The goal of this work is to minimize geometric inaccuracies in parts printed using a fused filament fabrication (FFF) additive manufacturing (AM) process by optimizing the process parameters settings. This is a challenging proposition, because it is often difficult to satisfy the various specified geometric accuracy requirements by using the process parameters as the controlling factor. To overcome this challenge, the objective of this work is to develop and apply a multi-objective optimization approach to find the process parameters minimizing the overall geometric inaccuracies by balancing multiple requirements. The central hypothesis is that formulating such a multi-objective optimization problem as a series of simpler single-objective problems leads to optimal process conditions minimizing the overall geometric inaccuracy of AM parts with fewer trials compared to the traditional design of experiments (DOE) approaches. The proposed multi-objective accelerated process optimization (m-APO) method accelerates the optimization process by jointly solving the subproblems in a systematic manner. The m-APO maps and scales experimental data from previous subproblems to guide remaining subproblems that improve the solutions while reducing the number of experiments required. The presented hypothesis is tested with experimental data from the FFF AM process; the m-APO reduces the number of FFF trials by 20% for obtaining parts with the least geometric inaccuracies compared to full factorial DOE method. Furthermore, a series of studies conducted on synthetic responses affirmed the effectiveness of the proposed m-APO approach in more challenging scenarios evocative of large and nonconvex objective spaces. This outcome directly leads to minimization of expensive experimental trials in AM.

## Introduction

### Objective and Hypothesis.

The goal of this work is the optimization of additive manufacturing (AM) process parameters at which parts with the least geometric inaccuracy are obtained. This goal is a key milestone in ensuring the commercial viability of AM. Despite extensive automation, the poor geometric consistency of AM parts prevents their use in mission-critical components in aerospace and defense applications [1]. Currently, the cumbersome factorial-based design of experiments (DOE) tests are used to find the optimal set of AM process parameters that will minimize the geometric inaccuracy of the part, often assessed in terms of the geometric dimensioning and tolerancing (GD&T) characteristics. To overcome this challenge, the objective of this work is to develop and apply a multi-objective optimization approach to balance between multiple requirements, and thereby produce parts with the least geometric inaccuracy with the fewest trials. The central hypothesis is that decomposing the multi-objective problem of minimizing the geometric inaccuracy into a series of simpler single-objective optimization problems leads to a reduction of experimental trials compared to conventional full factorial DOE method. This hypothesis is tested against experimental data from the fused filament fabrication (FFF) process. This work leads to an understanding of how to balance different GD&T characteristics specified for a part using FFF process parameters, namely, infill percentage ($If$) and extruder temperature ($te$), as the primary control.

### Motivation.

The motivation for this work, stemming from our previous studies, is demonstrated in Fig. 1 [25]. At the outset, we note that the intent of this example is not to claim that there is an inadequacy with the standardized geometric GD&T characteristics but to illustrate the difficulty in balancing between several tight geometric accuracy requirements for an AM part (in terms of GD&T characteristics).

Fig. 1
Fig. 1
Close modal

For instance, Fig. 1 shows the flooded contour plot for four different parts fabricated by FFF process. A flooded contour plot translates the geometric deviations of the part to the corresponding spatial locations in terms of colors. Each flooded contour plot in Fig. 1 is constituted from 2 million three-dimensional (3D) coordinate data points for a benchmark test artifact part called circle–square–diamond used in this research, see Sec. 3.1 for details. It is visually evident in Fig. 1 that printing under different infill percentages ($If$)—an FFF process parameter—results in different part geometric accuracies. It is apparent that the parts in Figs. 1(a)1(c) with , respectively, are better overall than the last one shown in Fig. 1(d) with $If=100%$ in terms of geometric accuracy; however, it is often difficult to find a set of process parameters that will globally minimize all the specified geometric inaccuracies.

The geometric accuracy for parts shown in Fig. 1 was specified in terms of GD&T characteristics, such as circularity, flatness, cylindricity, etc. (see Sec. 3.1 for details). Experimental data plotted in Fig. 2 reveal the difficulty in balancing between the geometric accuracy requirements based on adjusting a few process parameters. Specifically, in Fig. 2, it is evident that both concentricity and flatness deviations for parts made by varying infill ($If$) and extruder temperature ($te$) cannot be minimized simultaneously. The combination of high $If$ and low $te$ tends to result in low deviations in concentricity (area in the upper left corner of Fig. 2(a)). However, the same process parameter set results in high level of geometric deviations in terms of part flatness (area in the upper left corner of Fig. 2(b)). Accordingly, this work addresses the following open research question in AM: What experimental plan is required to optimize process parameters with respect to multiple geometric accuracy requirements?

Fig. 2
Fig. 2
Close modal

### Research Challenges and Overview of the Proposed Approach.

AM offers the unique opportunity to create complex geometries and tailored surface morphologies with multiple materials; enable rapid repairs and replacement of parts in battlefield environments; simplify the overall prototyping cycle, and significantly shorten the logistical supply chain [6]. Despite these paradigm-shifting capabilities, poor process repeatability remains an imposing impediment to the commercial-scale exploitation of AM capabilities. The uncertainty associated with the morphology of the subsurface and microstructure, surface finish, and geometry of the finished part is a major concern for use of AM parts in strategic industries important to the national interest, such as aerospace and defense [1].

