K-Means Module Division Method of FDM3D Printer-Based Function–Behavior–Structure Mapping

Abstract

Product performance, function, cost, and the level of module generalization are all significantly influenced by product modular design, but different goods require different division indicators and techniques. The purpose of this study is to provide a set of appropriate modular division techniques for FDM 3D printers. This research offers an ecologically friendly module division index and uses module clustering as the module division principle in accordance with the current industrial development trend and the fundamental requirements of FDM 3D printer consumers in the current market. The K-means algorithm is used to use the Jaccard similarity coefficient as the metric of similarity of the DSM clustering process to realize the module division of the FDM 3D printer after studying the function–behavior–structure mapping model of the 3D printer. Additionally, the elbow method–cluster error variance and average contour coefficient evaluation systems were built, respectively, in order to verify the viability of the FDM 3D printer module division method and obtain the best module division results. By analyzing these two systems, it was discovered that when the FDM 3D printer was divided into three modules, the in-cluster error variance diagram obviously had an inflection point, and the average profile coefficient and other modular approaches that need to be adjusted to their respective goods can use this division method as a theoretical foundation and point of reference.

1. Introduction

Another name for 3D printing is “additive manufacturing.” The method of fusing materials together to create products based on 3D models is known as additive manufacturing, according to the American Society for Testing and Materials (ASTM). Current applications of 3D printing technology range from the creation of soft pneumatic fixtures (SPG) [1] to manned space technology [2], touching every facet of daily life for humans.

Unlike subtractive manufacturing, which layers materials one on top of the other, 3D printing layers materials on top of each other. Fused deposition manufacturing technology [3], selective laser sintering technology [4], photocuring molding technology [5], and others are the primary divisions made in accordance with the various printing techniques and materials. In one of them, the hot-melt raw materials are heated and melted before being extruded through the printing nozzle. The printing nozzle also travels simultaneously in accordance with the slice’s route data. Under the motion of the printing head, the extruded molten material is covered on the printing platform, melts, and then builds up layer by layer until it eventually forms a print [6]. As a result, the FDM3D printer has the qualities of low cost, straightforward construction, and no pollution. This 3D printer is frequently used.

Product development that is based on the structural characteristics of the product, the product family, or the product platform modular planning can be supported by the scientific approach of product modular division. The benefits of a modular design can be seen at every level of the product life cycle as opposed to traditional product architecture [7,8]. Modularizing products in a reasonable way can help with product development and improvement, as well as improve the product portfolio’s ability to adapt to different market sectors [9]. A modular architecture is considered crucial in producing sustainable products, and modularization is a recognized strategy in academia and industry [10,11]. Many electromechanical products are currently developed and designed using modular design principles and methodologies, including coffee makers [12], industrial manufacturing robots [13], crawler cranes [14], and wind turbines [15]. Full life cycle product design has traditionally been a research hotspot for modular design theory and practice, which serves as the primary support technology for designing complicated electromechanical systems [16].

Scholars from numerous nations are currently undertaking in-depth research on product modularity and provided a wide range of useful solutions. Heuristic approaches and clustering methods are two broad groups into which these techniques can be separated [17]. Heuristic approaches primarily use various intelligent optimization algorithms to find the best solution or approximation solution of a particular fitness function depending on the driving force of some modules. Simulated annealing method [18], genetic algorithm [19], group genetic algorithm [20], and strength Pareto evolutionary algorithm 2 (SPEA2) [21] are a few of these algorithms. For instance, xu et al. proposed a set of modular design and configuration design methods that clustered components into standard modules that can be changed or replaced, and combined module types according to customer requirements to create a complete system in order to address the problems of high cost and long delivery cycles faced by traditional filament winding machines. The group genetic algorithm (GGA) method is used for modular optimization in order to maximize interaction and stability between module components [22]. In order to achieve effective clustering of software modules, Bahman Arasteh et al. proposed a method of combining the grey wolf optimization algorithm with the genetic algorithm [23]. This method addresses the issues of low success rate, limited stability, and poor modular quality of traditional product modularization. According to the link between the components, the clustering method, such as fuzzy c-means [24], k-means [25], hierarchical clustering [26], etc., determines the product’s modularization scheme. Simon Li et al., for instance, presented a modular approach based on the Design Structure Matrix (DSM) clustering to create product modules that combine and isolate protection information and lower the chance of protected information leaking [27]. Based on flow analysis, design structure matrix (DSM), and fuzzy clustering, Li suggested an integrated product modularization technique that can identify the modular architecture of flexible platforms with minimal computational effort [28]. In addition to the two primary methodologies mentioned above, some researchers additionally employ atomic theory [29] and the community discovery algorithm [30] for the division of product modules. Ren et al. [31,32] developed a low-cost modular sensing device based on the principle of photoelectric sensing by utilizing the concept of modularity in order to gather comprehensive and accurate human walking gait and attitude information. R.F. [33] suggests a new modular soft actuator-based soft robot shoulder external with two degrees of freedom that can simulate humeral motion.

