Finite Element Analysis and Machine Learning in the Loop Additive Manufacturing Process Parameter Optimization to Acquire the Desired Dynamics of Manufactured Parts

Abstract

In this study, a constrained optimization procedure was presented that adjusts processing parameters of powder bed fusion (PBF) process to acquire desired dynamic behavior of produced parts, particularly in case of geometrical design of part to be produced have already confirmed and not possible to change. This procedure includes the combination of machine learning (ML) and finite element method (FEM) in the loop to correlate between processing parameters and dynamic response of parts. PBF production technique has leading processing parameters (laser power, scanning speed, hatch distance, and beam diameter) which have effect on mechanical properties of produced parts, for example, stiffness, elasticity, or mass density. Among different mechanical properties there are some properties which are related with dynamic behavior of parts. Thus, understanding the effect of processing parameters on mechanical properties is a crucial step to predict dynamic behavior of part especially where part design has already completed and not allowed to change in geometry. To do that, mechanical properties were acquired by the experimental data and semi-analytical equations were created accordingly by ML models. Then, the equations were used in FEM to get dynamic responses such as natural frequencies and mode shapes of produced parts in an optimization procedure. Overall, experimental data were correlated with numerical FEM model in the loop of optimization cycle to find desired dynamic behavior from produced parts. Eventually, it is proofed that natural frequencies are function of material properties but mode shapes were not affected by changing material constants. In addition, proposed method showed that dynamic behavior of PBF produced parts (elasticity modulus, relative density, etc.) can be modified by optimizing processing parameters of production process, including FEM and ML in the loop of constrained optimization problem where the final part design has already fixed.

Keywords

optimization machine learning additive manufacturing neural networks finite element modeling

Introduction

Powder bed fusion (PBF) is one of the prominent additive manufacturing (AM) production methods which enables to produce high mechanical strength parts with an intricate design. In PBF method, the powder bed of the material is spread in a thin layer over the build plate, and the laser beam selectively melts the powder layer according to the part’s design. The melted metal solidifies quickly, creating a strong bond between the layers and gradually building up the part’s geometry.¹ One of the distinctive properties of this method is that the quality of the produced part is highly dependent on its processing parameters such as power of laser source, scanning speed, scanning distance, and beam diameter of laser. From this point of view, PBF process will have correlation between processing parameters and produced part’s mechanical properties. Mechanical properties such as elasticity modulus and mass density influence the dynamic behavior of parts. As long as the parameter–structure relationship established well enough, dynamical response of parts can be modified by optimizing the value of processing parameters of parts. This suggests that processing parameter optimization by using different methods would be an effective way to reach the desired performance from produced parts even though some trade-offs such as low level of tensile strength and hardness may happen.

The optimization process for PBF typically involves identifying the key process parameters and their effects on the part quality. Machine learning (ML) is an option to create relationship between processing parameters and mechanical properties as long as the models were trained and tested well with reliable dataset.² There are several key challenges associated with optimizing the PBF process. First, the process parameters are often interdependent, meaning that changing one parameter can affect the behavior of other parameters. Second, the optimization process is highly dependent on the specific application and the desired part properties. Finally, the optimization process is often iterative and can require significant experimentation and data collection to identify the optimal settings.^3,4

Recent studies showed that optimizing processing parameters are effective on the produced part and PBF process itself. There are significant number of studies including different optimization algorithms for increasing the potential performance of the PBF process, and most of them focused on optimizing processing parameters to obtain high mechanical strength parts. As the leading technology improves, popular requirements on parts such as mechanical durability and process robustness have gained popularity in the related research field. Criales et al. developed multi-objective genetic algorithm and particle swarm optimization model to enhance mechanical properties and process-related issues such as build rate and energy consumption by optimizing processing parameters.⁵ Shin et al. investigated an optimization problem of a novel alloy (Ti-5Al-5V-5Mo-3Cr) for producing high-density parts by creating datasets systematically and used random search algorithm to optimize processing parameters.⁶ Gheysen et al. studied the optimization of processing parameters to improve mechanical strength and melt pool dynamics by using design of experiments and response surface methodology methods.⁷ Lin et al. and Wu et al. studied melt pool quality and surface roughness optimization of high-performance materials (Inconel 718 and molybdenum) by using artificial neural network(ANN), and considerable results were yielded in the prediction of performance metrics.^8,9 In some studies, the optimization of processing parameters was experimentally handled to get higher mechanical strength.^10,11 Letenneur et al. created an experimental optimization model by using melt pool dimensions and verified the model for different materials such as Inconel 625, Ti6Al4V, and AlSi10Mg alloys.¹² Literature also shows that indirect process metrics (e.g., scanning strategy, build orientation, and powder bed evenness) have effects on the optimization problems.¹³ Ponticelli et al. investigated ML-based optimization models to optimize mechanical strengths of parts by changing part orientation.¹⁴ Shi et al. studied optimization to improve surface roughness and micro-hardness of Ti6Al4V materials through experimental results by introducing different scanning strategies.¹⁵ Furthermore, dynamic strength of PBF produced parts can be improved by design-based optimization methods. Cheng et al. studied a novel algorithm to optimize the topology of additively manufactured parts to maximize its dynamic response in terms of natural frequencies.¹⁶ Avinkrishnan et al. studied design-based dynamic optimization that shows structure stiffness can be improved by adding crumple into surfaces. Optimized crumple pattern led to an increase in natural frequency of produced plane.¹⁷ Chen et al. optimized hexachiral structures with advancement of natural frequencies around 12%; however, mode shapes have not changed in terms of displacements.¹⁸ Even though the performance of those studies is high enough in terms of dynamic response optimization, changing part design may be unfeasible. Parts may have strict geometrical constraints and fixed geometry because of assembly limits in general. Especially, geometrical design of part to be produced may have already confirmed and not possible to change. And yet, the part to be produced needs to comply with the mechanical and dynamical constraints.

