Analyzing 3D printing variables for small-scale clay objects using fuzzy cognitive maps

Abstract

In this article, we explore the first application of fuzzy cognitive maps (FCMs) for analyzing the variables of small-scale clay 3D printing processes, addressing an evident gap in methodologies within additive manufacturing (AM). While clay, as a sustainable and recyclable material, presents unique opportunities for customization in manufacturing, its inherent fragility poses significant challenges, including volumetric changes and structural instability. The proposed FCM methodology integrates advanced modeling techniques that extend beyond traditional statistical approaches, enabling a dynamic analysis of interrelated process variables. By systematically characterizing these factors, the study identifies critical parameters influencing print quality, contributing to the development of standardized best practices for clay AM. The findings underscore the potential of FCMs to enhance decision-making in process management and lay the groundwork for future research into optimizing 3D printing technologies with ceramic materials. This research ultimately aims to facilitate sustainable and efficient production methods in the context of circular economy principles. Furthermore, it may support the creation of guidelines, protocols, and standards that define optimal parameters and procedures for 3D printing with clay, thereby facilitating reproducibility and consistent quality across diverse production environments.

Keywords

FCM applications 3D printing analysis additive manufacturing production process improvement smart manufacturing

Introduction

Additive manufacturing (AM), more commonly known as 3D printing, is a digital manufacturing technique in which three-dimensional objects are created from a digital model by depositing materials layer by layer.^{1, 2} This approach contrasts with traditional subtractive manufacturing methods, in which materials are removed from a larger block to achieve the desired shape,³ and has had a transformative effect on manufacturing processes in a wide range of industries by offering a flexible and efficient alternative to conventional production.⁴ While polymers and metals have been the predominant materials in the research and development of additive technologies,^{5, 6} the use of ceramic materials, such as clay, has gained interest due to their diverse applications.^7–10

The use of this material in the creation of customized structures with complex geometries is facilitated, making it an attractive option for manufacturing in the context of the circular economy.¹¹ Clay is a natural, recyclable, and low-environmental-impact material. However, due to its inherent fragility, it is susceptible to volumetric changes, which can result in structural instability.^{12, 13}

The quality of parts printed using this emerging technology depends on several factors, including plasticity, material preparation, final print dimensions, machine parameters, part strength, shrinkage during drying, and fire resistance.^13–15 Therefore, developing advanced techniques for process characterization plays a crucial role in understanding the physical and chemical behavior of clay. Ang et al.¹⁴ and Beroya et al.¹⁶ have analyzed the mineral components of clay, providing information about its structural properties and its molding potential using X-ray diffraction (XRD) and Fourier-transform infrared spectroscopy (FTIR).

Xie et al. and Zhao et al.¹⁷ have examined the distribution of particles, the structure of materials, and rheology using optical and electron microscopy. According to the literature, it is necessary to combine molding and modeling tests with experimental validation in order to refine clay formulations and optimize the manufacturing process.^{18, 19} On the other hand, Alonso Madrid et al. have investigated the behavior of clay when exposed to different forms of protection during the 3D printing process, such as contact with air, cement, plastics, or metal foils²⁰. In addition, the effects of prototype geometry and printing process variables (layer height, overlap, etc.) on the quality of the printed material are analyzed. It is concluded that clay, as a natural and recyclable material, has great potential in AM within sectors such as architecture and construction, due to its low cost and versatility. Al-Noaimat et al. have examined the effects of replacing high percentages of Portland cement with calcined clay and limestone on the printability, stiffness, and early hydration of 3D printing concrete²¹. It is shown that, although the flowability of the material decreases, the buildability is significantly improved. However, a decrease in compressive strength is also evident, derived from the dilution effect caused by cement substitution.

Bourgault et al. have proposed an innovative approach to designing trajectories for 3D printing in clay using a tool called CoilCAM, which mimics the experience of hand modeling with clay²². Through an action-oriented CAM system, users are allowed to design integrated shapes and textures, connecting computational logic with craft sensibility. Chen et al.²³ have analyzed how impression parameters, especially interlayer time and nozzle distance, affect interlayer adhesion in cementitious mixtures based on calcined clay and limestone. Using uniaxial tensile testing and microtomography, it is identified that extended time intervals significantly reduce bond strength, due to increased local porosity. Numerical simulations support these findings and show how these factors directly impact structural integrity.

