Support Equilibrium Design for Laser Additive Manufacturing of Lightweight Component Based on Geometric Deformation Minimization

Abstract

This article proposes a support equilibrium design methodology for laser additive manufacturing of lightweight components based on geometric deformation minimization (GDM). Controlling geometric deformation commonly induced by material residual stress is especially significant for lightweight components, as slight deviation will be amplified, thereby inducing severe imbalance that makes premature failures. Aiming to investigate the impacts of support structures on photocurable manufactured parts and further conduct support equilibrium designs, a mechanically constrained volume shrink model was constructed jointly considering the chemical reaction kinetics and evolution of material properties. The deformation and stress distributions of manufactured parts with various conceptual support structures were confirmed, and the geometric deformation were mapped to the designed ideal model to generate the deformed manifold. The bidirectional fluid–solid coupling simulation, as well as buckling response analysis were conducted to verify the effectiveness of GDM in terms of improving working performance of lightweight components under variable working conditions. In addition, the balance analysis of mass distribution was conducted by calculating the offset distance of gravity center to examine the ability of the GDM-based support equilibrium design in reducing imbalance phenomena. The physical experiment is conducted on unmanned aerial vehicle (UAV) parts to verify GDM via digital light processing and microscopic images. The geometric deformation during manufacturing process is reduced by 13.08%, and the average centroid shift is improved by 27.44% based on GDM, which effectively improved the working performance of lightweight components.

Keywords

support equilibrium design laser additive manufacturing lightweight components geometric deformation minimization (GDM)digital light processing (DLP)

Introduction

The three-dimensional printing (3DP) technology, which is also referred to as additive manufacturing (AM) or rapid prototyping (RP), has made significant strides and achieved widespread applications in numerous fields, including bioengineering,¹ aerospace,² and automotive.³ 3DP enables the fabrication of conceptual manifold structures, which are commonly obtained by computer-aided design (CAD) through mostly layer-by-layer construction. The characteristic of layer-wise manufacturing endows 3DP the ability of fabricating intricate structures, such as lattice structures commonly designed for lightweight components, which is a significant superiority in comparison with conventional subtractive manufacturing technologies.⁴ With the rapid developments and increasingly broad applications of 3DP in industrial fields, demands on fabricating with advanced composite materials, more complex products, and higher quality have emerged, which present strict challenges.^5,6

Nevertheless, the manufacturing defects, such as unintended porosity, cracks, geometric deformation, and so on, significantly weaken mechanical responses of lightweight components⁷ and persist as a limitation of stable application of 3DP products.⁸ As one of the most common anomalies, geometric deformation, which makes printed structures deviate from a designed model and may cause failure of products, is closely related to manufacturing parameters whose incentive varies with 3DP methods.⁹ Keval et al.¹⁰ devoted to analysis component distortion induced by wire arc AM using surface deviation. The average error of distortion confirmed with thermomechanical numerical computation was acceptable, which provides reliable reference to predict manufacturing quality. Severe deformation is more prone to occur for components with large aspect ratios, and Eann et al.¹¹ alleviated severe out-of-plane deformation with enveloping and buttress support structures. The thermal deformation can also be effectively mitigated through carefully designing the deposition patterns for scanning and melting AM technologies.^12,13

Residual stress during manufacturing process is the main factor leading to warpage and deformation.¹⁴ Many works have been devoted to arranging support structures, namely, support equilibrium design in this work, which has been proven to be effective in equilibrating unbalanced manufacturing stress and reduce geometric deformation. Patiparn et al.¹⁵ predicted the distortion of laser powder bed fusion manufactured parts with numerical models and found that large distortion arose near the interface between the tibial tray and the support structure due to the different stiffness between the solid bulk and the support structures. Allaire et al.¹⁶ proposed some new criteria for optimizing structures of supports by jointly considering the volume, accessibility, and surface loads. Numerical experiments verified the effectiveness of the optimized supports in reducing displacements of the manufactured products. Aiming to absorb temperature-induced residual stresses with support structure, Eugen et al.¹⁷ designed supports by allocating stackable unit cells based on stress distribution confirmed via thermomechanical finite element process simulation. Vincent et al.¹⁸ proposed a comprehensive methodological approach to optimize the support structure for laser powder bed fusion (PBF) considering removal of supports, which was helpful for reducing machining costs in terms of time and cutting tool degradation.