On the contrary, there are some studies that combine both ML and finite element method (FEM) for different aims such as physical system modeling and fatigue lifetime prediction of PBF produced parts.^19–21 Besides, there are also a few studies that connect the AM applications with physics-informed ML algorithms which include physical governing equations to the model. Specifically, Zhu et al. demonstrated that the physics-informed neural network framework accurately predicts temperature and melt pool dynamics during metal AM processes with only a moderate amount of labeled datasets, demonstrating its potential for advanced manufacturing.²² Du et al. proposed that combining physics-informed ML, mechanistic modeling, and experimental data can reduce common defects in AM, improving part quality, reliability, and serviceability.²³ Wenzel et al. utilized the ML method to optimize input parameters and predict system responses in fused filament fabrication, increasing system reliability and improving print bed adhesion in AM.²⁴ Razvi et al. demonstrated in their review article that physics-informed ML can improve parameter optimization and anomaly detection in AM, with potential future research directions.²⁵ However, combination of ML modeling and FEM module with the purpose of improving additively manufactured part’s dynamic behavior without changing design has not encountered before in the literature. Therefore, the main contribution of this study is improving the dynamic response, for example, natural frequencies of PBF produced parts by optimizing processing parameters by combining ML modeling and FEM module in the loop with preserving the part shape. The study also contributes that material properties such as elasticity and mass density have no effect on mode shapes of parts. Analytical proof was verified by a numerical study that processing parameters were optimized in the light of aforementioned reasons to improve produced part’s dynamic behavior.

This article was structured as follows. In the next section, problem statement of the study was organized that investigates how material constants were effective on the dynamic properties and methodology of the study was presented. In the third section, ML structure and relation with proposed optimization procedure were explained. In the fourth section, optimization theory was introduced by different constrained optimization algorithms. Optimization flowchart was also presented and implemented for optimizing processing parameters in this section. In the fifth section, a simulation part was modeled with its boundary conditions (BC). Performance of the optimization results including mechanical properties and dynamic responses for the part were represented by surface maps and plots. Optimized parameters were presented. Finally, in the last section results were concluded and commented for future perspectives and suggestions.

Problem Statement

Mechanical properties such as stiffness and fatigue strength are the key properties that characterize the part for a given application; for example, metallic parts need to be more stiffer due to the strength requirements in operations of the automotive, aerospace, etc., parts which are exposed to consistent forces needed to have higher fatigue resistance. In addition, part density and stiffness are considerable metrics to evaluate a part with regard to dynamic response. For instance, high endurance to undesired vibrations would be a necessity for a structural part if needed.

In this context, unlike the conventional production methods, PBF method has a significant role that allows modifying material properties of produced parts by means of process parameters and/or build geometry. In case of no allowance to change part geometry, for example, mass-produced parts with assembly constraints, process parameters will be the key factor to improve part performance for specific applications. If the relationship of process and structure is established correctly, dynamic behavior of parts would be related with processing parameters which then guide to enhance mechanical properties. Therefore, changing material properties such as relative density and elasticity modulus would yield versatile dynamic characteristics in the part. That is quite important to prove analytically which then will lead to express the further optimization problem. According to the reference of equation of motion with undamped forced element:

M \ddot{x} + K x = F (t)

(1)

Here, M and K refer to mass and stiffness matrices of a part, respectively. To find the mode shapes and natural frequencies, the following eigenvalue problem must be solved:

[K - w^{2} M] u = 0

(2)

Here, w defines the natural frequency of the element and u defines the mode shape vector of corresponding element. The solution of the above problem gives ith natural frequency and mode shape pairs:

{\{{w_{i}}^{2}, u_{i}\}}_{i = 1}^{n}

(3)

Once this eigenvalue problem is solved it can be shown that mode shapes are orthogonal with respect to mass and stiffness matrices:

{u_{i}}^{T} K u_{j} = 0, i \neq j

(4)

{u_{i}}^{T} M u_{j} = 0, i \neq j

(5)

When i = j, it can be shown that the following triple matrix multiplications give modal stiffness and modal mass:

{u_{i}}^{T} K u_{j} = k_{i i}, i = j

(6)

{u_{i}}^{T} M u_{j} = m_{i i}, i = j

(7)

