Deformation Evolution and Perceptual Prediction for Additive Manufacturing of Lightweight Composite Driven by Hybrid Digital Twins

Abstract

This paper proposes a deformation evolution and perceptual prediction methodology for additive manufacturing of lightweight composite driven by hybrid digital twins (HDT). In order to improve manufacturing quality of irregular lightweight composite through boosting conceptual design in aeronautic and aerospace engineering, the HDT meaning hybridization of physical and digital domains, including deformation and energy efficiency can be built, where the essential parameters can be perceptually predicted in advance, by virtue of the fusion of physical sensors and digital information. The long short term memory (LSTM) can be employed to void vanishing gradient problem and improve predicting precision via Recurrent Neural Networks, thereby laying a foundation for the HDT. The diverse manufacturing requirements of different regions are integrated into the parameters designing phase by attaching region weights confirmed via empiricism and in-service simulation. The effects of slicing strategy and external support structures on manufacturing quality are considered from the perspective of improving dimensional accuracy. The manufacturing efficiency and comprehensive costs are accounted as consideration factors, which are perceptually predicted via LSTM. The designed manufacturing parameters through HDT were virtually examined by evaluating the deformation and equivalent stress distributions of fabricated lightweight component with composite material through AM process simulation. The physical experiments were conducted to verify the HDT-based pre-designing and optimization method of manufacturing parameters via fused deposition modeling (FDM). The energy consumption of actual manufacturing process was measured via digital power meter and applied to evaluate accuracy of perceptual prediction outcomes. The dimensional accuracy and distortion distribution of the manufactured lightweight prototype made with composite material were measured through the coordinate measuring machine (CMM) and 3D optical scanner. The proposed method demonstrates effectiveness in improving manufacturing quality and accurately predicting energy consumption, which have been verified with a three-way solenoid valve element, in which the maximum deformation was reduced by 39.78% and the mean absolute percentage error for perceptual prediction was 3.76%.

1 Introduction

The incorporation of lightweight components through conceptual design assumes a critical function in augmenting the mechanical properties of systems, while simultaneously facilitating efficient energy conservation across various fields such as aviation, maritime, and automotive [1]. The broad application of composite materials, which characterized with low cost, light weight, high strength-to-weight ratio, renewability, low density, and energy requirements for processing, has significantly increased the performances of lightweight structures (e.g., sandwich, honeycomb and periodic lattice structures) which could concurrently satisfy mechanical and low weight requirements [2, 3]. However, the intricate geometric structures and unmanageable manufacturing quality exist as obstructs in stable application of lightweight components fabrication.

Additive manufacturing (AM), which alternatively known as three-dimensional printing (3DP) or rapid prototyping (RP), offers potential solutions when conventional subjective manufacturing methods reach technological limits [4]. AM has been adopted as an essential industrial manufacturing technology, due to its capability of fabricating intricate shapes, decreasing the complexity of the production, enabling customized producing, improving resource efficiency and reducing material wastes [5]. Several salient AM techniques based on various forming principles, including vat photopolymerization, material jetting, material extrusion, powder bed fusion, etc., have been developed in the previous decades [6]. With the application of 3DP, it puts forward higher requirements for low energy and material consumption and high quality especially for complex aeronautical parts with grooves, convex shoulders, matching holes and inner cavity microfluidic channels. Nevertheless, the numerous fabricating and material types, the lack of quality control methods and deficiency of product characterization techniques during AM processes induce the incapability of achieving designed performances of fabricated products [7]. The challenges concerning surface roughness, anisotropic behavior, accuracy of part shapes and diverse manufacturing defects have persisted as obstacles in widespread adoption of AM [8].

Methods about controlling the manufacturing quality of AM products with in-situ monitoring and detecting techniques has been intensively investigated [9, 10]. The available manufacturing variables could be tracked, measured, and controlled by equipping a closed-loop control system to reduce mistakes during AM processes [11, 12]. However, the effectiveness of those in-process and post-process methods may lapse when immoderate manufacturing parameters were applied. Thus, rigorously designing and optimization of manufacturing process parameters before conducting physical fabricating are of great significance, especially for lightweight components with high quality requirements and vulnerable geometric structures [13].

The crux of pre-designing and optimization before actual fabricating relies on revealing interaction mechanism between manufacturing parameters and forming quality [14]. To enhance the quality in manufacturing process, Tronvoll et al. [15] predicted the reduction in ultimate tensile strength of transversely printed specimens relative to solid or longitudinally printed specimens, based on void sizes identified from microscopy images of the central plane of a tensile specimen. Mohamed et al. [16] investigated the effects of fused deposition modeling (FDM) fabrication conditions on the dimensional accuracy by applying second-order definitive screening design method. The interaction relation of build factors on the part dimensional accuracy was built through constrcuting and training artificial neural network. Addressing the inherent stair-step effect, induced due to the layer-wise slicing, which normally causes negative impacts on geometric accuracy and surface roughness, adaptive slicing and scanning path planning algorithms have been proposed to improve manufacturing accuracy and simultaneously ensure efficiency [17, 18].

The heating and rapid cooling cycles, which recurrently occur during AM processes, are responsible for the generation of non-uniform thermal gradient and the consecutive phase change of forming material [19]. Thus, the researches about thermal engineering and thermoforming were conducted to stabilize dimension accuracy, achieving tighter bond between fabricated layers and better mechanical properties [20, 21]. Some other works integrated the energy efficiency of AM processes into the scope of consideration and constructed implicit relationships between energy consuming and manufacturing variables [22]. The integration of considerable energy prediction with the process planning process contributes to increasing the success rate of AM, saving resource consumption and improving manufacturing cost-effectiveness [23].