Despite offering valuable insights, these current studies have limitations that restrict their applicability and adaptability. Most are based on experimental or statistical approaches that treat variables in isolation or in controlled combinations, without capturing the complexity and dynamic interdependence of the 3D printing process. In addition, little consideration is given to adaptability and real-time decision-making, as well as a lack of integration between qualitative knowledge (i.e., manual experience) and quantitative modeling. Some focus heavily on specific materials or conditions, reducing their generalizability to broader contexts.

On the other hand, fuzzy cognitive maps (FCMs) are a soft computing tool that have been proven to perform well in analyzing the interrelationships between process variables.^{24, 25} According to Dickerson and Kosko,²⁶ FCMs are a combination of fuzzy logic and neural networks, and a graphical representation used to illustrate causal reasoning with structure and perform direct and inverse correlation analysis between related events.²⁷

FCMs have been used in a wide range of knowledge areas such as medicine,²⁴ government improvement,²⁸ control systems,²⁹ time-series predictions,³⁰ urban systems,³¹ waste management,³² and urban planning.³³ The reader is referred to Karatzinis and Boutalis for a detailed analysis of several FCM applications. However, the literature review reveals no evidence of FCM applications in AM with clay, indicating a clear gap in current research. This article addresses this gap by proposing the integration of advanced modeling techniques—specifically FCMs—as a novel approach to analyze AM processes with clay, as FCMs are particularly suited to model the complex, nonlinear, and interdependent nature of these systems, where multiple parameters such as material properties, environmental conditions, and machine settings dynamically interact. By enabling the integration of expert knowledge and empirical data, FCMs facilitate the visualization and analysis of causal relationships between variables, leading to deeper understanding, improved prediction, and optimized decision-making. This methodological option goes beyond traditional descriptive statistical approaches, providing a flexible and interpretable framework that is particularly valuable in scenarios where traditional modeling techniques may be limited.

The main objective of this research is to develop a novel methodology based on FCMs for the evaluation and optimization of the parameters involved in clay 3D printing. It seeks to formulate an analytical framework to identify and quantify the interrelationships between the process variables that affect the final quality of the printed products. In this context, the following research questions are posed: (a) What are the critical parameters that significantly impact the print quality of ceramic objects and (b) what strategies can be implemented to improve reproducibility and structural integrity in AM processes using clay? The findings of this article will contribute to the formulation of standardized operating practices, thereby improving scalability and sustainability in the production of ceramic materials using advanced AM techniques. The proposed methodology was applied in a product design research laboratory that includes this type of emerging technology in its portfolio of services. A reminder of the article structure is presented as follows: the next section describes general aspects of FCMs. Then, the methodological steps are shown and explained in detail, followed by the results and their respective analysis. Finally, conclusions are drawn.

Fuzzy cognitive maps’ conceptualization

FCM was first introduced by Kosko in 1986³⁴ as a modeling tool that combines graph theory, fuzzy logic, and neural networks based on human knowledge and experience to design and control complex and dynamic systems. These maps demonstrate the causal relationships between different variables in the system, allowing situations to be modeled where relationships are nonlinear and may be uncertain or imprecise^35–37. They are used to identify how changes in one variable can affect other variables, thus facilitating impact analysis and decision-making in interrelated systems.^38–40 According to Pelta and Cruz, FCMs are represented as neural networks with interpretive functions, consisting of a set of neurons and their causal relationships. The activation value of these relationships determines the strength of the dependencies and their effect on the network. The strength of the relationship between two nodes, $C_{i}$ and $C_{j}$ , is quantified by a numerical weight $W_{i j} \in [- 1, 1]$ and is represented by a weighted causal arc.

The numerical weights employed in this system are indicative of the magnitude and direction of influence exerted by one node on another within the system.^{26, 34, 41} The structural design of FCMs facilitates the dynamic modeling of complex systems, where nonlinear interactions and uncertainty are prevalent. The capacity of FCMs to capture and analyze these causal relationships renders them a valuable instrument for the simulation and optimization of processes across diverse fields of study.⁴²

In the context of FCMs, three distinct types of causal relationship are delineated, reflecting the influence of one node on another. The sign of the weight of $W_{i j}$ determines the nature of the influence between two nodes, whether it is positive, negative, or nonexistent. Dikopoulou et al. state that when $W_{i j} > 0$ , there is a positive causality between $C_{i}$ and $C_{j}$ concepts; if $W_{i j} < 0$ , it means negative or inverse causality; and if $W_{i j} = 0$ , there is no relationship between the concepts. To evaluate the validity of the FCM model in the representation of the system studied, an inference process must be used,^{27, 42} in which several parameters are considered, such as the inference rule, the transfer function, and the detection criterion, which is determined by Kosko’s activation rule. However, to improve the accuracy and stability of the models in more complex applications, a modification of this activation function was introduced and is presented in Equation (1).

