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Research Papers

# Offline Predictive Control of Out-of-Plane Shape Deformation for Additive ManufacturingOPEN ACCESS

[+] Author and Article Information
Yuan Jin

Mork Family Department of Chemical
Engineering and Materials Science,
University of Southern California,
Los Angeles, CA 90089
e-mail: yuanjin@usc.edu

S. Joe Qin

Mork Family Department of Chemical
Engineering and Materials Science,
University of Southern California,
Los Angeles, CA 90089
e-mail: sqin@usc.edu

Qiang Huang

Associate Professor and Gordon S. Marshall
Early Career Chair in Engineering
Daniel J. Epstein Department of Industrial and
Systems Engineering,
University of Southern California,
Los Angeles, CA 90089
e-mail: qiang.huang@usc.edu

Manuscript received July 26, 2015; final manuscript received April 10, 2016; published online July 25, 2016. Editor: Y. Lawrence Yao.

J. Manuf. Sci. Eng 138(12), 121005 (Jul 25, 2016) (7 pages) Paper No: MANU-15-1375; doi: 10.1115/1.4033444 History: Received July 26, 2015; Revised April 10, 2016

## Abstract

Additive manufacturing (AM) is a promising direct manufacturing technology, and the geometric accuracy of AM built products is crucial to fulfill the promise of AM. Prediction and control of three-dimensional (3D) shape deformation, particularly out-of-plane geometric errors of AM built products, have been a challenging task. Although finite-element modeling has been extensively applied to predict 3D deformation and distortion, improving part accuracy based purely on such simulation still needs significant methodology development. We have been establishing an alternative strategy that can be predictive and transparent to specific AM processes based on a limited number of test cases. Successful results have been accomplished in our previous work to control in-plane (x–y plane) shape deformation through offline compensation. In this study, we aim to establish an offline out-of-plane shape deformation control approach based on limited trials of test shapes. We adopt a novel spatial deformation formulation in which both in-plane and out-of-plane geometric errors are placed under a consistent mathematical framework to enable 3D accuracy control. Under this new formulation of 3D shape deformation, we develop a prediction and offline compensation method to reduce out-of-plane geometric errors. Experimental validation is successfully conducted to validate the developed 3D shape accuracy control approach.

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## Introduction

AM directly builds products based on computer-aided design (CAD) models through layered fabrication processes [1,2]. Without using fixtures, jigs, or mold tools, AM enables building parts with complex contours, cavities, and lattice structures, which makes it a promising direct manufacturing technology.

However, geometric accuracy control is still one of the major bottlenecks for AM technology, particularly for control of 3D geometric errors [2,3]. The layer formation and interlayer bonding involve complicated mechanisms. Although numerical finite-element modeling has been extensively used to predict 3D deformation and distortion in AM processes [49], improving part accuracy based purely on such a simulation approach is far from being effective and seldomly used in practice [10,11]. A shrinkage compensation factor has been commonly used in practice to apply compensation uniformly to the entire product or to apply different factors to the CAD model for each section of a product [12]. However, it is time-consuming to establish a library of compensation factors for all the part shapes. Furthermore, it is proved in our work [13] that the shrinkage compensation factor is optimal only if the geometric deformation is the same everywhere.

A large body of the literature has been on experimental investigation of optimal process settings and on establishing empirical process models to minimize geometric errors. For instance, Taguchi and design of experiments methods have been applied to investigate stereolithographic (SLA) processes [1416] or fused deposition modeling processes [1720]. Factors such as layer thickness, part build orientation, raster angle, raster width, and raster to raster gap (air gap) are varied to evaluate their impacts on geometric accuracy. However, process performance is highly dependent on the products being experimented and the predictability for new complex shapes is limited.

Tong et al. [21,22] established polynomial regression models to analyze the product deformation in the x, y, and z directions separately. The mapped 3D error models were then applied to compensate 3D error. In their study, the volumetric error was reduced around 35%, showing that software error compensation was an effective way to improve geometric accuracy of AM products. However, they also mentioned that the out-of-plane deformation was not obviously reduced in their experimental studies.