Efforts are ongoing in industry and academia to understand the causal AM process interactions that govern part quality. Thermomechanical models have been proposed as an avenue for predicting the part properties, e.g., geometric shrinkage of AM parts [7,8]. However, the current research is in its embryonic stage and largely restricted to simple geometries such as cubes and cylinders. These elementary models cannot, as yet, capture the complex process interactions that affect parts with intricate geometries. Accordingly, the data-driven approaches for compensating for AM part geometry distortions have been recently proposed [911]. However, these studies are also restricted to elementary shapes and are limited to the modeling of uniaxial geometry deviations.

Hence, in the absence of practically applicable physical models, DOE methods are employed to identify, quantify, and optimize key process parameters with respect to mechanical properties, such as surface roughness, fatigue, and tensile strength, among others, in AM processes. These methods traditionally involve identifying existing patterns in experimental data, sampling the process space at predefined levels, and subsequently developing surrogate statistical models to approximate targeted objective functions.

For most practical cases, several geometric accuracy requirements must be satisfied together. This is a multi-objective process optimization problem with a multitude of open research challenges as follows:

• Understanding how to balance different GD&T characteristics specified for the part using process parameters as the primary control.

• The correlations among responses of interest are not typically known a priori. There is often a tradeoff between different responses. A process parameter set that produces favorable results for one geometric characteristic of part (e.g., concentricity) may be detrimental for another geometric characteristic (e.g., flatness).

• Developing a reliable empirical model for multiple responses requires a large number of experimental runs. Hence, building parts with various AM processing parameters and subsequent testing for each response mandate significant investment of both time and resources.

This work addresses these challenges by forwarding a multi-objective accelerated process optimization (m-APO) approach to find the AM process parameters that minimize part geometric inaccuracies with fewer experimental trials compared to existing DOE methodologies. This method, presented in Sec. 3, consists of the following sequential steps:

1. (1)

The concept of scalarization is used to convert the multi-objective problem into a sequence of single-objective subproblems.

2. (2)

The accelerated process optimization (APO) method—developed in a previous study [12]—is used to solve the single-objective subproblems.

3. (3)

A stopping criterion is defined for the current subproblem.

4. (4)

Subproblems are chosen to uncover intermediate sections of the Pareto front (Pareto front is the set of nondominated or noninferior solutions in the objective space).

By applying the proposed multi-objective optimization scheme, experimental results from previous subproblems are leveraged as prior data for the remaining subproblems to accelerate the multi-objective process optimization. Information captured from previous subproblems facilitates an experimental design for accelerated optimization of the remaining subproblems. This eschews conducting experiments again for each subproblem, and this subsequently reduces the experimental burden. The practical utility of the proposed methodology is validated in Sec. 4.1 for the geometric accuracy optimization of parts made using FFF polymer AM process. The applicability of the approach to a broader set of challenging nonconvex optimization domains is demonstrated in a series of simulation studies in Sec. 4.2.

The remainder of this paper is organized as follows: In Sec. 2, we review the existing literature addressing the problem of geometric accuracy optimization in AM processes and the multi-objective optimization techniques. In Sec. 3, the proposed approach is described in depth. In Sec. 4, the proposed approach is demonstrated in the context of the FFF process, as well as simulated cases. Finally, in Sec. 5, conclusions are summarized and directions for future work are discussed.

## Literature Review

We divide the literature review into two subsections: (1) the existing literature in AM process optimization specific to geometric accuracy, and (2) relevant multi-objective optimization approaches.

### Existing Literature in Geometric Accuracy Optimization in AM Processes.

We summarize some of the existing research efforts pertaining to geometric accuracy in AM processes. Bochmann et al. [13] studied the cause of imprecision in FFF systems with respect to surface quality, accuracy, and precision. They found that the magnitude of errors significantly varies in $x,y$, and $z$ directions. Mahesh et al. [14] proposed a benchmark part incorporating critical geometric features for evaluating the performance of rapid prototyping systems with respect to geometric accuracy. The proposed benchmark includes geometric features such as freeform surfaces and pass–fail features.

El-Katatny et al. [15] measured and analyzed the error in major geometric characteristics of specific landmarks on anatomical parts fabricated by the FFF process. Weheba and Sanchez-Marsa [16] determined the optimal process settings for stereolithography (SLA) process with respect to surface finish, flatness, and deviations of diameter measures from nominal values. The second-order empirical models were developed for the different characteristics, but only a single set of process parameters was given as the optimal design. Indeed, there should be tradeoffs for the different responses. Tootooni et al. [2,3] and Rao et al. [17] used a spectral graph theory methodology to quantify and assess the geometric accuracy of FFF parts using deviations of 3D point cloud coordinate measurements from design specifications. Although the proposed indicator facilitates comparing the geometric accuracy of parts, it does not propose a relationship between process parameters and geometric accuracy in terms of GD&T characteristics. Huang et al. [911] developed a framework to model part shrinkage in SLA, thereby optimizing shrinkage for better geometric accuracy. This work is limited to elementary geometric shapes and does not determine an optimal range of process parameters for the best geometric accuracy of parts.

Experimental data from our initial screening studies (see Sec. 3.1) show that the optimization of geometric accuracy for AM parts is a multi-objective optimization problem (i.e., the correlations among geometric characteristics are negative). As an example of multi-objective optimization in AM, Fathi and Mozaffari [18] optimized the directed energy deposition process in terms of three response characteristics—clad height (deposition layer thickness), melt pool depth, and dilution—in a two-stage manner. First, empirical models that represent the relationship between the key process parameters, i.e., laser power, powder mass flow rate, and scanning speed, and the two response characteristics were developed using a smart bee algorithm and a fuzzy inference system. Then, in the second stage, nondominated sorting genetic algorithm (NSGA-II)—a well-known multi-objective optimization evolutionary algorithm—is employed to achieve the best Pareto points in the objective space. Although this work was able to handle a multi-objective process optimization problem, it required several experimental runs (50 experiments) to establish a set of viable empirical models. As mentioned previously, the prohibitive experimental effort remains an intrinsic challenge with the conventional DOE methods.

