3D model retrieval method of mechanical parts based on B-Rep

Abstract

This study proposes a boundary representation (B-Rep) based method for three-dimensional CAD model retrieval of mechanical parts. To reduce computational complexity and improve matching precision, auxiliary geometric features that do not influence the global structure such as chamfers, fillets, casting curvatures, tool withdrawal grooves and grinding wheel grooves are removed during preprocessing. The refined model is decomposed into individual surfaces, from which comprehensive geometric and topological attributes, including surface type, outer-loop and inner-loop features, area, perimeter distribution, concave and convex vertices, and adjacency relationships, are extracted. Surface-level correspondences are computed using a shape and adjacency matching strategy, and overall similarity is assessed by aggregating matched surfaces at the part level. Experimental evaluation on a representative dataset demonstrates that the method achieves high retrieval accuracy and robustness, while maintaining moderate preprocessing and computational requirements. These results indicate that the proposed approach effectively balances retrieval performance with overall development effort, offering a practical solution for efficient and reliable 3D CAD model management.

Introduction

Boundary representation (B-Rep), as a fundamental principle of entity modeling, has received increasing attention from domestic and foreign scholars.^1,2 B-Rep-based 3D model retrieval has emerged as a prominent research focus.^3–5 Oberbichler et al. ⁶ introduced a consistent and efficient strategy for formulating geometrically nonlinear finite elements for isogeometric analysis (IGA) and isogeometric B-Rep analysis (IBRA), utilizing the adjoint method. Boris et al. ⁷ proposed a suite of methods to generate truss lattices that fill the interior of B-Rep models, whether polygonal or (trimmed) NURBS-based, of any shape, using 3D sphere packing. Andreas et al. ⁸ suggested an isogeometric B-Rep mortar-based mapping method, which was further extended for transforming fields between a B-Rep model and a low-order discrete surface representation of the geometry. This typically occurs when the finite volume method (FVM) or the finite element method (FEM) are employed. Junhao et al. ⁹ proposed a graph structure descriptor, termed B-Rep graph, to preprocess B-Rep data and extract precise topological and geometric features from 3D-CAD models. Concurrently, an innovative and efficient neural network, FuS–GCN, was developed, based on graph convolutional networks (GCNs), to handle this graph data. Maquart et al. ¹⁰ presented a universal framework to construct 3D structured volumetric meshes of complex geometry and arbitrary topology. They utilized the boundary of the triangulated solid 3D model provided from boundary-representation in computer aided design (B-Rep CAD) models. Jia et al. ⁹ proposed a structure correspondence framework for CAD retrieval, segmenting models into local structural cells, extracting descriptors, and matching them efficiently to generate similarity rankings. Zhang et al. ¹¹ introduced a framework to reconstruct editable parametric CAD models from dumb B-Rep models, using UV-graph representation, a neural network (eCAD-Net) for sequence prediction, and feature matching to produce CAD operations. Quan et al. ¹² proposed a self-supervised contrastive GNN framework (GC-CAD) for CAD retrieval, extracting geometric and topological features from parameterized CAD files and generating retrieval-ready representations without manual labels. Yiping et al. ¹³ proposed and implemented local dynamic update methods for 3D geological bodies based on the B-Rep structure model for new local drilling and profile data. They used two local dynamic update methods. The dynamic update method of the B-Rep structure model for local new borehole data was employed to update the 3D geological body structure model to align with the new borehole constraints.

In summary, B-Rep model retrieval research primarily concentrates on local similarity matching of the model by identifying common subgraphs. Currently, descriptors extracted by global statistical features can effectively represent the feature distribution of CAD models. These descriptors usually exhibit invariance to scaling, rotation, and translation transformations and demonstrate robustness to issues such as cracks, overlaps, and noise in the model. ¹⁴ The present study extends this line of research by focusing on global similarity matching of mechanical parts based on B-Rep. Specifically, CAD models are decomposed into a series of relatively regular surfaces, from which attribute information is extracted. Optimal surface correspondences between models are then identified, ensuring consistency in both shape and topology. Finally, the overall similarity between two models is evaluated by integrating surface-level similarities.

Dividing parts into surface and extracting surface attributes

The basic idea of B-Rep denotes that an entity can be represented by a collection of surfaces, where each surface is defined by its bounding edges, each edge by its vertices, and each vertex by three-dimensional coordinates. ¹ B-Rep emphasizes the feature of the solid surface and records all the geometric and topological information of the object. In this method, the information on the surfaces, edges, and points is recorded by layers, and the relation between the layers is established.

Presently, most CAD software in the market uses B-Rep as the main technology for describing their product information. Since the surfaces of mechanical parts are typically regular, B-Rep-based retrieval can effectively enhance both the efficiency and accuracy of part retrieval.

Simplifying parts and dividing model surface

The part model must be simplified before extracting the surfaces of the mechanical part model, and certain features for processing plan, such as chamfers, fillets, curving of castings, tool withdrawal groove, and grinding wheel groove, must be removed. The existence of these features does not affect the overall shape and topology of the model but increases calculation time and influence the efficiency of retrieval. In addition, the similarity of the models depends mainly on their backbone structure. Therefore, the features of the processing plan should be filtered out before the model surface is extracted. In Figure 1(b), the chamfer, fillets, and rounded corners of Part I are removed to obtain Model I.

Figure 1.

Simplifying parts and dividing model surface: (a) Part I, (b) Model I and (c) Surface partition of Model I.

The feature modeling process of mechanical parts involves superposing and cutting basic geometries, such as prism, pyramid, cylinder, cone, and sphere. Therefore, the shape of mechanical parts is relatively regular and simple. The surface is mainly composed of planes and regular curved surfaces and has obvious boundary contour line. Based on the principle of B-Rep, dividing a CAD model of mechanical parts into multiple surfaces is straightforward. In Figure 1(b), Model I is divided into 10 surfaces, and the results are illustrated in Figure 1(c).

Extracting surface attributes

In this study, the classification and definition of surface attributes strictly follow the core principles of the boundary representation (B-Rep) model, in which a solid is composed of closed surfaces and their topological relationships.^5,10,14 Accordingly, the attributes are naturally categorized into two types: shape attributes, which characterize the intrinsic geometric properties of a surface, and adjacency attributes, which describe the topological and connectivity relationships between surfaces. This classification is not arbitrarily defined but is consistent with mainstream approaches in the field of CAD model retrieval.^5,10 When selecting specific attribute descriptors, our goal is to effectively capture the key geometric and topological features of mechanical parts, ¹⁰ while balancing descriptive comprehensiveness with computational efficiency. The final attribute set (as shown in Figure 2) encompasses critical indicators capable of distinguishing common mechanical features such as planes, cylinders, holes, and bosses, thereby ensuring the completeness of representation (Supplemental Appendix).