In a series of work [13,2328], Huang and coworkers intended to develop a new predictive learning strategy with the ability of learning from a limited number of tested shapes and deriving compensation plans for new and untried shapes. This new alternative strategy is motivated by the fact that AM has to build products with huge varieties. It is imperative to establish a methodology independent of shape complexity and specific AM processes. Huang et al. first established a generic, physically consistent approach to model and predict in-plane (x–y plane) shape deformation along product boundary and derive optimal compensation plans. Huang et al. [25,26] extended the work in Ref. [13] from a cylindrical shape to polyhedrons. However, no detailed discussion has been provided for methodology extension to the 3D case.

This work is therefore motivated by extending the methodological framework in Refs. [13,25,26] from in-plane error to out-of-plane error prediction and offline compensation. Following Introduction, Sec. 2 presents a novel formulation of spatial error representation to place both in-plane and out-of-plane errors under a consistent mathematical framework. Section 3 presents the detailed modeling procedure to establish the out-of-plane error model and its experimental validation using SLA process. Optimal compensation of out-of-plane errors and its experimental validation are illustrated in Sec. 4. Conclusion is given in Sec. 5.

## Spatial Deformation Formulation for AM—A Unified Representation of 3D Geometric Errors

###### From In-Plane Deformation Representation to 3D Spatial Deformation Representation.

In our previous study, the in-plane shape deformation along boundary of an AM printed product is defined as $Δr(θ,r0(θ),z)=r(θ,r0(θ),z)−r0(θ,z)$ in the polar coordinate system (PCS) (Fig. 1), where $r0(θ)$ represents the nominal radius of the product design point at location θ and z denotes the height of a specific layer in the vertical z direction. The essence of this representation is to decouple the geometric shape complexity from the deformation modeling by transforming in-plane geometric errors from the Cartesian coordinate system into a functional profile defined in the PCS. This transformation helps to visualize deformation patterns and model complexity. For example, Fig. 2 illustrates the in-plane geometric shape deformation $Δr(θ,r0(θ)|z0)$ for four cylindrical products built by SLA process [13].

Denote $Δr(θ,r0(θ)|z0)=f(θ,r0(θ)|z0)$, the in-plane shape deformation along its boundary at a given layer z0. Huang et al. [13] decomposed the predicted in-plane deformation $f(θ,r0(θ)|z0)$ into three components Display Formula

(1)$f(θ,r0(θ)|z0)=f1(r0(θ)|z0)+f2(θ,r0(θ)|z0)+f3(θ|z0)+ϵ$

where f1 represents the volumetric deformation independent of locations; f2 is the location-dependent deformation affected by local geometric shape features; f3 represents the deformation not captured by the first two terms, which might include some high-order terms, and ϵ is the noise term.

To incorporate the out-of-plane deformation, according to Huang [29], we revise the previous $(r,θ,z)$ representation to a spherical coordinate system (SCS) $(r,θ,ϕ)$ to depict both the in-plane and out-of-plane shape deformation in a unified mathematical formulation. The main reason for this change is that it facilitates the representation of the out-of-plane error in the same way as the in-plane error.

To illustrate this idea, we first define the in-plane and out-of-plane shape deformation in the SCS using the simple cylindrical shape. As shown in Fig. 3, for an arbitrary point $P0(r0,θ0,ϕ0)$ at a given height $ϕ=ϕ0$ or $z=r cos(ϕ0)$, the horizontal cross section view of the product passing P0 is given as $(r sin(ϕ0),θ|ϕ0)$, whose shape deformation $Δr(θ,r0(θ,ϕ0)|ϕ0)$ represents the in-plane geometric error and its model formulation has been developed in our previous work [26].

Denote the in-plane deformation model $Δr(θ,r0(θ,ϕ)|ϕ)$ as $h(r,θ|ϕ)$. Let us define the expectation of the in-plane deformations $h(r,θ|ϕ)$ over all $ϕ$Display Formula

(2)$∫ϕh(r,θ|ϕ)dϕ$

Remark. Model (2) represents the average in-plane deformation over all the layers. Our previous models developed for cylindrical and polyhedron shapes can be viewed as the average in-plane deformation when the out-of-plane deformation is not considered.