Different from 3DP technologies such as fused deposition modeling (FDM), selective laser melting (SLS), or electron beam freeform (EBF) where the deformation is commonly induced by the fast, intense, and insistent heating-cooling cycles, the distortion for vat photopolymerization 3DP technologies (e.g., digital light process [DLP]) is more relevant with volume shrink of printing materials throughout the entire photopolymerization process.¹⁹ To reveal the deformation mechanism of vat polymerization, many efforts have been poured in to construct physical models and predict the deformation of photocured components. Mathew et al.²⁰ optimized printing quality of DLP by assessing the effect of print angle on geometries and reduced deviation between fabricated parts with designed models. Some other works attempted to improve printing accuracy by adjusting the shape of light sources. Andrew and James²¹ investigated the impacts of digital light beam shape on DLP print accuracy by demonstrating an alternative model of beam distribution, which was able to predict small-scale features and avoid formation errors. Montgomery et al.²² enhanced printing accuracy by using pixel-level grayscale control to create round features from sharp pixels. The influence of support structures on the characteristics of flat and cylindrical surfaces has also been investigated.²³ Nevertheless, the impacts were verified with limited geometries, and the method is difficult to be universally applied, owing to the lack of theoretical models.

Besides, most works applied support equilibrium design aiming to reduce deformation at the stage of manufacturing but neglected the impacts on service phase where the functionality of mechanical components was utilized.²⁴ The support equilibrium design is especially important for lightweight components with high-speed working conditions, such as mechanical compressors, blades, turbines, and so on. The residual unbalanced forces will be significantly increased at high speeds, which may lead to malfunction of the machinery or fatigue failure of those components.²⁵ Under high-speed working conditions, the interactions between mechanical components with environmental media (e.g., air flows) are non-negligible factors, which have important effects on working performance.²⁶

Addressing the above problems, based on previous works,^27–29 the support equilibrium design method was proposed based on geometric deformation minimization (GDM), aiming to reduce printing deformation throughout the whole photocurable AM process and improve working performance of mechanical components in service stage. The theoretical model of deformation mechanism during photocuring process was constructed to evaluate the influences of external support structures on manufacturing distortion. In addition, the working performance of deformed components under high-speed rotation conditions was examined via fluid–solid coupling simulation and nonlinear buckling response analysis. The physical experiment is conducted to verify the effectiveness of the proposed GDM method through DLP printing.

Theoretical Fundamentals of Support Equilibrium Design-Based Geometric Deformation Minimization

Mechanical constrained volume shrink model of photocurable AM process

Different from AM technologies with melting and solidification processes, geometric deformation of vat polymerization 3DP methods are mainly induced by changes in material properties when exposed to lights (ultraviolet rays). To equilibrate unbalanced manufacturing stress and reduce geometric deformation of lightweight components during manufacturing process, the mechanically constrained volume shrink model was constructed jointly considering the chemical reaction kinetics and evolution of material properties.

The DLP printing process is conducted by applying a patterned image via digital light projector flashes onto photosensitive liquid resin. A thin layer is cured and sticks together with the previous printed layer after exposing to ultraviolet light for a period of time. There were four main steps for typical photopolymerization reaction including photodecomposition [Equation (1)], initiation [Equation (2)], propagation [Equation (3)], and termination [Equation (4)].³⁰

\begin{matrix} P \to 2 R^{*} \end{matrix},

(1)

\begin{matrix} R^{*} + M \to P^{*}, \end{matrix}

(2)

\begin{matrix} P_{n}^{*} + M \to P_{n + 1}^{*}, \end{matrix}

(3)

\begin{matrix} P^{*} + P^{*} \to P_{d}, \end{matrix}

(4)

Exposing to digital light, the photoinitiator molecules $P$ are first decomposed into active radicals $R^{*}$ in the photodecomposition stage. In the initiation step, the active radicals further combinate with monomers $M$ and become active monomers $P^{*}$ . The active monomers will react with each other to be polymer chains $P_{n}^{*}$ and crosslink with polymer chains, which indicate the propagation process. The active monomers vanish in termination stage when two monomers react with each other and become dead polymers $P_{d}$ .

The material properties of photosensitive resin are significantly associated with the degree of curing $D$ , which could be characterized with the ratio of concentration unconverted functional monomers $C_{M} (mol \cdot m^{- 3})$ with its initial concentration $C_{M_{0}} (mol \cdot m^{- 3})$ as in Equation (5).

\begin{matrix} D = 1 - \frac{C_{M}}{C_{M_{0}}}, D \in (0,1), \end{matrix}

(5)

The dynamic change of unconverted functional monomers concentration was induced by the irradiation of lights. The illumination model was simplified as a one-dimensional attenuation model based on the Beer’s law as in Equation (6).

\begin{matrix} \frac{δ I (z, t)}{δ z} = - A (z, t) I (z, t), \end{matrix}

(6)

where

I (z, t) (W \cdot m^{- 2})

is the light intensity at coordinate of point in thickness direction

z

and time

t

, and

A (z, t) (m^{- 1})

is the local depletion rate of light intensity due to the absorbance of the species.