Figure 2.

Surface attribute classification.

All attributes are automatically extracted through a self-developed B-Rep model parsing system. This workflow ensures the consistency, objectivity, and repeatability of the attribute data. The main steps are as follows:

Model parsing and topology reconstruction: The system reads the B-Rep data of the CAD file and constructs a complete topology graph structure containing surfaces, edges, loops, and vertices. ¹⁵

Shape attribute extraction: For each surface, the system automatically calculates its geometric features.

• Type: Determined by analyzing the geometric equation of the surface (e.g. a first-degree equation indicates a plane, while second-degree equations indicate cylindrical surfaces, conical surfaces, etc.).

• Area and loop length distribution (LD): The surface area is calculated using numerical integration methods; the loop length distribution is obtained by traversing and calculating the lengths of each edge in the outer loop of the surface.

• Convexity/concavity vertex information (AD/TD): The system calculates the angle between the normal vectors of two surfaces sharing an edge at their common vertex. This angle is used to determine the convexity or concavity of the vertex, and the results are tallied.

• Number of inner loops (NI): Obtained by directly querying the topological structure of the surface for the count of its inner loops (i.e. holes).

• Adjacency attribute extraction: For each surface, the system analyzes its topological connections.

• Adjacent surfaces (AS): The system automatically finds and records all surfaces that share an edge with the current surface.

• Adjacency length distribution (AL): For each pair of adjacent surfaces, the total length of their shared edges is calculated.

According to the above rules, the surface AS of Model I in Figure 1 is listed in Table 1.

Table 1.

Attribute information of the surface shape of Model I.

Surface	T	N	A (mm2)	LD (mm)	AD/TD	NI
S 1	CYS	4	116	(23.32, 5, 23.32, 5)	(0, 4)	0
S 2	PLS	4	125	(25, 5, 25, 5)	(0, 4)	0
S 3	PLS	4	388	(35, 25, 23.32, 10)	(0, 4)	1
S 4	PLS	6	150	(20, 5, 15, 10, 5, 15)	(1, 5)	0
S 5	PLS	5	563	(35, 15, 23.32, 25, 5)	(0, 5)	1
S 6	PLS	4	100	(20, 5, 20, 5)	(0, 4)	0
S 7	PLS	4	525	(35, 15, 35, 15)	(0, 4)	0
S 8	PLS	4	175	(35, 5, 35, 5)	(0, 4)	0
S 9	PLS	4	700	(35, 20, 35, 20)	(0, 4)	0
S 10	HOS	2	157	(31.42, 31.42)	(0, 0)	0

For example, the AR of surface $S_1$ of Model I in Figure 1(c) is listed in Table 2. Its adjacent surfaces are $S_2$ , $S_3$ , $S_4$ , and $S_5$ . The AS of adjacent surface include the T, N, A, LD, AD/TD, and NI, as displayed in Table 1. $L_{S_1{S}_2}$ , $L_{S_1{S}_3}$ , $L_{S_1{S}_4}$ , and $L_{S_1{S}_5}$ represent the adjacent edges of $S_1$ and $S_2$ , $S_1$ and $S_3$ , $S_1$ and $S_4$ , $S_1$ and $S_5$ , correspondingly, and their length values are summarized in Table 2.

Table 2.

AR of surface $S_1$ .

Surface	AS	Shape attributes of AS						AL distribution
		T	N	A (mm2)	LD (mm)	AD/TD	NI
$S_1$	$S_2$	PLS	4	125	(25, 5, 25, 5)	(0, 4)	0	$L_{S_1{S}_2}=5$
	$S_3$	PLS	5	388	(35, 25, 23.32, 10)	(0, 4)	1	$L_{S_1{S}_3}=23.32$
	$S_4$	PLS	6	150	(20, 5, 15, 10, 5, 15)	(1, 5)	0	$L_{S_1{S}_4}=5$
	$S_5$	PLS	6	563	(35, 15, 23.32, 25, 5)	(0, 5)	1	$L_{S_1{S}_5}=23.32$

AL: adjacency length; AR: adjacent attributes; AS: adjacent surface.

Searching for the optimal matching surface

To enhance the clarity of presentation, the general procedure for identifying the optimal matching surface between two models is first outlined. The overall idea is to compare each surface in the query model with all candidate surfaces in the target model, evaluate their similarity using both shape attributes (AS) and adjacency attributes (AR), and then select the most similar pairs. Specifically, the algorithm proceeds in three stages:

• Attribute-level similarity calculation: For each surface, compute shape similarity (based on type, area, loop information, etc.) and adjacency similarity (based on neighboring surfaces and adjacency lengths).

• Surface-level matching: Integrate AS and AR similarity to obtain an overall surface similarity score, and identify the optimal matching surface pairs between the two models.

• Model-level evaluation: Aggregate the similarity of matched surface pairs, weighted by their areas, to derive the overall similarity between models.

This general workflow provides the conceptual framework of our approach, while the subsequent subsections present the detailed equations and application to example models.

The similarity degrees of the AS and AR of the model are computed, and the optimal matching surface of the two models is searched according to the similarity degrees of the two attributes. The surface sets of the two models are assumed as $P=\{S_i\}$ , $i=1,2...m$ and $Q=\{S_j\}$ , $j=1,2,...,n$ . The similarity degrees of the surface attributes of two models are calculated by comparing each surface of Model P with each surface of Model Q through the traversal method and finding the optimal matching surface with the highest similarity degree. The optimal matching surface among similar surfaces is searched, and if the two surfaces are of different types, then the similarity degree is not calculated.