However, most works conducted parameters designing and optimization by equally treating the integrated component and aimed to improve the overall manufacturing quality. The mechanical parts, according to its shape, can be divided into shaft hole class, disk cover class, fork class, box class, standard parts, and more. Those various structures may undertake different functions under service conditions and accordingly propose heterogeneous requirements. Thus, conducting customized manufacturing parameters designing for different structures is a more reasonable way for improving the quality and conducting ecological manufacturing.

Digital twin (DT) technology, which embeds a virtual mapping of the reality constantly synchronized with the real operating scenario for hybrid fusion, maps the structure, state, characteristics, and functionality of physical entities into virtual entities within a computerized environment. By utilizing programmatic logic control and leveraging the digital evolutionary information of virtual entities, DT being developed at present enables the prediction and control of physical entities, facilitating interactive feedback between the virtual and physical entities [24]. The construction of DT involves multidimensional, multiscale, and multimodal representations, encompassing geometric, physical, behavioral, and rule-based methods. DT techniques emphasize the interaction between the virtual and physical realms, enabling real-time updates and dynamic evolution, resulting in a dynamic and realistic mapping of the physical world.

The visualization and perceptualization of multi-domain information in complex manufacturing processes pose significant challenges in constructing digital twin evolutionary models for AM process [25, 26]. It is important to emphasize that the role of DT extends beyond visualizing manufacturing processes, as it enables performance evolution under different operating conditions, allowing precise and quantitative prediction and control of form and behavior [27]. Scime et al. [28] decomposed the manufacturing process into discrete digital workflow operations. The DT techniques were conducted to perform inspections on complex individual parts and used artificial intelligence algorithms to predict component performance for hybrid AM process. Phua et al. [29] evaluated the unique characteristics of manufacturing processes and identified, certified, and optimized defects in the production of complex parts. A DT structure composed of four levels was proposed, including modeling, sensing, hardware control, and intelligent control. Pantelidakis et al. [30] developed a DT ecosystem for simulating virtual environments in additive manufacturing, enabling testing, process monitoring, and remote management. The ecosystem incorporated sensor data components and virtual representation components, achieving synchronization between physical devices and digital systems with robust historical data collection and analysis capabilities.

Admittedly, the research on constructing DT models for customized AM processes is still in its early stages. The technical support in areas such as the integration of digital-physical systems, data analysis, and standardized interactive interfaces is still being explored [31]. Furthermore, most of the DT methods for manufacturing processes remain at the level of building virtual mapping based on multiple heterogeneous sensing information sources, without delving into the virtual evolution of mapped digital objects which requires hybridization of visual twinning, functional twinning and mechanism twinning [32].

Aiming at high quality and efficiency fabricating for mechanical elements with multiple features, based on the previous systematic work [32,33,34,35,36], this paper proposes a deformation evolution and perceptual prediction method for pre-designing and optimizating manufacturing parameters of AM process using hybrid digital twins (HDT), meaning hybridization of physical and digital domains, including deformation and energy efficiency. The physical experiment is conducted with the proposed HDT method via FDM which contributed to verify the effectiveness of the parameter optimization method in improving manufacturing quality before conducting actual fabricating. The HDT method is verified to be effective in improving the dimensional stability of additive manufactured lightweight structures. The method has potential to be integrated with fluid vibration analysis to improve the flow channel performances with regard to energy efficiency, carbon peak and carbon neutrality.

2 Perceptual Prediction of Lightweight Composite Through Conceptual Design via LSTM

2.1 Long Short Term Memory (LSTM) via Recurrent Neural Networks (RNN)

The singular spectrum analysis (SSA) can be employed for physical signal decomposition to extract different components of original sequence to enhance the predictive power of digital twins. The long short-term memory (LSTM) neurons keep a context of memory within their pipeline to allow for tackling sequential and temporal problems without the issue of the vanishing gradient affecting their performance. This is where the LSTM neural network come to the rescue.

Let f_t be the forgetting gate that controls what information should be discarded from the cell state, as formulated in Eq. (1); i_t is the input gate, which determines how much new information should be added to the cell state (also known as memory) at the current time step, as demonstrated in Eq. (2); o_t is the output gate, which determines the information that should be output at the current time step, as indicated in Eq. (3). Although substantial similarities exist among these different gates, variations can be noted in their respective weight matrices W and bias b. x_t is the current input, while h_t-1 is the hidden state from the previous time step. $\tilde{C}_{t}$ represents the candidate value for the current cell state. Guided by the forgetting gate and input gate, cell state C_t-1 and $\tilde{C}_{t}$ are weighted to derive the current cell state value C_t, as dipicted in Eqs. (4), (5). According to the output gate, the hidden state h_t at the current time step can be obtained by adopting activated C_t, as expressed in Eq. (6).

$$f_{t} = \sigma (W_{f} \cdot [h_{t - 1} ,x_{t} ] + b_{f} ),$$

(1)

$$i_{t} = \sigma (W_{i} \cdot [h_{t - 1} ,x_{t} ] + b_{i} ),$$

(2)

$$o_{t} = \sigma (W_{o} \cdot [h_{t - 1} ,x_{t} ] + b_{o} ),$$

(3)

$$\tilde{C}_{t} = \tanh (W_{C} \cdot [h_{t - 1} ,x_{t} ] + b_{C} ),$$

(4)

$$C_{t} = f_{t} \cdot C_{t - 1} + i_{t} \cdot \tilde{C}_{t} ,$$

(5)

$$h_{t} = o_{t} \cdot \tanh (C_{t} ).$$

(6)

