Feature-constrained automatic geometric deformation analysis method of bridge models toward digital twin

ABSTRACT

It is very important to construct digital twin scenes, which can accurately describe the dynamically changing geographical environment and improve the level of refined management in bridge construction. This article proposes a feature constrained automatic diagnostic analysis method for geometric deformation of bridge digital twins. The geometric deformation feature library of bridge twins was first created to accurately describe structural relationships and behavior characteristics. Secondly, line surface feature constraints were used to extract geometric deformation information from bridge digital twins. Then, a geometric deformation diagnosis algorithm was designed based on an improved Hausdorff method. Finally, a case study was conducted to implement experimental analysis. The experimental results show that the method proposed in this paper can automatically extract the geometric morphology and rapidly calculate line and surface deformations for point cloud bridge digital twins. It achieves an efficiency improvement above 90% and with millimeter-level accuracy, which effectively enhances the diagnostic analysis capabilities for geographical digital twin models.

KEYWORDS:

• Feature-constrained
• bridge digital twins
• geometric deformation
• automatic diagnostic analysis
• improved Hausdorff algorithm

1. Introduction

Throughout the entire lifecycle of a bridge, the construction phase exhibits intricate and dynamic characteristics, particularly for medium- to large-scale bridges. where project scale and complexity are elevated. Consequently, the challenges faced by construction management become notably pronounced (Li et al. 2022; Song et al. 2023). Unlike other architectural structures, bridge structures endure a higher level of complexity in terms of load-bearing capacity. Even minor deviations and slippages during the construction process can lead to adjustments in the overall structural load-bearing capacity, thereby significantly affecting its load-carrying capacity (Li et al. 2023; Zhang et al. 2023). Therefore, the implementation of deformation control during the bridge construction phase is important to prevent structural damage and uphold the construction quality of the bridge.

To address the aforementioned requirements, the application of digital twin technology provides management personnel with a more convenient tool, facilitating the efficient implementation of deformation control for bridges. Digital twinning, as an innovative technology that combines multiphysics modeling with data-driven analytics, can establish a mapping from the real world to the virtual world, accurately depicting the dynamic changes in construction scenes. Consequently, digital twinning has extensive applications in the control of bridge construction progress and quality (Chen, Videiro, and Soares 2022; Lai et al. 2020; Wu et al. 2023; Zhang et al. 2018). A bridge digital twin represents a virtual manifestation of the physical entity of a bridge and its associated environment, supporting the bidirectional interaction and dynamic updating of information between bridge entity perception data and virtual entities, but does not only represent a reflection of a physical entity based on past and current data (Botín-Sanabria et al. 2022; Hu et al. 2021; Ladj et al. 2021; Sepasgozar 2021). Integrating mechanistic models and expert knowledge enables the simulation, analysis, and prediction of changes in bridge engineering performance and behavioral processes, thereby facilitating comprehensive management applications throughout the lifecycle of bridge engineering (Guo and Fang 2023; Zhuang, Liu, and Xiong 2018). Through digital twin technology, the entire construction process information of a bridge can be integrated into a shared platform. This technology replicates the real construction process in a virtual geographic environment, allowing for early prediction and warning of potential issues during construction and the formulation of countermeasures to enhance construction management efficiency (Jiang et al. 2021; Kang, Chung, and Hong 2021; Lei et al. 2022). However, in the current state of bridge construction, bridge digital twins emphasize the visualization of construction scenes, and they have demonstrated sufficient capabilities in scene description and visualization. Nevertheless, the absence of model diagnostic functionality constrains the efficient feedback on construction quality, significantly impeding the improvement of construction management. Therefore, there is an urgent need to enhance the diagnostic capabilities of bridge twins by incorporating other deformation diagnostic technologies and methods, which will enable the application of results in practical bridge management and maintenance decision-making, supporting deformation control and construction management for bridges (Song et al. 2023; Tao et al. 2022).

Currently, commonly employed methods for deformation detection in bridge structures include traditional instrument-based measurement methods, finite element analysis, photogrammetry, and 3D laser scanning technology (Bianchi et al. 2023; Ghahremani, Enshaeian, and Rizzo 2022; Glisic and Inaudi 2012; Qin et al. 2021). Traditional instrument-based measurement methods involve the collection of deformation data using geodetic methods or sensor installations. While these methods offer high precision, it is challenging to apply them to medium- and large-scale bridge projects. Additionally, these methods only capture discrete points on the surface, making it difficult to assess the overall shape and dimensions of the entire bridge structure (Beshr and Zarzoura 2021; Jiménez-Martínez et al. 2023). Furthermore, the integration of sensor network-acquired bridge monitoring data with finite element analysis allows for the continuous autonomous updating of models, enabling autonomous monitoring and state assessment. However, finite element methods are often limited to discrete nodes and elements, making it difficult to capture the overall shape and deformation of structures. Moreover, these methods rely on accurate material parameters and boundary conditions, and uncertainties in parameter acquisition and estimation significantly affect the accuracy of analysis results (Mahmoodian et al. 2022; Yang, Xu, and Neumann 2019). Digital photogrammetry combines computer vision, digital image processing, pattern matching, and other technologies to extract geometric and physical information from images, allowing for efficient and dynamic monitoring of the instantaneous deformation of bridge structures (Deng et al. 2020; Khuc and Catbas 2017). However, image calibration and distortion correction processes introduce a substantial amount of complex geometric calculations and image processing, resulting in noticeable deficiencies in efficiency and reliability. In summary, the above-mentioned monitoring methods face challenges in balancing reliability, completeness, and processing efficiency, rendering them unable to meet the application requirements for geometric deformation diagnostic analysis in bridge digital twin systems.