If equation 2 is premultiplied by ${u_{i}}^{T}$

{u_{i}}^{T} [K - {w^{2}}_{i} M] u_{i} = 0

(8)

and using equations 4–7 it yields

[k_{i i} - {w^{2}}_{i} m_{i i}] = 0

(9)

Since modal stiffness k_ii and modal mass m_ii are scalar values, natural frequency can be calculated using modal mass and modal stiffness values as follows:

{w_{i}}^{2} = \frac{k_{i i}}{m_{i i}}

(10)

For the FEM model of a structural system, mass matrix is a function of density and geometry. If geometry is not allowed to change which is fixed, then mass matrix is a linear function density $ρ$ :

M = M (ρ) = ρ M_{ρ}

(11)

On the contrary, stiffness matrix is a linear function of material properties of elasticity modulus E:

K = K (E) = E K_{E}

(12)

So if we update equation 2 with mass and stiffness matrices given in equations 11 and 12 we have:

[E K_{E} - w^{2} ρ M_{ρ}] u = 0

(13)

If we apply the equations 4 –7 with updated mass and stiffness matrices:

{u_{i}}^{T} E K_{E} u_{j} = 0, i \neq j

(14)

{u_{i}}^{T} ρ M_{ρ} u_{j} = 0, i \neq j

(15)

{u_{i}}^{T} E K_{E} u_{j} = {E k}_{i i}, i = j

(16)

{u_{i}}^{T} ρ M_{ρ} u_{j} = ρ m_{i i}, i = j

(17)

Now let us repeat modal analysis for the structural system with the same geometry but with different material properties of two identical parts ( $ρ_{1}, ρ_{2}, E_{1}, E_{2}) .$ Then we have the following eigenvalue problem:

[(E_{1} K_{E}) - w^{2} (ρ_{1} M_{ρ})] u = 0

(18)

[(E_{2} K_{E}) - w^{2} (ρ_{2} M_{ρ})] u = 0

(19)

With reference to equations 9 and 10, natural frequencies can be related as follows:

\frac{{k^{1}}_{i i}}{{k^{2}}_{i i}} = \frac{E_{1}}{E_{2}}, \frac{{m^{1}}_{i i}}{{m^{2}}_{i i}} = \frac{ρ_{1}}{ρ_{2}} ≫ {{(w}_{1})}^{2} = \frac{{k^{1}}_{i i}}{{m^{1}}_{i i}} {≫ {(w}_{2})}^{2} = \frac{{k^{2}}_{i i}}{{m^{2}}_{i i}} ≫ {(\frac{w_{1}}{w_{2}})}^{2} = \frac{E_{1} ρ_{2}}{E_{2} ρ_{1}}

(20)

Equation 20 shows two different part’s natural frequencies were related with their material constants. It also proves that material constants such as E or ρ changes natural frequencies of the element. On the contrary, mode shapes are not the function of material properties. This would explain the motivation of this study that if material constants can be varied in PBF production, then enhancing natural frequencies of a part would be possible.

In the light of these information, the methodology of this study can be explained as follows. The experimental data of previously PBF produced parts which includes processing parameters and mechanical properties were extracted from the literature.^26,27 Processing parameters were selected as inputs which are laser power, scanning speed, hatch distance, and laser beam diameter. Mechanical properties were selected as outputs which are tensile strength, micro-hardness, elasticity modulus, surface roughness, and relative density. Then, ML models were created by Matlab Optimization Toolbox to define mechanical properties as semi-analytical functions of processing parameters to be further used in optimization phase. ML models utilized experimental data that include the processing parameters as inputs and mechanical properties as outputs. Then each model was trained by feedforward backpropagation algorithm.²⁸ Overall, experimental data were correlated with numerical FEM model in the loop of optimization cycle to find desired dynamic behavior from produced parts. Different optimization algorithms were implemented in optimization process and the best fitted algorithm selected as outcome of optimization in terms of performance and time.

Machine Learning Modeling

The dataset was collected from the literature studies and used in the training of ANN models. Each mechanical property has independent ANN model. The algorithm structure has one hidden layer which have 10 neurons which is shown with connections in Figure 1. Training, validation, and test data percentages were split as 70%,15%, and 15%, respectively. Levenberg–Marquardt algorithm is selected in the training process. Performance function was selected as mean square error. Trained ANN models created functions for each mechanical property. All functions were connected through optimization process by Quadratic Programming which were implemented in Matlab environment.

FIG. 1.

Representative ANN structure of proposed study: All inputs have connections for each output which specifically shows the laser power connecting through tensile strength.

Constrained Multi-Objective Optimization

A constrained optimization problem aims to get a solution of the following function with constraint criteria:

Find x with respect to minimize f(x)

Subject to

g_{i} (x) \leq 0, i = 1, 2, \dots m

(21)

h_{j} (x) = 0, j = 1, 2, \dots n

(22)

Equations 21 and 22 stand for the set of inequality and equality constraints, respectively, which both set cover the feasible region in optimization problem. Most common and highly preferred optimization algorithms include Sequential Quadratic Programming (SQP) with different computing techniques which are SQP, Active-Set, Interior Point, and SQP-Legacy.