2.2 Evaluation Indexes of Perceptual Prediction

The six effect evaluation indexes, namely mean absolute error (MAE), mean absolute percentage error (MAPE), root-mean-square error (RMSE), coefficient of determination (R²), maximum error (ME) and median absolute deviation (MAD), are used to measure the prediction effect. They reflect the difference between the predicted sequence and the original sequence. R² reflects the reliability level of the predicted sequence. And the nearer it approximates to one, the more reliable the predicted sequence is. These indexes can reflect the precision of prediction. The calculation formulas are:

$${\text{MAE}} = \frac{1}{n}\sum\limits_{i = 1}^{n} {\left| {\hat{y}_{i} - y_{i} } \right|} ,$$

(7)

$${\text{MAPE}} = \frac{1}{n}\sum\limits_{i = 1}^{n} {\left| {\frac{{\hat{y}_{i} - y_{i} }}{{y_{i} }}} \right|} ,$$

(8)

$${\text{RMSE}} = \sqrt {\frac{1}{n}\sum\limits_{i = 1}^{n} {(\hat{y}_{i} - y_{i} )^{2} } } ,$$

(9)

$${\text{R}}^{2} = \frac{{\sum\nolimits_{i = 1}^{n} {(\hat{y}_{i} - \overline{y})^{2} } }}{{\sum\nolimits_{i = 1}^{n} {(y_{i} - \overline{y})^{2} } }},$$

(10)

$${\text{ME}} = \max \,\left( {\left| {\hat{y}_{i} - y_{i} } \right|} \right),$$

(11)

$${\text{MAD}} = \left| {\widehat{med} - med} \right|,$$

(12)

where $\hat{y}_{i}$ ($i = 1,2, \ldots ,n$) is the ith elements of predicted sequence. $y_{i}$ is the ith elements of original sequence. $\widehat{med}$ is the median of the predicted sequence. $med$ is the median of the original sequence.

It should be noted that the high-precision prediction method lays a foundation for the hybrid digital twin, where both physical and non-physical information can be fully integrated, including hybridization of deformation and energy efficiency, etc.

3 Deformation Evolution for Additive Manufacturing of Lightweight Composite

3.1 Preprocessing of Multiple Lightweight Features Mechanical Elements

The mechanical multiple-featured components, pertained to the mechanical mechanism, are characterized by physically separated or closely uncoupled entities, which exhibit volatile types of fit along with specific clearances. The defining trait of these components lies in their extensive, elongated cantilever structure, leading to challenges in their fabrication process. The inhomogeneous mechanical bearing distributions of different structures also propose diverse manufacturing requirements for various regions.

The lightweight mechanical features components can be represented by mathematical set:

$${\varvec{F}} = \left\{ {{\varvec{F}}_{1} ,{\varvec{F}}_{2} , \ldots ,{\varvec{F}}_{i} , \ldots ,{\varvec{F}}_{n} } \right\}, \, i = 1,2, \ldots ,n,$$

(13)

where each component F_i is a closed manifold, which can be whatever format, including constructive solid geometry (CSG), boundary representation (B-Rep), etc.

Normally, the regions of lightweight structures require higher manufacturing quality and lower deformation. Thus, the region weights ${{\varvec{\omega}}} = \left\{ {\omega_{{p \in {\varvec{F}}}} } \right\}$ would be attached to the manifold according the priority of structures. Due to diverse requirements in dimensional accuracy and mechanical strength of various mechanical structures, the weights could be obtained by conducting in-service behavior simulation and manually confirming.

The mechanical function of the aircraft engine mount is to install the engine reliably on aircraft, and to make the engine working normally in all flight conditions, which transforms the thrust produced by the rotating propellers to the flight power effectively. The engine mounting, which contains rib, plate, hook and local symmetry, is one of the most critical force transmission parts in the aircraft structure, as depicted in Figure 1. Once the failure or destruction happens, it will endanger the safety of the flight, and then may lead to disastrous consequences. Thus, it requires light weight, large stiffness and high positioning performance, leading the difficulty of both conceptual design and subsequent detailed design.

Figure 1

Conceptual design of aircraft engine mount with multiple mechanical features

In Figure 1, the dimension scale of the mount in the XYZ direction is 1.4098: 0.5804: 1, whose center of mass locates at (50.0029%, 61.2647%, 49.2480%). The specific surface area is 2.0420 mm⁻¹, with height-diameter ratio 0.7057.

3.2 Theoretical Fundamentals of in-Service Condition Simulation for Lightweight Component

The mechanical performance requirements for satisfying functional behavior, such as bearing capacity of external forces, can be confirmed by conducting structural simulation under service conditions. The structural responses of mechanical elements can be figured out by solving the governing equations as Eqs. (14)–(16).

$$\sigma_{ij,j} + f_{i} = \rho \ddot{u}_{i} ,$$

(14)

$$\tau_{ij} = \frac{{u_{i,j} + u_{j,i} }}{2},$$

(15)

$$\sigma_{ij} = 2\kappa \tau_{ij} + \lambda \delta_{ij} \tau_{kk} ,$$

(16)

where $i,j,k \in \{ X,Y,Z\}$ are the direction indicator symbols, $\sigma_{ij,j}$ is the derivative of the stress tensor component $\sigma_{ij}$ (Pa), $f_{i}$ (kg/(m²∙s²)) is the external force per unit volume, $\rho$ (kg/m³) is the density, $\ddot{u}_{i}$ (m/s²) is the acceleration component, $\tau_{ij}$ and $\tau_{kk}$ represent the component of strain tensor, $u_{i,j}$ and $u_{j,i}$ are the derivatives of displacements $u_{i}$ and $u_{j}$ along directions j and i, $\kappa$ and $\lambda$ are the Lame constants, $\delta_{ij}$ is the component of Kronecker delta tensor.