On the other hand, the out-of-plane shape deformation, which is the error in the vertical direction, can be represented in the vertical cross section determined by angle θ (Fig. 4). Any point P0 on the boundary of the vertical cross section $θ=θ0$ is given as $(r,ϕ|θ0)$.

Consistent with the in-plane deformation $Δr(θ,r0(θ,ϕ)|ϕ)$, the out-of-plane deformation model $Δr(ϕ,r0(θ,ϕ)|θ)$ is denoted as $v(r,ϕ|θ)$. Let us define the expectation of the out-of-plane deformations $v(r,ϕ|θ)$ over all θ in the vertical cross section as Display Formula

(3)$∫θv(r,ϕ|θ)dθ$

which can be interpreted as the average out-of-plane deformation in a similar fashion.

Remark. Geometrically, model $v(r,ϕ|θ)$ is essentially equivalent to $h(r,θ|ϕ)$. This suggests that the mathematical formulation developed in our previous work for the in-plane errors can be borrowed under this new formulation in the SCS. In this way, 3D geometric deformation can be described in a unified mathematical framework [29,30].

Remark. It should be noted that $v(r,ϕ|θ)$ and $h(r,θ|ϕ)$ may differ even the horizontal and vertical cross section views share the same shape. Different from the in-plane deformation whose representation is along the radial direction, the out-of-plane deformation is defined along the vertical direction. The difference in representation leads to different error patterns. Furthermore, the vertical deformation is influenced by extra factors, such as layer interactions and gravity, resulting in different deformation patterns. Thus, other than presenting a new formulation for 3D geometric errors, another major effort of this paper is to further establish the out-of-plane deformation model in order to obtain the 3D representation.

The 3D error of any point P0 on the shape boundary is therefore decomposed into two orthogonal components: in-plane and out-of-plane errors. The orthogonality ensures that the model building and error compensation can be conducted separately for two error components. Since our previous work [13,2327] has addressed the issue of in-plane error control, this work will focus more on the method of prediction and control of out-of-plane errors.

To summarize, this novel formulation of spatial deformation enables a consistent mathematical formulation of both in-plane and out-of-plane errors, which readily incorporates our previous work on predictive modeling and compensation of in-plane errors.

###### Predictive Modeling of Out-of-Plane Errors.

Predictive modeling of out-of-plane deformation aims to achieve effective prediction based on a limited number of test shapes. Since the out-of-plane deformation shares the same mathematical formulation with the in-plane deformation, we first review the predictive in-plane error model $f(θ,r0(θ))$ developed in Refs. [25,26] for cylindrical and polyhedron shapes (in PCS) Display Formula

(4)$f(θ,r0(θ))=g1(θ,r0(θ))+g2(θ,r0(θ))+g3(θ,r0(θ))+εθ$

where g1 represents the basis model for the cylindrical shape, $g2(θ,r0(θ))$ is the so-called cookie-cutter function added to the cylindrical basis in order to carve out any polygon shape from a circumcircle, $g3(θ,r0(θ))$ is an optional higher order term, and $εθ$ is the noise term.

Two kinds of cookie-cutter functions have been proposed: the square wave model Display Formula

(5)$g2(θ,r0(θ))=β2r0αsign[cos(n(θ−ϕ0)/2)]$

and the sawtooth cookie-cutter model Display Formula

(6)$g2(θ,r0(θ))=β2r0αI(θ−ϕ0)sawtooth(θ−ϕ0)$

where the sawtooth function is defined as Display Formula

(7)$sawtooth(θ−ϕ0)=(θ−ϕ0)MOD(2π/n)$

Although this formulation paves the way to model the out-of-plane deformation in the vertical cross section given θ0, the out-of-plane error modeling faces unique issues. The first issue is the definition of the deformation in the vertical section. Figure 5 illustrates our definition of the out-of-plane error $Δz$ in a PCS defined vertical cross section view.

For representation convenience, we define Display Formula

(8)$ϕ′=(π/2−ϕ) MOD 2π$

In the PCS defined vertical cross section view, the nominal shape boundary is represented as $r0(ϕ′)$ at angle $ϕ′$, while the corresponding actual shape boundary is represented as $r(ϕ′,r0(ϕ′))$. The out-of-plane deformation $Δz$ is therefore defined as Display Formula

(9)$Δz(ϕ′|θ0)=(r(ϕ′,r0(ϕ′)|θ0)−r0(ϕ′)|θ0)sin(ϕ′|θ0)$

with this definition, $Δz$ can adopt the same formulation developed for the in-plane deformation.