On the other hand, 3D laser scanning technology, due to its noncontact nature, high precision, and ability to capture global information, enables the rapid acquisition of the three-dimensional shape and surface characteristics of an entire bridge. Without the need for calibration and distortion control, this technology provides reliable technical support for the deformation diagnostic analysis of bridge digital twins (Cha, Park, and Oh 2019; Hosamo and Hosamo 2022; Zhou et al. 2021). Several point cloud processing software tools, such as CloudCompare, 3Dreshaper, and Geomagic Studio, are available on the market for preprocessing raw data. However, the use of other methods is often needed for further deformation analysis and interpretation (Kermarrec, Paffenholz, and Alkhatib 2019). Many scholars have conducted research on the deformation diagnosis of road, tunnel, and bridge structures using point cloud data. The main idea is to perform structural deformation diagnostic analysis through multiperiod point cloud data comparison and change detection. This analysis is typically achieved through methods such as cloud-to-cloud (C2C), cloud-to-mesh (C2M), and multiscale model-to-model cloud (M3C2) comparisons, resulting in accurate structural deformation results (Barnhart and Crosby 2013; Graves, Aminfar, and Lattanzi 2022; Lague, Brodu, and Leroux 2013; Truong-Hong, Lindenbergh, and Nguyen 2022). However, many of these methods directly calculate corresponding offsets after registration and overlay, which demands high registration accuracy and complex preprocessing. Furthermore, due to difficulties in unifying the sampling density and total number of points between the reference point cloud and the target point cloud, the most reliable Hausdorff algorithm cannot establish precise correspondence between two identical positions in a two-period point cloud, resulting in deviations from the true values in displacement calculations (Graves, Aminfar, and Lattanzi 2022; Ziolkowski, Szulwic, and Miskiewicz 2018).

To address the abovementioned challenges, this study primarily focuses on the analysis of structural geometric states during the bridge construction phase, proposing an innovative approach of a feature-constrained, geometric deformation automatic diagnostic analysis method that utilizes a laser point cloud to facilitate the geometric deformation diagnosis and analysis of bridge digital twins. By analyzing the construction processes and structure of the bridge, a geometric deformation feature library is constructed. Based on a differential approach, precise geometric features of the bridge components are extracted to summarize the three-dimensional spatial pose of the entire bridge during the construction phase. This approach reduces preprocessing complexity while synchronizing point cloud density and optimizes the accuracy of point cloud comparisons. Additionally, the Hausdorff-based, cloud-to-cloud displacement estimation algorithm is optimized with an interpolation fitting method to calculate accurate corresponding point relationships and obtain more reliable displacement values, thereby enhancing displacement estimation accuracy. This method is expected to achieve the accurate summarization of the geometric shape of bridge construction, with the optimization of processing efficiency and automation of geometric deformation calculations, to enhance the accuracy of deformation descriptions, supporting effective improvement in the deformation diagnostic capabilities of bridge digital twins. Building upon this foundation, the aim is to establish a correlation between geometric state analysis results and adjustments in bridge construction, forming a feedback loop. This loop influences decision-making in the physical world of bridge construction, facilitating a bidirectional interaction between the digital model of the bridge and its real-world counterpart.

2. Methods

2.1. Overall framework

Figure 1 illustrates the overall framework of this study, which consists mainly of three parts: the construction of a geometric deformation feature library, feature extraction, and deformation diagnostic analysis. First, by analyzing the structural characteristics of a bridge, the stress distribution and the relationships among components are clarified, and the bridge construction procedures, main structures, and deformation diagnosis items are examined. Based on this analysis, the geometric deformation information feature library of the bridge digital twin is constructed. Second, fully considering the geometric characteristics of bridge components (Truong-Hong and Lindenbergh 2022; Yan and Hajjar 2021), precise exploration of geometric properties that can encapsulate the bridge’s structural morphology is conducted, enabling the extraction of geometric features of bridge components. Last, the extracted feature point cloud that summarizes the bridge’s geometric shape is overlaid with the reference point cloud. Based on an improved Hausdorff algorithm, feature parameters and displacement deviations are calculated, and analysis results are generated. This study is aimed at establishing an efficient and high-precision workflow for bridge digital twin deformation calculation, enhancing the deformation diagnostic analysis capabilities of digital twin models and providing technical support for construction quality management.

Figure 1. Overall framework.

2.2. Construction of the bridge digital twin geometric deformation feature library

In bridge construction, it is often necessary to perform deformation diagnostics on its structure to ensure the safety of construction projects, especially when excessive deformations can have an impact. Moreover, as bridge construction evolves, the focus areas and monitoring tasks may change accordingly. Therefore, this study fully considers the bridge structure and construction phases, clarifying deformation monitoring tasks at different stages, thereby establishing a geometric deformation feature library for bridge digital twins. This approach enables the utilization of corresponding geometric feature extraction methods based on the specific diagnostic requirements for different components.