Surface AS similarity calculation

As mentioned in Section 2.2, the surface AS include T, N, A, LD, AD/TD, and NI. The similarity calculations of these attributes are expressed in equations (1) to (5):

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{Sim}(N_{\mathrm{Pi},\mathrm{Qj}})=\{\begin{array}{cc}0\hfill & N_{\mathrm{Pi}}\ne N_{\mathrm{Qj}}\hfill \\ \multicolumn{1}{c}{1}& N_{\mathrm{Pi}}=N_{\mathrm{Qj}}\hfill \end{array}}\end{array}$$ (1)

where $\mathrm{Sim}(N_{\mathrm{Pi},\mathrm{Qj}})$ is the similarity of the N between surface $S_i$ of Model P and surface $S_j$ of Model Q, $N_{\mathrm{Pi}}$ is the number of edges of the outer loop of surface $S_i$ of Model P, and $N_{\mathrm{Qj}}$ is the number of edges of the outer loop of surface $S_j$ of Model Q. The number of edges has a significant effect on the shape similarity. If the N of the two surfaces is equal, then the similarity is 1; otherwise, 0.

$$\mathrm{Sim}(A_{\mathrm{Pi},\mathrm{Qj}})=\frac{min(A_{\mathrm{Pi}},A_{\mathrm{Qj}})}{max(A_{\mathrm{Pi}},A_{\mathrm{Qj}})}$$ (2)

where $\mathrm{Sim}(A_{\mathrm{Pi},\mathrm{Qj}})$ is the similarity of the A between surface $S_i$ of Model P and surface $S_j$ of Model Q; $A_{\mathrm{Pi}}$ is the A of surface $S_i$ of Model P, and $A_{\mathrm{Qj}}$ is the A of surface $S_j$ of Model Q. The A reflects the magnitude of the quantity; therefore, the size of the area has a minimal effect on shape similarity.

The LD of surface $S_i$ of Model P is set as $L{D}_{\mathrm{Pi}}(x_1,x_2,\cdots ,x_m)$ . The LD of surface $S_j$ of Model Q is $L{D}_{\mathrm{Qj}}(x_1\prime ,x_2\prime ,\cdots ,x_n\prime )$ ; if $m\ne n$ , then adding zero enables the number of elements of the two arrays equal. If the shapes of the two surfaces are similar, then the distributions of the length of the edges are also similar. The cosine of the vectors of two arrays is large when the two arrays are similar. Thus, the similarity degree of LD is evaluated using cosine similarity, ¹⁶ as defined in equation (3). This equation quantitatively reflects the correlation between LD feature vectors and serves as the basis for similarity assessment in subsequent analysis.

$$\mathrm{Sim}\left(L{D}_{\mathrm{Pi},\mathrm{Qj}}\right)=\frac{\sum _{i=1}^{max(m,n)}{x}_i{x\prime }_i}{\sqrt{\sum _{i=1}^m{x_i}^2}\sqrt{\sum _{i=1}^n{{x\prime }_i}^2}}$$ (3)

Figure 3(a) and (b) illustrate the simplified Model II and its surface partition. The surface AS information of Model II is presented in Table 3.

Figure 3.

Model II and its surface partition: (a) Model II, (b) Surface partition of Model II.

Table 3.

Attribute information of the surface shape of Model II.

Surface	T	N	A (mm2)	LD (mm)	AD/TD	NI
$S_1\prime$	CYS	4	220	(31.42, 7, 31.42, 7)	(0, 4)	0
$S_2\prime$	PLS	4	278.5	(31.42, 10, 20, 10)	(0, 4)	1
$S_4\prime$	PLS	6	170	(20, 15, 7, 10, 13, 5)	(1, 5)	0
$S_4\prime$	PLS	8	528.5	(31.42, 15, 5, 5, 30, 5, 5, 15)	(2, 6)	1
$S_5\prime$	PLS	6	170	(20, 5, 13, 10, 7, 15)	(1, 5)	0
$S_6\prime$	PLS	4	260	(20, 13, 20, 13)	(0, 4)	0
$S_7\prime$	PLS	8	250	(30, 5, 5, 5, 20, 5, 5, 5)	(2, 6)	0
$S_8\prime$	PLS	4	100	(20, 5, 20, 5)	(0, 4)	0
$S_9\prime$	PLS	4	100	(20, 5, 20, 5)	(0, 4)	0
$S_{10}\prime$	PLS	4	600	(30, 20, 30, 20)	(0, 4)	0
$S_{11}\prime$	PLS	4	100	(20, 5, 20, 5)	(0, 4)	0
$S_{12}\prime$	PLS	4	100	(20, 5, 20, 5)	(0, 4)	0
$S_{13}\prime$	HOS	2	220	(31.42, 31.42)	(0, 0)	0

In Table 1, the LD of surface $S_2$ of Model I is $L{D}_{S2}(25,5,25,5)$ , and the LD of surface $S_4$ is $L{D}_{S4}(20,5,15,10,5,15)$ . In Table 3, the LD of surface $S_3\prime$ of Model II is ${\mathrm{LD}}_{S3}\prime (20,15,7,10,13,5)$ , and the LD of surface $S_5\prime$ is ${\mathrm{LD}}_{S_5}\prime (20,5,13,10,7,15)$ . In equation (3), the similarity degree of LD between surface $S_2$ of Model I and surface $S_3\prime$ of Model II is:

$$\begin{array}{c}\mathrm{Sim}\left(L{D}_{S_2,S_3\prime }\right)\hfill \\ =\frac{25\times 20+5\times 15+25\times 7+5\times 10+0\times 13+0\times 5}{\sqrt{{25}^2+5^2+{25}^2+5^2}\times \sqrt{{20}^2+{15}^2+7^2+{10}^2+{13}^2+5^2}}\hfill \\ =0.713\hfill \end{array}$$

Correspondingly, the similarity degree of LD between $S_2$ and $S_5\prime$ is $\mathrm{Sim}\left(L{D}_{S2,S\prime 5}\right)=0.802$ , and the similarity degree of LD between $S_4$ and $S_5\prime$ is $\mathrm{Sim}(L{D}_{S4,S\prime 5})=0.996$ . From the similarity degree of LD, the similarity is lower between surfaces $S_2$ and $S_3\prime$ than between surfaces $S_2$ and $S_5\prime$ . That is, surface $S_2$ is similar to surface $S_5\prime$ . However, the similarity is higher between surfaces $S_4$ and $S_5\prime$ than between surfaces $S_2$ and $S_5\prime$ . Therefore, surfaces $S_4$ and $S_5\prime$ are similar in terms of the LD.

As mentioned in Section before, the AD/TD of the outer-loop is a 2D array. Therefore, the cosine similarity is also used to calculate the AD/TD similarity of the outer-loop, as defined in equation (4).

$$\mathrm{Sim}(\mathrm{AD}/T{D}_{\mathrm{Pi},\mathrm{Qj}})=\frac{x_1{x}_2+y_1{y}_2}{\sqrt{x_1^2+y_1^2}\times \sqrt{x_2^2+y_2^2}}$$ (4)

where $\mathrm{Sim}(\mathrm{AD}/T{D}_{\mathrm{Pi},\mathrm{Qj}})$ is the similarity degree of the AD/TD between surface $S_i$ of Model P and surface $S_j$ of Model Q, $x_1$ is the number of concave vertexes of $S_i$ , $y_1$ is the number of convex vertexes of $S_i$ , ${\mathrm{x}}_2$ is the number of concave vertexes of $S_j$ , and ${\mathrm{y}}_2$ is the number of convex vertexes of $S_j$ .