The deformation gradient $F$ of curing resin was decomposed into three parts including the volumetric shrinkage $G$ , thermal expansion $F^{T}$ , and mechanical deformations $F^{M}$ as formulated in Equation (7).

\begin{matrix} F = F^{M} F^{T} G . \end{matrix}

(7)

The unconstrained reaction shrinkage $G$ during the photopolymerization process was assumed to be isotropic and could be formulated as the function of degree of curing $D$ . In addition, the thermal expansion distortion could be obtained based on the temperature increment.

\begin{matrix} G = \sqrt[3]{(1 - Λ (D)} I, \end{matrix}

(8)

\begin{matrix} F^{T} = α Δ T I, \end{matrix}

(9)

where

Λ (\cdot)

is the material volume shrinkage ratio function, which could be confirmed through material calibration,³¹

I

is the identity tensor,

α

is the thermal expansion coefficient, and

Δ T

is the temperature increment.

The material residual stress during the photopolymerization process was mainly induced by the constraints of the build plate or other solidified layers which contributed to the mechanical deformation gradient $F^{M}$ . The mechanical deformation gradient could be confirmed by constructing the constitutive equation as in Equation (10).

\begin{matrix} σ_{n} = f_{0} σ (F_{1 \to n}) + \sum_{i = 1}^{n} Δ f_{i} σ (F_{i \to n}), \end{matrix}

(10)

where

σ (\cdot)

represents the neo-Hookean material model,

Δ f_{n}

represents the crosslink fraction at time

n

, and

F_{i \to n}

is the total mechanical deformation gradient from time

i

n

For lightweight design of thin-walled artifact structure $M$ in $R^{3}$ , the specific surface of unit volume $δ$ can be defined as a positive real number.

\begin{matrix} δ = \frac{S_{object}}{V_{object}}, δ \in R^{+} . \end{matrix}

(11)

The manufacturing progress rate is denoted by the normalized height. The normalized height $h_{n}$ of the $n^{t h}$ layer is defined as in Equation (12). The normalized layer volume $V_{n}$ is calculated according to Equation (13) to evaluate the structure proportions of layers. In addition, the layer average deformation $d_{a, l}$ and stress $s_{a, l}$ are formulated as in Equation (14) to quantitatively describe the distributions along the normalized height.

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

(12)

\begin{matrix} V_{n} = \frac{V_{l}}{V_{b}}, V_{n} \in (0,1], \end{matrix}

(13)

\begin{matrix} d_{a, l} = \frac{\sum_{i = 1}^{N_{l}} d_{i}}{N_{l}}, s_{a, l} = \frac{\sum_{i = 1}^{N_{l}} s_{i}}{N_{l}}, \end{matrix}

(14)

where

z_{l}

is the cumulative layer height of

l

layers,

z_{b}

is the total height of the 3D manifold,

V_{l}

is the volume of

l^{t h}

layer,

V_{b}

is the total volume of the manifold,

d_{i}

and

s_{i}

are deformation and stress values of sampling points, and

N_{l}

represents the number of sampling points in

l^{t h}

layer.

Fluid–solid coupling simulation for lightweight components under high-speed working condition

Considering the interactions between lightweight components with environmental media under high-speed working conditions, the effectiveness of GDM-based support equilibrium design method in improving working performances was verified. The calculation of the rotor flow field adopted the unsteady, compressible, and viscous Navier–Stokes (N-S) governing equations.^32,33 The momentum equation can be described as in Equation (15). The continuity equation can be formulated with the differential form as in Equation (16).

\begin{matrix} \frac{\partial}{\partial t} (ρ u_{i}) + \frac{\partial}{\partial x_{i}} (ρ u_{i} u_{j}) = - \frac{\partial P}{\partial x_{i}} + \frac{\partial}{\partial x_{j}} (μ \frac{\partial u_{i}}{\partial x_{j}} - ρ \bar{u_{i} u_{j}}) + S_{i}, \end{matrix}

(15)

\begin{matrix} \frac{\partial ρ}{\partial t} + \frac{\partial}{\partial x_{i}} ({ρ u}_{i}) = 0, \end{matrix}

(16)

where

ρ (kg \cdot m^{- 3})

is the density,

u_{i} (m \cdot s^{- 1})

represents the velocity in

x_{i}

direction,

P (Pa)

is the static pressure,

μ (Pa \cdot m)

is the viscosity,

- ρ \bar{u_{i} u_{j}}

is the Reynolds stress term, and

S_{i} (N \cdot m^{- 3})

represents the effects of external force.

