Meta-modeling of high-fidelity FEA simulation for efficient product and process design in additive manufacturing

Abstract

Finite element analysis (FEA) has been widely adopted to identify potential defects in additive manufacturing (AM) processes. For personalized product realization, it is necessary to validate a number of heterogeneous product and process designs before or during manufacturing by using FEA. Multi-fidelity FEA simulations can be readily implemented with different capabilities in terms of simulation accuracy. However, due to its complexity, high-fidelity FEA simulation is time-consuming and decreases the efficiency of product realization in AM, while low-fidelity FEA simulation has fast computation speed yet limited capability. Hence, our objective is to improve the capability of FEA by providing an efficient data-driven model. In this research, a Gaussian process-constrained general path model is proposed to approximate the high-fidelity FEA simulation results based on low-fidelity results voxel-by-voxel. The proposed model quantifies the heterogeneous discrepancies between low- and high-fidelity FEA simulation results by incorporating the product design information (e.g., Cartesian coordinates of deposition sequence) and process design information from inputs of FEA simulation (e.g., input heat). Therefore, it enables the validation of new product and process designs based on the simulation results with the desired capability in a timely manner. The advantages of the proposed method are illustrated by FEA simulations of the fused deposition modeling (FDM) process with two levels of fidelity (i.e., low- and high-fidelity).

Introduction

In recent years, additive manufacturing (AM) becomes a driving force for personalized product realization [1], such as customized brackets in aircraft and automotive manufacturing [2], and biomedical devices conforming to patient anatomy [3]. Despite advancements in AM for facilitating personalized manufacturing, it is important to validate a number of heterogeneous products and process designs in a timely manner. In the literature, there are usually two ways to validate AM designs: i) traditional run-to-run studies to physically quantify the quality/functional performance of the product and process designs through the design of experiments (DOE) [4], [5], [6]; and ii) high-fidelity simulations to predict the corresponding quality/functional performance of designs [7], [8], [9]. The DOE approach can be inefficient and expensive as different customized AM designs have different underlying mechanisms [10]. For each individual design, one needs to physically conduct DOE to collect sufficient samples to estimate the model. Alternatively, simulations have been widely adopted to identify potential defects in AM processes by simulating the physical mechanisms of the manufacturing processes [7], [8], [9]. However, a high-fidelity FEA simulation can be computationally expensive and thus cannot be easily used to validate AM designs in a timely manner. On the other hand, a low-fidelity simulation provides affordable computation time. The accuracy of low-fidelity simulations might not be satisfactory due to their low meshing resolutions.

In this work, the objective is to improve the accuracy of the low-fidelity FEA simulation results by predicting high-fidelity simulation results to facilitate and accelerate the AM design validation with heterogeneous product and process features (e.g., different geometries or process settings). Let us take the example of the thermal distribution analysis of the fused deposition modeling (FDM), which is a material extrusion AM process. An infrared camera cannot capture the external spatial-temporal thermal distribution of the product since the extruder will block the vision of infrared camera during the process. In addition, the internal thermal distribution is not measurable. However, the thermal distribution is very important in the FDM process as it is closely related to the quality/defect of AM products, such as residual stress [11] and geometric deviation [12]. In such a case where sensing capability is limited, FEA simulation can help the validation of product and process designs as well as understanding the process-quality relationship in AM processes [13]. There are different levels of fidelity for FEA simulations, as compared briefly in Table 1 for a 3D transient thermal field evolution FEA simulation in FDM with the same design.

A high-fidelity FEA simulation can be computationally intensive (more than 20 hours for a single product with one specific process design), which cannot be affordable to validate a large number of personalized AM designs with heterogeneous geometries and process settings. Thus it is useful to develop a model that can approximate the high-fidelity FEA simulation results based on the low-fidelity FEA results. Such a model can boost the capability of low-fidelity FEA simulation and efficiently facilitate the personalized product and process design validation (i.e., identifications of manufacturability or functionality for varied geometries under different process setting combinations) in AM.