A_{i}^{(k + 1)} = f [A_{i}^{k} + \sum_{j = 1, j \neq 1}^{N} W_{i j} \times A_{j}^{k} + θ_{i}]

(1)

where $A_{i}^{(k + 1)}$ represents the value of the concept $C_{i}$ at simulation stage $k + 1$ , $A_{k}^{j}$ refers to the value of the concept $C_{i}$ at time $k$ , $W_{i j}$ is the weight of the intersection of the concepts $C_{i}$ and $C_{j}$ , $k$ is each iteration over the modeled system, $θ_{i}$ is an additional threshold or bias term that more precisely adjusts the activation of each node, and $f (\cdot)$ is the threshold activation function used to assign the activation value of each node within the interval $[0, 1]$ , given by:

f (A_{i}) = \frac{1}{1 + e^{- λ (A_{i} - h)}}

(2)

where λ is a positive number that determines the slope of the continuous function $f (\cdot)$ and $h$ indicates the displacement. It has been reported by some authors that, in certain models, these parameters are closely associated with the convergence of the model, thereby rendering the predictions more reliable due to the model’s fit with the real system. This setting allows a more precise setting of activations, which is useful when modeling compressed systems such as clay 3D printing. In this context, precision is crucial to ensure that the interactions between the different parameters of the system are effectively modeled, allowing for a more accurate and higher quality print. However, the inference process in FCMs can result in a variety of behaviors that reflect the dynamics of the system and are essential for the interpretation and application of the results obtained by FCMs, such as:

The state vector may stabilize in a stationary vector reaching a fixed attractor point.

The state vector may stabilize periodically at the same value, indicating a repetitive cycle.

The state vector may change chaotically.

Materials and methods

The methodology for analyzing 3D printing variables using FCM is based on six steps (see Figure 1).

Figure 1.

Methodological steps.

The first step is to define the variables of the model, considering the process and output variables that are frequently monitored. In the second step, data are collected on the behavior of the process variables and a correlation analysis is performed to determine the weights of the causal relationships. The resulting correlation matrix is converted into adjacency matrices that serve as inputs to the FCM model. The third is to construct the FCM of the process, whereby the variables of the model are grouped in a map and linked by weighted arcs to the values of the adjacency matrix. In the fourth step, the actual dynamic behavior of the model is evaluated by carrying out the inference process using Equations (1) and (2). Finally, in the fifth step, the convergence analysis of the FCM is carried out, ranking the variables of the process that influence the response variable according to the final value of their vectors.

Results

In the following sections, we present the results after applying the methodology to analyze the variables of the printing process. The results were carried out using the open-source R language^® tool for estimating FCM inference in decision-making with the FCM package, presented by Dikopoulou et al.⁴¹

Definition of the process variables

The selection of the variables was made with reference to the parameters set by the manufacturer for the equipment and considering the needs to characterize the variables for 3D printing with clay as stated by Tüylü and Güneri.¹¹ Table 1 shows the 3D printing process parameters that influence the quality of the final product. The variables are coded from C1 to C8, and the corresponding units of measurement are given. C1–C3 are the operating variables of the printing process, C4 is an input parameter, and C5–C8 are the quality characteristics of the printed parts.

Table 1.

Variables of the 3D printing process.

ID	Variable	Units
C1	Layer height	millimeters (mm)
C2	Printing speed	$\frac{milimeters (mm)}{seconds (s)}$
C3	Nozzle diameter	millimeters (mm)
C4	Material preparation time	hours (h)
C5	Outer diameter	millimeters (mm)
C6	Inner diameter	millimeters (mm)
C7	Height	millimeters (mm)
C8	Thickness	millimeters (mm)

The quality of the printed part is defined through a dimensional approach.^{43, 44} The expected values of these variables depend on a specific preparation time and process parameters (see Table 2). Compliance with the operating range of the material preparation time facilitates that the raw material has the appropriate characteristics for the extrusion process with the operating ranges of C1–C3.

Table 2.

Variables and operation range.