The $k - ϵ$ model was implemented to enclose the momentum equation by calculating the Reynolds stress through turbulent kinetic energy $k$ and the turbulent dissipation rate $ε$ as in Equations (17-18).

\begin{matrix} \frac{\partial ρ k}{\partial t} + \frac{\partial}{\partial x_{i}} (k ρ u_{i}) = \frac{\partial}{\partial x_{i}} [(μ + \frac{μ_{t}}{σ_{k}}) \frac{\partial k}{\partial x_{i}}] + G - ρ ε, \end{matrix}

(17)

\begin{matrix} \frac{\partial ρ ε}{\partial t} + \frac{\partial}{\partial x_{i}} (ε ρ u_{i}) = \frac{\partial}{\partial x_{i}} [(μ + \frac{μ_{t}}{σ_{ε}}) \frac{\partial ε}{\partial x_{i}}] + c_{1 ε} \frac{ε}{k} G - c_{2 ε} ρ \frac{ε^{2}}{k}, \end{matrix}

(18)

where

μ_{t}

is the turbulent viscosity,

G

is the production of turbulence due to shear, and

σ_{k}

σ_{ε}

c_{1 ε}

, and

c_{2 ε}

are model constants, which are confirmed via experiments.

The static pressure was confirmed and applied to the blade surfaces to achieve the coupling simulation. The dynamic response of the lightweight structure induced by the air flows was described as in Equation (19).

\begin{matrix} M \frac{d^{2} u}{d t^{2}} + C \frac{d u}{d t} + K u + τ = 0, \end{matrix}

(19)

where

M

is the mass matrix,

C

is the damping matrix,

K

is the stiffness matrix,

u

is the blade velocity, and

τ

is the load exerted by the air flow on the blade interface.

The mass distribution balance analysis of lightweight components was conducted by calculating the offset distance of gravity center $δ_{c}$ . Assuming uniform density distribution of manufactured parts, the offset of gravity center was induced by asymmetrical shrinkage which further induce volume deviations and unbalance mass distribution. Considering the rotary working condition, the unbalanced mass will be amplified in proportionally to square of distance. Thus, the offset distance of gravity center $δ_{c}$ was calculated by weighted average of coordinates $x_{i}$ where the weights were calculated by multiplying the square of distance to original point $r_{i}$ and volume $V_{i}$ of sampling points as in Equation (20).

\begin{matrix} δ_{c} = \sum_{i = 1}^{N} r_{i}^{2} V_{i} x_{i} . \end{matrix}

(20)

Structure buckling response of lightweight components

The buckling response is the sudden change in shape (deformation) of a structural component under load, such as the bowing of a column under compression or the wrinkling of a plate under shear, on the basis of which, the strength, stiffness and stability of the structure design could be evaluated and further assured.

Buckling response analysis is mainly employed to explore the elastic-plastic stability of lightweight structures under specific loads and determine the critical load of structural instability. The eigenvalue buckling analysis equation is:

^{τ} K τ \emptyset = 0,

(21)

where τ is the moment when the load reaches the critical load (s),

^{τ} {}^{τ}K

is the nonlinear tangent stiffness matrix at the critical load moment, and

\emptyset

is the displacement eigenvector.

Structural stability problems can be divided into linear stability analysis and nonlinear stability analysis. The classical elastic theory establishes the equilibrium equation at the initial position of the structure and does not consider the change of the equilibrium condition after the structural deformation. Eigenvalue buckling analysis does not consider the influence of geometric nonlinearity on the equilibrium equation and geometric equation. It can be used to predict the theoretical buckling strength of an ideal linear structure. At the same time, it provides a reference load value for nonlinear buckling analysis. The buckling analysis formula for linear stability analysis is:

\begin{matrix} ^{τ} K = K_{e} + p_{c r} (K_{u} (u) + K_{G} (σ)), \end{matrix}

(22)

where

K_{e}

is the elastic stiffness matrix of the structure,

K_{u}

is the initial displacement matrix,

K_{G}

is the initial stress matrix, and

p_{c r}

is the critical load (N).

Then, the linear stability analysis can be reduced to solving the linear eigenvalue problem:

\begin{matrix} [K_{e} + p_{c r} (K_{u} (u) + K_{G} (σ))] \emptyset = 0 . \end{matrix}

(23)

When the material is in an elastic state, $K_{u}$ can be ignored. Then, the characteristic equation of classical stability analysis is obtained:

\begin{matrix} (K_{e} + p_{c r} K_{G} (σ)) \emptyset = 0 . \end{matrix}

(24)

A series of eigenvalues $p_{1}$ , $p_{2}$ , and so on can be calculated. The minimum eigenvalue $p_{1}$ is the critical load of the linear stability analysis of the structure, and the corresponding displacement mode $\emptyset_{1}$ is the buckling mode of structural instability.

When the structure is in a large deformation state, the stiffness matrix of the structure is the nonlinear function of the load amplitude and the displacement vector. The process of nonlinear stability analysis is as follows.

Calculate the stiffness matrix corresponding to each incremental step, if any:

\begin{matrix} \det (^{τ} K) > 0, det (^{t + Δ t} K) < 0 . \end{matrix}

(25)

It indicates that in the interval (t, t + Δt) at some point within $τ (t < τ < t + Δ t)$ has:

\det^{τ} K = 0,

(26)

where K is a singular matrix and

τ

is the equilibrium point of the structure transits from stable to unstable states. This equilibrium point may be a bifurcation critical point or an extreme critical point. The load corresponding to this point may be the critical load of bifurcation instability or extreme instability.