The aforementioned techniques each put forth their unique product modularization techniques from various perspectives and design specifications. On the utilization of material resources across the entire product design process, there is, however, comparatively little research. In addition to considering the needs of the client, product modularization should consider the effects on society, the environment, and the economy. This research will present an index for module partition based on environmental sustainability and use the design dependence matrix DSM clustering approach for FDM3 D printer module partition in an effort to address this issue.

2. FDM3D Printer Module Division Principles and Indicators

2.1. Principle of Module Division

The standard 3-dimensional Cartesian coordinate system is chosen as the coordinate system for the base of the UR10 robot, which is shown in Figure 1. The UR10 robot can be installed in upright or upside-down modes on a sliding table. The difference between the two installation modes lies in whether the robot is upright or upside down relative to the sliding table. When a larger workpiece is to be processed by the UR10 robot, the UR10 robot is installed upside down and is not in the same direction as the z-axis of the standard 3-dimensional Cartesian coordinate system.

The FDM3D printer module division uses the functional analysis approach to examine the FDM printer’s functional structure as well as the product’s signal flow, energy flow, and material flow mode. The fundamental concept of modularity must be created prior to product modularization in order to make the purpose of FDM printer modularization clear and enable the smooth advancement of the modularization process. Currently, the three main principles for the division of functional modules in mechanical products are as follows:

(1)

The principle of similarity

Different products should be analyzed and studied, similarities should be found between them, and they should be divided into basic functions or common modules.

(2)

Hierarchical compression principle

The use of large-grained product standard modules can compress product levels, simplify product structure, and shorten product manufacturing processes and cycles.

(3)

The principle of module clustering

The modular design should make it have a highly functional integration module, which is conducive to simplifying the product structure and facilitating reuse.

Since the principle of similarity is typically used for modular analysis between different products or similar products but different models, such as customized products industrial steam turbines, for different models of products, if their main parameters are close, their output section, input section, and support section have a greater degree of similarity, but the flow section is different, so the output section, input section, and support section of the parts can be combined to establish a modular system. The modular division of products with a medium to a high level of complexity typically employs the hierarchical compression approach. The principle of module clustering is used for modularization with a high level of functional integration, i.e., the correlation degree of information, function, and structure within the product module is as large as possible, and the correlation degree of information, function, and structure between different modules is as small as possible. This study adopts the notion of module clustering as the fundamental premise of module division with a focus on the structural and functional aspects of FDM3D printers.

The structure interaction principle, energy interaction principle, matter interaction principle, signal interaction principle, and force interaction principle are the five components of the modular clustering principle.

Table 1. Basic principles of module division.

First-Order Principle	Secondary Principle	Content
Module clustering principle	Structural interaction principle	The greater the interaction between two modules, the more they should be separated into the same module
	Principle of energy interaction
	Principle of material interaction
	Signal interaction principle
	Interaction principle of forces

displays the specific tenets.

2.2. Divide Indicators Based on Environmentally Sustainable Modules

The modular classification of environmental sustainability indicators, which can be used to represent the above three aspects, is proposed to consider the potential impact of modular products on three dimensions (social, environmental, and economic):

(1)

Maintainability

Parts with the same maintenance frequency, maintenance requirements, maintenance time, and maintenance complexity are divided into the same module for centralized maintenance.

(2)

Reusability

Reusable parts are placed in the same module so that some parts can be reused after the entire product is scrapped, which can reduce waste on a large scale.

(3)

Recyclability

Parts and components of compatible or homogeneous materials (e.g., toxic or hazardous) are grouped together in the same module to facilitate product disassembly, material sorting, and material recycling.