Main idea of the SQP algorithm is to model the problem at a given approximate solution, x_k, by quadratic programming subproblem. The solution is then used in this subproblem to get a better approximation x_k+1. Objective function generally has quadratic terms, and it models the Lagrangian equations. Finding an appropriate choice of quadratic subproblem will yield as Newton-based methods to the constrained optimization problem.²⁹ Another algorithm is called the Active-set which is a crucial algorithm since it only uses the constraints which will have an impact on the optimization’s output. An estimate of the algorithm will provide with a subset of inequalities to monitor while looking for the solution, which lowers the search complexity in quadratic programming.³⁰ One of the drawbacks of the algorithm is their inability to handle both equality and inequality constraints concurrently due to the usage of only active constraints. Interior Point algorithm is an alternative algorithm among SQP algorithms which can be applied to solve the QP subproblem. This algorithm is competitive with other algorithms for general problems, particularly in early iterations when the active sets are very variable from iteration to iteration and there is little value in using hot-start knowledge.³¹ SQP legacy is similar to SQP algorithm which uses more memory than SQP and it is slower than that in terms of optimization performance. Accuracy and convergence performance are generally similar to SQP algorithm.³² For the case of this study, multi-objective optimization is applicable for each algorithm at Matlab software with implementing fmincon function. Besides, ML and FEM will be integrated into the optimization algorithm with constraints, for example, limiting processing parameters and mechanical properties. Each algorithm will be simulated in the software to compare the performances. Eventually, optimized processing parameters will be yielded for better mechanical properties and dynamic behavior of produced parts.

Process parameter optimization—SQP

In the optimization process, ML models and FEM module were fed into the loop of optimization process to create desired outputs of which are mechanical properties and dynamic responses of part.

Optimization process was handled by constrained nonlinear optimization technique. Different optimization algorithms such as SQP, Active set, Interior-point, and SQP-legacy were defined and used to reach desired target for each output. Optimization process was finished when the error function reached the threshold value and defined constraints satisfied. Flowchart of optimization process is summarized in Figure 2. According to the chart, each mechanical property was defined as the function of processing parameters in the optimization phase. ML model was generated to create mapping functions between processing parameters and mechanical properties. At the same time, FEM module was implemented to get natural frequencies and mode shapes of parts in the function of elasticity modulus and relative density. Each function was used in the multi-objective optimization process by weighted sum method which then returned with a single cost function of both mechanical and dynamic properties. In the figure lp, ss, lbd, and hd stand for laser power, scanning speed, laser beam diameter, and hatch distance, respectively. Basic representation of the cost function is shown in equations 28 and 29.

f_{cost} = \sum_{n = 1}^{m} α_{n} f_{n} \sum_{n = 1}^{m} α_{n} = 1, α_{n} \in (0,1)

(28)

f_{cost} = α_{1} {[\frac{(σ_{ref} - σ)}{σ_{ref}}]}^{2} + α_{2} {[\frac{({R a}_{ref} - R a)}{{R a}_{ref}}]}^{2} + α_{3} {[\frac{(E_{ref} - E)}{E_{ref}}]}^{2} + α_{4} {[\frac{({h r}_{ref} - h r)}{{h r}_{ref}}]}^{2} + α_{5} {[\frac{(ρ_{r e f} - ρ)}{ρ_{ref}}]}^{2} + α_{6} [\sum_{i = 1}^{6} {[\frac{(ω_{(n, ref (i))} - ω_{i})}{ω_{(n, ref (i))}}]}^{2}]

(29)

FIG. 2.

Simple representation of presented procedure’s flowchart: ML creates the functions of mechanical properties, FEM creates the function of natural frequencies, and optimization cycle finds the desired output from created correlation of multiple objectives.

where $α$ _n is the weighting coefficient of nth order objective function f_n, which expresses the relative importance of the nth objective. As it is seen in equation 29, cost function includes weights, function output values, and function reference values. An objective function with a greater value of weight indicates that it is more significant than other objective functions. Function reference values $σ_{ref}, E_{ref}, {h r}_{ref}, {R a}_{ref}, ρ_{ref}, ω_{ref}$ define tensile strength, elasticity modulus, hardness, surface roughness, relative density, and natural frequency of part, respectively. After optimization was ended, optimized and non-optimized models were compared in terms of the refined natural frequencies along with mechanical properties. (Non-optimized model was referenced by using the Inconel 718 cast material properties.) Optimization algorithms were implemented to a simulation model. Dynamic response of the model was both simulated in ANSYS and Matlab software to verify the FEM model’s numerical results.

Simulation Results

PBF production technique includes different novel alloys and material types in the process. Inconel 718 is one of the well-suited powder materials due to its excellent high-temperature strength and fatigue resistance, which allows it to withstand the high thermal loads generated during the PBF process.³³ On the contrary, rocket fuel engine nozzles are one of the most important part in aerospace industry. These parts are subjected to random vibration and thermal loads during launch or accelerating stages, and this affects the structure of part dynamically. From this point of view, dynamic response can be performed in a part to understand those dynamic effects by using modal analysis which explains to what extent the dynamic forces effect the part from unexpected vibrations.³⁴ For that reason, an Inconel 718 rocket nozzle model was used as numerical study for improving dynamic behavior optimization.