Figure 2 illustrates the construction method of the bridge geometric deformation feature library. Starting with multiple data sources, such as design data, monitoring data, domain-specific data, and expert knowledge, feature analysis and behavior analysis are conducted based on the bridge's design materials and construction plans. This process clarifies the geometric characteristics of the construction objects and their sequencing in the construction process. From this analysis, key deformation information related to the bridge is extracted. The bridge geometric deformation feature library is then constructed based on the fundamental concepts of bridge main components, deformation diagnostic knowledge, and construction procedures.

Figure 2. Bridge geometric deformation feature library construction method.

Using suspension bridges as an example, the typical construction sequence includes foundation construction, anchorage and tower construction, cable system installation, and bridge deck construction. Based on this finding, the main structure and deformation monitoring tasks are decomposed. The construction process determines different monitoring intervals for various bridge components. Therefore, it is necessary to determine the diagnostic objects based on the current construction progress, to extract deformation diagnostic knowledge, and to instantiate it into the deformation feature library, which includes storing model class data for the bridge components’ shapes and information class data parameterized for storage. Figure 3 illustrates the construction process of the geometric deformation feature library tailored for suspension bridge construction scenes. The standard component models within it will serve as a reference segmentation template for the subsequent segmentation of the bridge model. Deformation diagnosis and analysis is performed on individual components. Depending on the diagnostic requirements for deformation tasks, corresponding geometric feature extraction methods are invoked to generate diagnostic reports and provide construction quality assessments. The diagnostic results will be fed back to the digital twin model, assisting construction personnel in better understanding and managing the real situation, thus providing data support for decision-making.

Figure 3. Construction of geometric deformation feature library for suspension bridge construction scene.

2.3. Feature-constrained bridge digital twin geometric deformation information extraction

Figure 4 provides an overview of the process for extracting geometric features of bridges. To enhance the efficiency of feature extraction for components, first, it is necessary to ensure that the point cloud data do not have significant gaps after preprocessing. Second, based on the deformation feature library, standard models of bridge components are invoked, and the point cloud data are superimposed to obtain the intersection, facilitating the segmentation process and obtaining a clean point cloud of components. Third, selective feature line or surface extraction methods are employed based on the morphological characteristics of the components. Last, point cloud axes or surfaces that comprehensively represent the geometric morphology of the components are obtained, enabling step-by-step extraction of bridge geometric features and deformation analysis from the component level onward.

(1)	Line feature extraction

Figure 4. Feature-constrained bridge geometry information extraction process.

For the extraction of feature lines, the main approach involves obtaining a series of cross-sectional contour lines that are orthogonal to the point cloud axis line and then calculating the centroid of each unit profile and fitting it as a line, as illustrated in Figure 5. Extracting the axis lines of components is intended to describe linear features such as the tower body, cables, stiffening beams, and bridge deck. Furthermore, this approach facilitates the calculation of inclinations, deflections, local deformations, and the generation of diagnostic reports.

Figure 5. Schematic diagram of line feature extraction method based on segment division.

First, the process is divided into microsegments based on the axis line. Determining the division direction and step size, equidistant planes $N_i$ that are orthogonal to the axis line are constructed. For each microsegment, with the unit tangent direction to the loading route denoted as $\overrightarrow{m}=(A,B,C)$, the equation of the current plane is derived. However, for each division unit, since the number of point cloud points intersecting the plane is limited, directly identifying the intersection points may result in a sparse point set, which makes it challenging to represent the actual contour information and may further hinder centroid calculation. Therefore, it is necessary to expand the point cloud volume within the plane. Using the plane as a reference, it is extended forward and backward by the distance ‘d’ to create a slice. All inner points of the slice are orthogonally projected onto the plane. The true coordinates of any projected point $P_{\mathrm{op}}$ are calculated using Equation (1(1) $\{\begin{array}{c}{x}_{\mathrm{op}}={\displaystyle \frac{(B^2+C^2)x_m-A(B{y}_m+C{z}_m+D)}{A^2+B^2+C^2}}\\ y_{\mathrm{op}}={\displaystyle \frac{(A^2+C^2)y_m-B(A{x}_m+C{z}_m+D)}{A^2+B^2+C^2}}\\ z_{\mathrm{op}}={\displaystyle \frac{(A^2+B^2)z_m-C(A{x}_m+B{y}_m+D)}{A^2+B^2+C^2}}\end{array}$(1) ), thereby obtaining a series of point cloud cross-sectional contour lines. (1) $\{\begin{array}{c}{x}_{\mathrm{op}}={\displaystyle \frac{(B^2+C^2)x_m-A(B{y}_m+C{z}_m+D)}{A^2+B^2+C^2}}\\ y_{\mathrm{op}}={\displaystyle \frac{(A^2+C^2)y_m-B(A{x}_m+C{z}_m+D)}{A^2+B^2+C^2}}\\ z_{\mathrm{op}}={\displaystyle \frac{(A^2+B^2)z_m-C(A{x}_m+B{y}_m+D)}{A^2+B^2+C^2}}\end{array}$(1) Here, $P_m(x_m,y_m,z_m)$ represents any point within the slice concerning the plane $N_i$, and $P_{\mathrm{op}}(x_{\mathrm{op}},y_{\mathrm{op}},z_{\mathrm{op}})$ represents the coordinates of the projected point on the plane for points within the slice.