(4)

Interchangeability

Parts with the same or similar functions are divided into the same module so that when subsequent products fail again, interchangeable parts can be found in time for a replacement.

3. Module Partition Method

3.1. Customer Demand Survey and Analysis and Product Modularization Possibility Evaluation

Each client has a unique set of wants that are included in the collection of customer needs. Because it might be challenging for customers to express their needs in a professional and clear manner, it is beneficial to standardize demand information expression. The template for client requirements is shown in

Table 2. Customer requirements template.

Customer Characteristics		Description
Customer demand	Performance	Identify and complement customer needs
	Economy
	Reliability
	Convenience
	Function
	Appearance
Customer demand weight Customer evaluation of product requirements of the same type		Priority of requirements
		Satisfaction degree
		Current needs are similar to customers

.

The degree of product modularity cannot be identified in the early stages of design, so it is necessary to confirm whether the product can be entirely or partially modularized. There are six questions in

Table 3. Modularity possibility questionnaire.

Problem	Yes	No
Are the components connected closely?		√
Are the majority of the parts reversible?		√
Can a system or subsystem be modified without affecting the product’s overall functionality?	√
Are certain components reusable?	√
Can the modules be isolated from one another?	√
Can I modify the functionality of some products without disassembling them entirely?	√

. The work team will ultimately determine whether the product has a modular architecture if all the answers to the questions are broadly consistent with the table. However, these questions can be used to examine the probability of product modularity.

3.2. FBS-Based Product Overview Design

The function–behavior–structure (FBS) model is a model-based approach to product overview design that starts by understanding client wants before transforming rather ambiguous requirements into concrete structures [34]. The schematic diagram is shown in Figure 1 using a hierarchical way to describe the product according to the function.

(1)

Function, an attribute that meets customer needs, that is, functions necessary for modular products in order to meet the performance, parameters, service life, etc., required by customer needs;

(2)

Behavior, in accordance with functional requirements, through a certain technical solution or principle to achieve the corresponding function;

(3)

Structure, the structural modules required to realize various technical solutions or principles while showing the relationship between entities.

To prevent missing any relationships and creating mistakes, the FBS model is used to realize the order of the whole before the part, function before structure, and suitably split the structural modules with one or several functions.

3.3. DSM Construction Based on FBS

The Design Structure Matrix (DSM), commonly referred to as the Dependency Structure Matrix, is a straightforward method for managing and undertaking complicated system analysis. Users can model, visualize, and analyze dependencies between architectures of any designed system using DSM, which is represented as a square matrix with an equal number of rows and columns. Users can then suggest improvements to the system or suggest a synthesis of it [35].

DSM is superior to other system modeling techniques in two key ways: (1) it offers a clear and succinct manner to depict complex systems, and (2) it is suited for sophisticated analysis such as clustering (to encourage modularity) and sequencing (to reduce process costs and scheduling hazards). To split or rank each structure in the functional structure model produced by FBS into a module with a high degree of integration, DSM clustering is employed. It analyzes the correlation degree of each structure.

A simple DSM example is shown in Figure 2, A–H represents the element, and the diagonal marker symbol of 3a in the Figure indicates the presence or absence of an association; the specific rules are shown in a, b, and c in Figure 3. The number, impact, and intensity of associations, which can be valued by numerical values and colors, can be added to DSM marker symbols in a variety of ways, as shown in Figure 3.

At present, there are two types of DSM-based product modules, one of which is to consult design experts to complete the correlation analysis between product parts, which is a large workload and has great subjectivity. The other is to build DSM based on the FBS model. The associations derived from the FBS model show more indirect dependencies, and this association information is more obscure and not easy to find. The FBS model can be used to create a DSM matrix in the following ways:

(1)

The FBS model creates the weighted directed graph between components, as shown in Figure 4. A directed graph G = (V,E) representing the product’s weights has node sets V = [v1, v2,..., vn] and edge sets E = [e1, e2,..., em]. A matrix R is created by the weighted directed graph, which also describes the degree of reliance between product components.

$$\mathit{R}=\left[\begin{array}{cccc}{r}_{11}& r_{12}& \cdots & r_{1n}\\ r_{21}& r_{22}& \cdots & r_{2n}\\ ⋮& ⋮& & ⋮\\ r_{n1}& r_{n2}& \cdots & r_{nn}\end{array}\right]$$

The 5-scale weights are used to define the interdependence between the components, and

Table 4. Dependency weight table.