Optimization specifications were explained as follows: Optimization targets were determined with thanks to the experimental dataset of reference literature studies. Natural frequency values were also assigned according to the engine nozzle working vibration responses.³⁵ Initial processing parameters were assigned as the center of each parameter value range. Weights were selected according to the importance of mechanical properties which were explained in previous section. Constraint and tolerance equations (equations 30 and 31) explain the range of processing parameters and mechanical properties limits. They were defined by the reference literature studies.^{26,27,36–40}

Optimization Targets:

σ_{ref} = 800 M P a, {Ra}_{ref} = 8.9 µ m, {hr}_{ref} = 3310 MPa, ρ_{ref} = 234 GPa, ρ_{ref} = 8220 \frac{k g}{m^{3}}

ω_{ref} = \{700,700,1000,1000,1600,2700\} H z

Initial Processing Parameters:

lp = 170W, ss = 1250 mm/sec, lbd = 0.075 mm, hd = 0.135 mm

Weights:

α₁ = 0.1 α₂ = 0.1 α₃ = 0.1 α₄ = 0.1 α₅ = 0.2 α₆ = 0.4

Constraints and Tolerances:

ρ - ρ_{\max} < 0, - ρ + ρ_{\min} < 0, ρ_{\min} = 1000 \frac{k g}{m^{3}}, ρ_{\max} = 10000 \frac{k g}{m^{3}}

(30)

E - E_{\max} < 0, - E + E_{\min} < 0, E = 10 GPa, E_{\max} = 400 GPa

(31)

100 < lp < 370 (Watt), 500 < ss < 2000 (mm/sec), 40 < lbd < 100 (µm), 80 < hd < 150 (µm),

Simulations were executed in a Workstation with 6 cores Intel Xeon processor, 3.60 GHz processing frequency, and 48 GB RAM. The analysis was done in both Matlab 2023a and Ansys Workbench 2023 software to verify the FEM model results with identical BCs. Results show consistency for both analyses. As for the FEM parameters, both ANSYS and Matlab FEM models used 10-noded tetrahedron element type which has 4 and 7 mm maximum element size, respectively, with more than 42000 nodes both. In the simulations, Inconel 718 (Elasticity modulus: 165 GPa and Density: 8220 kg/m³) cast material’s mode shapes and natural frequencies were extracted. Natural frequency results of non-optimized and optimized models between ANSYS and Matlab software can be seen in Table 1. The table shows that natural frequencies were manipulated according to the dynamic requirements by only changing processing parameters of the process. Besides, ANSYS software increased the reliability of proposed method for the sake of validation. Different algorithms were implemented for comparing the performances in terms of time, cost function, mechanical response values, and natural frequencies which can be seen in Tables 2 and 3 with first 6 modes. Minimum value of cost function indicates that each response reached closely to the desired target. According to that, SQP algorithm has the best performance and it was highlighted in a box.

Table 3.

Optimization Algorithms’ Natural Frequency Results

Algorithm	ω ₁ (Hz)	ω ₂ (Hz)	ω ₃ (Hz)	ω ₄ (Hz)	ω ₅ (Hz)	ω ₆ (Hz)
Non-optimized	552	552	814	815	1418	2193
Sqp	661	662	1012	1013	1681	2730
Activeset	612	612	980	980	1510	2344
Interior point	609	610	975	975	1496	2312
Sqp legacy	651	651	1010	1010	1685	2725

Table 1.

Simulation Results of Optimized and Non-Optimized Natural Frequencies of SLM Part for Both Software—(MATLAB and ANSYS)

$ω$ n(Hz)	Non-Optimized results			Optimized results
$ω$ n(Hz)	MATLAB	ANSYS	Delta (%)	MATLAB	ANSYS	Delta (%)
$ω$ ₁	549.7	552.2	0.44	661.3	664.7	0.51
$ω$ ₂	550.4	552.8	0.37	662.2	664.5	0.34
$ω$ ₃	841	814.7	3.1	1012.8	979.3	3.3
$ω$ ₄	841.9	815.8	3.2	1013.9	980.6	3.2
$ω$ ₅	1397.2	1418.5	1.5	1681	1705.1	1.4
$ω$ ₆	2269	2193.4	3.3	2730	2636.6	3.4

Table 2.