Expressed $^{τ} K$ in linear interpolation form of ${}^{t}K$ and $^{t + Δ t} K$ :

^{τ} K =^{t} K + \frac{t_{c r} - t}{Δ t} (^{t + Δ t} K -^{t} K),

(27)

where

t_{c r} (s)

is the moment when the critical load is reached.

The characteristic equation of nonlinear stability analysis is expressed as:

\begin{matrix} [{}^{t}K + λ (^{t + Δ t} K - {}^{t}K)] \emptyset = 0, \end{matrix}

(28)

\begin{matrix} λ = \frac{t_{c r} - t}{Δ t} . \end{matrix}

(29)

A series of eigenvalues can be obtained by calculation $λ_{1}$ , $λ_{2}$ , and so on and corresponding characteristic displacement mode $\emptyset_{1}$ , $\emptyset_{2}$ , and so on. Minimum characteristic value $λ_{1}$ corresponding to the load amplitude $p_{c r}$ is the critical value of nonlinear stability of the structure, and $\emptyset_{1}$ is its corresponding buckling mode.

Confirming Geometric Deformation Through Constrained Volume Shrink Model

The mechanical rotor blade that was an essential part for unmanned aerial vehicle (UAV) was chosen as the numerical instance due to its high-speed working condition and high-aspect-ratio geometric structure, which arise strict demands in manufacturing quality. Figure 1a and 1b display the overall structure of UAV, including blade, motor, fuselage, and undercarriage, from various views. The corresponding wall thickness distributions and blade manifolds are displayed in Figure 1c–1f. For UAV shell, the total surface area $S_{object}$ is 6447.5422 cm², total volume $V_{object}$ is 769.0479 cm³, and specific surface area $δ$ is 0.8384. The rotor radius $R$ of the blade was 123 mm, the root length $L$ was 25.8685 mm, the surface was 59.6070 ${cm}^{2}$ , and the volume was 2.994 ${cm}^{3}$ .

FIG. 1.

The overall UAV and blade structures from various views. (a) and (b) are the overall UAV structures. (c) and (d) are load-bearing shell. (e) and (f) show the corresponding blade structures. UAV, unmanned aerial vehicle.

Figure 2 shows load-bearing response contour of thin-walled UAV (shown in Fig. 1).

FIG. 2.

Load-bearing response contour of thin-walled UAV.

The printing schemes for blade components with various support structures are displayed in Figure 3. For the support in (a), the volume was 1.052 ${cm}^{3}$ and the support proportion was 0.2600. For the support in (b), the volume was 3.358 ${cm}^{3}$ and the support proportion was 1.3645. The designed manufacturing parameters could be applied to virtual printing and visualized. The theoretical manufacturing situation with various conceptual support structures was visualized by employing 3D visualized computing digital twins, as shown in Figure 3c and 3d. Note that the layer thickness was relational to manufacturing precision but uninfluential to manufacturability; thus, it was magnified for visualization purposes, and it can be much smaller in physical manufacturing.

FIG. 3.

The printing schemes for blade components with various conceptual support structures. (a) and (b) illustrate different support structures. (c) and (d) visualize the virtual printing process of corresponding printing schemes.

The mechanically constrained volume shrink model was conducted via finite element analysis (FEA) simulation. The room temperature is set to $25 ° C$ , and the mesh size is $0.1 mm$ in the manufacturing direction and 0.7 $mm$ in directions parallel to the slices. The mesh type was C3D8T for conducting thermal–structure coupling analysis. The initial curing rate was set to 0.001 and light intensity was 5.0 $mW \cdot {cm}^{- 2}$ . For the material properties, the density is $1.14 g \cdot {cm}^{- 3}$ , the thermal conductivity is $0.2 W \cdot m^{- 1} \cdot K^{- 1}$ , the specific heat is $400 J \cdot {kg}^{- 1} \cdot K^{- 1}$ , and the thermal expansion coefficient is 3.3e–5 ${° C}^{- 1}$ . More technical details can be referred to in previous work.²⁹

The deformation and residual stress under various manufacturing parameters, which were mainly induced due to constrained shrinkage during photopolymerization process, were simulated and displayed in Figure 4 (all metadata of Fig. 4 are shared with the readers for verification or calculation in Supplementary Data S1). The average manufacturing deformation of the blade in (a) was 0.0367 and the maximum deformation was 0.4023, which occurred in relative position (0.5144, 0.6364, 0.6883). The average manufacturing Mises stress of the blade in (b) was 28.6727, and the maximum stress was 488.511, which occurred in relative position (0.4615, 0.5568, 0.7208). The average manufacturing deformation of the blade in (c) was 0.0319, and the maximum deformation was 0.4014, which occurred in relative position (0.2885, 0.3295, 0.4221). The average manufacturing stress as in (d) was 28.5441, and the maximum stress was 488.69, which occurred in relative position (0.4712, 0.4318, 0.5812). With the support structure designed through GDM, the manufacturing deformation reduced and the Mises stress slightly increased due to more mechanical constraints.