The manufacturing quality requirements of structure surfaces is increasing due to the broad applications of lightweight components in fluid instruments. The effects of geometric structures and surfaces on fluid dynamic evolution can be revealed by conducting fluid analysis, through computing hydrodynamic equations, including continuity equation, momentum equation and bernoulli equation. The fluid motion could be expressed by three-dimensional Navier-Stokes (N-S) equations for mass and momentum conservation as in Eqs. (17) and (18). The spatiotemporal evolution mechanism of fluid in flow channels could be revealed by solving the N-S equations and could be further integrated with vibration coupling analysis.

$$\frac{{\partial u_{x} }}{\partial x} + \frac{{\partial u_{y} }}{\partial y} + \frac{{\partial u_{z} }}{\partial z} = 0,$$

(17)

$$\frac{{\partial {\varvec{U}}}}{\partial t} + ({\varvec{U}} \cdot \nabla ){\varvec{U}} = \frac{1}{\rho }(f - \nabla p) + \nu \Delta {\varvec{U}},$$

(18)

where ${\varvec{U}} = (u_{x} ,u_{y} ,u_{z} )$ (m/s) represents the velocity components in XYZ directions respectively, t (s) is the time, $\rho$ (kg/m³) is the density, f (N) is the external force, $\nu$ (m²/s) is the kinematic viscosity.

The $k - \varepsilon$ model is implemented to solve the N-S equations based on three-dimensional grid. The turbulent kinetic energy k and the turbulent dissipation $\varepsilon$ are proposed to estimate turbulent viscosity as in Eq. (19). Two extra transport equations are included to represent the turbulent flow properties in the mathematical expression as in Eq. (20).

$$\frac{\partial \rho k}{{\partial t}} + \frac{\partial }{{\partial x_{i} }}(\rho ku_{i} ) = \frac{\partial }{{\partial x_{i} }}\left[ {\left( {\mu + \frac{{\mu_{t} }}{{\sigma_{k} }}} \right)\frac{\partial k}{{\partial x_{i} }}} \right] + G - \rho \varepsilon ,$$

(19)

$$\frac{\partial \rho \varepsilon }{{\partial t}} + \frac{\partial }{{\partial x_{i} }}(\varepsilon \rho u_{i} ) - \frac{\partial }{{\partial x_{i} }}\left[ {\left( {\mu + \frac{{\mu_{t} }}{{\sigma_{\varepsilon } }}} \right)\frac{\partial \varepsilon }{{\partial x_{i} }}} \right] = c_{1\varepsilon } \frac{\varepsilon }{k}G - c_{2\varepsilon } \frac{{\varepsilon^{2} }}{k},$$

(20)

$$\mu_{t} = \rho C_{\mu } \frac{{k^{2} }}{\varepsilon },$$

(21)

where $x_{i}$ represents the directional components, $u_{i}$ (m) is the directional displacements along XYZ directions, $i \in \{ X,Y,Z\}$, $\mu$ (N∙s/m²) is the dynamic viscosity, $\mu_{t}$ (Pa∙s) is the turbulent viscosity which could be estimated via Eq. (21), G is the production of turbulence due to shear, $\sigma_{k}$, $\sigma_{\varepsilon }$, $c_{1\varepsilon }$, $c_{2\varepsilon }$, $C_{\mu }$ are the constants which are confirmed via experiments.

In order to quantify performance requirements, the structural analysis is implemented on the aircraft engine mount. Figure 2(a) shows that the lugs are the most deformed parts with stepped distribution characteristics, when the force of 1 kN is exerted on the centroid of both top and bottom of the aircraft engine mount. The tensile strength of the deformed material is 50 MPa. The maximum deformation is 279.523 times more than the minimum deformation. Figure 2(b) is the distribution of equivalent stress, and shows the stress concentrating on the back of the component mainly. The maximum equivalent stress is 8.988 times more than the minimum equivalent stress.

Figure 2

In-service behavior simulation outcomes of aircraft engine mount under extreme working condition: (a) The total deformation distribution, (b) The equivalent stress distribution

4 Hybrid Digital Twins (HDT) of Manufacturing Parameters Optimization for Lightweight Composite Component

4.1 Evaluation of Volume Error Due to Slicing Strategy

The discrepancy between the original manifold and the sliced layer represents a primary cause of distortion for AM processes, attributed mainly to the inherent staircase effect. It is worth noting that the sequencing of the slicing process directly influences the accuracy of the final geometry.

Note that it is a trade-off between layer thickness and fabricating time, due to the contradiction between printing speed and accuracy which are difficult to be ensured simultaneously. Generally, the regions with higher weights will attach more emphasis on accuracy and are allocated with thinner layers. And thicker layer will be attached to regions with lower weights. Thereby, carefully designing the thickness sequence $\mathcal{D}$ as defined in Eq. (22) is essential for ameliorating distortion.

$${\mathcal{D}} = \left\{ {d_{1} ,d_{2} , \ldots ,d_{l} , \ldots ,d_{L} } \right\},$$

(22)

where $d_{L} \in [d_{\min } ,d_{\max } ]$ represents the layer thickness of the lth layer, $d_{\min }$ and $d_{\max }$ are the minimum and maximum layer thickness due to limitation of printing equipment.