### Background in Multi-Objective Optimization.

Multi-objective optimization methods can be grouped into two main categories: (i) scalarization or aggregation-based methods and (ii) evolutionary algorithms [19]. Scalarization methods, which represent a classic approach, combine multiple objective functions into a single-objective problem, enabling the use of single-objective optimization methods to solve the problem [20]. These methods are not adaptable for the current multi-objective geometric accuracy optimization problem in AM because the individual objective functions are not explicitly known. In contrast, evolutionary algorithms iteratively generate groups of potential solutions that represent acceptable compromises between objective functions [21]. The disadvantage of this approach is that the objective functions require a large number of candidate solutions to be evaluated, i.e., many AM experiments for the current multi-objective geometric accuracy optimization problem.

Other multi-objective optimization approaches share similar disadvantages. For instance, Kunath et al. [22] applied a full factorial design of experiments for three process parameters to develop a set of regression models as the functional form of objective functions representing the binding properties of a molecularly imprinted polymer. Then by assigning predefined desirability values, i.e., weight coefficients, for dependent response variables, a range of process parameters resulting in the highest desirability values are introduced as optimal. Again, this approach presents challenges for most AM processes since many process parameters are involved and a large number of experiments may be required to fit regression models within tight confidence bounds. Therefore, there are critical research gaps and numerous technical challenges pertaining to process optimization for geometric accuracy of multiple AM part geometric characteristics.

## Methodology

### Description of the Experimentally Obtained Geometric Accuracy Data.

A systematic optimization approach for improving the geometric accuracy of AM parts is motivated by the experimental response data collected for a benchmark part. The presented experimental data in this work are generated in the authors' previous research [25]. The so-called circle–diamond–square part is designed as the benchmark part of interest for geometrical optimization (Fig. 3) [23,24]. This is based on the NAS 979 standard part used in the industry for assessing the performance of machining centers [25]. The circle–diamond–square is useful for assessing the performance of AM machines with respect to part geometric accuracy. For instance, the outside square (lowest layer) can be used to measure the straightness of an individual axis and the squareness across two axes in FFF. The diamond feature (middle layer) can be used to measure the rotation among two axes (i.e., from the bottom to middle layer). The large circle feature (top layer) can be used to measure the circular interpolation of two axes [23,26]. By axes we refer to the FFF machine gantry on which the nozzle is rastered.

Fig. 3
Fig. 3
Close modal

Five important GD&T characteristics are specified: flatness, circularity, cylindricity, concentricity, and thickness (see Fig. 4). These GD&T characteristics are chosen because they are independent of the feature size—also called regardless of feature size characteristics. More detail about GD&T characteristics can be found in Ref. [27]. We did not specify positional tolerances on the part because matching a datum surface from laser scanned point cloud data was found to be exceedingly error prone. Additionally, we concede another weakness with this work; it is likely that the GD&T characteristics specified for the circle–square–diamond test artifact might entail that the part is over tolerated or constrained and probably beyond the capability of the desktop FFF machine used in this work. Our rationale is to use this test artifact, albeit as an extremely contrived case, to explore a larger theme—understanding how to balance different GD&T characteristics specified for the part using FFF process parameters, namely, infill percentage ($If$) and extruder temperature ($te$) as the primary control. This is the primary contribution of this work.

Fig. 4
Fig. 4
Close modal

The aim is to minimize the magnitude of deviations within these five GD&T characteristics of parts from the targeted design specifications. A NextEngine (Santa Monica, CA) HD 3D laser scanner was used to capture part geometric data, and the qa scan 4.0 software (Santa Monica, CA)2 was used to estimate the deviations from the targeted design specifications [25]. Figure 2 (Sec. 1.2) shows examples of contour plots of absolute value for deviations within flatness and concentricity versus two controllable FFF process parameters: infill percentage (If) and extruder temperature (te).

In practice, the laser scanning is a heuristic process that requires adhering to a carefully attuned procedure, particularly, in the manner in which the part is aligned to the computer-aided design (CAD) model to obtain consistent results. The alignment step requires matching of at least four landmark points from the raw point cloud data with CAD model. Several trials are conducted and herewith summarized is a method that showed the least variability. Four points each on the square and diamond portions are used to align the part as depicted in Fig. 5. Additionally, laser scanning was conducted on a sturdy, vibration-free table in a darkened closure, and by coating the part with a thin layer of antireflective gray modeling paint.

Fig. 5
Fig. 5
Close modal

The data scatter plot matrix for the GD&T characteristics, called correlation matrix, is shown in Fig. 6. The slope of lines represents the Pearson correlation coefficient ($ρ$) for pairs of GD&T characteristics. It is evident from Figs. 2 and 6 and Table 1 (see Sec. 4.1) that flatness and concentricity are not positively correlated ($ρ=−0.21$). Similarly, thickness and concentricity are negatively correlated ($ρ=−0.63$). In other words, it is not possible to simultaneously optimize GD&T characteristics. Hence, optimization for geometric accuracy is best considered as a multi-objective optimization problem, and the set of process parameters setups should consider the tradeoff between multiple geometric characteristics.

