## 1 Introduction  
The development of smart cities leads to the need for better urban 3D modeling technologies to digitally represent urban geographic entities, and building information modeling (BIM) has emerged as an essential tool for this purpose due to its ability to provide comprehensive information about a city and support micro-spatial analysis [1; 2; 3]. However, due to the development of computer multimedia technology, BIM data are becoming more accessible and are susceptible to infringement and leakage during distribution and use [4; 5; 6]. Therefore, the secure use of BIM data is becoming increasingly important. Digital watermarking technology has emerged as a cutting-edge solution for copyright protection by hiding copyright information in data to carry out copyright identification, infringement traceability, and content authentication [7; 8; 9; 10; 11]. The robust watermark tightly integrates copyright information with the data to resist a wide range of attacks, making robust watermarking technology for BIM data important content in the field of data security [12; 13; 14]. The existing research on robust watermarking algorithms for BIM data is limited, but there are numerous studies on 3D mesh models and point cloud models, which are visualized 3D models like BIM data, that could provide a reference for watermarking research on BIM data. The current watermarking algorithms for 3D models can be grouped into three categories: transform domain-based watermarking algorithms, global feature-based watermarking algorithms, and local feature-based watermarking algorithms. Robust watermarking algorithms based on the transform domain generally use frequency analysis methods such as discrete cosine transform, wavelet analysis, and streaming harmonic bases to embed the watermark by modifying coefficients obtained from model changes [15; 16; 17; 18; 19]. For example, [PERSON] et al. proposed a robust 3D mesh watermarking algorithm based on mesh saliency and wavelet transform. The watermark is inserted by quantifying the wavelet coefficients using quantization index modulation (QIM) according to the mesh saliency of the 3D semiregular mesh [18]. While this approach has good imperceptibility, since such algorithms embed watermark information into data transformation coefficients, watermark embedding extraction relies on carrier data transformation, which greatly increases the computational cost of the algorithms. BIM data, on the other hand, comprise fine models with a large number of elements, thus requiring more computationally efficient watermarking algorithms. Robust watermarking algorithms based on global features embed watermarking information into global feature quantities in the geometric domain, such as histogram offset and region error expansion [20; 21; 22; 23]. For example, [PERSON] et al. divided the mesh in a 3D mesh model into several segments based on the shape diameter function (SDF) and then used the histogram mapping function to embed the watermark into each segment [23]. This type of algorithms exhibits robustness against various forms of attacks owing to the inherent stability of global statistical characteristics. Furthermore, in contrast to transform domain algorithms, they demonstrate lower computational demands. However, they are vulnerable to attacks that destroy the overall structure of the model. Robust watermarking algorithms based on local features use the relative positions of adjacent points, vertex order, vertex coordinate, etc. as data features to embed watermarking information [24; 25; 26; 27; 28]. For example, [PERSON] et al. cluster the vertices in a 3D mesh model by proposing an intelligent fuzzy based clustering scheme and embedded the watermark based on surface-based segmentation [27]. This type of algorithm can select the most suitable watermark embedding location after structural analysis of the 3D model, so they are more flexible than global feature-based algorithms and transform domain algorithms.