The volume error metric $L_{d} (\user2{\mathcal{D}})$ is deduced based on both the sliced scheme ${\varvec{N}}$ and voxelized theoretical manifold ${\varvec{M}}$, so as to evaluate the distortion caused by slicing. The volume error evaluation function could be defined with the weighted Hausdorff distance between sliced scheme and theoretical manifold as follows:

$$L_{d} (\user2{\mathcal{D}}) = H({\varvec{N}},{\varvec{M}}) = \max (h({\varvec{N}},{\varvec{M}}),h({\varvec{M}},{\varvec{N}})),$$

(23)

$$h({\varvec{N}},{\varvec{M}}) = \mathop {\max }\limits_{{n \in {\varvec{N}}}} \, \mathop {\min }\limits_{{m \in {\varvec{M}}}} \left\| {n,\omega_{m} \cdot m} \right\|,$$

(24)

where $H( \cdot )$ represents the weighted Hausdorff distance, n and m are the voxels of voxelized sliced scheme N and theoretical weighted manifold M, ${{\varvec{\omega}}}_{{\varvec{m}}}$ is the weights of voxels of M which can be confirmed through ${{\varvec{\omega}}}$.

The manufacturing progress rate can be denoted by the normalized height. The normalized height $h_{n}$ of the lth layer is defined as:

$$h_{n} = \frac{{z_{l} }}{{z_{b} }}, \, h_{n} \in (0,1],$$

(25)

where $z_{l}$ is the cumulative layer height of l layers and $z_{b}$ is the total height of the manifold.

4.2 External Supports Design Considering Manufacturing Quality and Utilization Rate

The inclusion of external supports, denoted as $\user2{\mathcal{S}}$, is vital to AM processes in order to ensure the printability of mechanical components.

$$\user2{\mathcal{S}} = \left\{ {{\varvec{s}}_{1} ,{\varvec{s}}_{2} , \ldots ,{\varvec{s}}_{l} , \ldots ,{\varvec{s}}_{L} } \right\},$$

(26)

where ${\varvec{s}}_{l}$ represents the external support regions in lth layer.

Simultaneously, the attachments between the supports and the printing structures impart a capacity to withstand distortion to some degree. Conversely, the removal of support structures will compromise the surface integrity of the final product. Therefore, strategically organizing the placement of support distributions will significantly contribute to guaranteeing the quality of the manufacturing process.

The proportion distribution of mechanical features is defined by the volume ratio $V_{c}$ as in Eq. (27).

$$V_{c,l} = \frac{{A(F_{l} ) \cdot d_{l} }}{{\sum\limits_{u = 1}^{L} {A(F_{u} ) \cdot d_{u} } }},$$

(27)

where $V_{c,l} \in (0,1)$ is the volume ration of lth layer, $A(F_{l} )$ (mm²) is the area of lightweight features of lth layer.

The evaluation function of support structures could be formulated as in Eq. (28). The proportion limitation of support structures is achieved by introducing restriction of support volume proportion $p_{s} \in (0,1)$ to increase the material utilization rate.

$$p_{s} = \frac{{V_{\user2{\mathcal{S}}} }}{{V_{\user2{\mathcal{S}}} + V_{F} }},$$

(28)

$$L_{S} (\user2{\mathcal{S}}) = \sum\limits_{l = 1}^{L} {V_{c,l} A({\varvec{s}}_{l} )} - \eta_{1} \omega_{F} A(\user2{\mathcal{S}}_{F} ) - \eta_{2} p_{s} ,$$

(29)

where $V_{\user2{\mathcal{S}}}$ (mm³) is the volume of external support structures, $V_{{\varvec{F}}}$ is the volume of lightweight mechanical component, $A( \cdot )$ (mm²) represents the area magnitude of corresponding sets, $\user2{\mathcal{S}}_{F}$ is the support set whose connection point reside on light weight mechanical feature surfaces, $\eta_{1}$ and $\eta_{2}$ are the hyperparameters for the proportion of penalty terms for supports connecting with feature surfaces and supports proportion.

4.3 Manufacturing Efficiency and Comprehensive Costs Constrains

In addition to control the distortion, the printing efficiency and costs are factors to be considered when designing the manufacturing parameters. The relative manufacturing efficiency $L_{t} (\user2{\mathcal{D}},\user2{\mathcal{S}},V_{F} )$ can be calculated by Eq. (30). For scanning printing process, the cost time for forming layers could be reckoned as Eq. (31). Note that the comprehensive manufacturing cost, formulated in Eq. (32), could be calculated by analyzing the manufacturing process of 3DP.

Specifically, the energy cost C_energy mainly includes servo energy and digital light irradiation energy. The material cost C_material represents the consumed material for fabricating elements including both support and original parts. The associated cost C_associated means the cost except C_energy and C_material. Those comprehensive costs could be rapidly predicted via LSTM as explained before. More details about the comprehensive costs could be referred in Ref. [22].

$$L_{t} (\user2{\mathcal{D}},\user2{\mathcal{S}},V_{F} ) = {\text{card}}(\user2{\mathcal{D}}) \cdot t_{a} + \sum\limits_{l = 1}^{L} {t_{l} } ,$$

(30)

$$t_{l} = \frac{{F_{l} }}{{\mathcal{V}_{F} }},$$

(31)

$$L_{C} (\user2{\mathcal{D}},\user2{\mathcal{S}},T_{n} ,V_{F} ) = \frac{{C_{energy} + C_{material} + C_{associated} }}{{V_{t} }},$$

(32)

where ${\text{card}}(\user2{\mathcal{D}})$ is the cardinality of $\user2{\mathcal{D}}$, t_a (s) is the constant time interval for device adjusting before starting new-phase printing, t_l (s) is the manufacturing time of the lth layer, F_l (mm) is the length of scanning trajectories of the lth layer, $\mathcal{V}_{F}$ (mm/s) is the printing velocity, $V_{t}$ (mm³) is the volume of target mechanical element.

Figure 3 counts in the geometric information of aircraft engine mount part. Regarding layered area in Figure 3(a), the max is 300.1687 mm², when h_n = 98%, the min is 51.4544 mm², when h_n = 46%, the mean is 104.3634 mm², the standard deviation is 86.2198 mm², the variance is 7433.8517, the 2nd central moment is 7285.1747.