The NI similarity degree between surface $S_i$ of Model P and surface $S_j$ of Model Q is defined in equation (5). If the NI of the two surfaces is the same, then the similarity is 1; otherwise, 0.

$$\mathrm{Sim}\left(N{I}_{\mathrm{Pi},\mathrm{Qj}}\right)=\{\begin{array}{cc}0\hfill & N{I}_{\mathrm{Pi}}\ne N{I}_{\mathrm{Qj}}\hfill \\ \multicolumn{1}{c}{1}& N{I}_{\mathrm{Pi}}=N{I}_{\mathrm{Qj}}\hfill \end{array}$$ (5)

where $\mathrm{Sim}(N{I}_{\mathrm{Pi},\mathrm{Qj}})$ is the similarity degree of the NI between surface $S_i$ of Model P and surface $S_j$ of Model Q, $N{I}_{\mathrm{Pi}}$ is the NI of $S_i$ , and $N{I}_{\mathrm{Qj}}$ is the NI of $S_j$ . The effect of the NI on similar shapes is also significant.

The shape similarity between surface $S_i$ of Model P and surface $S_j$ of Model Q is obtained after calculating equations (1) to (5), as expressed by equation (6).

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{Sim}\left(A{S}_{\mathrm{Pi},\mathrm{Qj}}\right)={\delta }_0\mathrm{Sim}\left(N_{\mathrm{Pi},\mathrm{Qj}}\right)+{\delta }_1\mathrm{Sim}\left(A_{\mathrm{Pi},\mathrm{Qj}}\right)+{\delta }_2\mathrm{Sim}\left(L{D}_{\mathrm{Pi},\mathrm{Qj}}\right)}\\ +{\delta }_3\mathrm{Sim}(\mathrm{AD}/T{D}_{\mathrm{Pi},\mathrm{Qj}})+{\delta }_4\mathrm{Sim}\left(N{I}_{\mathrm{Pi},\mathrm{Qj}}\right)\hfill \end{array}$$ (6)

Where $S_i$ is the shape similarity between surface $S_i$ of Model P and surface $S_j$ of Model Q. $\mathrm{Sim}(A_{\mathrm{Pi},\mathrm{Qj}})$ , $\mathrm{Sim}(A_{\mathrm{Pi},\mathrm{Qj}})$ , $\mathrm{Sim}(N{I}_{\mathrm{Pi},\mathrm{Qj}})$ , $\mathrm{Sim}(N{I}_{\mathrm{Pi},\mathrm{Qj}})$ , and $\mathrm{Sim}(N{I}_{\mathrm{Pi},\mathrm{Qj}})$ are presented as equations (1) to (5), correspondingly. ${\delta }_0$ , ${\delta }_1$ , ${\delta }_2$ , ${\delta }_3$ , and ${\delta }_4$ , are the weighting coefficients of N, A, LD, AD/TD, and NI, respectively; and ${\delta }_0+{\delta }_1+{\delta }_2+{\delta }_3+{\delta }_4=1$ . According to the influence of different weighting coefficients on the shape similarity of the model, Expert scoring and the analytic hierarchy process (AHP) ¹³ are employed to determine the weighting coefficients in the prototype retrieval system. ${\delta }_0$ =0.20, ${\delta }_1$ =0.10, ${\delta }_2$ =0.30, ${\delta }_3$ =0.20, and ${\delta }_4$ =0.20.

If the T of two surfaces is the same, then the similarity of their corresponding AS can be calculated using equations (1) to (6). However, if the Ts of the two surfaces are different, then the similarity must be calculated and recorded as 0. The surfaces of Model I in Figure 1 and Model II in Figure 3 are compared, and the AS similarity is defined as equation (7).

$$\begin{array}{cccccccccccccc}& S_1\prime \hfill & S_2\prime \hfill & S_3\prime \hfill & S_4\prime \hfill & S_5\prime \hfill & S_6\prime \hfill & S_7\prime \hfill & S_8\prime \hfill & S_9\prime \hfill & S_{10}\prime \hfill & S_{11}\prime \hfill & S_{12}\prime \hfill & S_{13}\prime \hfill \\ \multicolumn{1}{c}{S_1}& 0.987\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{S_2}& 0\hfill & 0.489\hfill & 0.414\hfill & 0.170\hfill & 0.441\hfill & 0.979\hfill & 0.402\hfill & 0.999\hfill & 0.999\hfill & 0.977\hfill & 0.999\hfill & 0.999\hfill & 0\hfill \\ \multicolumn{1}{c}{S_3}& 0\hfill & 0.989\hfill & 0.258\hfill & 0.435\hfill & 0.237\hfill & 0.489\hfill & 0.212\hfill & 0.776\hfill & 0.776\hfill & 0.789\hfill & 0.776\hfill & 0.776\hfill & 0\hfill \\ \multicolumn{1}{c}{S_4}& 0\hfill & 0.257\hfill & 0.950\hfill & 0.202\hfill & 0.999\hfill & 0.452\hfill & 0.431\hfill & 0.452\hfill & 0.452\hfill & 0.451\hfill & 0.452\hfill & 0.452\hfill & 0\hfill \\ \multicolumn{1}{c}{S_5}& 0\hfill & 0.488\hfill & 0.263\hfill & 0.403\hfill & 0.256\hfill & 0.292\hfill & 0.225\hfill & 0.273\hfill & 0.273\hfill & 0.292\hfill & 0.273\hfill & 0.273\hfill & 0\hfill \\ \multicolumn{1}{c}{S_6}& 0\hfill & 0.792\hfill & 0.420\hfill & 0.173\hfill & 0.443\hfill & 0.984\hfill & 0.403\hfill & 1.0\hfill & 1.0\hfill & 0.983\hfill & 1.0\hfill & 1.0\hfill & 0\hfill \\ \multicolumn{1}{c}{S_7}& 0\hfill & 0.794\hfill & 0.436\hfill & 0.178\hfill & 0.447\hfill & 0.996\hfill & 0.401\hfill & 0.996\hfill & 0.996\hfill & 0.995\hfill & 0.996\hfill & 0.996\hfill & 0\hfill \\ \multicolumn{1}{c}{S_8}& 0\hfill & 0.786\hfill & 0.406\hfill & 0.167\hfill & 0.437\hfill & 0.972\hfill & 0.401\hfill & 0.998\hfill & 0.998\hfill & 0.971\hfill & 0.998\hfill & 0.998\hfill & 0\hfill \\ \multicolumn{1}{c}{S_9}& 0\hfill & 0.785\hfill & 0.449\hfill & 0.178\hfill & 0.443\hfill & 0.968\hfill & 0.391\hfill & 0.979\hfill & 0.979\hfill & 0.998\hfill & 0.979\hfill & 0.979\hfill & 0\hfill \\ \multicolumn{1}{c}{S_{10}}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0.954\hfill \end{array}$$ (7)

Surface AR similarity calculation

As mentioned in Section 2.2, the AR of the surface includes the AS of adjacent surface and AL. $\mathrm{Sim}(A_{\mathrm{Pi},\mathrm{Qj}})$ is the AR similarity between surface $S_i$ of Model P and surface $S_j$ of Model Q. The method of calculating $Sim(A{R}_{Pi,Qj})$ is discussed as follows.

