwhile, an advanced point-cloud learning method [4850] will be adapted in our proposed method, which makes the 3D thermal field modeling more efficient and more accurate. We are interested in merging the proposed model with comprehensive UQ and process optimization [5658].

Funding Data

• Michigan Institute of Data Science (MIDAS).

• National Science Foundation (NSF) (Grant No. CMMI-1662864; Funder ID: 10.13039/100000001).

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