The obtained data points for the component's radial outer contour are relatively large and contain many duplicate points. Andrew's algorithm is employed to calculate the convex hull, reducing it to a series of characteristic points that form the outer contour of the cross-sectional projection. Second, the cross-sectional contour is triangulated, and the centroid of the convex polygon is calculated. Since the plane in which the polygon resides is a three-dimensional space plane at this point, it is not conducive to centroid calculation. Therefore, this plane needs to be rotated to be parallel to the xoy plane and transformed into a two-dimensional problem for calculation. Delaunay triangulation is applied to the rotated polygon, establishing coordinate indices for each vertex, calculating the area of each triangular unit, and constructing a vertex index catalog. Consequently, the centroid $(x_c,y_c)$ of the cross-sectional polygon is determined based on the evenly distributed weight of the triangle areas, as shown in Equation (2(2) $\{\begin{array}{c}{x}_c={\displaystyle \frac{{\sum }_{i=1}^{n-2}(s_i\ast {\overline{x}}_i)}{{\sum }_{i=1}^{n-2}{s}_i}}\\ y_c={\displaystyle \frac{{\sum }_{i=1}^{n-2}(s_i\ast {\overline{y}}_i)}{{\sum }_{i=1}^{n-2}{s}_i}}\end{array}$(2) ). Last, an inverse transformation is applied to the two-dimensional plane points to obtain the true three-dimensional coordinates of the cross-sectional centroid. (2) $\{\begin{array}{c}{x}_c={\displaystyle \frac{{\sum }_{i=1}^{n-2}(s_i\ast {\overline{x}}_i)}{{\sum }_{i=1}^{n-2}{s}_i}}\\ y_c={\displaystyle \frac{{\sum }_{i=1}^{n-2}(s_i\ast {\overline{y}}_i)}{{\sum }_{i=1}^{n-2}{s}_i}}\end{array}$(2) Here, ${\overline{x}}_i$ and ${\overline{y}}_i$ represent the centroid coordinates of any triangle with index i, where ${\overline{x}}_i=(x_{i1}+x_{i2}+x_{i3})/3$, and ${\overline{y}}_i=(y_{i1}+y_{i2}+y_{i3})/3$. $s_i$ represents the area of this triangle, calculated as $s_i=(x_{i1}{y}_{i2}+x_{i2}{y}_{i3}+x_{i3}{y}_{i1}-x_{i1}{y}_{i3}-x_{i2}{y}_{i1}-x_{i3}{y}_{i2})/2$

By iterating through all the microsegments to obtain centroids, the obtained cross-sectional centroid representation consists of discrete points that are oriented in the same direction as the initial microsegment partition and have equal spacing. These points can be fitted to obtain the target axis line that reflects the actual line shape.

(2)	Surface feature extraction

For the extraction of feature surfaces based on virtual grids, the main approach is to project the components in any coordinate plane direction, perform virtual grid segmentation, group point cloud data, and extract surface points based on the nearest neighbor point concept. Continuous surfaces and their expressions are obtained by interpolation or fitting methods, as illustrated in Figure 6. This step allows for the extraction of surface features of components such as piers and bridge decks, enabling the calculation of deformation feature values including flatness, local displacement, and the output of diagnostic results in subsequent steps.

Figure 6. Illustration of surface feature extraction method based on virtual grid.

First, for the target point cloud, an orthographic projection onto a coordinate plane is performed. This step involves projecting along parallel lines perpendicular to the coordinate plane, resulting in a 2D point set, denoted as A. Based on the projection results, a virtual grid is established. The coordinates of the two-dimensional point set determine the grid's range, and the grid division unit is determined based on the precision requirements. Ultimately, the grid is divided into an $m\times n$ grid of small cells, and the theoretical center point coordinates $C_{\mathrm{ij}}(i=0,1,\dots ,m-1j=0,1,\dots ,n-1)$ for each small cell are calculated. The 2D point set A is then grouped into various small grids according to the grid boundaries, which are expressed as the array $Z{A}_{m\times n}$, where the 2D point set located in the $C_{\mathrm{ij}}$ small grid should have the relationship $Z{A}_{\mathrm{ij}}=\{(x,y)|(x,y)\mathrm{ϵA}\}$. Next, we iterate through all grid units and search for the nearest point to the theoretical center point in each small grid, which can be approximately estimated as the actual center point, forming the point set $R_{\mathrm{ij}}$. At this point, the obtained point sets are in 2D, so the z-coordinate values of the actual center points of the grid need to be restored based on the indices. This step produces a uniformly distributed point set that can capture the regional characteristics. At this stage, continuous surfaces can be fitted using interpolation methods to replicate the surface morphology of bridge components. Components with planar surface features can be approximated as planes and expressed through equations.