Fig. 6
Fig. 6
Close modal
Table 1

Signed correlation among GD&T characteristics of benchmark part samples

### Multi-Objective Process Optimization.

The aim of this section is to elucidate the mathematical foundation for multi-objective process optimization in AM. We illustrate the case with two objective functions. The proposed methodology is extensible to multi-objective cases.

#### Scalarization of Multi-Objective Optimization and Pareto Front.

The methodology developed herein is a generalization of an existing APO methodology, developed in the form of maximization [12]. A minimization problem can be expressed in the form of maximization by multiplying the objective function by a negative sign (and vice versa).

Suppose the problem is to maximize two objective functions (or response variables $Y1$ and $Y2$). The bi-objective maximization problem is expressed as follows:
(1)

where $Y(s)$ denotes the vector of objective functions $(Y1(s),Y2(s))′$, $s$ is the vector of process parameters (e.g., infill percentage (If) and extruder temperature (te)), and $S$ denotes the design space, which includes all possible values of $s$. The objective space, i.e., the set of all possible response vectors $Y$ corresponding to the design space, is denoted by $C={(Y1(s),Y2(s))′∈ℝ2:s∈S}$.

For most AM applications, the functional expressions of $(Y1(s),Y2(s))$ are unknown; the empirical relationship between geometric accuracy responses and AM process parameters is yet to be understood and quantified. Moreover, the correlation between $Y1(s)$ and $Y2(s)$ is also unknown. A higher value of $Y1(s)$ may result in a lower value of $Y2(s)$ or converse. In other words, the optimized process parameters for $Y1$ may not necessarily result in favorable $Y2$ due to the possible low or even negative correlation between two response variables. For instance, concentricity and flatness shown in Fig. 2 are negatively correlated. Consequently, improving the response value of flatness will result in worsening concentricity. Therefore, the optimal solution to the multi-objective optimization problem is nonunique. Our objective is to develop a systematic and sequential DOE procedure that efficiently identifies sets of optimal solutions as a tradeoff between such contradictory response behaviors.

For this purpose, the optimization problem is converted into a sequence of single-objective problems by defining weight coefficients for each objective. In this way, the bi-objective optimization problem can be presented as a sequence of single-objective optimization problems, each defined by the subproblem index variable $h$, with $h=1,2,3,…,m$ as follows:
(2)

For each subproblem with index $h$, $γkh$ denotes weight coefficient corresponding to the $kth$ objective function. $k=1$ and $k=2$ in the previous formulation, satisfying the constant $γ1h+γ2h=1$. Different weight coefficients correspond to different subproblems and will accordingly lead to different optimal solutions.

For example, consider a subproblem with $γ11=0.8$ and $γ21=0.2$. In this case, the single-objective optimization problem is expressed in the form of . The weight coefficients ($γkh$) are graphically shown in Fig. 7 by the tangent of a line, which represents the desired search direction for the current single-objective maximization function (subproblem h). Changing the corresponding weight coefficients changes the single-objective function being optimized. For example, consider a second subproblem with $γ12=0.2$ and $γ22=0.8$; the optimum solution for the problem is not the same as that in the first subproblem. In real-world applications, simultaneously achieving the best individual solution for two negatively correlated (or uncorrelated) objectives is intractable. Hence, in these cases, the optimum solution is a subset of objective space $C$ which can recognize and identify the best tradeoff among the value of $Y1$ and $Y2$. In what follows, we discuss in detail the approach to identify the sets of process parameter setups which results in an optimal objective value with different weight coefficients.

Fig. 7
Fig. 7
Close modal

We focus on identifying the Pareto optimal solutions associated with the multi-objective optimization problem. A Pareto optimal solution is not dominated by any other feasible solution and represents the best compromise between multiple objective functions. We define each member of Pareto optimum as a design point $s*∈S$ if and only if there is no other $s∈S$ such that $Yk(s)≥Yk(s*)$ for $k=1,2$. Here, $s*$ is called a nondominated design point, and its corresponding response vector in the objective space is a Pareto point, $Yk(s*)$. Regarding geometric accuracy optimization, a Pareto point indicates an optimal design point where there is no other solution that results in better values in term of any geometric responses. The Pareto optimum set is denoted by $E$. In the bi-objective optimization problem shown in Eq. (1), the Pareto front representing response vectors of the Pareto set in the objective space $C$ is defined by $H$, that is, $H={(Y1(s),Y2(s))′∈ℝ2:s∈E}$. Given two controllable process parameters for the purpose of demonstration, the terms design space, objective space, nondominated design point, the Pareto point, and Pareto front for a bi-objective optimization problem are illustrated in Fig. 7.

#### Multi-Objective Accelerated Process Optimization.

Our approach is to solve the bi-objective optimization problem presented in Eq. (2) by obtaining a well-distributed set of Pareto points and thereby approximate the Pareto front with reduced number of experimental runs. Conventional scalarization divides the multi-objective optimization problem into individual single-objective problems and optimizes them individually. The proposed m-APO method, in contrast, leverages a knowledge-guided optimization approach based on the similarity among different subproblems. The proposed methodology is initially developed by preliminary studies [28,29] to deal with multi-objective AM process optimization problems.