Figure 3

Additive manufacturing area and material volume of aeronautical aircraft engine mount

Regarding layered volume in Figure 3(b), the max is 53.9097 mm³, occurred when h_n = 96%, the min is 7.2447 mm³, occurred when h_n = 46%, the mean is 15.1453 mm³, the standard deviation is 13.6905 mm³, the variance is 187.4289, the 2nd central moment is 183.6803.

The effectiveness of obtained manufacturing parameters were virtually verified using visualized hierarchical digital twin technique. As displayed in Figure 4, the slicing visualization of aircraft engine mount with 25 and 50 layers are visualized. We construct the digital twins based virtual prototype according to the cross-sectional contours of the layered mount.

Figure 4

The digital twins based virtual prototype of the layered mount with (a) 25 layers and (b) 50 layers

4.4 Manufacturing Parameters Optimization and Verifying via HDT

The general energy function $L({\varvec{x}})$ which is constructed for evaluating the designed manufacturing parameters from aspects including distortion distribution, manufacturing efficiency and comprehensive costs is formulated as follows. The NSGA-II (Non-dominated Sorting Genetic Algorithm-II), which is endowed with fast convergence speed and strong local search ability, is adopted to solve the constructed nonlinear multi-objective optimization model.

$$\begin{gathered} {\text{Find}}:{\varvec{x}} = \left[ {\user2{\mathcal{D}},\user2{\mathcal{S}},\mathcal{T}_{n} ,\mathcal{V}_{F} } \right], \\ {\text{Minimize}}:L({\varvec{x}}) = \{ L_{\user2{\mathcal{D}}} ,L_{\user2{\mathcal{S}}} ,L_{t} ,L_{C} \} , \\ {\text{Subject to:}}\left\{ \begin{aligned} d &_{\min } \le d \le d_{\max } , \\ \mathcal{T} &_{\min } \le \mathcal{T}_{n} \le \mathcal{T}_{\max } , \\ \mathcal{V} &_{\min } \le \mathcal{V}_{F} \le \mathcal{V}_{\max } , \\ \end{aligned} \right. \\ \end{gathered}$$

(33)

where x is the parameter of fusing process, T_n (℃) is the nozzle temperature, $\mathcal{V}_{F}$ (mm/s) is the printing velocity, d (mm) is the layer thickness, T_min (℃) and T_max (℃) are the minimum and maximum available temperature, $\mathcal{V}_{\min }$ (mm/s) and $\mathcal{V}_{\max }$ (mm/s) are the minimum and maximum available printing velocity.

The overall flowchart of the proposed method can be illustrated in Figure 5. Generally, the regions weights are attached to structures via working condition simulation. The available manufacturing parameters are preliminarily confirmed and adopted to construct optimization paradigm. The optimal parameters are further examined via HDT to verify its feasibility.

Figure 5

Overall schematic of the proposed method

5 Numerical Instances of Three-Way Solenoid Valve with Lightweight Features

5.1 Designed Prototype of Lightweight Three-Way Solenoid Valve

Figure 6 illustrates an instance of the virtual modelling of large conceptual prototype with multiple features which could be implemented using geometry modeling kernel of Parasolid, ACIS 3D Toolkit and STEP (standard for the exchange of product model data). Note that the axis-aligned bounding box (AABB) of the three-way solenoid valve manifold is [− 65, − 67.94, − 8.00, 65.00, 34.97, 97.00], and the side length x_b, y_b, z_b of the AABB along XYZ directions are 130.00, 102.90, 105.00, respectively.

Figure 6

Body of a three-way solenoid valve with multiple features: (a) Front view, (b) Back view, (c), (d) Full section views

The concept of ‘thin-walled’ is relative which can be quantized by ratio of thickness to overall size, for example, 5% in mechanical and aerospace fields. The wall thickness distribution analysis of the three-way solenoid valve housing is illustrated in Figure 7. The redder the color tends to be, the more thin-walled the structure. The surface area of the valve structure is 6705.12 ${\text{mm}}^{2}$. The number of statistical samples for wall thickness analysis is 13876 and the average thickness is 8.35 mm. From the intuitive color distribution, thin-walled materials account for a large proportion and the valve housing could be regarded as thin-walled structure.

Figure 7

Material thickness distribution of an irregular thin-walled with internal channels called three-way valve: (a) Wall thickness distribution from front view, (b) Back isometric view

5.2 Fluid Velocity and Static Pressure Distribution of the Extracted Designed Flow Channel

The valves are significant elements in hydraulic systems which provide systems various flexible functions like cutting off fluids, changing flow directions, maintaining system pressure, controlling the flows and so on. There are multiple features, including internal channels and small orifices, in three-way valve as shown in Figure 8, which greatly increase the manufacturing difficulty. The prototype of three-way valve could be rapidly manufactured with polylactic acid (PLA) via FDM to verify the effectiveness of designed structures and reliability of manufacturing parameters which is meaningful before mass manufacturing.

Figure 8

Sketch of the structure of (a) three-way solenoid valve with multiple thin-walled features and (b) the extracted flow channel structure

The fluid simulation was conducted to confirm the weights of inner flowing channel surfaces. The effects of surface on fluid velocity and pressure loss could be extracted to translate the requirements on manufacturing accuracy. The fluid inlet velocity was set to 5 m/s for working condition 1 and 10 m/s for working condition 2. The outlets were both set as outflows. The fluid material was liquid water whose density is 998.2 kg/m³ and viscosity is 1.003×10³ kg/(m·s). The turbulent kinetic energy is 0.09 m²/s² and turbulent dissipation rate is 78.72 m²/s².