Optimization Algorithms’ Performances and Corresponding Results

Algorithm	Time	F_cost	$σ$ (MPa)	HR (MPa)	$ρ$ (kg/m³)	E (GPa)
Non-optimized	—	—	896	1355	8220	165
Sqp	2 h 47 min	0.029	653	2013	7964	231
Activeset	3 h 25 min	0.084	669	2554	7897	222
Interior point	1 h 25 min	0.23	673	2750	8040	194
Sqp legacy	2 h 55 min	0.031	655	2030	7873	231

Various algorithms yield different scale of performance for the optimization process. Before and after optimization cycle processing parameters and corresponding mechanical responses are shown in Table 4. The optimized parameters modified natural frequencies and elasticity modulus, and yet there are variations in other mechanical properties. Besides, change of mechanical properties was plotted in 2D and 3D and represented as trajectory lines during optimization process in Figures 3 and 4, respectively. According to that, elasticity modulus and relative density responses have optimized during the dynamic response change. However, there is not much difference on the values of micro-hardness and tensile strength responses which might be caused by the selected weights. In Figure 4, mode shapes of optimized and non-optimized models were compared by mass normalized Modal Assurance Criterion (MAC) matrix.⁴¹ For optimized and non-optimized mode shapes, namely, $u_{x}$ and $u_{y},$ mass normalized MAC matrix can be found with respect to equations 32 –34. It can be inferred from Figure 5 that diagonal terms were valued as 1 which explains that mode shapes are identical and even optimization was applied to the part which was also proved analytically in the Problem Statement section.

m_{x x} = \sqrt{({\{u_{x}\}}^{T} M \{u_{x}\})}

(32)

m_{y y} = \sqrt{({\{u_{y}\}}^{T} M \{u_{y}\})}

(33)

MAC (x, y) = \frac{{\{u_{x}\}}^{T} M \{u_{y}\}}{m_{x x} m_{y y}}

(34)

FIG. 3.

Step-by-step trajectory lines of optimization process for mechanical responses: (a) Tensile strength steps are following to find reasonable spot; (b) Young’s modulus steps are going through the maximal value.

FIG. 4.

Trajectory lines of optimization process for the most effected mechanical responses in 3D surface: (a) Young’s modulus reaches maximal value; (b) relative density optimizes its value in medium levels.

FIG. 5.

Modal Assurance Criterion (MAC) matrix of (a) non-optimized, (b) optimized, and (c) comparison with mode shapes: All matrix show that mode shapes of part are identical before and after optimization and there are no similarities between each other.

Table 4.

Comparison Table of Processing Parameters and Mechanical Responses: Before and After Optimization Procedure

Processing parameters	Before	After	Mechanical responses	Before	After
Laser Power (Watt)	170	130	Tensile Strength (MPa)	674	653
Scanning Speed (mm/s)	1250	1015	Surface Roughness (mm)	0.094	0.092
Laser Beam Diameter (mm)	0.075	0.085	Micro-Hardness (MPa)	2042	2013
Hatch Distance (mm)	0.135	0.15	Density (kg/m³)	8110	7964
			Elasticity Modulus (GPa)	194	231.4

Figure 6 represents the change of mechanical properties of produced parts during the optimization of SQP algorithm. According to that, micro-hardness and surface roughness responses keep their values roughly in the same level. It can be inferred from the figure that, there is a gain–loss balance between mechanical properties and dynamic behavior in multiple objective optimization process. Therefore, goal-oriented optimization process will yield better results for desired targets. Optimization process also includes the change of natural frequencies step by step. The change can be seen in Figure 7 for each iterations. It can be understood from those graphs that, there is a dramatic change for six natural frequencies by comparison with non-optimized model. First six natural frequencies of engine nozzle model were refined to the level of nearly 22% with the algorithm which is shown in Figure 8.

FIG. 6.

Change of mechanical properties during SQP optimization process: tensile strength, relative density, and surface roughness are in decreasing trend; Young’s modulus is in increasing trend.

FIG. 7.

Variation of natural frequencies during SQP optimization process: Each frequency has linear behavior and increasing trend.

FIG. 8.

Natural frequency and mode shape results after optimization for structural nozzle part: First 6 natural frequencies were modified that met the requirements; mode shapes have not shown any change as expected.

For the case of nozzle implementation, those results would be adequate for real applications according to the literature. For many nozzle types, mechanical durability, heat and pressure resistance primarily matter. Results also indicate that optimized model would handle those scenarios successfully and met with desired performances in accordance with Inconel 718 nozzle studies.^21,40–42

Conclusion

In this study, a novel constrained optimization procedure was proposed for optimizing processing parameters of PBF produced parts to acquire optimal dynamic behavior. This procedure included the combination of ML and FEM in the loop to correlate between processing parameters and dynamic responses which was then used in a constrained optimization process. The most remarkable advantage of dynamic property optimization was that we would modify the natural frequencies of a structural part by only adjusting processing parameters in the process to the desired levels whose design constraints have already confirmed and not allowed to change geometrically. Originality of this study was that mechanical properties were acquired by the experimental data, and semi-analytical equations were created by ML models accordingly. Then, the equations were used in FEM to get dynamic responses such as natural frequencies and mode shapes in an optimization procedure.

The optimization procedure was implemented by different algorithms such as SQP, Active-set, Interior Point, and SQP-legacy. Each algorithm was evaluated in terms of their converging performance. The implementation of the procedure was presented with a numerical study which is PBF produced Inconel 718 engine nozzle model. In addition, it was analytically proven that mode shapes were not affected by varying material properties, but natural frequencies were affected. The analytical proof was also verified by the numerical study with MAC matrix. Numerical study showed that multi-objective optimization process usually had a gain–loss balance between mechanical and dynamic properties of parts which gains performance of one property with loses from another property’s performance.