There are a few challenges in modeling the relationship between low- and high-fidelity FEA simulation results. First, along the deposition sequence, the length of thermal history for each location varies in an AM process. For example, Fig. 1 demonstrates a 3D transient FEA for the thermal field evolution results of three locations (i.e., A, B and C) on a square product which is modified based on the NIST standard part [14] and built by PLA in a FDM process [13]. Arrows represent the deposition sequence of the FDM process. In Fig. 1, it can be observed that each location has a different length of thermal history. This is because that each location on the product is deposited at a different time point following the pre-defined deposition sequence. According to the sequence shown in Fig. 1(a), when location A has been deposited on the platform, B and C have not been deposited yet. This inconsistency of thermal history sequences length among different locations restricts many existing data-driven methods [15], [16] with the assumption of the same sequence length. Second, the discrepancies between low- and high-fidelity FEA simulation results are heterogeneous among different locations over time. As shown in Fig. 1(a), in terms of geometries in one layer, the Euclidean distance between point A and point B is the same as that between point A and point C. However, it can be found that the discrepancy patterns between low- and high-fidelity FEA simulation results vary along the deposition sequence. For example, the predicted thermal evolution histories from low-fidelity FEA simulation results in Fig. 1(b) and (c) have relatively low errors by comparing with high-fidelity results. But for Fig. 1(d), it can be observed at the beginning of the predicted thermal evolution history, the accuracy of the low-fidelity simulation is very inaccurate. Moreover, in the middle of these three predicted thermal histories, it can be seen that for Fig. 1(b) and (c), the prediction results from the high-fidelity simulation are larger than the low-fidelity simulation, while for Fig. 1(d), it is opposite. These heterogeneous discrepancies are significant since the numerical value of the difference between low- and high-fidelity simulation is larger than the simulation accuracy (i.e., 5 Kelvin (K) in terms of root-mean-squared-error) [17]. Therefore, even if two locations have the same Euclidean distance with the same reference point, the discrepancies are heterogeneous and cannot be modeled by a stationary process. This heterogeneity is difficult to be quantified by functional data analysis methods with the assumption of thermal history sequences having similar patterns in temporal correlation [18]. Third, one-of-a-kind products in highly personalized manufacturing provide limited historical data to effectively support the model estimation by using most of the existing data-driven models.

Therefore, the authors propose a Gaussian process-constrained general path (GPGP) model to improve the capability of low-fidelity FEA simulation in terms of simulation accuracy and computational efficiency. The GPGP model can effectively predict the high-fidelity FEA simulation results based on the low-fidelity simulation by modeling the heterogeneous discrepancies between low- and high-fidelity results via general path models. As shown in Fig. 1, the AM product can be decomposed into individual locations, and data at each location are treated as one sample in this study. For different locations, general path models are estimated with Gaussian process model-based constraints to quantify correlation among locations. When two paths have similar covariates, they tend to have similar discrepancy patterns, which lead to highly correlated model coefficients (see Section 4 for validation). In this manner, the estimated GPGP model (trained from the historical data in other designs) can make a prediction on the high-fidelity FEA simulation results for a new design based on low-fidelity simulation with the use of the product design and process design features.

There are several advantages of the proposed method. First, the proposed method is to model discrepancies between low- and high-fidelity FEA simulation results at each individual location regardless of lengths. It can mitigate the dependency on geometric characteristics of products to improve the generality of the model. Second, we use the general path model to parameterize discrepancies between low- and high-fidelity FEA simulation results at each location through one or several paths. The general path model considers a general function form (e.g., polynomial function) with few coefficients to approximate a sequence [19]. Note that the discrepancy between low- and high-fidelity simulation within each path can be roughly approximated by a general function form (e.g., a polynomial function). Therefore, the heterogeneity of discrepancies among locations is decomposed into individual discrepancies within each path and further modeled by a parametric model. Third, recall that the simulation results shown in Fig. 1(b) and (c), and define the low- and high-fidelity simulation results from two locations as two paths. Since the discrepancies between the two paths are similar, it also leads to similar model coefficients of the general path models. This similarity can be represented by product design information (e.g., Cartesian coordinates of deposition sequence) and process design information from inputs of low-fidelity FEA simulation (e.g., input heat). To enforce similar correlation structures among general path models and correlation structure among covariates in different paths, Gaussian process models are adopted as constraints in the estimation of coefficients for individual general path models. In a short summary, the proposed GPGP model can facilitate the validation for heterogeneous AM designs by efficiently reducing the computational cost of FEA simulation with reasonable accuracy based on the historical simulation results. Benefit from the flexibility of FEA simulation, the proposed model can be potentially extended to other AM processes which have a similar mechanism (i.e., layer-wise deposition of material) such as selective laser melting (SLM).

The rest of the article is organized as follows. Section 2 demonstrates the state-of-the-art of FEA simulations in AM processes and statistical methods to improve the FEA simulation accuracy. Section 3 introduces the proposed GPGP model in detail. Section 4 demonstrates the proposed method via the case study of thermal field simulation in the FDM process. Lastly, Section 5 summarizes the contributions of this work and discuss future work.