Scale	Weight	Meaning
Higher	0.9, 0.8	Higher dependency intensity
High	0.7, 0.6	High dependency intensity
Medium	0.5, 0.4	Medium strength of dependence
Low	0.3, 0.2	Low dependency intensity
Lower	0.1	Lower dependency intensity

displays the 5-scale weight table.

Figure 4. Schematic diagram of weight digraph.

Figure 4. Schematic diagram of weight digraph.

(2)

Build DSM C for the product.

$$\mathit{C}=\left[\begin{array}{cccc}{c}_{11}& c_{12}& \cdots & c_{1n}\\ c_{21}& c_{22}& \cdots & c_{2n}\\ ⋮& ⋮& & ⋮\\ c_{n1}& c_{n2}& \cdots & c_{nn}\end{array}\right]$$

3.4. DSM Module Division Based on K-Means Algorithm

In reality, the module division of products based on DSM clusters the product DSM to make sure that the closely related components are grouped into one module, where the correlation strength between the components within the module is high, and the correlation strength with other components in the module is low.

Because of its simplicity and effectiveness, the K-means clustering method—an iterative unsupervised learning clustering algorithm invented by MacQueen—is the one that is most frequently employed. These are the precise steps [36]:

(1)

First, determine a K value; that is, the data pair is clustered and analyzed to obtain K clusters;

(2)

K initial clustering centers are randomly selected from the data objects;

(3)

Calculate the distance between each data object and each cluster center, and assign the object to the nearest cluster center according to the minimum distance criterion;

(4)

Calculate the centroid of the new cluster center, and then reassign it to the nearest cluster center according to the minimum distance criterion;

(5)

Compare the distance between the new cluster center and the cluster center of the previous iteration, and end the cluster if the distance is less than the set threshold; Otherwise, go back to step 3 and continue iterating until the clustering end condition is met.

Each row and column element of the product DSM corresponds to the structural layer components in the product FBS model, so the relationship between them is no longer represented as a spatial distance. Using Euclidean distance as a metric, the correlation between the elements in DSM cannot be accurately measured. The K-means algorithm uses Euclidean distance as a metric of similarity of the DSM clustering process.

For asymmetric binary qualities, the Jaccard similarity coefficient is frequently employed as a metric of similarity; the higher the Jaccard coefficient, the more similar the elements are. Given two vectors A and B, respectively, different row vectors in the product DSM, the Jaccard similarity coefficient is defined as the ratio of the magnitude of the intersection of A and B to the magnitude of the union of A and B, defined as follows:

$$J(A,B)=\frac{A\cap B}{A\cup B}=\frac{{\sum }_{i=1}^N\mathrm{m}\mathrm{i}\mathrm{n} (A_i,B_i)}{{\sum }_{i=1}^N\mathrm{m}\mathrm{a}\mathrm{x} (A_i,B_i)}$$

(1)

3.5. FDM3D Printer Module Division Process

The process of product modularization mainly consists of four steps: gathering customer needs, processing and analysis in line with those needs, breaking down the product into its component parts, determining whether there is a specific relationship between those components, and finally, creating a structural module library in accordance with these associations. Figure 5 displays all of the processes in the modular design methodology for the FDM 3D printer, along with additional explanations.

4. FDM3D Printer Module Division

4.1. FDM 3D Printer Product Requirements

Utilizing network analysis, market trend analysis, consulting with relevant technical personnel, and other methods, we can gather the essential requirements of FDM 3D printer users and then screen, hone, and categorize those needs. The primary client requirements for FDM 3D printers are displayed in

Table 5. Core requirements of FDM3D printer customers.

Requirement Type	Demand Classification	Core Requirement
Performance	Accuracy	XY axis positioning accuracy is high; high printing precision
	Noise	≤50 dB
	Speed	Faster printing speed
Economy	Three types	Within 1000 RMB; 1000~2000 RMB; 2000~4000 RMB
Reliability	Security	High structural strength, printing stability
	Intelligent control	Automatic leveling; power off continued to play; blocked material and no material alarm
Convenience	Operating interface	Chinese/English; full-color HD touchscreen
Function	Printing mode	TF card; USB connection
	Support format	STL; OBJ; Gcode
	Print size	Print in a variety of sizes to suit different environments
	Support consumables	PLA; ABS
	Slicing software	Cura; Simplify3D; Repetier-Host
Appearance	Style	The appearance is diverse, beautiful, and simple

.