For the real-world applications, a structural part which is not allowed to change in design could be improved in terms of its dynamic response by producing with PBF method. Specifically, in case the nozzle’s working frequency was known, natural frequencies of model could be modified to eliminate resonances by the presented method with preserving the topology. For future perspective, the optimization procedure would be implemented for different parts and different case studies which include structural stress and displacement analysis. Besides, optimized outcomes would be used and compared with an organized experimental study. Finally, optimization results would provide significant results for those who interested in this research field before the real experimental runs started.

Authors’ Contributions

C.B.T. and C.U.D.: Conceptualization; C.B.T.: Data curation; C.B.T. and C.U.D.: Formal analysis; C.B.T.: Investigation; C.B.T. and C.U.D.: Methodology; C.U.D.: Project administration; C.B.T.: Software; C.U.D.: Supervision; C.U.D.: Validation; C.B.T.: Visualization; C.B.T.: Writing—original draft; C.B.T. and C.U.D.: Writing—review and editing. All authors have read and agreed to the published version of the article.

Footnotes

Acknowledgments

The authors would like to acknowledge that this paper is submitted in partial fulfillment of the requirements for PhD degree at Hacettepe University. It was supported by the Hacettepe University Scientific Research Projects unit under BAP Project FDK-2022-20225. The authors would also like to thank the Additive Manufacturing Technologies Application and Research Center (EKTAM) research group for their technical support and Cem BATUR in particular for his guidance related with nozzle simulations and implementation.

Author Disclosure Statement

The authors declared that they have no conflicts of interest in this work.

Funding Information

No funding was received for this article.

References

Sun

, Brandt

, Easton

. Powder bed fusion processes: An overview. Las Addit Manuf, 2017; 55–77; doi: 10.1016/B978-0-08-100433-3.00002-6

Alridha

, Fadhil Abdalhasan

, Mazin Kareem Kadhim

. Training analysis of optimization models in machine learning. Int. J. of Nonli Analy Appli, 2021; 1453–1461.

Guo

, Agarwal

, Cooper

, et al. Machine learning for metal additive manufacturing: Towards a physics-informed data-driven paradigm. J. Manuf Sys, 2022; 62:145–163; doi: 10.1016/j.jmsy.2021.11.003

Poprawe

, Hinke

, Meiners

, et al. SLM production systems: Recent developments in process development, machine concepts and component design. Adv Prod Tech., 2015; 49–65.

Criales

, Arısoy

, Lane

, et al. Predictive modeling and optimization of multi-track processing for laser powder bed fusion of nickel alloy 625. Addit Manuf, 2017; 13:14–36.

Shin

, Lee

, Kühn

, et al. Optimizing laser powder bed fusion of Ti-5Al-5V-5Mo-3Cr by artificial intelligence. J. Allo Comp, 2021; 862:158018.

Gheysen

, Marteleur

, van der Rest

, et al. Efficient optimization methodology for laser powder bed fusion parameters to manufacture dense and mechanically sound parts validated on AlSi12 alloy. Mat. & Des., 2021; 199:109433; doi: 10.1016/j.matdes.2020.109433

Lin

, Tran

, Lo

, et al. Optimization of surface roughness and density of overhang structures fabricated by laser powder bed fusion. 3D Prin Addit Manuf, 2022; 10(4):732–748.

, Li

, Wang

, et al. Powder-bed-fusion additive manufacturing of molybdenum: Process simulation, optimization, and property prediction. Addit. Manuf., 2022; 58:103069.

10.

Yonehara

, Ikeshoji

, Nagahama

, et al. Parameter optimization of the high-power laser powder bed fusion process for H13 tool steel. Int J Adv Manuf Technol, 2020; 110(1–2):427–437.

11.

Wang

, He

, Song

, et al. Optimization of the process parameters of laser beam powder bed fusion GTD222 nickel-based superalloy based on two laser energy densities. Metals, 2022; 12(7):1154.

12.

Letenneur

, Kreitcberg

, Brailovski

. Optimization of laser powder bed fusion processing using a combination of melt pool modeling and design of experiment approaches: Density control. Jmmp, 2019; 3(1):21.

13.

Brika

, Zhao

, Brochu

, et al. Multi-objective build orientation optimization for powder bed fusion by laser. J. Manuf. Sci. Eng, 2017; 139(11).

14.

Ponticelli

, Venettacci

, Giannini

, et al. Fuzzy process optimization of laser powder bed fusion of 316L stainless steel. Prog Addit Manuf, 2022; 8(3):437–458; doi: 10.1007/s40964-022-00337-z

15.

Shi

, Li

, Jing

, et al. Combination of scanning strategies and optimization experiments for laser beam powder bed fusion of Ti-6Al-4V titanium alloys. Appl. Sci, 2022; 12(13):6653.

16.