2.4. Geometric deformation evaluation of bridge digital twins based on improved Hausdorff algorithm

Structural deformations will be determined by comparing the geometric features of the same components. Due to some uncontrollable implicit data operations in methods that calculate point clouds to the best-fitting plane and mesh surface, this study chooses to calculate deformations by directly comparing the distances between two points within the point cloud. The most commonly employed method for estimating point cloud distances is a cloud-to-cloud calculation based on the directed Hausdorff distance, which calculates the Euclidean distance ‘d’ between each point ‘p’ in the point cloud ‘P’ and its nearest neighbor point in the reference point cloud ‘R’ as a local distance. This method transforms the problem of cloud-to-cloud distance into point-to-point distance calculations (Chithra and Tamilmathi 2020), as shown in Equation (3(3) $d(p,R)=\underset{r\in R}{min}\Vert p-r{\Vert }_2$(3) ). (3) $d(p,R)=\underset{r\in R}{min}\Vert p-r{\Vert }_2$(3) This point cloud distance algorithm is an approximate estimation method that does not consider the point cloud surface but only searches for points. Therefore, this method has the advantage of being fast and direct. However, the method requires calculating the distance from the point of interest to all points in its neighborhood; thus, the setting of the neighborhood size and the density of the reference point cloud can significantly affect the calculation speed. Additionally, as shown in Figure 7, the accuracy of this algorithm is greatly influenced by the point cloud density. In cases where the densities of the two point clouds are different, the distance measurement will inevitably be affected. Especially when the density of the reference point cloud is lower than that of the comparison point cloud, the points searched may be random and may not accurately represent corresponding points in the comparison point cloud. Even if the density of the reference point cloud is higher than that of the comparison point cloud, excessive roughness can interfere with the search results. Points at ideal positions may be falsely detected as nearest neighbors due to local reverse offsets, leading to diagnostic errors.

Figure 7. Cloud-to-cloud estimation algorithm based on Hausdorff.

To address these issues with the point cloud-to-cloud displacement estimation algorithm, the geometric feature extraction methods mentioned above have synchronized sampling frequencies to ensure that both sets of feature point clouds have a similar point cloud density, which has initially improved the diagnostic accuracy. However, as shown in Figure 8, it is evident that the two-phase point clouds obtained after feature extraction still exhibit nonuniform distributions and differences in roughness, leading to the misalignment of corresponding point pairs. To overcome this challenge, this study introduces an interpolation-based approach to estimate the actual positions of point pairs with higher precision, facilitating a more comprehensive geometric deformation assessment of bridge twins using multiple evaluation parameters.

Figure 8. Illustration of Interpolation-Optimized Hausdorff Algorithm.

3. Experiment and analysis

3.1. Case description

This study selects a large suspension bridge located in the mountainous region of western China as the case study for bridge geometric deformation analysis. The bridge has a span exceeding 1000 meters and is situated in a complex geological and topographical area that is located in a high seismic intensity zone with a wide distribution of adverse geological conditions and poor stability. The bridge spans a V-shaped canyon with steep slopes and deep terrain, making it a challenging construction site. Furthermore, the region experiences a high-altitude mountainous climate with significant temperature variations between day and night, strong and continuous winds throughout the year, and complex meteorological conditions. These complex external factors lead to dynamic changes in the bridge construction process, making it difficult to predict scene behaviors and trends. This bridge project is considered a critical control point in the entire construction route. Therefore, there is an urgent need for quality control and deformation monitoring during the construction process of this bridge. The bridge is chosen as the case study area in this study due to its representativeness, typical characteristics, and significance. Figure 9 depicts the design model of the completed suspension bridge and its 3D environment.

Figure 9. Experimental case study.

3.2. Experimental environment construction

To test the proposed bridge geometric deformation automatic diagnostic analysis method, experiments on bridge geometric feature extraction and deformation evaluation were conducted using the integrated development environment (IDE) VS2019. The environmental configuration information involved in these experiments is listed in Table 1.

Table 1. Development environment configuration.

	Contents	Detailed information
Hardware	CPU	Intel(R) Core(TM) i5-10210U CPU @1.60GHz 2.11 GHz
	GPU	Intel(R) UHD Graphics
	Memory	16GB
Software	IDE	Visual Studio 2019
	System	Windows 10
	Algorithm library	PCL 1.12.1

Through on-site inspections and investigations of the construction site, as shown in Figure 10, it was determined that the bridge is currently in the tower construction phase and has not yet had the cable system installed. The main diagnostic object has been identified as the tower. Additionally, various text and data materials were collected to build a feature library that stores key deformation information, enhancing the algorithm's understanding and processing capabilities of the data. Figure 11 shows the results of feature library construction and is primarily divided into three levels. The data layer stores model data, geometric information, spatial information, and other relevant data about the diagnostic object. The method layer provides interfaces for deformation evaluation methods based on geometric features, allowing for direct calls according to specific feature requirements. The output layer offers parameterized result reports for users to visually assess and showcase the diagnostic results.

Figure 10. Diagnostic object.

Figure 11. Feature library construction result.