FIG. 4.

The deformation and stress distributions of blades with various support structures due to photopolymerization.

The layer average deformation and stress of manufactured blades with various conceptual support structures are illustrated in Figure 5. The normalized layer volume was displayed to indicate the material proportion of layers, and the maximum layer volume was in $h_{n}$ = 0.6753. For Figure 5a, the maximum layer average deformation of blade without GDM occurred when $h_{n} = 0.7086$ and $h_{n}$ = 0.7517 for blade with GDM. The standard deviation of layer average deformation for blade without GDM was 0.0297 and 0.0072 for blade with GDM. For Figure 5b, the maximum layer average stress of blade without GDM occurred when $h_{n} = 0.7053$ and $h_{n}$ = 0.9279 for blade with GDM. The standard deviation of layer average deformation for blade without GDM was 18.5248 and 4.8379 for blade with GDM.

FIG. 5.

The layer average deformation and stress of manufactured blades with various conceptual support structures.

Interactions Between Lightweight Blade Structure with Environmental Media

The performance of blades manufactured with various conceptual support structures was tested through fluid simulations. The external flow field was set to a cylinder with the 400 mm diameter and 400 mm length. The element size was set to 25 mm for external flow field, 15 mm for rotational moving domain, and 0.35 mm for blade. The rotation velocity of the blade was 5000 rev/min. The overall pressure distributions for blades manufactured with various support structures are illustrated in Figure 6. The maximum pressure difference for situation in (a) was 3360.4 Pa and 7171.9 Pa for situation in (b).

FIG. 6.

The overall pressure distributions for blades manufactured with various conceptual support structures. (a) is the blade manufactured without GDM, and (b) is the blade manufactured with GDM. GDM, geometric deformation minimization.

The air flow traces induced due to the rotation of blade are shown in Figure 7. The maximum velocity magnitude was $46.2380$ m/s for (a) and $48.2723 m / s$ for (b). And the maximum velocity along negative z direction was $28.3579 m / s$ for (a) and $31.4828 m / s$ for (b).

FIG. 7.

The air flow pathlines with velocity and pressure distributions. (a) and (b) are the blade without GDM. (c) and (d) are blades with GDM.

The velocity and static pressure distributions of air flows on blade surfaces were confirmed to more intuitively display the performance of blades as in Figure 8. The average velocity was 23.9812 m/s for blade in (a) and $23.8634 m / s$ for blade in (c).

FIG. 8.

The velocity and static pressure distributions of blade surfaces. (a) and (b) are velocity and static pressure distributions on blade surfaces that were manufactured without GDM. (c) and (d) are about blades manufactured with GDM.

Working Performance Evaluation via Bidirectional Fluid–Solid Coupling Simulation and Buckling Response Analysis

The solid–fluid coupling simulation was conducted to reveal the interactions between fluid field and blade structures. The solid–fluid coupling simulation for blades manufactured through various conceptual support structures was conducted by calculating 20 time-steps with the time interval of $5 \times 10^{- 5} s$ . A 5000 rev/min rotation load was applied to blade structures as the working condition. There were 17,907 elements for transient structure analysis and 614,313 elements for fluid flow simulation where the meshes of interfaces were refined to ensure accuracy. The iteration root mean square error changes are illustrated in Figure 9. There were 45 iterations for (a) and 42 iterations for (b). The convergence targets were set to 0.01 for both situations.

FIG. 9.

The iteration root mean square error for fluid–solid coupling simulations.

The deformation and equivalent stress distributions obtained via bidirectional fluid–solid coupling simulation and buckling response analysis as visualized in Figure 10 were confirmed to verify the effectiveness of the proposed method in improving working performance of blades. The distributions of the overall blade structures and cross sections were displayed. As for blades manufactured without GDM, the maximum deformation in (a) was 4.7524 mm, and the average deformation was 2.8230 mm. The maximum equivalent stress in (b) was 88.414 MPa, and the average stress was 8.3805 MPa. As for blades manufactured with GDM, the maximum deformation in (c) was 4.3611 mm, and the average deformation was 2.8143 mm. The maximum equivalent stress in (d) was 87.540 MPa, and the average stress was 8.1819 MPa. More detailed information about the fluid–solid coupling simulation is listed in Table 1.

FIG. 10.

The deformation and equivalent stress distributions of blades obtained via bidirectional fluid–solid coupling simulation and buckling response analysis, (a) and (b) are outcomes about blades manufactured without GDM. (c) and (d) are about blades manufactured with GDM.

Table 1.