Cheng

, Liang

, Belski

, et al. Natural frequency optimization of variable-density additive manufactured lattice structure: Theory and experimental validation. J. Manuf. Sci. Eng, 2018; 140(10):105002.

17.

Vijayachandran

, Waas

. Optimized creasing pattern for maximizing natural frequencies of a thin elastic plate. J. Sou. Vibr., 2023; 542:117334.

18.

Chen

, Yi

. Optimization of natural frequency for hexachiral structure based on response surface method. J Vibroeng, 2020; 22(7):1705–1714.

19.

Baturynska

, Semeniuta

, Martinsen

. Optimization of process parameters for powder bed fusion additive manufacturing by combination of machine learning and finite element method: A conceptual framework. Procedia Cirp, 2018; 67:227–232.

20.

Silva

, Beber

, Pitz

. Machine learning and finite element analysis: An integrated approach for fatigue lifetime prediction of adhesively bonded joints. Fatigue Fract Eng Mat Struct, 2021; 44(12):3334–3348.

21.

Kononenko

, Kononenko

. Machine learning and finite element method for physical systems modeling. arXiv preprint arXiv:1801.07337 2018.

22.

Zhu

, Liu

, Yan

. Machine learning for metal additive manufacturing: Predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Comput Mech, 2021; 67(2):619–635.

23.

, Mukherjee

, DebRoy

. Physics-informed machine learning and mechanistic modeling of additive manufacturing to reduce defects. Appl. Mat. Tod., 2021; 24:101123.

24.

Wenzel

, Slomski-Vetter

, Melz

. Optimizing system reliability in additive manufacturing using physics-informed machine learning. Machines, 2022; 10(7):525.

25.

Razvi

, et al. A review of machine learning applications in additive manufacturing. International design engineering technical conferences and computers and information in engineering conference. Vol. 59179. American Society of Mechanical Engineers, 2019.

26.

Kladovasilakis

, Charalampous

, Tsongas

, et al. Influence of selective laser melting additive manufacturing parameters in ınconel 718 superalloy. Materials, 2022; 15(4):1362; doi: 10.3390/ma15041362

27.

Balbaa

, Mekhiel

, Elbestawi

, et al. On selective laser melting of Inconel 718: Densification, surface roughness, and residual stresses. Mat & Des, 2020; 193:108818.

28.

Cilimkovic

. Neural networks and back propagation algorithm. Institute of Technology Blanchardstown, Blanchardstown Road North Dublin. 2015; 15:1.

29.

Boggs

, Tolle

. Sequential quadratic programming. Acta Numerica, 1995; 4:1–51.

30.

Gill

, Michael

, Saunders

, et al. On the performance of SQP methods for nonlinear optimization. Modeling and optimization: Theory and applications: MOPTA, Bethlehem, PA, 2014. Sel. Contri. Springer International Publishing, 2015.

31.

Bashir

, Liang

. Interior point method based sequential quadratic programming algorithm with quadratic search for nonlinear optimization. Int. J. Comp. Info. Sys. Ind. Manage. Appl., 2011; 3:51–60.

32.

Byrd

, Gilbert

, Nocedal

. A trust region method based on interior point techniques for nonlinear programming. Math Program, 2000; 89(1):149–185.

33.

Sutrisno

, Widowati

, Sunarsih

. Determining the best optimization method for large scale probabilistic supplier selection problem integrated with inventory management. E3S Web Conf, 2020; 202:14001.

34.

Maconachie

. The dynamic behaviour of AlSi10Mg lattice structures fabricated via selective laser melting. Diss. RMIT University, 2022.

35.

, Liu

, Han

, et al. Influence study of flow separation on the nozzle vibration response. Prop. Power Res., 2016; 5(2):108–117.

36.

Gradl

, Greene

, Protz

, et al. Additive manufacturing of liquid rocket engine combustion devices: A summary of process developments and hot-fire testing results. 2018 Joint propulsion conference.

37.

Hwang

J-R

, Zheng

J-Y

, Kuo

P-C

, et al. Process optimization of ınconel 718 alloy produced by laser powder bed fusion. Metals 12, 2022; 12(9):1494.

38.

Jia

, Gu

. Selective laser melting additive manufacturing of Inconel 718 superalloy parts: Densification, microstructure and properties. J. Alloys Comp, 2014; 585:713–721.

39.

Pastor

, Binda

, Harčarik

. Modal assurance criterion. Proc Eng, 2012; 48:543–548.

40.

Chen

, Zhao

, Wang

, et al. Theoretical analysis and verification on plastic deformation behavior of rocket nozzle using a novel tube upsetting-bulging method. Materials, 2023; 16(4):1680; doi: 10.3390/ma16041680

41.

Jia-Hui

, Ke

, Xiu-Cong

. Dynamic simulation of nozzle structure based on thermal-fluid-solid coupling analysis. J Phys: Conf Ser, 2023; 2472(1):12065.

42.

Violett

. Space shuttle main engine structural analysis and data reduction/evaluation. Main Inje LOX Inlet Anal, 1989; 5(NASA-CR-183668).