4.2. Establishment of FDM Printer FBS Model

The FDM 3D printer FBS model is formed by evaluating client needs. Although the construction of 3D printers appears complex, there is really a rather stable connection link between various parts. Figure 6 depicts the FBS model of the FDM3D printer.

4.3. Establishment and Clustering of DSM Model of FDM Printer

The weight of the FDM 3D printer component is established, as shown in Figure 7 (Figure 8 provides the significance of each letter in the illustration), in accordance with the interaction relationship between the FBS model and components of the FDM 3D printer, as shown in Figure 6, in accordance with the method of constructing DSM in accordance with the FBS model, and the correlation between the two components is calculated to complete the automatic creation of DSM. Figure 8 displays the DSM for the FDM 3D printer.

The K-means algorithm is used to cluster the 17 components of the FDM 3D printer DSM. Before clustering, the K values must be determined because their choice determines the number of final module divisions of the product. According to the theory of module division, the K value is either too little or too large, causing two elements with a strong correlation to be separated or two elements with a weak connection to be divided into different modules. The DSM is clustered and analyzed, and the clustering results are displayed in

Table 6. FDM3D printer module division results.

K = 3				K = 4				K = 5				K = 6
Module	Component	Number	Distance	Module	Component	Number	Distance	Module	Component	Number	Distance	Module	Component	Number	Distance
1	fuselage	A	0.695	1	fuselage	A	0.695	1	fuselage	A	0.682	1	fuselage	A	0.564
	Consumables holder	B	0.834		Consumables holder	B	0.834		Consumables holder	B	0.811		Consumables holder	B	0.651
	Power supply	J	0.613		Power supply	J	0.613		Power supply	J	0.613		Hotbed	N	0.429
	Hotbed	N	0.626		Hotbed	N	0.626		Hotbed	N	0.626	2	Power supply	J	0.050
	Display screen	O	0.556		Display screen	O	0.556		Display screen	O	0.556		Display screen	O	0.050
	Control components	P	0.519		Control components	P	0.519	2	Control components	P	0.000	3	Control components	P	0.000
2	Moving parts	C	0.588	2	Moving parts	C	0.588	3	Moving parts	C	0.588	4	Moving parts	C	0.613
	Limit switch	D	0.629		Limit switch	D	0.629		Limit switch	D	0.629		Limit switch	D	0.626
	Timing belt	E	0.504		Timing belt	E	0.411		Timing belt	E	0.411		Timing belt	E	0.556
	Coupling	F	0.676		Coupling	F	0.676		Coupling	F	0.608		Coupling	F	0.519
	Drive motor	G	0.474		Drive motor	G	0.576		Drive motor	G	0.588		Drive motor	G	0.588
	Synchronous gears	H	0.577		Synchronous gears	H	0.629		Synchronous gears	H	0.629		Synchronous gears	H	0.629
3	Fan	I	0.890	3	Fan	I	0.411	4	Fan	I	0.411	5	Fan	I	0.411
	Feeding assembly	K	0.465		Feeding assembly	K	0.576		Feeding assembly	K	0.576		Feeding assembly	K	0.576
	Heating devices	L	0.586		Sensor	L	0.559		Sensor	Q	0.559		Sensor	Q	0.559
	Nozzle	M	1.028	4	Heating devices	M	0.534	5	Heating devices	L	0.534	6	Heating devices	L	0.534
	Sensor	Q	0.836		Nozzle	Q	0.534		Nozzle	M	0.475		Nozzle	M	0.534

. Typically, the K value is mostly determined based on the number of product elements divided; thus, the ultimate decision of the K value is taken as K = 3, 4, 5, and 6, respectively.

By choosing the K value size, four distinct module division schemes of FDM3D printers are created, where modules represent FDM printers divided into several modules and which components are included in each module; this can be seen from the module division results in Table 6. Each sample’s distance from the cluster’s center is represented by the term “distance”.