Solid–Fluid Coupling Simulation Outcomes of Blades with Various Support Structures

Manufacturing parameters	Deformation (mm)			Equivalent stress (MPa)
Manufacturing parameters	Max	Min	Average	Max	Min	Average
Without GDM
Overall	4.7524	0.5929	2.8230	88.4140	0.0016	8.3805
Section 1	4.1262	2.4573	3.3709	18.2000	0.3415	7.0462
Section 2	2.8786	2.5675	2.7639	10.4490	0.2743	3.2270
Section 3	3.3067	2.9822	3.1368	0.4946	0.0523	0.3162
With GDM
Overall	4.3611	0.5628	2.8143	87.5400	0.0077	8.1819
Section 1	3.7262	2.4310	3.1863	17.7860	0.2988	6.8160
Section 2	2.7162	2.0370	2.3875	7.6895	0.1708	2.8685
Section 3	3.3040	2.3507	2.7882	0.9004	0.0239	0.3729

GDM, geometric deformation minimization.

The gravity center offset distance $δ_{c}$ of 3DP fabricated blade structures with and without GDM is confirmed as in Figure 11. The average gravity center offset distance without GDM was 0.0215 mm, and the largest offset distance was 0.2928 mm and occurred when $h_{n} = 0.65 %$ . The average gravity center offset distance with GDM was 0.0156 mm, and the largest offset distance was 0.1374 mm and occurred when $h_{n} = 28.57 %$ .

FIG. 11.

The gravity center offset distance of layers for UAV blade lightweight structure.

Physical Experiments of Manufacturing Lightweight Blade Component via Digital Light Processing

The physical experiment of additive manufacturing application is captured in Figure 12. The stereolithography apparatus is based on DLP. The power supply for the adopted printer is AC (alternating current) 220V. The ambient temperature is 25°C with 55% RH (relative humidity). The laser system is ultraviolet light-emitting diode with wavelength λ 405 nm and power 60 mW. The printing accuracy can reach ±0.035 mm. The layer thickness can be adjusted within 50–100 $μ m$ , and the stroke length along Z-axis is 200 mm. The printing velocity is 10–15 s per layer.

FIG. 12.

The physical experiment of manufacturing UAV blade structures via DLP. (a) is the overall view of the DLP equipment. (b) is the enlarged view of the build platform and fabricated blade components. (c) and (d) are printed blade components with various conceptual designed support structures and materials after postprocesses. DLP, digital light processing.

The forming material can be photosensitive materials such as photosensitive resin, light curing wax, and so on. For the blade components that were labeled from M1S1 to M1S4 as illustrated in Figure 12c, the blades were manufactured with standard white resin. For the blade components that were labeled from M2S1 to M2S4 as illustrated in Figure 12d, the blades were manufactured with tough transparency resin. The components labeled with M1S3 and M2S3 corresponding to the design scheme are illustrated in Figure 3a. The components labeled with M1S4 and M2S4 corresponding to the design scheme are illustrated in Figure 3b. The comparison of material properties is listed in Table 2, which were provided by sales manufacturers.

Table 2.

Material Properties of Resins Used to Print Blade Components

Forming material type	Standard resin (Fig. 12c)	Tough resin (Fig. 12d)
Curing wavelength ( $nm$ )	450
Viscosity ( $mPa \cdot s$ )	106.2	240.0
Tensile strength ( $MPa$ )	44–49	57–64
Tensile modulus ( $MPa$ )	1287–1429	1960–2200
Flexural strength ( $MPa$ )	66–72	65–76
Flexural modulus ( $MPa$ )	1780–1976	1890–2130
Elongation at break	17–19%	10–16%
Shore hardness	82D	77D

The impacts of various support structures and materials on the manufacturing quality were investigated through optical microscope, as shown in Figure 13. The adopted optical microscope can be widely applied in detections of semiconductor, flat panel display, circuit package, electric subgrade plate, metal/ceramic components, and precision die. The minimum graduation in fine movement of focusing system is 2 μm, and the size of objective table is 192 mm × 141 mm, which can be moved within the range 50 × 40 mm. The illumination system offers a 12V 50W halogen lamp as well as field stop and aperture stop. The resolution of microscopic images was 5600 × 5600 captured with a camera whose pixel point size is $2.2 μ m \times 2.2 μ m,$ and the frame rate can be 40 fps. In order to keep the optical path clean and dust-free, the dust-blowing ball adopts a one-way air valve made with copper and chrome plating, and the wind speed can reach 80 km/h. In order to enlarge the field of view of the electronic imaging sensor, the charge-coupled device adapter with build-in lens can be employed. The microscopic images in Figure 13a, 13c, 13e, and 13g correspond to blade components labeled from M1S1 to M1S4 as in Figure 12c. The microscopic images in Figure 13b, 13d, 13f, and 13h correspond to blade components labeled from M2S1 to M2S4 as in Figure 12d.

FIG. 13.