Assume that the number of adjacent surfaces of $S_i$ is c, and the set of adjacent surfaces of $S_i$ is ( $S_{i+1}$ , $S_{i+2}$ , …, $S_{i+k}$ , …, $S_{i+c}$ ). The number of adjacent surfaces of $S_j$ is d, and the set of adjacent surfaces of $S_j$ is ( $S_{j+1}$ , $S_{j+2}$ , …, $S_{j+h}$ , …, $S_{j+d}$ ). First, the min(c, d) pairs of optimal matching surfaces are found between ( $S_{i+1}$ , $S_{i+2}$ , …, $S_{i+k}$ , …, $S_{i+c}$ ) and ( $S_{j+1}$ , $S_{j+2}$ , …, $S_{j+h}$ , …, $S_{j+d}$ ) through the traversal method to calculate the AS similarity. Then, the similarity values of each pair of optimal matching surfaces are multiplied by the sum of their respective adjacent edge lengths. Finally, the sum of the multiplied result is divided by the sum of the circumference of the two surfaces $S_i$ and S _j . $\mathrm{Sim}(A{R}_{\mathrm{Pi},\mathrm{Qj}})$ is calculated as defined in equation (8).

$$\mathrm{Sim}(A{R}_{\mathrm{Pi},\mathrm{Qj}})=\frac{\sum _1^{min(c,d)}(L_{S_i,S_{i+k}}+L_{S_j,S_{j+h}})\mathrm{Sim}(A_{S_{i+k},S_{j+h}})}{\mathrm{Ci}{r}_{S_i}+\mathrm{Ci}{r}_{S_j}}$$ (8)

where $S_{i+k}$ is the adjacent surface of $S_i$ , $S_{j+h}$ is the adjacent surface of $S_j$ , k = 1…c, h = 1…d, and that $S_{i+k}$ and $S_{j+h}$ are the optimal matching surfaces for AS. $\mathrm{Sim}\left(A_{S_{i+k},S_{j+h}}\right)$ is the corresponding shape similarity. $L_{S_i,S_{i+k}}$ is the length of the adjacent edge of $S_{i+k}$ and $S_i$ , $L_{S_j,S_{j+h}}$ is the length of the adjacent edge of $S_{j+h}$ and $S_j$ , and $\mathrm{Ci}{r}_{S_i}$ and $\mathrm{Ci}{r}_{S_j}$ are the perimeters of $S_i$ and $S_j$ , respectively.

For example, the four adjacent surfaces of surface $S_2$ of Model I in Figure 1 are ( $S_2$ , $S_3$ , $S_4$ , $S_5$ ); the four adjacent surfaces of surface $S_1\prime$ of Model II in Figure 3 are ( $S_2\prime$ , $S_3\prime$ , $S_4\prime$ , $S_5\prime$ ). First, the optimal matching surfaces of the four pairs are searched by the similarity calculation of the AS of all the adjacent surfaces of surfaces $S_1$ and $S_1\prime$ . The similarity degrees of ( $S_2$ , $S_3$ , $S_4$ , $S_5$ ) and ( $S_2\prime$ , $S_3\prime$ , $S_4\prime$ , $S_5\prime$ ) from equation (7) are extracted and defined in equation (9).

$$\begin{array}{ccccc}& S_2\prime \hfill & S_3\prime \hfill & S_4\prime \hfill & S_5\prime \hfill \\ \multicolumn{1}{c}{S_2}& 0.489\hfill & 0.414\hfill & 0.170\hfill & 0.441\hfill \\ \multicolumn{1}{c}{S_2}& 0.989\hfill & 0.285\hfill & 0.435\hfill & 0.237\hfill \\ \multicolumn{1}{c}{S_2}& 0.257\hfill & 0.950\hfill & 0.202\hfill & 0.999\hfill \\ \multicolumn{1}{c}{S_2}& 0.488\hfill & 0.263\hfill & 0.403\hfill & 0.256\hfill \end{array}$$ (9)

The similarity degrees in equation (9) are arranged in several rows from high to low, and the information of each row is $S_i$ , $S_j$ , and their corresponding similarity degree. In the first row, a pair of optimal matching surfaces is found, that is, ( $S_4$ and $S_5\prime$ ), and all the rows that contain $S_4$ or $S_5\prime$ are deleted. Then, the first row of the new series is observed, a pair of optimal matching surfaces ( $S_3$ , $S_2\prime$ ) is found, and then all the rows that contain $S_3$ or $S_2\prime$ are deleted. The first row of the new series is continuously observed until the last line. The four pairs of optimal matching surfaces for AS similarity, that is, ( $S_4$ , $S_5\prime$ ), ( $S_3$ , $S_2\prime$ ), ( $S_2$ , $S_3\prime$ ), ( $S_5$ , $S_4\prime$ ) are found, and their similarity degrees are 0.999, 0.989, 0.414, and 0.403, respectively. The perimeter of surfaces $S_4$ and $S_1\prime$ and the length of each adjacent side are separately calculated. These attributes are integrated into equation (8), and the result is presented in equation (10).