To achieve prediction of arbitrary polygon or circular shapes in the vertical cross section, we still adopt the cookie-cutter concept proposed in our previous work. However, we have to make modification to accommodate the out-of-plane deformation.

Similarly, the out-of-plane deformation model is defined as Display Formula

(10)$Δz(ϕ′,r0(ϕ′)|θ0)=g1(ϕ′,r0(ϕ′)|θ0)+g2(ϕ′,r0(ϕ′)|θ0)+εϕ′|θ0$

where g1 is the cylindrical basis model, g2 is the revised cookie-cutter function to be developed in Sec. 3, and $εϕ′|θ0$ is the noise term.

Remark. This formulation works more efficiently for axial symmetric products such as prisms and cylinders, whose vertical cross sections along z axis are in the same shape. In this case, we normally need one predictive model for the out-of-plane deformation. Otherwise, for products with quite different vertical cross section shapes, several predictive models are required to cover every point on the shape boundary, making the generic model much more complicated. For brevity, this work illustrates the proposed method using symmetric products.

## Methodology Illustration and Experimental Validation

In this section, we demonstrate how the generic out-of-plane deformation model (10) can be applied to a rectangular prism. There are two reasons why we choose rectangular prism as the experimental investigation example:

1. (1)With the vertical cross section shape being polygons, in-plane deformation models developed in Refs. [25,26] for polygons can be directly applied.
2. (2)It is easy to get the measurement data from the prism part using the Micro-Vu vertex measuring machine, for all the six surfaces of the prism part.

###### Experiment Design and Observations From SLA Process.

A variant of SLA process (MIP-SLA) machine is used to build the test parts, where a digital micromirror device projects the designed pattern to the resin tank. The light exposure initiates the resin solidification process for each layer and the platform moves down vertically for the next layer. The size of the test part is 1.6 in. × 1.2 in. × 0.5 in. (L × W × H). The part is positioned at the center of the platform. Since we are interested in the deformation of every point on the part surfaces, including the bottom surface, we add the support under the part for the purpose of separating the test part from the printing base. The specific parameters in SLA process and the design for the part are listed in Table 1.

The printed part is shown in Fig. 6. A nonsymmetric sunk cross with line thickness of $0.02 in.$ is designed at the center of each surface to help identify the orientation of the test part and reduce measurement errors (errors caused by part shift and rotation during measurement). A Micro-Vu precision machine is used to measure and evaluate the geometric accuracy of all the six surfaces of the prism part. Same measurement procedure is followed for each surface to reduce the measurement errors. Figure 6 shows a side view of the measured surface, where the dotted line is the designed shape boundary and the solid line is the actual measured shape boundary. Deformation occurs everywhere on the boundary, especially the obvious deformation at the bottom.

In Fig. 7, the geometric deformations of the four-side surfaces are presented in the PCS (curves in solid lines). By comparing the deformation profiles of the four-side surfaces, it is clear that all the systematic deformation patterns follow the same trend, indicating the out-of-plane deformation is caused by the same factors with slightly different impact. For simplicity of method illustration, in Sec. 3.2, we apply a unified model to describe the out-of-plane deformation.

###### Predictive Modeling of 3D Geometric Errors.

This section demonstrates the proposed method of constructing the out-of-plane deformation model.

###### Cylindrical Basis Function.

The cylindrical basis function $g1(ϕ′,r0(ϕ′))$ in model (10) represents the deformation pattern for a cylinder. For the SLA process in the experiment, we have shown in Ref. [13] that $cos(2ϕ′)$ basis function captures the deformation pattern well. To test the generality of the identified basis and its applicability to the out-of-plane deformation, we still adopt its form Display Formula

(11)$g1(ϕ′,r0(ϕ′))=β0r0+β1r0 cos(2ϕ′)$

###### Modified Cookie-Cutter Function to Predict Polygon Shape Deformation in the Vertical Cross Section.

The two cookie-cutter models proposed in Ref. [25] are applied to regular polygons. They have to be modified when applying to irregular polygons according to their geometric characteristics, particularly, the transitions at the corners.