Each single-objective subproblem is solved using the APO method, which uses results from prior experiments to accelerate the process optimization [12]. APO balances two important properties simultaneously, i.e., optimization and space-filling. For optimization, more trial runs are needed in the regions of $s$ which potentially result in the maximum value of the response function $Zh(s)$. In contrast, to avoid being trapped in a local optimum, the space-filling aspect is also considered. In the APO approach, each design point is assigned a so-called positive charge denoted by $qh(sj)$. Selection of the charge function $qh(s)$ relies on the optimization objective. Considering the maximization objective in our case, the charge function $qh(s)$ should be inversely proportional to the weighted single-objective response values $Zh(s)$ from Eq. (2) [12,30]. Thus, higher magnitude charges are assigned to design points with lower $Zh(s)$ and vice versa. Analogous to the physical laws of static charged particles, the design points repel each other apart to minimize the total electrical potential energy among them. Hence, design points with lower $Zh(s)$, i.e., with a higher charge, strongly repel other design points. On the other hand, design points with higher $Zh(s)$, i.e., with lower electrical charge, accommodate more design points in their neighborhood. The resulting positions correspond to the minimum potential energy among charged particles. Accordingly, more design points with lower charge potentials (i.e., higher $Zh(s)$ values) are selected to sequentially maximize the objective function of interest in the current subproblem, i.e., $Zh(s)$.

The potential energy between any two design points $si$ and $sj$ is equal to $q(si)q(sj)/d(si,sj)$, where $d(si,sj)$ represents the Euclidean distance between $si$ and $sj$. Hence, total potential energy function within $hth$ subproblem including the nth new design is formulated as follows:
$Enh=∑i=1n−1∑j=i+1nqh(si)qh(sj)d(si,sj)$
(3)

The new design point can be obtained by solving .

At each step, Pareto points are identified based on the nondomination concept. Afterward, the appropriate weight coefficients for the next subproblem are chosen to lead the next subproblem optimization in such a way that maximizes the distribution of the Pareto points. Instead of solving each optimization subproblem individually and independently, experimental data obtained from previous subproblems are used as prior data to accelerate optimization process for the subsequent subproblems. For example, in Fig. 8, experimental data from subproblems 1 and 2 (represented by segments a-b-c-d and e-f-g, respectively) accelerate the optimization process for subproblem 3 (segment h-i). In other words, fewer numbers of experiments are needed to reach the Pareto point corresponding to the subproblem 3. This is due to the fact that experimental data obtained from previous subproblems contribute to designing experiments for the next subproblems. Hence, we do not need to design the experiments from scratch for each subproblem. This process is continued until the improvement in the resulting Pareto front is insignificant. The area dominated by Pareto points on the objective space is used to measure the efficiency of the resulting Pareto points. The proposed method accelerates the bi-objective optimization process by jointly solving the subproblems in a systematic manner. In fact, the method maps and scales experimental data from previous subproblems to guide remaining subproblems that improve the Pareto front while reducing the number of experiments required. The algorithm is described herewith in detail and summarized in Fig. 9.

Fig. 8
Fig. 8
Close modal
Fig. 9
Fig. 9
Close modal

Step 1: Decomposing master problem into subproblems. The master bi-objective optimization problem (see Eq. (1)) is decomposed into a sequence of single-objective functions, each of which is expressed as a convex combination of the objective functions (see Eq. (2)). We initialize the algorithm with optimizing two boundary subproblems with $(γ11=0,γ21=1)$ and $(γ12=1,γ22=0)$. The solution to the first two subproblems resulted in two end points of the Pareto front (i.e., points d and g in Fig. 8).

Step 2: Solving subproblems via accelerated process optimization. Using APO [12], we sequentially design experiments to optimize the constructed single-objective subproblems. Experimental data generated from previous subproblems are treated as prior data for subsequent subproblems. Assuming that weight coefficients $(γ1h,γ2h)$ are determined, all the design points represented in the response vector form, ($si,Yi$), are converted to the form of weighted single-objective response data as ($si,Zih$) in the framework of APO.

The design points and corresponding weighted single-objective response are incorporated and applied throughout the APO algorithm, i.e., $si$ and $Zih(si)=γ1h.Y1(si)+γ2h.Y2(si)$. All the experimental data attained during the optimization process of prior subproblems (i.e., subproblems $1,2,…,h−1$) are transformed and fed into the APO of the hth subproblem as prior data to accelerate optimization process of the current subproblem by predicting the weighted single-objective responses in a more accurate manner. The new design point can be obtained by solving , where $Enh$ is the total energy function defined in Eq. (3). A detailed discussion about the computation of the energy function and predicting the single-objective response values for the new untested design points can be found in Ref. [12].

Step 3: Defining stopping criteria for subproblems. To define the stopping criteria, we use the hypervolume ($HV$) metric as a measure of the Pareto points' contribution [31]. By definition, $HV$ is the volume in the objective space dominated by resulting Pareto points; a higher $HV$ results in better coverage of the Pareto front and thus provides the better solution. In Fig. 10, light gray area is the $HV$ associated with gray Pareto points. $ΔHV$, which is the contribution of new Pareto point, is represented by the dark gray rectangle area. The algorithm is repeated designing experiments for the current subproblem until $ΔHV$ is less than a prespecified threshold $(i.e.,ΔHV<ε1)$. The proposed algorithm stops continuing introducing further subproblems and designing more experiments when we do not observe significant improvement in $ΔHV$$(i.e.,ΔHV<ε2)$.