ID	Type	Operating range
C1	Process	[0.3, 0.5]
C2	Process	[10, 40]
C3	Process	[13, 15]
C4	Input	[1, 2]
C5	Product	[48, 53]
C6	Product	[28, 29]
C7	Product	[22, 24]
C8	Product	[2.3, 3.2]

Correlation and adjacency matrix

In order to determine the interaction values between clay 3D printing model variables (see Table 1), data collected in 2024 from a product design research laboratory that includes this type of technology in its portfolio of services were used. The data were collected using a Moore 1 3D printing machine, and to ensure reproducibility and repeatability, a single trained operator was used in a controlled environment, which had an ambient temperature between 26°C and 28°C and humidity between 60% and 65%, following a single standardized procedure. The information collected comprises the print data for each object on the values of the input parameters, process operation, and print quality characteristics until a 325 × 8 array is obtained. Descriptive statistical analyses were performed on the collected data (see Table 3), calculating key measures of central tendency and dispersion, including mean, standard deviation, and the 25th, 50th (median), and 75th percentiles.

Table 3.

Statistical summary of variables C1–C8.

	Mean	Std. dev.	25th percentile	50th percentile	75th percentile	Confidence interval
C1	0.41	0.10	0.30	0.50	0.50	[0.11, 0.71]
C2	25.00	15.05	10.00	25.00	40.00	[0.00, 75.15]
C3	13.06	1.00	12.00	14.00	14.00	[10.60, 16.06]
C4	1.50	0.50	1.00	1.50	2.00	[0.00, 3.00]
C5	50.50	2.51	48.00	50.50	53.00	[42.97, 58.03]
C6	28.50	0.50	28.00	28.50	29.00	[27.00, 30.00]
C7	23.00	1.00	22.00	23.00	24.00	[20.00, 26.00]
C8	2.62	0.21	2.30	2.60	2.70	[1.99, 3.25]

In addition, 95% confidence intervals for the mean were estimated for each variable. These intervals were computed under the assumption that the data follow a normal distribution, employing the standard formula for a confidence interval of the mean:

I C_{95 %} = μ \pm 3 σ

(3)

where $μ$ is the mean and $σ$ is the standard deviation. This analysis offers a more robust evaluation of the precision of the mean estimates by constructing a confidence interval within which the true population mean is expected to fall with 95% probability, under the assumption of a normally distributed dataset. After this, a multiple correlation analysis was performed, considering the measurement results of the variables of each impression and part obtained, thereby yielding the adjacency matrix (see Table 4). The construction of the FCM adjacency matrix is based on experimental data from the clay 3D printing process through a learning process based on the causal relationships observed between the variables of the system.

Table 4.

Adjacency matrix of the 3D printing process.

	C1	C2	C3	C4	C5	C6	C7	C8
C1	0.00	−0.06	−0.01	−0.07	−0.07	−0.06	0.07	−0.08
C2	−0.06	0.00	0.07	0.01	0.00	0.00	−0.01	−0.10
C3	−0.01	0.07	0.00	0.00	0.00	0.00	−0.06	0.07
C4	−0.07	0.01	0.00	0.00	−0.01	0.00	0.01	0.07
C5	−0.07	0.00	0.00	−0.01	0.00	0.01	0.01	0.04
C6	−0.06	0.00	0.00	0.00	0.01	0.00	−0.01	−0.26
C7	0.07	−0.01	−0.06	0.01	0.01	−0.01	0.00	−0.20
C8	−0.08	0.00	0.00	0.00	0.00	0.00	0.00	0.00

Elaboration of the FCM process

Figure 2 presents the constructed FCM synthesizes the system’s variables and their weighted interdependencies. The mapping structure, which is derived from the adjacency matrix (see Table 4), encodes the fuzzy relational dynamics within the model framework.

Figure 2.

FCM of the 3D printing process.

Table 5 shows the centrality values of the variables represented in the FCM in Figure 2. These values are obtained as the weighted sum of the incoming and outgoing arcs at each node. This reflects their degree of connectivity within the network, with the centrality of a variable being a key indicator of its relative influence on the dynamics of the modeled system. It provides a measure of the robustness of the FCM and facilitates the identification of the variables with the greatest impact on the response variable.⁴⁵

Table 5.

Centrality of the FCM process.

	Entries (a)	Outputs (b)	Centrality (c) = \| a + b \|	Type
C1	−0.28	−0.13	0.41	Ordinary
C2	−0.29	0.01	0.08	Ordinary
C3	0.26	0.00	0.26	Ordinary
C4	0.02	−0.01	0.01	Ordinary
C5	0.05	0.01	0.06	Ordinary
C6	−0.32	0.01	0.31	Ordinary
C7	−0.20	0.01	0.19	Ordinary
C8	0.00	−0.46	0.46	Receiver

Inference process

The inference process was developed using what-if simulations and the modified Kosko’s activation rule with own memory of Equation (1). As demonstrated in Figure 3, the variables rapidly converge to a steady state. Due to the characteristics of the modeled process, the fixed point or fixed value of the data was established as a detection criterion. This facilitated the identification of optimal parameters, with the objective of enhancing the accuracy and repeatability of manufacturing.