Assume the position of the ith corner is $(ϕ′c,i,rc,i)$ in the PCS and let n be the number of the polygon sides in the vertical cross section. We propose a one-to-one mapping from $(ϕ′c,i,ϕ′c,i+1)$ to $((i−3/2)π,(i−1/2)π)$, which is the domain for trigonometric function $cos θ$ that all the points in this domain are positive or negative. The mapping is proposed as Display Formula

(12)$ϕ′↦θ: where θ=ϕ′−ϕ′c,iϕ′c,i+1−ϕ′c,i+i−32$

Using this relationship, the modified square wave cookie-cutter function is defined as Display Formula

(13)$squarewave(ϕ′)=rc,isign[cos(π(ϕ′−ϕ′c,iϕ′c,i+1−ϕ′c,i+i−1)−π2)] for ϕ′c,i≤ϕ′<ϕ′c,i+1,0≤i

Similarly, the one-to-one mapping from $(ϕ′c,i,ϕ′c,i+1)$ to (−1, 1) is Display Formula

(14)$ϕ′↦θ: where θ=2(ϕ′−ϕ′c,i)MOD(ϕ′c,i+1−ϕ′c,i)(ϕ′c,i+1−ϕ′c,i)−1$

And the sawtooth wave cookie-cutter is modified as Display Formula

(15)$sawtooth(ϕ′)=rc,i[1−2(ϕ′−ϕ′c,i)MOD(ϕ′c,i+1−ϕ′c,i)(ϕ′c,i+1−ϕ′c,i)] for ϕ′c,i≤ϕ′<ϕ′c,i+1,0≤i

For the prism part studied in the experiment, Fig. 8 shows its modified square wave cookie-cutter function (middle) and the modified sawtooth wave cookie-cutter function (bottom). The top curve shows the nominal radius $r(ϕ′)$ as a reference for catching the transition at corners.

Another significant generalization of the cookie-cutter model $g2(ϕ′,r0(ϕ′)|θ0)$ from Ref. [25] is that we allow the mixed selection of square wave and sawtooth wave functions in model (10). The cookie-cutter function in Ref. [25] is only applied to in-plane deformation in the horizontal cross section view. It is expected that a single type of square wave or sawtooth wave function can be applied to each polygon side. In the vertical section view, however, deformation patterns of the left and right sides are expected to be quite different from those of the top and bottom sides. This rationale leads to a mixed selection among cookie-cutter functions for different sides in the vertical cross sections.

This mixed selection is through an indicator function defined as Display Formula

(16)$1[ϕ′c,i,ϕ′c,i+1](ϕ′)=12×(sign[cos(π(ϕ′−ϕ′c,iϕ′c,i+1−ϕ′c,i+i−1)−π2)]+1)for ϕ′c,i≤ϕ′<ϕ′c,i+1,0≤i

For the four-side surfaces of the rectangular prism, the indicator functions are shown in Fig. 9. They are the indicator function $1h(ϕ′)$ for left and right sides (middle) and the indicator function $1v(ϕ′)$ for top and bottom sides (bottom) with the top curve showing the nominal radius $r(ϕ′)$ as a reference for the transition at corners.

Thus, the out-of-plane deformation model in a specific vertical cross section is derived as Display Formula

(17)$Δz(ϕ′)=β0r0+β1r0 cos(2ϕ′)+β2r0squarewave×1h(ϕ′)+β3r0sawtooth×1v(ϕ′)+εϕ′$

###### Model Estimation.

To estimate model (17), we take advantage of the observation that similar deformation patterns exist among the four-side surfaces shown in Fig. 7. So, we make use of the front and back side surface data to obtain the predictive model for the vertical cross section given $θ0=0$ and use the left and right side surface data to obtain the predictive model for the vertical cross section given $θ0=π/2$. The maximum likelihood estimation of model parameters for the two out-of-plane deformation models is listed in Table 2. The fitted model results are shown as the dashed line in Fig. 7.