Fig. 10
Fig. 10
Close modal
Step 4: Determining weight coefficients of subsequent subproblems. Based on the resulting Pareto points obtained at the end of each subproblem, weight coefficients for the next subproblem, $γh$, are calculated as follows. Note that upper case letters represent the unknown variables in this paper, while lower case letters are used for known variables. Assuming that after solving the $(h−1)th$ subproblem, the Pareto set $Φh−1={(s1*,y1*),(s2*,y2*),…,(sm*,ym*)}$ including $m$ nondominated design points and corresponding actual response vectors are obtained. Thereafter, all the existing optimal parameter setups are sorted in increasing order of $y1(s)$ and labeled as $Ψh−1={s(1)*,s(2)*,…,s(m)*}$. At this stage, the Euclidean distance between all of the neighboring Pareto points is calculated as follows:

Then, the maximum gap on the existing Pareto front is determined by $Δ=maxj=1,…,(m−1)δj$. If two neighboring Pareto points corresponding to $Δ$ are $sa*$ and $sb*$, where $y1(sa*), the weight coefficients for the next subproblem is computed as $γh=ch(y2(sa*)−y2(sb*),y1(sb*)−y1(sa*))$, where $ch$ is a constant leading to $γ1h+γ2h=1$. Accordingly, we can achieve a uniform coverage of Pareto front in an efficient manner.

## Experimental and Numerical Studies

We now apply and demonstrate the proposed approach to experimental and simulated data. We first apply our method to a real-world case study for minimizing the deviation in geometric characteristics of parts produced using FFF AM process. Since the experimental data include measurement of five GD&T characteristics, we first use the principal component analysis procedure to reduce the dimension of objective space to the first two PCs, which account for 88.15% of total data variation. Subsequently, the proposed optimization method is applied to minimize absolute values of the first two PCs. The results show that m-APO methodology achieves all true Pareto points in the objective space with 20% fewer experiments compared to a full factorial DOE plan.

To further validate our methodology and test its robustness, we also conducted numerical studies with a different number of input parameters and characteristics of objective space and Pareto fronts [3134].

### Experimental Case Study: Multigeometric Characteristic Optimization of Parts Fabricated by FFF System.

The aim of this section is to apply the proposed m-APO method for optimizing the geometric accuracy of AM parts. Samples are fabricated using a polymer extrusion AM process called FFF. They are made with acrylonitrile butadiene styrene thermoplastic on a desktop machine (MakerBot Replicator 2X). A schematic of the FFF process is shown in Fig. 11. Based on the initial screening designs, we take two important controllable process parameters including percentage infill (If) and extruder temperature (tf) [25]. Extruder temperature is the temperature at which the filament is heated in the extruder. Infill relates to the density of the part, for instance, 100% infill corresponds to a completely solid part. The target is to minimize absolute deviations concerning five major GD&T characteristics, namely, flatness, circularity, cylindricity, concentricity, and thickness, from the targeted design specifications. Since the m-APO methodology is expressed in the form of maximization problem, the response values in the case study are multiplied by a negative sign.

Fig. 11
Fig. 11
Close modal

The 20 experimental data used in the present study are resulted from the previously published works [25], which use a full factorial DOE plan. Factors and levels corresponding to this design are illustrated in Table 2. The correlation among deviations within GD&T characteristics is illustrated in Table 1 in terms of the Pearson correlation coefficient ($ρ$) for pairs of GD&T characteristics. The higher correlation coefficients are more noticeable compared with low coefficients. The correlation between cylindricity and circularity is extremely high ($ρ=0.93$) in that both contribute to describing the circular feature of the test part in different ways. There is no any discernible pattern among the other GD&T correlations. In other words, the GD&T characteristics are positively correlated (e.g., $ρ=0.93$ for cylindricity and circularity), negatively correlated (e.g., $ρ=−0.63$ for concentricity and thickness), and nearly uncorrelated (e.g., $ρ=0.06$ for cylindricity and thickness).

Table 2

Levels of the process parameters applied to generate experimental data based on full factorial DOE plan

 $Te$ (°C) 220 225 230 235 240
 $Te$ (°C) 220 225 230 235 240
 If (%) 70 80 90 100
 If (%) 70 80 90 100

Principal component analysis is first applied to reduce the dimension of the objective space from five to two. Accordingly, the proposed bi-objective process optimization can be directly applied to the geometric accuracy optimization problem for FFF system. The principal component analysis results (see Table 3) show that 88.15% of variability within the parts' geometric characteristics data is captured by the first two principal components (i.e., $PC1$ and $PC2$). Hence, the first two PCs can sufficiently describe the data variations with the very negligible loss of information. All the five variables contribute to the $PC1$ by positive coefficients (Table 4). Flatness, circularity, and cylindricity are about equally important to $PC1$ with the largest weight. Although concentricity's contribution is negligible in $PC1$, thickness plays a significant role in this component. Hence, $PC1$ can be considered as representative of average deviations within all GD&T characteristics. However, we see a unique pattern within $PC2$ 's coefficients. The variables that are related to the analogs features—i.e., circularity, cylindricity, and concentricity—contribute to $PC2$ with negative coefficients. Hence, $PC2$ can be inferred as the difference of deviations within two clusters of geometric characteristics: roundness characteristics (circularity, cylindricity, and concentricity) and others (flatness and thickness).

Table 3

The 88.15% of variability within the parts' geometry characteristics data is captured by the first two principal components (i.e., $PC1$ and $PC2$)

$PC1$$PC2$$PC3$$PC4$$PC5$
Standard deviation1.5761.38660.589810.467130.16190
Proportion of variance0.4970.38450.069580.043640.00524
Cumulative proportion0.4970.88150.951120.994761
$PC1$$PC2$$PC3$$PC4$$PC5$
Standard deviation1.5761.38660.589810.467130.16190
Proportion of variance0.4970.38450.069580.043640.00524
Cumulative proportion0.4970.88150.951120.994761
Table 4

Illustrating principal components' coefficients. All the five variables contribute in the $PC1$ by positive coefficients. Flatness, circularity, and cylindricity are about equally important to $PC1$ with the largest weight.