Regarding the output level, the diagnostic report provides primarily calculation methods for deformation evaluation metrics, including local displacement values, fitting equations, deflection, inclination, smoothness, and other deformation assessment indicators. For sets of linear feature points, an approximate spatial linear equation can be directly obtained through fitting. The angle between two lines, denoted as $\phi$, can be calculated. Assuming that the expression of any fitted line's equation is ${\displaystyle \frac{x-x_0}{m}={\displaystyle \frac{y-y_0}{n}={\displaystyle \frac{z-z_0}{p}=t}}}$, then its direction vector is represented as $\left(\begin{array}{c}m\\ n\\ p\end{array}\right)$. The formula for calculating the angle $\phi$ between two lines is shown in Equation (4(4) $\phi =\arccos \left({\displaystyle \frac{|m_1{m}_2+n_1{n}_2+p_1{p}_2|}{\sqrt{m_1^2+n_1^2+p_1^2}\sqrt{m_2^2+n_2^2+p_2^2}}}\right)$(4) ). The deflection of an object is defined as the transverse displacement of the centroid $(x_c,y_c,z_c)$ along a line perpendicular to the axis when the object undergoes bending deformation. The calculation formula for deflection $\gamma$ per unit of line feature is given in Equation (5(5) $\gamma =\sqrt{{(x_c-\mathrm{mt}-x_0)}^2+{(y_c-\mathrm{nt}-y_0)}^2+{(z_c-\mathrm{pt}-z_0)}^2}$(5) ). (4) $\phi =\arccos \left({\displaystyle \frac{|m_1{m}_2+n_1{n}_2+p_1{p}_2|}{\sqrt{m_1^2+n_1^2+p_1^2}\sqrt{m_2^2+n_2^2+p_2^2}}}\right)$(4) (5) $\gamma =\sqrt{{(x_c-\mathrm{mt}-x_0)}^2+{(y_c-\mathrm{nt}-y_0)}^2+{(z_c-\mathrm{pt}-z_0)}^2}$(5) For planar features, the unit average displacement value and overall displacement standard deviation can be calculated based on the local displacement values of each point to assess the degree of displacement. Additionally, after fitting the points of the planar feature into a spatial plane equation, the distance from each point to the spatial plane is calculated to obtain the distribution of plane deviation values. This information is then used to calculate the standard deviation of planarity, which is denoted as ${\epsilon }_i$, as shown in Equation (6(6) ${\epsilon }_i=\sqrt{{\displaystyle \frac{\left(\sum d_i^2-{\left(\sum d_i\right)}^2\right)/N_{\mathrm{all}}}{N_{\mathrm{all}}-1}}}$(6) ). Here, $d_i$ represents the distance from each point to the spatial plane, and $N_{\mathrm{all}}$ denotes the total number of points. (6) ${\epsilon }_i=\sqrt{{\displaystyle \frac{\left(\sum d_i^2-{\left(\sum d_i\right)}^2\right)/N_{\mathrm{all}}}{N_{\mathrm{all}}-1}}}$(6)

3.3. Experimental results and analysis

3.3.1. Engineering bridge geometric deformation evaluation experiment and analysis

The geometric features of the tower component consist mainly of the linear tower pillar and the cylindrical surface. For the tower pillar, a feature line extraction method is applied with a sampling precision of 1 meter, which is at the millimeter level. The slice thickness for projecting points is set to 30% of the sampling interval to increase the number of cross-sectional samples while reducing data interference from points with weaker correlations. This process yields a series of cross-sectional centroids, as shown in Figure 12. The extracted axis lines effectively reflect the linear trend of the tower body. The noticeable deviations in centroids are primarily caused by unevenness on the point cloud surface due to external construction equipment, which slightly affects the calculation results.

Figure 12. Axis extraction results.

Based on the results of axis line extraction, analysis and assessment of construction deviations were conducted. The results were compared with the design parameters, and one of the sections was selected for specific numerical calculations, as shown in Table 2. To better reflect local deformations, the table compares the actual axis line's displacements in the x and y directions relative to the design axis line at the same z-coordinate. Both sets of centroids were fitted as spatial lines. The equation for the design axis line is represented as $Z=-35.197\mathrm{X}-96.642\mathrm{Y}+9254.702$, while the equation for the actual axis line is $Z=-35.532\mathrm{X}-49.013\mathrm{Y}+5451.338$. The angle between the two lines is 0.278 degrees, indicating minimal tilting deformation in the tower body. Furthermore, the table lists the deflection magnitude γ per unit of the axis line in actual engineering. The reference values for deflection calculation are derived from preliminary data collected in a similar temperature and weather environment to eliminate random errors. This step creates an isovariable environment to simulate a zero-status bridge in an attempt to reproduce the deformation that occurred after it was subjected to forces.

Table 2. Analysis results of axis deformation.

	Actual axis line		Design axis line			Deviation		Deflection
Z	Xa (m)	Ya (m)	Xb (m)	Yb (m)	ΔX (m)	ΔY (m)	Δ(m)	(m)
115	39.908	80.071	39.879	80.128	0.029	−0.057	0.064	0.105
116	39.705	80.137	39.748	80.192	−0.043	−0.055	0.070	0.056
117	39.737	80.042	39.767	80.081	−0.030	−0.038	0.049	0.015
118	39.695	80.055	39.784	80.058	−0.089	−0.002	0.089	0.018
119	39.727	80.078	39.715	80.102	0.012	−0.024	0.026	0.072
120	39.715	79.990	39.715	80.034	0.001	−0.043	0.043	0.010
121	39.753	80.077	39.757	80.077	−0.004	0.001	0.004	0.120
122	39.757	80.030	39.815	80.024	−0.058	0.006	0.058	0.101
123	39.820	79.993	39.862	80.004	−0.042	−0.011	0.043	0.125
124	39.774	79.961	39.744	79.958	0.029	0.002	0.030	0.088

In the original data, irregular concavities are observed on one side of the tower body. Surface extraction and evaluation of its smoothness were conducted. The section extraction method was invoked with a grid unit side length of 0.5 meters, resulting in a 13 × 78 grid. This grid was used to sample the surface points in the middle part of the tower body, yielding a series of surface points, as shown in Figure 13. The extracted surface points have a relatively uniform point cloud density and effectively reflect the regional characteristics of the surface.