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{Sim}\left(A{R}_{S_1,S_1\prime }\right)=\frac{\sum _1^4(L_{S_1,S_{1+k}}+L_{S_1\prime ,{S\prime }_{1+h}})\mathrm{Sim}(A_{S_{1+k},S_{1+h}\prime })}{\mathrm{Ci}{r}_{S_1}+\mathrm{Ci}{r}_{S_1\prime }}}\\ =\frac{(5+7)0.999+(23.32+31.42)0.989+(5+7)0.414+(23.32+31.42)0.403}{76.83+56.64}\hfill \\ =0.698\hfill \end{array}$$ (10)

According to equation (8), the AR similarity of the two surfaces is computed by traversing between Model I in Figure 1 and Model II in Figure 3, and the results are recorded in equation (11).

$$\begin{array}{cccccccccccccc}& {S\prime }_1\hfill & {S\prime }_2\hfill & {S\prime }_3\hfill & {S\prime }_4\hfill & {S\prime }_5\hfill & {S\prime }_6\hfill & {S\prime }_7\hfill & {S\prime }_8\hfill & {S\prime }_9\hfill & {S\prime }_{10}\hfill & {S\prime }_{11}\hfill & {S\prime }_{12}\hfill & {S\prime }_{13}\hfill \\ \multicolumn{1}{c}{S_1}& 0.698\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{S_2}& 0\hfill & 0.597\hfill & 0.675\hfill & 0.572\hfill & 0.675\hfill & 0.520\hfill & 0.475\hfill & 0.407\hfill & 0.660\hfill & 0.513\hfill & 0.660\hfill & 0.407\hfill & 0\hfill \\ \multicolumn{1}{c}{S_3}& 0\hfill & 0.905\hfill & 0.695\hfill & 0.632\hfill & 0.695\hfill & 0.467\hfill & 0.431\hfill & 0.537\hfill & 0.637\hfill & 0.702\hfill & 0.637\hfill & 0.537\hfill & 0\hfill \\ \multicolumn{1}{c}{S_4}& 0\hfill & 0.663\hfill & 0.785\hfill & 0.778\hfill & 0.785\hfill & 0.493\hfill & 0.701\hfill & 0.479\hfill & 0.662\hfill & 0.685\hfill & 0.662\hfill & 0.479\hfill & 0\hfill \\ \multicolumn{1}{c}{S_5}& 0\hfill & 0.860\hfill & 0.881\hfill & 0.914\hfill & 0.881\hfill & 0.415\hfill & 0.611\hfill & 0.511\hfill & 0.632\hfill & 0.449\hfill & 0.632\hfill & 0.511\hfill & 0\hfill \\ \multicolumn{1}{c}{S_6}& 0\hfill & 0.473\hfill & 0.749\hfill & 0.574\hfill & 0.749\hfill & 0.505\hfill & 0.601\hfill & 0.550\hfill & 0.771\hfill & 0.547\hfill & 0.771\hfill & 0.550\hfill & 0\hfill \\ \multicolumn{1}{c}{S_7}& 0\hfill & 0.512\hfill & 0.793\hfill & 0.668\hfill & 0.793\hfill & 0.775\hfill & 0.656\hfill & 0.678\hfill & 0.763\hfill & 0.688\hfill & 0.763\hfill & 0.678\hfill & 0\hfill \\ \multicolumn{1}{c}{S_8}& 0\hfill & 0.687\hfill & 0.754\hfill & 0.704\hfill & 0.754\hfill & 0.622\hfill & 0.936\hfill & 0.553\hfill & 0.703\hfill & 0.591\hfill & 0.703\hfill & 0.553\hfill & 0\hfill \\ \multicolumn{1}{c}{S_9}& 0\hfill & 0.483\hfill & 0.580\hfill & 0.537\hfill & 0.580\hfill & 0.549\hfill & 0.558\hfill & 0.689\hfill & 0.752\hfill & 0.680\hfill & 0.752\hfill & 0.689\hfill & 0\hfill \\ \multicolumn{1}{c}{S_{10}}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0.696\hfill \end{array}$$ (11)

Surface similarity calculation

The AS similarity between surface $S_i$ of Model P and surface $S_j$ of Model Q, that is, $\mathrm{Sim}\left(A{S}_{\mathrm{Pi},\mathrm{Qj}}\right)$ is calculated using equations (1) to (6). In addition, the AR similarity between surfaces $S_i$ and $S_j$ , that is, $\mathrm{Sim}\left(A{R}_{\mathrm{Pi},\mathrm{Qj}}\right)$ , is calculated using equation (8). Therefore, the similarity between surface $S_i$ of Model P and surface $S_j$ of Model Q, that is, $\mathrm{Sim}(P_i,Q_j)$ , is expressed in equation (12).

$$\mathrm{Sim}(P_i,Q_j)={\beta }_1\mathrm{Sim}(A{S}_{\mathrm{Pi},\mathrm{Qj}})+{\beta }_2\mathrm{Sim}(A{R}_{\mathrm{Pi},\mathrm{Qj}})$$ (12)

where ${\beta }_1$ and ${\beta }_2$ are the weighting coefficients of AS and AR, respectively, and ${\beta }_1+{\beta }_2=1$

In the prototype retrieval framework, the weighting parameters were assigned as β₁ = 0.5 and β₂ = 0.5. The similarity degrees between the paired surfaces of Model I (Figure 1) and Model II (Figure 3) were subsequently computed according to equations (7), (11), and (12), the results of which are expressed in equation (13).

$$\begin{array}{cccccccccccccc}& {S\prime }_1\hfill & {S\prime }_2\hfill & {S\prime }_3\hfill & {S\prime }_4\hfill & {S\prime }_5\hfill & {S\prime }_6\hfill & {S\prime }_7\hfill & {S\prime }_8\hfill & {S\prime }_9\hfill & {S\prime }_{10}\hfill & {S\prime }_{11}\hfill & {S\prime }_{12}\hfill & {S\prime }_{13}\hfill \\ \multicolumn{1}{c}{S_1}& 0.843\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{S_2}& 0\hfill & 0.543\hfill & 0.545\hfill & 0.371\hfill & 0.558\hfill & 0.750\hfill & 0.439\hfill & 0.703\hfill & 0.830\hfill & 0.745\hfill & 0.830\hfill & 0.703\hfill & 0\hfill \\ \multicolumn{1}{c}{S_3}& 0\hfill & 0.947\hfill & 0.477\hfill & 0.534\hfill & 0.466\hfill & 0.478\hfill & 0.322\hfill & 0.657\hfill & 0.707\hfill & 0.746\hfill & 0.707\hfill & 0.657\hfill & 0\hfill \\ \multicolumn{1}{c}{S_4}& 0\hfill & 0.460\hfill & 0.868\hfill & 0.490\hfill & 0.892\hfill & 0.473\hfill & 0.566\hfill & 0.466\hfill & 0.557\hfill & 0.568\hfill & 0.557\hfill & 0.466\hfill & 0\hfill \\ \multicolumn{1}{c}{S_5}& 0\hfill & 0.674\hfill & 0.572\hfill & 0.659\hfill & 0.569\hfill & 0.354\hfill & 0.418\hfill & 0.392\hfill & 0.453\hfill & 0.371\hfill & 0.453\hfill & 0.392\hfill & 0\hfill \\ \multicolumn{1}{c}{S_6}& 0\hfill & 0.633\hfill & 0.585\hfill & 0.374\hfill & 0.596\hfill & 0.745\hfill & 0.502\hfill & 0.775\hfill & 0.886\hfill & 0.765\hfill & 0.886\hfill & 0.775\hfill & 0\hfill \\ \multicolumn{1}{c}{S_7}& 0\hfill & 0.653\hfill & 0.615\hfill & 0.423\hfill & 0.620\hfill & 0.885\hfill & 0.529\hfill & 0.837\hfill & 0.880\hfill & 0.841\hfill & 0.880\hfill & 0.837\hfill & 0\hfill \\ \multicolumn{1}{c}{S_8}& 0\hfill & 0.737\hfill & 0.580\hfill & 0.436\hfill & 0.596\hfill & 0.810\hfill & 0.669\hfill & 0.776\hfill & 0.851\hfill & 0.781\hfill & 0.851\hfill & 0.776\hfill & 0\hfill \\ \multicolumn{1}{c}{S_9}& 0\hfill & 0.634\hfill & 0.515\hfill & 0.358\hfill & 0.512\hfill & 0.775\hfill & 0.475\hfill & 0.834\hfill & 0.866\hfill & 0.840\hfill & 0.866\hfill & 0.834\hfill & 0\hfill \\ \multicolumn{1}{c}{S_{10}}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0.825\hfill \end{array}$$ (13)