$PC1$$PC2$$PC3$$PC4$$PC5$
Flatness0.50460.2895−0.7181−0.38240.0481
Circularity0.5854−0.15980.4859−0.18310.6016
Cylindricity0.5607−0.31610.17100.0901−0.7404
Concentricity0.0650−0.6677−0.46590.50000.2877
Thickness0.28950.58950.04250.74960.0687
$PC1$$PC2$$PC3$$PC4$$PC5$
Flatness0.50460.2895−0.7181−0.38240.0481
Circularity0.5854−0.15980.4859−0.18310.6016
Cylindricity0.5607−0.31610.17100.0901−0.7404
Concentricity0.0650−0.6677−0.46590.50000.2877
Thickness0.28950.58950.04250.74960.0687
Using the first two $PC$ s, the geometric accuracy optimization problem is formulated as follows:
$MinPC(s)=(|PC1(s)|,|PC2(s)|)′s.t. s∈S$

where $PC(s)$ denotes the vector of first two principal components of deviations within GD&T characteristics of the part, $s$ is the vector of process parameters, and $S$ is the design space.

After conducting 20 experiments using full factorial DOE plan, we attain three Pareto points in the objective space (red dots in Fig. 12). Note that the Pareto set in this case study naturally forms a convex Pareto front. After choosing a random initial experiment (blued dot in Fig. 12), we iteratively apply m-APO. The m-APO methodology leads to the same Pareto points after 16 experimental runs, which translates to a 20% reduction of experiment runs compared with the full factorial design. Optimal process parameters and GD&T characteristics corresponding to the Pareto points are presented in Table 5.

Fig. 12
Fig. 12
Close modal
Table 5

Optimal process parameters and corresponding GD&T values achieved in the case study

$te$ (°C)$If$ (%)FlatnessCircularityCylindricityConcentricityThickness
220900.18690.39050.50110.20610.1861
230900.18230.37830.44070.17330.2604
240900.18870.35000.46240.20010.1910
$te$ (°C)$If$ (%)FlatnessCircularityCylindricityConcentricityThickness
220900.18690.39050.50110.20610.1861
230900.18230.37830.44070.17330.2604
240900.18870.35000.46240.20010.1910

### Numerical Simulation Studies for Nonconvex Pareto Front.

As presented in the FFF case study (Sec. 4.1), the m-APO methodology is effective for a convex bi-objective problem. In the numerical simulation studies, the aim is to evaluate the robustness of m-APO in the case of more challenging nonconvex Pareto fronts, which is usually challenging for multiple objective optimizations. To simulate various experimental conditions, three different combinations of design space structures and Pareto front characteristics are considered: (a) nonconvex Pareto front and well-distributed objective space, (b) nonconvex Pareto front and congested objective space, and (c) high dimensional design space. The ultimate goal of the simulation study is achieving a set of high-quality uniformly spread Pareto points representing the true ones. We note that in reality, the functional form of objectives, i.e., $Y1(s)$ and $Y2(s)$, is unknown, and here we just present them to simulate the real experimentation.

We measure the efficiency of the m-APO methodology using general distance ($GD$) and proportional hypervolume ($PHV$) defined as follows:

1. (1)
General distance quantifies the difference between the true Pareto points and those obtained with m-APO. Assuming that at the end of simulation $N$ Pareto points are obtained, $GD$ is calculated as follows [35,36]:
$GD=∑i=1Nτi2N$
where  $τi$  represents the minimum Euclidean distance between $i$ th Pareto point from m-APO and true Pareto points. Smaller values in $GD$ indicate that the Pareto points obtained from m-APO are closer to true ones; and in an ideal case, $GD=0$.
2. (2)
Proportional hypervolume is the ratio of the hypervolume of the Pareto points obtained using m-APO and the hypervolume of the true Pareto points
$PHV=HV(Paretopointsobtainedfromm-APO)HV(trueParetopoints)$
By definition, $PHV$ falls within [0, 1]. In an ideal case, $PHV=1$.

We benchmark the m-APO method against full factorial DOE by comparing $GD$, $PHV$ within fix number of experiments. The results show that the m-APO method achieves significantly higher $PHV$ and lower $GD$ compared to full factorial DOE.

#### Case A: Nonconvex Pareto Front and Well-Distributed Objective Space.

The m-APO method is applied to an equally spaced discrete design space from a commonly used bi-objective optimization test problem, OKA1. This test problem is applied to evaluate the performance of ParEGO algorithm proposed in Refs. [31] and [32], and benchmark it against nondominated sorting genetic algorithm (NSGA-II). The relationship between process parameters and two objective functions are as follows:
$Maxf(s)=−(Y1(s),Y2(s))$
$Y1(s)=s1′$
$s1′=cos(π12)s1−sin(π12)s2$
$s2′=sin(π12)s1+cos(π12)s2$

where $s1∈[6 sin (π/12),6 sin (π/12)+2π cos (π/12)];s2∈[−2π sin (π/12),6 cos (π/12)]$. A design space which includes 342 design points is chosen to construct a well-distributed objective space. This objective space with normalized values consisting of 11 true Pareto points is illustrated in Fig. 13. Because many design points with a different set of process parameters result in same points in objective space, hence, visually the number of design points in the objective space appears to be less than 342.