Figure 9 depicts the transient fluid velocity and static pressure distributions of the designed irregular thin-walled internal flow channels under various working conditions. And more quantitatively data are listed in Table 1. The maximum and average static pressure of working condition 2 are significantly higher than condition 1. The standard deviations of velocity magnitude are 4.06 m/s and 4.4985 m/s under condition 1 and 2, respectively. The standard deviations of static pressure are 5.44 × 10⁴ Pa and 3.06 × 10⁵ Pa under condition 1 and 2, respectively. The eddy occurred around outlet 2 greatly reduce the exit velocity and static pressure.

Figure 9

Fluid velocity and static pressure distribution of the extracted designed flow channels: (a), (b) Fluid velocity and static pressure distributions under working condition 1, respectively, (c), (d) Simulation distributions under working condition 2

Table 1 Quantitatively comparison of three-way fluid flow channel simulation outcomes under various flow working conditions

5.3 Layer Average and Standard Deviation of Deformation and Mises Stress

The finite element analysis (FEA) of AM process was conducted. The room temperature was set to 25 ℃, and the substrate was preheated to 45 ℃. The mesh size was set to $2\text{ mm}$. The glass fiber reinforced PLA was adopted as the manufacturing material. The density is 1.12 g/mm³, isotropic thermal conductive is 0.2 W/(m∙K), the coefficient of thermal expansion is 6.8×10⁻⁵ K⁻¹.

The simulation outcomes of deformation distribution and mises stress are displayed in Figure 10. The regions with higher distortions transferred into top by comparing Figure 10(a), (c). The standard deviation of deformation for (a) is 0.2816 mm and 0.2479 mm for (c). The standard deviation of deformation for (b) is 53.61 mm and 53.15 mm for (d). The effectiveness of HDT for releasing residual stress is more significant when referring to the stress distribution of the valve prototype as illustrated in (c) and (d). More quantitative data about Figure 10 are listed in Table 2. Generally, the average distortion and stress are reduced with HDT which was helpful to improve the manufacturing quality for fabricated lightweight element.

Figure 10

Deformation and stress distributions generated based on manufacturing simulation: (a) Manufacturing deformation under manufacturing parameters without HDT, (b) Manufacturing residual stress distribution without HDT, (c) Deformation with HDT, (d) Residual stress with HDT

Table 2 Comparison of manufacturing distortion and stress with various manufacturing parameters

Figure 11 illustrates the deformation distribution of flow channels under various manufacturing parameters. (a) is about the channel deformation fabricated without HDT. The maximum deformation is 0.2420 mm, average is 0.0698 mm and standard deviation is 0.0336 mm. (b) is about the channel deformation fabricated with HDT. The maximum deformation is 0.0238 mm, average is 0.0182 mm and standard deviation is 0.0041 mm.

Figure 11

Deformation distribution of flow channels: (a) Channel deformation of manufactured three-way solenoid valve without HDT, (b) Channel deformation of manufactured three-way solenoid valve with HDT

5.4 Virtual Printing Visualization of Additive Manufacturing Process via HDT

The designed manufacturing parameters could be applied to virtual printing and visualized. Figure 12 illustrates the variation of cross section areas and volumes of sliced layers. The total area is 1.75 × 10⁵ mm², the maximum layer area is 1.03 × 10⁴ mm², minimum layer area is 943.60 mm², the average is 3497.5 mm² and standard deviation is $2673.4$. The total volume is 6.71 × 10⁴ mm³, the maximum layer volume is 1.09 × 10⁴ mm³, minimum layer volume is 1216.1 mm³, average is 4529.7 mm³ and standard deviation is 3298.8 mm³.

Figure 12

Variation of cross-sectional areas and volumes of sliced layers

The theoretical manufacturing situation with various parameters are visualized by employing 3D visualized computing digital twins as in Figure 13. Note that the layer thickness is magnified for visualization purposes, and it can be much smaller in physical manufacturing.

Figure 13

Virtual printing visualization of AM process with various manufacturing parameters: (a) Normalized maximum deformation of layers which is manufactured without HDT, (b) Normalized maximum deformation of layers with HDT

6 Physical Experiment of Three-Way Solenoid Valve Mechanical Elements

The physical experiment of additive manufacturing application with toxicity isolation protection for the experimenter can be captured in Figure 14. The deposition and fusion of fiber-reinforced filament material was based on FDM. The power supply of the manufacturing device is 200 W/AC (alternating current) 220 V. The ambient temperature is 25 ℃ with 55% RH (relative humidity). The air pressure is between 86 kPa and 106 kPa. The working temperature of electrothermal responsive nozzle is 180−250 ℃ whose extrusion port diameter can be adjusted within 0.2−1.2 mm. The maximum available manufacturing dimension can reach to 200 × 200 × 200 mm. The printing velocity is accessible within 0−100 mm/s, and the layer thickness is feasible to vary from 0.1 mm to 0.3 mm.

Figure 14

Physical experiments of the AM process for fabricating lightweight component via FDM

The typical graphical user interface (GUI) of output electrical energy consuming can be seen in Figure 15 from the high precision power meter, assisted with special metal shielding box jig. The power meter can be configured to test equipment with direct current or alternating current. The ambient temperature is 18 ℃−28 ℃ (64.4 ℉−82.4 ℉). The measuring voltage range is 0.015 V−600 V. And the accuracy is sum of 0.1% read value and 0.1% range. The measuring current range is 0.000025 A−20 A. And the accuracy is sum of 0.1% read value and 0.1% range. The residual noise is 0.5% range. The synchronizing frequency is 45 Hz−6 kHz. The measuring sequence can be saved with time label to analyze the running state of measured equipment. During the real-time monitoring process of electrical power, the max can reach 364.87 W, the min is 15.40 W, the average is 129.48 W.