Searching for the optimal matching surfaces

Models P and Q have $m$ and $n$ surfaces, respectively. According to equation (12), The $min(m,n)$ surface similarity degrees can be obtained between any two surfaces of the two models. The $min(m,n)$ pairs of optimal matching surfaces are selected. A high similarity degree equates to a significant similarity of the two surfaces because the $min(m,n)$ surface similarity degrees are obtained, independent of the path and sequence of comparison. The method of selecting $min(m,n)$ pairs is divided into the following steps.

• Step 1: Arrange the $m\times n$ similarity degrees from high to low, and form a temporary data, as shown in Table 4.

• Step 2: Take the data from the first row in Table 4, that is, find a pair of optimal matching surfaces ( $S_{\mathrm{Pi}}$ , $S_{\mathrm{Qj}}$ ). Place them in a new table (with a same data structure as Table 4) and delete all the rows containing $S_{\mathrm{Pi}}$ and $S_{\mathrm{Qj}}$ in Table 4. That is, according to the similarity principle of “a large similarity degree equates to two objects being similar”; one surface of Model P corresponds to only one surface of Model Q, thereby indicating that no one-to-many situation occurs.

• Step 3: Repeat Step 2 until Table 4 is empty. The $min(m,n)$ pairs of optimal matching surfaces are found and recorded in the new table.

Table 4.

Temporary data table.

Surface of Model P	Surface of Model Q	Similarity degree
$S_{P1}$	$S_{Q1}$	$\mathrm{Sim}(P_1,Q_1)$
$S_{P1}$	$S_{Q2}$	$\mathrm{Sim}(P_1,Q_2)$
…	…	…
$S_{P1}$	$S_{\mathrm{Qn}}$	$\mathrm{Sim}(P_1,Q_n)$
$S_{P2}$	$S_{Q1}$	$\mathrm{Sim}(P_2,Q_1)$
…	…	…
$S_{P\mathrm{m}}$	$S_{\mathrm{Qn}}$	$\mathrm{Sim}(P_{\mathrm{m}},Q_n)$

According to the steps above and equation (13), 10 pairs of optimal matching surfaces are found between Model I in Figure 1 and Model II in Figure 3, that is,

$$\begin{array}{c}\mathrm{Sim}(S_1,S_1\prime )=0.843,\mathrm{Sim}(S_2,S_8\prime )=0.703,\mathrm{Sim}(S_3,S_2\prime )=0.947,\hfill \\ \mathrm{Sim}(S_4,S_5\prime )=0.892,\mathrm{Sim}(S_5,S_4\prime )=0.659,\mathrm{Sim}(S_6,S_9\prime )=0.886,\hfill \\ \mathrm{Sim}(S_7,S_6\prime )=0.885,\mathrm{Sim}(S_8,S_{11}\prime )=0.851,\mathrm{Sim}(S_9,S_{10}\prime )=0.886,\mathrm{and}\mathrm{Sim}(S_{10},S_{13}\prime )=0.825.\hfill \end{array}$$

Model similarity calculation

The similarity of the AS and AR of these surfaces are calculated first using the surface sets of the two models. Then, the similarity of the surfaces for finding several pairs of the optimal matching surfaces and the overall similarity of the model are calculated.

Based on the principle of B-Rep, the model is divided into several surfaces that contain all model AS and AR. The shape and topology of the two models can be matched well by comparing the similarity of the surface. The similarity of the two models is as presented in equation (14):

$$\mathrm{Sim}(P,Q)=\frac{\sum _1^{min(m,n)}\mathrm{Sim}(P_i,Q_j)\times (A_{\mathrm{Pi}}+A_{\mathrm{Qj}})}{\sum _{i=1}^m{A}_{\mathrm{Pi}}+\sum _{j=1}^n{A}_{\mathrm{Qj}}}$$ (14)

where $\mathrm{Sim}(P,Q)$ is the similarity between Models P and Q. $\mathrm{Sim}(P_i,Q_j)$ is calculated according to equation (12), which is the surface similarity of the pair of the optimal matching surfaces ( $S_{\mathrm{Pi}}$ , $S_{\mathrm{Qj}}$ ). $(A_{\mathrm{Pi}}+A_{\mathrm{Qj}})$ represents the sum of the A of the optimal matching surfaces ( $S_{\mathrm{Pi}}$ , $S_{\mathrm{Qj}}$ ). $A_{\mathrm{Pi}}(i=1,2,\cdots ,m)$ represents the A of each surface of Model P, and $A_{\mathrm{Qj}}(i=1,2,\cdots ,n)$ represents the A of each surface of Model Q.