4.4. Evaluation of FDM3D Printer Module Division Results

Before DSM clustering, it was unclear whether dividing the FDM3D printer into multiple modules would be appropriate, so different K values were chosen to create four different division structures for the FDM3D printer modules. Evaluation indicators of cluster analysis were introduced to assess the effectiveness of the four-module division. The specific indicators are as follows:

(1)

The sum of squares due to error (SSE)

The sum of squared errors of the distance from the samples in each K value corresponding to the cluster to the cluster center is calculated after selecting different K values, and the K value corresponding to a sudden slowdown in the rate of decline is theoretically considered to be the best K value, and the SSE formula is as follows:

$$SSE=\sum _{i=1}^k\sum _{p\in C_i}^m\left|p-m_i\right|$$

(2)

In the formula, $C_i$ represents the ith cluster,$p$ represents the object in the cluster, and $m_i$ represents the cluster center of the ith cluster.

(2)

Silhouette Coefficient (SC)

The contour coefficient is used to evaluate the clustering results and aims to compare the similarity of a sample to the cluster it is in with the similarity to other clusters. The contour coefficient ranges between (−1 and 1); the closer it is to 1, the better the clustering result, and vice versa. The contour coefficient is calculated by the following formula:

$$S\left(i\right)=\frac{b\left(i\right)-a\left(i\right)}{\mathrm{m}\mathrm{a}\mathrm{x}\left\{a\left(i\right),b\left(i\right)\right\}}$$

(3)

$$S=\frac{1}{n}{\sum }_{i=1}^{n}s\left(i\right)$$

(4)

Formula: $a_{\left(i\right)}$ represents the average distance from sample $i$ to other samples in the same cluster, and $b_{\left(i\right)}$ represents the average distance from sample $i$ to other samples of the cluster, whichever is minimum.

The SSE of DSM clustering is calculated for k = 2,3,...,10. The SSE results are shown in Figure 9, and the average contour coefficients of the four module partition results are shown in Figure 10.

According to Figure 9, there is a noticeable inflection point in the cluster error variance at k = 3 when compared to k = 4, 5, and 6. Additionally, it is evident that the average profile coefficient value at k = 3 is the highest. The results of the delineation are mainly as follows: Module 1 contains the body, consumable support, power supply, hotbed, display, and control components, mainly reflecting the FDM 3D printer in the support, power supply, and control functions; Module 2 mainly includes mechanical drive system components such as moving parts, drive motors and limit switches; Module 3 includes nozzles, feeds, fans and other components related to consumables extrusion, and the color DSM of the module delineation results is shown in Figure 11. By coloring the nodes of the FBS model, the results of the block division are shown in Figure 12.

In Figure 11, three individually colored rectangles are used to show the corresponding three modules. It can be seen that most of the large value (0.6, 0.9) associations are located within the module rectangle, and most of the small value (0, 0.3) associations are scattered outside the module rectangle. In Figure 12, the functions and behaviors associated with only one module or with several modules can be reflected, as can be seen from the fact that the ‘support’ function is implemented almost exclusively by the red module, the ‘energy’ function is implemented exclusively by the red module, and the forming function is implemented primarily by the blue and green The forming function is mainly performed by the blue and green modules.

5. Conclusions

(1)

By analyzing and comparing the three mainstream module classification principles in the market, the module clustering principle was determined as the module classification principle for FDM3D printers, and the environmental sustainability indicators for module classification were proposed, taking into account the possible impact of modular products on society, environment, and economy;

(2)

The FDM 3D printer DSM model was established through the analysis of customer needs and market research, and on the basis of the module clustering principle. The K-Means algorithm was then used to examine the product DSM clustering, in which K = 3, 4, 5, and 6 were chosen for cluster analysis, and four distinct division results were obtained;

(3)

The elbow method–intracluster error variance (SEE) and average profile coefficient (SC) were used as evaluation indicators to evaluate the division results in order to obtain the best module division results and confirm the practicability of the module division method. The results revealed that when K = 3, the image of intra-cluster error variance (SEE) obviously showed the inflection point, and the FDM3D printer was best divided into three modules, which verified the practicability.

The approach put forth in this work can serve as a guide for product modular division, and future research on the adaptability, precision, and economic assessment approaches to module combination design is possible.