Figure 13. Surface feature extraction results.

After surface extraction, a series of points summarizing the surface characteristics of the tower body were obtained. The points were overlaid with reference data, and point cloud comparison analysis based on the improved Hausdorff algorithm was performed. The result is the plane deviation value distribution shown in Figure 14, where the blue areas protrude and the red areas are depressions. This result indicates the presence of multiple depressions on the surface, which is consistent with the on-site situation, and the credibility of the detection results. Based on the unit local displacement values, the unit average displacement value was calculated to be 6.208 mm, and the overall displacement standard deviation was 0.021, indicating relatively uniform displacement with minimal variation. In addition, the flatness parameter was calculated to be 13.3996, attributed mainly to the presence of several depressions on the tower body surface, which had an impact on the tower body's flatness.

Figure 14. Comparison result of surface features.

3.3.2. Efficiency experiment and analysis

Algorithm efficiency, as an important reference for effectiveness assessment, is a crucial factor in determining whether a method can be effectively applied in practical production. In the experiments, the processing speed of the proposed method was recorded for different data volumes and the efficiency improvement compared to the Hausdorff algorithm. Table 3 shows the speed test of the feature line extraction method for different data scales and with various sampling intervals controlled. The extraction speed can be maintained within 5 s when the data scale is in the hundreds of thousands, and when the data scale reaches millions, the processing time remains within one minute for medium to low sampling accuracies. Figure 15 illustrates the approximate linear relationship between processing time and data scale. Figure 16 shows the relationship between sampling intervals and computation time, indicating that in most cases, the processing time remains relatively low. However, when the data scale reaches millions and the sampling interval is less than 0.4 meters, the processing time significantly increases.

Figure 15. Relationship between data scale and time consumption.

Figure 16. Relationship between sampling interval and time consumption.

Table 3. Feature line extraction speed test results.

Data scale(points)	Sampling interval(m)
	Time-consuming (ms)
	1	0.8	0.6	0.5	0.4	0.1
102359	405	378	616	545	624
236877	522	829	1066	1253	1484
240688	589	825	1080	1247	1473
176445	561	635	781	990	1131	4408
4671192	12531	16420	18860	22688	27631	105460
6246912	16514	19756	26586	30405	36786	137388

Table 4 shows the speed tests for the feature surface extraction method at different data scales. The experiments indicate that even when the total data size reaches one million points, fast computing speeds can still be achieved. Figure 17 demonstrates that the processing time roughly increases linearly with the growth of the data scale, but its impact on the processing efficiency is weaker than the effect of increased sampling accuracy on the computation time. Figure 18 demonstrates a significant increase in processing time when the sampling interval is less than 0.1, while in other cases, good processing efficiency is maintained.

Figure 17. Relationship between data scale and time consumption.

Figure 18. Relationship between sampling interval and time consumption.

Table 4. Feature surface extraction speed test results.

Data scale(points)	Sampling interval (m)
	Time-consuming (ms)
	1	0.8	0.5	0.3	0.1	0.05
19709	45	48	72	107
46676	96	91	133	166
33926	84		87		733	2850
944431	2580		2682		2798	4802
1170233	2511		2630		3330	5369

Table 5 shows the overall processing time statistics for the proposed method under moderate precision sampling intervals. It is evident that even when handling data volumes reaching theoretical millions, the proposed method achieves a favorable processing speed. The table also provides a comparison of the overall processing time between the proposed algorithm and the Hausdorff algorithm. The results reveal that, when the data volume is on the order of hundreds of thousands, the proposed method's overall computation time is consistently below 3 s, whereas direct comparison using the Hausdorff algorithm consumes over 30 s. As the data volume increases to millions, the proposed method's overall computation time remains below 5 min, while the Hausdorff algorithm requires more than 30 min of processing time. The proposed method significantly reduces the total processing time compared to the Hausdorff algorithm, with an efficiency improvement exceeding 90%, effectively addressing the substantial time consumption issue associated with direct comparison methods for large data volumes.

Table 5. Comparison results with the Hausdorff algorithm.