###### Model Evaluation.

We use the relative total area change as a criterion to evaluate the predictive model. The total area change is defined as the absolute area outlined by the deformation profile. It quantifies the total deformation of the printed part in a vertical cross section. It is defined as Display Formula

(18)$ΔS(r(ϕ′))=∫02πr0(ϕ′)|Δr(ϕ′)|dϕ′$

And the relative total area change is proposed as Display Formula

(19)$RΔS=ΔSmodel/ΔSmeasurement$

If $RΔS$ is close to 1, it indicates that the model predicts the average deformation well. The relative total area change of the four-side surfaces is listed in Table 3. As can be seen, the out-of-plane deformation models capture the major deformation patterns in four vertical cross sections.

## Offline Optimal Compensation and Experimental Validation

To control the geometric error of AM built products, we adopt the offline compensation strategy by adjusting CAD design. The amount of adjustment is based on the predicted geometric shape deformations and original design. Following the compensation approach proposed in Ref. [13], we denote $f(ϕ′,r0(ϕ′),x(ϕ′))$ as the profile deformation at angle $ϕ′$ when compensation $x(ϕ′)$ is applied at angle $ϕ′$. As derived in Ref. [13], the optimal compensation function is Display Formula

(20)$x*(ϕ′)=−g(ϕ′,r0(ϕ′))1+g′(ϕ′,r0(ϕ′))$

where $x*(ϕ′)$ satisfies the objective that $E[f(ϕ′,r0(ϕ′)x*(ϕ′)|ϕ′)]=0$, $g(·,·)$ is the deformation model, and $g′(·,·)$ is the first-order derivative with respect to $r0(ϕ′)$.

The out-of-plane deformation model for the prism is substituted into the compensation function to obtain the optimal compensation policy at each point along the boundary Display Formula

(21)$x*(θ)=−β0r0+β1r0 cos(2ϕ′)+β2r0squarewave×1h(ϕ′)+β3r0sawtooth×1v(ϕ′)1+β0r0+β1r0 cos(2ϕ′)+β2r0squarewave×1h(ϕ′)+β3r0sawtooth×1v(ϕ′)$

By adjusting CAD design based on Eq. (21), another prism part is printed for validation of compensation effectiveness. As shown in Fig. 10, the out-of-plane deformation before compensation is represented by the dotted line, while the out-of-plane deformation after compensation is represented by the solid line. The nominal value 0 is also plotted. Obviously, the out-of-plane deformation is in general decreased under compensation.

Relative total area change is used as a criterion to evaluate the impact of compensation, which is proposed as Display Formula

(22)$RΔS=ΔSafter/ΔSbefore$

A small value of $RΔS$ indicates that the compensation reduces the geometric error of AM products. The relative total area change of the four-side surfaces is summarized in Table 4. As can be seen, compensation methodology performs well for out-of-plane geometric shape control with approximately 50% deduction of the geometric errors.

## Conclusion

This study extends our previous work on control of in-plane shape deformation to out-of-plane shape deformation of AM built products. This is accomplished by adopting a novel spatial formulation of 3D geometric errors, which places the in-plane and out-of-plane errors under a unified and consistent framework. Under the SCS, the in-plane error and out-of-plane error are two orthogonal components depicted in horizontal and vertical cross sections, respectively. Thanks to the mathematical consistency, the out-of-plane error modeling is able to be derived from the previously developed in-plane error modeling framework.

Deriving out-of-plane deformation models, however, is not a trivial extension of in-plane error modeling due to complicated effects of interlayer bonding. With proper definition of out-of-plane error, we revise the previously proposed cookie-cutter modeling framework. The revised model provides the flexibility of mixed selection of cookie-cutter functions in order to accommodate the unique characteristics of errors in the vertical direction. Furthermore, the optimal compensation methodology is applied in an SLA process and the out-of-plane deformation is reduced by approximately 50%. The error can be further reduced when more training data are available to understand the effect of interlayer bonding effects.

The study provides a methodological prospect of fully integrating both in-plane and out-of-plane errors for 3D deformation reduction. Not limited to SLA process, the developed methodology can be applied to a wide category of AM processes.

## Acknowledgements

This work was supported by the U.S. National Science Foundation with Grant No. CMMI-1544917. We thank Professor Yong Chen and Mr. Kai Xu for AM experimental support.