The microscopic images of various conceptual support structures and materials. (a), (c), (e), and (g) are manufactured with white photosensitive resin as in Figure 12c. (b), (d), (f), and (h) are manufactured with transparent photosensitive resin as in Figure 12d.

An innovation comparison between those methods published in the literature, and the proposed method is summarized in Table 3.

Table 3.

Innovation Comparison Between the Published Papers and the Proposed GDM Method

Innovation concerns	Others’ published papers	The proposed GDM method (Ours)
Support equilibrium design considering geometric deformation	• The scanning approach was optimized to achieve uniform thermal distribution and reduce manufacturing deformations.¹³• The geometric accuracy was improved by optimizing printing angle.²⁰	• The effects of support structures in equilibrating and reducing geometric deformation were investigated.• The support structures were equilibrated by jointly considering manufacturing deformation and working performances of mechanical components (“Mechanical constrained volume shrink model of photocurable AM process” and “Fluid–solid coupling simulation for lightweight components under high-speed working condition” sections).
Deformation induced during manufacturing process	• The support structures were repurposed to reduce deformation for metal AM technologies.¹⁴• The distortion of components manufactured via LPBF was numerically investigated.¹⁵	• The mechanical constrained volume shrink model of photocuring process was constructed according to the reaction kinetics of polymerization [Equations (1–4)].• The manufacturing deformation induced by volumetric shrinkage, thermal expansion, and mechanical deformations during photocuring process were jointly considered [Equation (7)].
Working performance in service stage	• Investigate the impacts of residual stresses induced during manufacturing.¹¹• Considered the effects of heat source scanning patterns on thermo-mechanical behavior merely in manufacturing stage.¹²	• The interaction between mechanical components and environment media was considered via fluid–solid simulation, as well as buckling response analysis (Fig. 10).• The deformation and stress distributions for components under high-speed working conditions were simulated. The maximum deformation was decreased by 8.23%, and the maximum stress was decreased by 0.99% (Table 1).
Experiment verification	• The effects of optimized support structures on reducing thermal deformations were verified via numerical simulations.¹⁶• The numerical and physical experiments were conducted merely in manufacturing stage.¹⁷	• Numerical experiments were conducted and figured out the impacts of support structures on working performances. The largest gravity center offset distance was reduced by 53.07% (Figs. 8 and 11).• Physical experiments were conducted to verify the effectiveness of GDM via DLP and microscopic images (Figs. 12 and 13).

AM, additive manufacturing; DLP, digital light processing; LPBF, laser powder bed fusion.

Conclusion

Support equilibrium design method for laser AM of lightweight component based on GDM is proposed

Addressing the increasingly higher manufacturing quality demands, especially for lightweight component with high-speed working conditions, the GDM-based support equilibrium design method was thereby proposed. Aiming to reduce geometric deformation and improve the working performance of lightweight component in service stage, the manufacturing distortion was confirmed and deformed lightweight blade was generated to be tested under working conditions. The performance-oriented support equilibrium design method is superior in ensuring the functionality of mechanical parts.

GDM was conducted to investigate the impacts of support structures on working performance of lightweight mechanical components in service stage

The geometric deformation of photocurable AM process was calculated by constructing mechanically constrained volume shrink model. The deformation and stress distributions under various conceptual support morphologies were quantitatively confirmed. The deformed manifold was generated by mapping deformation to the designed digital manifold, and the interaction between lightweight blade structures with air flows under rotation conditions was investigated through fluid simulation.

Effectiveness of GDM in reducing geometric deformation and improving working performance of lightweight components was verified with both numerical and physical experiments

The geometric deformation during manufacturing process was simulated and predicted through mechanical constrained volume shrink model. The effectiveness of GDM-based method in improving working performance of lightweight blade components under high-speed rotary working conditions was examined with solid–fluid coupling simulation and buckling response analysis. The lightweight UAV blade components with various conceptual support structures and materials were physically manufactured via DLP. In addition, the impacts of support structures and materials on the manufacturing quality were investigated through optical microscope.

In the future, the proposed GDM method could be applied to support equilibrium design for 3DP fabricated mechanical components made with reinforced composite materials under various extreme working conditions such as vibration, repeated impact, load-bearing distribution and drastic temperature variations.

Authors’ Contributions

J.X. initiated the essential innovations and finalized the article. L.W. carried out the analytic mathematical programming and theoretical computations. Z.T. participated in the mechanical experiment. K.W. verified the practical superiority of energy consumption. S.Z. guided the team research as principal investigator. J.T. made constructive suggestions to the work as Chief Scientist.

Footnotes

Author Disclosure Statement

No competing financial interests exist.

Funding Information

The work presented in this article is funded by the National Key Research and Development Project of China (No. 2022YFB3303303), and Open Fund of State Key Laboratory of Materials Processing and Die & Mould technology of China (No. P2024-001).

Supplemental Material

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