Figure 3.

Note: Inference rule = Modified from Kosko (mk), activation function = sigmoidal (s), λ = 3, and ε = 0.000001.

In the analysis of the FCM process, the height of the layer, the speed of printing, and the time to prepare the material are identified as the primary variables, as illustrated in Figure 2. It is imperative to acknowledge that the centrality of these variables is a static measure used to evaluate the congruence of the model with the system, based on the collected data. However, to determine the process variables that influence the quality of the product, it is essential to carry out a process of inference and convergence analysis.²⁷ This process incorporates the dynamic component of the interaction between variables, allowing a more accurate and contextualized assessment of their impact on the quality of the final product.

Convergence analysis

In Table 6 the values of the variables in the final step of the simulation are presented in a descending order, thus facilitating the identification of the variables that exert the greatest influence on the process and the quality of the manufactured components. The analysis of the results indicates that the final state vectors of the process variables converge to values that exceed the significance threshold, suggesting their collective impact on the response variable. However, the identification of three distinct ranges of influence is also possible.

Table 6.

Final sorting of the process variables.

Order	Variable	Value
1	C1	0.9939
2	C2	0.9915
3	C3	0.9839
4	C4	0.9615
5	C5	0.8515
6	C6	0.8815
7	C7	0.9711
8	C8	0.8081

In the initial range, layer height (C1) and printing speed (C2) are identified as the most influential parameters affecting the dimensional quality characteristics of the printed part, with maximum influence values of 0.9939 and 0.9915, respectively, as in the case of Alonso Madrid et al.²⁰ In real-world applications, precise adjustment of these parameters is essential to minimize structural defects and ensure the dimensional stability of printed objects. This highlights the need for careful calibration of the deposition system to avoid undesirable effects such as collapse, deformation, nongeometric forms, or discontinuities in the extrusion process.⁴³ These findings align with an existing theoretical reference that mentions the need for the exploration of parameters in 3D printing processes best suited for each application and material type to ensure the success of the AM process.¹¹

In the second range, three variables demonstrate a medium-high influence on the dimensional quality characteristics of the final product: the nozzle diameter (C3) with a maximum value of 0.9839, the material preparation time (C4) with 0.9615, and the height of the printed object (C7) with 0.9711. In practical terms, this underscores the importance of implementing calibration and material conditioning strategies that enhance process reproducibility, including the optimization of material preparation and nozzle configuration to ensure uniform and consistent extrusion. Furthermore, variables such as outer diameter (C5), inner diameter (C6), and thickness (C8) exhibit moderate influence, with maximum values of 0.8515, 0.8815, and 0.8081, respectively. Optimizing these parameters can significantly improve the dimensional and mechanical integrity of components produced by AM. In practice, this may involve refining object design and print settings to ensure consistent and robust structural properties. These findings from the application of FCM offer valuable insights for the development of standardized operational practices, contributing to greater scalability and sustainability in ceramic production using advanced AM techniques. This may further support the creation of guidelines, protocols, and standards that define optimal parameters and procedures for 3D printing with clay, thereby facilitating reproducibility and consistent quality across diverse production environments.

Conclusions

In this article, we present a novel methodology for analyzing clay 3D printing parameters using FCMs. Based on the results, it was found that the variables with the greatest impact on the final quality of the printed objects were the layer height, the printing speed, and the material preparation time. The proposed methodology constitutes a new approach to clay AM processes, allowing factors that contribute to variability to be identified and their interrelationships to be quantified.

The methodology may be further extended by incorporating additional simulation and optimization models to address specific challenges in clay-based manufacturing. This includes the integration of artificial intelligence and machine learning techniques to predict and optimize printing parameters, enhancing the efficiency and quality of the process.

As with any modeling technique, the study is limited by the availability and subjectivity of expert knowledge and the specific context in which the model was applied. Future research could aim to validate and refine the FCM approach through real-time experimentation, wider expert participation, and application to diverse AM systems.

Ultimately, this research aims to facilitate sustainable and efficient production methods within the framework of principles of the circular economy. It may also support the creation of guidelines, protocols, and standards that define optimal parameters and procedures for 3D printing with clay, thereby facilitating reproducibility and consistent quality across diverse production environments

Footnotes

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Jean P. Morán-Zabala

Jhon A. Restrepo-Henao

Alexander A. Correa-Espinal

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