ID	Data source	Data scale (points)	Hausdorff (s)	Proposed method(s)				Efficiency Improvement
				Sampling interval (m)	Step1	Step2	Total time
1a	Design data	102359	30.5491	0.5	0.545	0.075	1.873	99.38%
	Target data	236877			1.253
1b	Historical data	240688	47.693		1.247	0.051	2.551	94.65%
2a	Design data	176445	>2700	0.1	4.408	0.196	110.064	>95.92%
	Target data	4671192			105.46
2b	Historical data	6246912	>2700		137.388	0.225	243.103	>91.00%
3a	Design data	19709	46.597	0.5	0.072	0.367	0.57	98.78%
	Target data	46374			0.131
3b	Historical data	46676	49.524		0.133	0.365	0.629	92.67%
4a	Design data	33926	>1800	0.1	0.733	119.129	122.66	>93.19%
	Target data	944431			2.798
4b	Historical data	1170233	>1800		3.33	116.964	123.098	>93.11%

To further elucidate the superiority of our proposed method, a comparative analysis of processing efficiency is conducted against other commonly used point cloud methods, namely C2M and C3M2, as depicted in Table 6. It is crucial to note that, unlike the Hausdorff method, the computational efficiency of the C2M algorithm is significantly influenced by the data volume of the reference data, as observed in experiments 2a and 2b, 4a and 4b. This characteristic renders the algorithm less robust. The C3M2 algorithm involves the selection of core points, which is not the primary focus of this paper. For the purpose of speed testing, 0.1% of the total data volume was randomly chosen to simulate core points, and it is evident that when the number of core points approaches 1000, the algorithm's processing time becomes substantial, as illustrated in experiment 4a. It is apparent that this algorithm exhibits considerable complexity and time consumption in distance calculations for each core point, demonstrating favorable computational efficiency only when core point selection is below 100, which is evidently impractical in real-world applications.

Table 6. Efficiency comparative analysis with alternative algorithms.

	Consuming time (s)
ID	Hausdorff	C2M	C3M2	Our method
1a	30.549	23.950	96.630	1.873
1b	47.693	77.567	151.999	2.551
2a	>2700	245.895	>2700	110.064
2b	>2700	>1800	>2700	243.103
3a	46.597	9.105	3.523	0.57
3b	49.524	15.967	4.774	0.629
4a	>1800	169.582	1540.221	122.66
4b	>1800	>1800	>2700	123.098

The experimental results demonstrate that our proposed method achieves a diagnostic efficiency improvement of over 90% compared to the Hausdorff method. Additionally, when compared to other point cloud comparison methods such as C2M and C3M2 algorithms, our method ensures efficient processing across different data scales, exhibiting favorable stability and robustness. It effectively supports the geometric deformation diagnostic analysis of bridge digital twin scenes with high temporal and spatial resolutions.

4. Conclusions

To enhance the precision management of bridge construction and improve the diagnostic analysis capability of the geospatial digital twin model, this study presents a feature-constrained geometric deformation automatic diagnostic analysis method. The method consists of three steps: 1) A bridge digital twin model geometric deformation feature library is constructed. The diagnostic object is identified based on the current construction progress, and data-property information association storage for digital twin objects is established. 2) Corresponding feature extraction methods are utilized based on the large-scale external geometric features of the bridge. This study focuses primarily on line and surface features, and a distributed and uniform feature point set is obtained. 3) The improved Hausdorff algorithm is employed to calculate local feature deformations. A diagnostic report that includes parameters describing deformations such as deflection, inclination, flatness, and local deviation is generated. This report provides an intuitive description of overall and local deformations. To validate this method, a case study is conducted using a suspension bridge as an example. The diagnostic results demonstrate minimal deviations between the actual bridge construction and the design model, which is consistent with the real-world scene. This method serves as an example for improving the diagnostic analysis capability of geospatial digital twin models and contributes to the development of digital earth construction. The contributions of this study are summarized as follows:

First, a feature-constrained geometric deformation diagnostic analysis method for bridge digital twin models is proposed. By analyzing the bridge structure and construction processes, the diagnostic objects and deformation monitoring tasks are clarified for different construction stages. This analysis leads to the establishment of a digital twin model geometric deformation feature library for bridges. Deformation information is integrated into the library to enhance the algorithm's understanding and processing capabilities for data. Second, a method for extracting bridge digital twin model geometric deformation information based on line and surface features is designed. This method employs differential principles to achieve precise geometric feature extraction and accurately summarizes the geometric morphology of the bridge, resulting in a well-distributed and uniform set of feature points. Last, an improved Hausdorff algorithm is employed using interpolation concepts to establish precise correspondences between two point clouds. This approach achieves millimeter-level computational accuracy and generates parameterized diagnostic reports, providing comprehensive quantitative assessments. Compared to the Hausdorff algorithm, the overall diagnostic efficiency is improved by more than 90%, effectively enhancing the diagnostic analysis capabilities of the digital twin models.

The method proposed in this study focuses primarily on the external geometric features of bridges but does not consider the structural stress conditions. Therefore, for some critical structural elements in bridges, such as anchorages, which lack large-scale geometric shapes, this method may not provide deformation state diagnostics and descriptions. To further improve this method, in future work, we plan to integrate finite element analysis and mechanical analysis to delve deeper into bridge deformation diagnostics, especially for components with complex stress conditions. Additionally, the diagnostic results presented in this study are mostly in the form of charts and other parameter indicators. We will further enhance the integration of diagnostic results with 3D models by combining diagnostic reports and deformation analysis results with 3D models, to achieve visual representation of information on the 3D model, enabling better support for bridge construction management, planning, and decision-making processes.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Additional information

Funding

This work was supported by National Natural Science Foundation of China (Grant Nos. U2034202, 42271424 and 42171397) and Technology research and development plan project of Xi'an-Chengdu passenger dedicated line Shaanxi Co., Ltd. (Xikang High Speed Rail Contract (2021) No. 24).