The surface similarity degrees of the 10 pairs of optimal matching surfaces searched in Section 3.4 and area data of Models I and II in Tables 1 and 3, respectively, are calculated by using equation (14); the similarity between Model I in Figure 1 and Model II in Figure 3 is:

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{Sim}(\mathrm{ModelI},\mathrm{ModelII})=\frac{\sum _1^{10}\mathrm{Sim}(\mathrm{Model}{I}_{\mathrm{si}},\mathrm{ModelI}{I}_{\mathrm{sj}})}{\sum _{i=1}^{10}{A}_{\mathrm{ModelIi}}+\sum _{j=1}^{13}{A}_{\mathrm{ModelIIj}}}}\\ \times (A_{\mathrm{ModelIi}}+A_{\mathrm{ModelIIj}})\hfill \\ =\frac{4586.3}{6096}=0.752\hfill \end{array}$$

The similarity obtained through calculation between Models I and II is 0.752.

Case verification

To verify the feasibility of the proposed algorithm, a prototype system for mechanical part model retrieval was developed using the API of the SolidWorks software platform. The experimental dataset comprised ∼500 three-dimensional models of mechanical parts. All these models were created by the authors of this study to support teaching activities within the university, such as design cases from classic textbooks (e.g. Course Design of Mechanical Design) and typical parts from practical instructional projects (e.g. reducers and fixture designs), ensuring the dataset’s engineering practicality and pedagogical representativeness.

The dataset encompassed a variety of common mechanical part types to reflect diversity, primarily including: shaft-type parts (e.g. transmission shafts, stepped shafts), gear-type parts (e.g. spur gears, helical gears), disc-cover parts (e.g. end covers, flanges), housing-type parts (e.g. gearbox housings), as well as standard and commonly used parts (e.g. bolts, nuts, bearing blocks). The models exhibit a spectrum of geometric complexity, ranging from structurally simple fasteners to housing-type parts and special-shaped parts characterized by complex surfaces, internal cavities, and intricate hole systems.

Within this model library, similarity-based retrieval was implemented. The similarity degrees between three query parts belonging to distinct categories and all parts in the library were calculated. The five parts with the highest similarity degrees were retained and ranked in descending order, with the results summarized in Table 5.

Table 5.

Matching results of model similarity.

	Target parts	Five most similar parts in the library
No.	Part a	0256	0236	0223	0268	0279
Graphs
Similarity degree		0.837	0.791	0.753	0.687	0.642
No.	Part b	0129	0131	0121	0141	0137
Graphs
Similarity degree		0.825	0.773	0.754	0.663	0.626
No.	Part c	0038	0033	0021	0041	0037
Graphs
Similarity degree		0.746	0.681	0.660	0.583	0.546

The retrieval results show that target part a and part no. 0256 has the highest similarity. The same observation is obtained for target part b and part no. 0129 and target part c and part no. 0038. Furthermore, other similar parts retrieved from the library and the target parts belong to a certain kind of parts, and the design reuse value of the same kind of parts is high.

Experimental analysis

This study conducted a systematic and comprehensive experimental evaluation of the proposed algorithm. In terms of retrieval efficiency, based on a test library consisting of ∼500 parts, the results demonstrate that the average retrieval time of the algorithm is significantly superior to that of the comparison methods. Specifically, the proposed algorithm requires only 1.11 min on average, whereas the feature tree-based and optimal matching-based algorithms take 1.28 and 1.53 min, respectively (Table 6). This efficiency advantage primarily stems from the innovative preprocessing mechanism, which intelligently identifies and filters out nonessential geometric features such as chamfers and fillets. By preserving the integrity of the model’s primary topological structure, this mechanism effectively reduces computational complexity and enables faster retrieval.

Table 6.

Average retrieval time of similar part.

Algorithm name	Algorithm in this study	Algorithm based on extended feature tree	Algorithm based on optimal matching
Time consumed (min)	1.11	1.28	1.53

With respect to retrieval accuracy, the comparative analysis of the P–R curves (Figure 4) clearly shows that the proposed algorithm consistently maintains stable advantages across most recall levels. When the recall reaches 0.6, the precision of the proposed method remains as high as 0.48, which is significantly better than that of the two comparison methods. This accuracy advantage can be attributed to the algorithm’s multidimensional representation strategy, which not only extracts fundamental geometric attributes but also incorporates structural analysis of adjacency relationships. Combined with a global traversal matching mechanism, this strategy ensures both high retrieval accuracy and sustained recall performance.

Figure 4.

P–R curve.

More importantly, the algorithm exhibits excellent scalability when handling models of varying complexity. The preprocessing stage functions as a “complexity regulator,” ensuring that the core matching process always operates on optimized backbone features of the model. This design effectively curtails the rapid growth of computation time as model complexity increases. In contrast, the optimal matching method is more prone to combinatorial explosion due to its pairwise comparison mechanism, while the feature tree-based algorithm suffers from heavier parsing overhead when dealing with large-scale features.

In terms of robustness, the algorithm demonstrates outstanding adaptability to both scale variations and geometric noise. For uniform scaling, the normalization-based attribute comparison mechanism provides inherent scale invariance. For geometric noise, the preprocessing module filters out secondary features, thereby mitigating the adverse effects of minor design modifications and manufacturing tolerances. This capability ensures that models with identical functions but differing local details are correctly identified as highly similar, which significantly enhances the algorithm’s practicality and reliability in industrial scenarios such as variant design and cross-platform retrieval.

Overall, the experimental results demonstrate that the proposed algorithm, by organically combining model preprocessing with multidimensional feature matching, not only overcomes the computational bottlenecks faced by traditional methods when applied to complex models but also achieves sustained improvements in retrieval accuracy. The advantages in performance, scalability, and robustness make it a highly efficient, reliable, and broadly applicable solution for 3D CAD model retrieval.

Conclusion

This study proposes a retrieval method for mechanical part models based on B-Rep. By simplifying the model and filtering out non-structural features, the method reduces computational complexity while preserving essential geometric and topological information. The model is decomposed into multiple surfaces, from which comprehensive attributes are extracted to effectively represent both shape features and adjacency relationships. Compared with traditional methods, the proposed approach offers several advantages: it improves retrieval accuracy by jointly considering shape and topology attributes; it achieves high efficiency with fast computation and short processing time, thereby contributing to a reduction in the overall development effort and design workflow time; and it demonstrates robustness when handling models with relatively regular surfaces. These characteristics render the method particularly suitable for mechanical parts with simple or regular geometries and highlight its practical value for CAD model management and reuse in engineering applications. However, its applicability to complex models, especially freeform surfaces, remains limited.

Supplemental Material

sj-docx-1-ade-10.1177_16878132251409281 – Supplemental material for 3D model retrieval method of mechanical parts based on B-Rep

Supplemental material, sj-docx-1-ade-10.1177_16878132251409281 for 3D model retrieval method of mechanical parts based on B-Rep by Yulun Chi, Yijun Tian, Xinyu Cui and Wenbo Zhu in Advances in Mechanical Engineering