Fig. 13
Fig. 13
Close modal

#### Case B: Nonconvex Pareto Front and Congested Objective Space.

An equally spaced discrete design space is selected based on another bi-objective test problem. This is a more challenging case in that the objective space includes very congested points at the middle farther from the Pareto front. This test problem is constructed to test the performance of an adaptive weighted-sum method for solving bi-objective optimization problems [33]. A discretized design space within 441 design points is chosen. The objective space consists of 19 true Pareto points (Fig. 14). The functional form of objective functions is as follows:

Fig. 14
Fig. 14
Close modal
$Maxf(s)=(Y1(s),Y2(s))$
$Y1(s)=3(1−s1)2e−s12−(s2+1)2−10(s15−s13−s25)(e−s12−s22)−3e−(s1+2)2−s22+0.5(2s1+s2)$
$Y2(s)=3(1+s2)2e−s22−(1−s1)2−10(−s25+s23+s25)(e−s22−s12)−3e−(2−s2)2−s12$

where $s1,s2∈[−3,3]$.

#### Case C: High Dimension Design Space.

To test the performance of our methodology in cases with more than two process parameters, we test a bi-objective problem with four process parameters, SK2 [34]. The design space includes 625 design points, and the objective space consists of five true Pareto points as illustrated in Fig. 15. The functional form of objective functions is as follows:

Fig. 15
Fig. 15
Close modal
$Maxf(s)=(Y1(s),Y2(s))$
$Y1(s)=−(s1−2)2−(s2+3)2−(s3−5)2−(s4−4)2+5$
$Y2(s)= sin s1+sin s2+sin s3+sin s41+(s12+s22+s32+s12)/100$

where $s1,s2,s3,s4∈[−3,3]$.

#### Simulation Results: Pareto Front Estimation.

The performance of the m-APO methodology is compared with full factorial DOE. The estimated Pareto front achieved by each method with 25 experiments is depicted in Figs. 1618, overlaid with the true Pareto fronts. We report that the m-APO method quickly converges toward the true Pareto points much faster than the full factorial DOE.

Fig. 16
Fig. 16
Close modal
Fig. 17
Fig. 17
Close modal
Fig. 18
Fig. 18
Close modal

Table 6 illustrates the improvement in terms of the performance measures ($GD,PHV)$ achieved by applying the m-APO methodology compared with the full factorial DOE. Because a smaller $GD$ is preferable to a larger $GD$, the $GD$ improvement is reported by a negative sign; and since a larger $PHV$ is preferred, an improvement in $PHV$ is reported as a positive number. We observe a significant $PHV$ and $GD$ improvement in all cases by applying the m-APO methodology compared with the full factorial DOE. It is therefore concluded that the proposed methodology outperforms the full factorial DOE in multi-objective process optimization cases. This is because conventional DOE methods are performed simultaneously—as opposed to the sequential approach developed in this work. Furthermore, this work forwards an approach to balance multiple and potentially negatively correlated (or uncorrelated) geometric accuracy requirements, while conventional empirical approaches are not capable of this tradeoff.

Table 6

The improvement achieved in simulation studies by m-APO in terms of decreasing GD and increasing PHV for three test problems

Test problem specificationsGD (%)PHV (%)
Nonconvex Pareto front and well-distributed objective space−5542
Nonconvex Pareto front and congested objective space−5724
High dimension design space−9329
Test problem specificationsGD (%)PHV (%)
Nonconvex Pareto front and well-distributed objective space−5542
Nonconvex Pareto front and congested objective space−5724
High dimension design space−9329

## Conclusions

This work presented an approach invoking the concept of m-APO to optimize AM process parameters such that parts with least geometric inaccuracy were obtained. The proposed m-APO technique decomposes a multi-objective optimization problem into a series of simpler single-objective optimization problems. The essence of the approach is that prior knowledge is used to determine the parameter settings for the next trials. This sequential approach guides experiments toward optimal parameter settings quicker than conventional design of experiments. In other words, instead of conducting experimental trials in the vicinity of process parameter setups where poor results are more probable, the m-APO methodology suggests experimentation at process parameter setups more inclined to favorable outcomes.

This approach is tested against both experimental datasets obtained from FFF AM process and numerically generated data. The specific outcomes are as follows:

• The proposed approach was able to effect a tradeoff among geometric accuracy requirements and reached the optimal process parameter settings with $20%$ fewer trials compared to full factorial experimental plans.

• We further tested the performance of the proposed approach to accommodate various simulated cases, such as nonconvex Pareto front, well-distributed objective space, congested objective space, and increased number of process parameters. The results indicate that the proposed methodology outperforms the full factorial designs for such complex cases. The performance metrics—GD and PHV—obtained from the proposed approach significantly superseded the full factorial design; there were $55−93%$ and $24−42%$ improvement in GD and PHV, respectively, in the simulated test cases.

The results presented in this work are practically important. Given the time- and cost-intensive nature of AM experimental trials, a prudent approach to balance the tradeoff between multiple geometric accuracy requirements is needed in practice. In contrast, this work answers the following research question in the context of AM process optimization: What approach is required to balance between multiple geometric accuracy requirements with the minimal number of experimental trials?

The gap in the current work is that it is demonstrated in the case of nonfunctional polymer AM parts. The authors are currently researching functional metal AM parts with m-APO as a means for in situ control.

## Acknowledgment

The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.

One of the authors (PKR) acknowledges the work of Dr. M. Samie Tootooni and Mr. Ashley Dsouza. The experimental data for this work are from their doctoral dissertation and MS thesis research, respectively [4,5].

## Funding Data

• Army Research Laboratory (Cooperative Agreement No. W911NF-15-2-0025).

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