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Huang, Q. , Nouri, H. , Xu, K. , Chen, Y. , Sosina, S. , and Dasgupta, T. , 2014, “ Statistical Predictive Modeling and Compensation of Geometric Deviations of Three-Dimensional Printed Products,” ASME J. Manuf. Sci. Eng., 136(6), p. 061008.
Song, S. , Wang, A. , Huang, Q. , and Tsung, F. , 2014, “ Shape Deviation Modeling for Fused Deposition Modeling Processes,” 2014 IEEE International Conference on Automation Science and Engineering (CASE), Taipei, Taiwan, Aug. 18–22, pp. 758–763.
Song, S. , Wang, A. , Huang, Q. , and Tsung, F. , 2016, “ Shape Deviation Modeling and Compensation for Fused Deposition Modeling Processes,” IEEE Trans. Autom. Sci. Eng. (to be published).
Huang, Q. , 2016, “ An Analytical Foundation for Optimal Compensation of Three-Dimensional Shape Deformation in Additive Manufacturing,” ASME J. Manuf. Sci. Eng., 138(6), p. 061010.
Jin, Y. , Qin, S. J. , and Huang, Q. , 2015, “ Out-of-Plane Geometric Error Prediction for Additive Manufacturing,” 2015 IEEE International Conference on Automation Science and Engineering (CASE), Gothenberg, Germany, Aug. 24–28, pp. 918–923.
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Huang, Q. , Nouri, H. , Xu, K. , Chen, Y. , Sosina, S. , and Dasgupta, T. , 2014, “ Predictive Modeling of Geometric Deviations of 3D Printed Products: A Unified Modeling Approach for Cylindrical and Polygon Shapes,” 2014 IEEE International Conference on Automation Science and Engineering (CASE), Taipei, Taiwan, Aug. 18–22, pp. 25–30.
Huang, Q. , Nouri, H. , Xu, K. , Chen, Y. , Sosina, S. , and Dasgupta, T. , 2014, “ Statistical Predictive Modeling and Compensation of Geometric Deviations of Three-Dimensional Printed Products,” ASME J. Manuf. Sci. Eng., 136(6), p. 061008.
Song, S. , Wang, A. , Huang, Q. , and Tsung, F. , 2014, “ Shape Deviation Modeling for Fused Deposition Modeling Processes,” 2014 IEEE International Conference on Automation Science and Engineering (CASE), Taipei, Taiwan, Aug. 18–22, pp. 758–763.
Song, S. , Wang, A. , Huang, Q. , and Tsung, F. , 2016, “ Shape Deviation Modeling and Compensation for Fused Deposition Modeling Processes,” IEEE Trans. Autom. Sci. Eng. (to be published).
Huang, Q. , 2016, “ An Analytical Foundation for Optimal Compensation of Three-Dimensional Shape Deformation in Additive Manufacturing,” ASME J. Manuf. Sci. Eng., 138(6), p. 061010.
Jin, Y. , Qin, S. J. , and Huang, Q. , 2015, “ Out-of-Plane Geometric Error Prediction for Additive Manufacturing,” 2015 IEEE International Conference on Automation Science and Engineering (CASE), Gothenberg, Germany, Aug. 24–28, pp. 918–923.

## Figures

Fig. 1

Shape deformation representation under polar coordinates

Fig. 2

In-plane error of cylindrical parts with r0 = 0.5 in., 1 in., 2 in., and 3 in. [13]

Fig. 3

In-plane deformation representation

Fig. 4

Out-of-plane deformation representation

Fig. 5

Out-of-plane deformation definition

Fig. 6

Prism case study: (a) printed rectangular prism part and (b) measured data of the prism right surface

Fig. 7

Prism case study: deformation profiles and model predictions. (a) Front and back side surfaces and (b) left and right side surfaces.

Fig. 8

Fig. 9

Modified indicator function

Fig. 10

Prism case study: before and after compensation. (a) Front and back side surfaces and (b) left and right side surfaces.

## Tables

Table 1 SLA process settings and test part design parameters
Table 2 Model estimation for side surface
Table 3 Relative total area change: model evaluation
Table 4 Relative total area change: before and after compensation

## Discussions

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