Figure 15

Typical measuring GUI of output digital power meter during physical manufacturing process

The perceptual prediction outcomes of energy consumption, whose measurement unit is Joule, during AM process via LSTM is demonstrated in Figure 16. Figure 16(a) is about the predicted and actually measured energy consumption comparison. The max measured value is 27100.3807 at 55.3265%, min is 1669.3562 at 79.3814%, mean is 9830.3177, standard deviation is 5399.6881, variance is 29156631.6520, median is 8046.3131, mode is 1669.3562. The max predicted value is 28113.1359 at 55.3265%, min is 1633.3202 at 85.2234%, mean is 10170.8601, standard deviation is 5588.9314, variance is 31236153.7420, median is 8347.0305, mode is 1633.3202. Generally, slice layers with larger area and volume require more energy for melting material and filling internal regions. And the energy consumption will accordingly increase.

Figure 16

Perceptual prediction of lightweight composite component manufacturing process via LSTM: (a) Manufacturing process varying characteristic obtained by perceptual prediction and actually measuring, (b) Error of energy consumption prediction, (c) Calibration curve of the perceptual prediction

Figure 16(b) reveals the prediction error, where the max is marked with fuchsia facing up triangle. The MAE of the perceptual prediction via LSTM is 370.05, MAPE is 0.0376, RMSE is 423.0072, R² is 1.0713 and MAD is 300.7174. The ME is 1074.8432 at 7.3310%, min is 62.3879 at 79.3814%.

Figure 16(c) illustrates the calibration curve of the perceptual prediction. The linear equation y=1.0346x+0.6081 can be derived to visually characterize the perceptual prediction accuracy distribution. The distance between two parallel lines, namely $\sqrt 2$ maximum absolute error (ME) among all cases, is 1520.0578, only 5.6090% of the maximum value.

The printing accuracy and distortion distribution were measured through coordinate measuring machine (CMM) and 3D optical scanner as represented in Figure 17. The CMM of HEXAGON Bridge−SF−04.05.04 captured in Figure 17(a) was applied to measure and evaluate the manufacturing accuracy of the multiple features for manufactured mechanical elements. The measuring range of the CMM is 355 × 514 × 353 mm with the absolute maximum permissible error of 3.1+3.0×L/1000 μm, where L denotes measuring length. The maximum permissible probing error is 3.2 μm, the unidirectional repeat accuracy is 0.35 μm and the reposition accuracy is 1.5 μm.

Figure 17

Physical experiments for measuring distortion distribution of manufactured mechanical elements with various measuring equipment: (a) CMM device for measuring printing accuracy of functional features, (b) Adoption of 3D optical scanner for distortion distribution measurements with testing specimen

The scan accuracy of 3D optical scanner indicated in Figure 17(b) can reach 0.045 mm, scan speed ≥ 10 f/s, 3000000 points/s for sophisticated scanning mode. For rapid scanning mode, the accuracy is ± 0.1 mm, scan speed 15−30 f/s, 1500000 points/s. The directional error is ≤ 0.3 mm/m, scanning resolution ≥0.2 mm which can be adjusted after scanning, scan range 150×120 mm−250×200 mm, depth of field ±100 mm, light source is tricolor LED. The actually fabricated three-way solenoid valve element via FDM and correspondingly obtained point clouds are displayed in Figure 18.

Figure 18

Actually fabricated three-way solenoid valve and corresponding optical scanning output: (a) Fabricated three-way solenoid valve element though FDM, (b) Point clouds captured though 3D optical scanner

7 Conclusions

• (1) A deformation evolution and perceptual prediction method for AM of lightweight composite driven by hybrid digital twins, meaning hybridization of deformation and energy efficiency, was put forward.

  Immoderate manufacturing parameters may induce failure of lightweight mechanical elements, especially catastrophic for elements with multiple functional features like internal channels, variable wall thickness. The manufacturing parameters pre-designing method was thereby developed by optimizing associated factors based on HDT, aiming to control the manufacturing quality of these susceptible elements to ensure the mechanical performance. The HDT-based consolidated method was especially suitable for fabricating of complex lightweight structures with diversified manufacturing requirements, thereby speeding up the efficiency of conceptual design.

• (2) A multi-objective optimization scheme was constructed, and the virtual verification paradigm for driven by HDT was developed.

  The quality and efficiency associated manufacturing parameters, incorporating slicing sequences, support structures, material and energy consumption, were considered to construct the multi-objective optimization scheme with the aim of ameliorating distortion of lightweight component while ensuring manufacturing efficiency. The HDT-based simulation paradigm of fused deposition modeling process was developed for virtually visualizing the manufacturing process and verifying the designed parameters. The numerical instance with three-way solenoid valve verified that the manufacturing efficiency can be maintained while improving the dimensional accuracy via HDT-based method.

• (3) The physical experiments were conducted to verify HDT-based method via FDM and measuring equipments.

  The three-way solenoid valve prototype was physically manufactured with composite material based on HDT through FDM technology to verify the effectiveness. The printing accuracy was measured via coordinate measuring machine (CMM) and the distortion distribution was confirmed by 3D optical scanner. Compared with other traditional methods, the proposed HDT method conducted in FDM can overcome the inhomogeneous distortion with more optimal manufacturing parameters which contributes to higher geometry fidelity and better mechanical performance assurances of lightweight mechanical element prototype.

  In the future, the proposed designing method will be planned to be more suitable for AM of complex artifacts via HDT consisting of more physical sensors and virtual components. Future work further involves the long-term dimensional stability and fatigue of industrial light curing 3D fabricated large functional artifacts, which might identify future avenues for multiple industries.