## 6 Conclusions  
This paper presents a digital watermarking algorithm for protecting the copyright of BIM data. The algorithm is based on the local features of the model, which is characterized by adding little redundancy, not changing the original topological relationships of the BIM data, and ensuring the availability of the data. In the proposed method, movable element perturbation and invisible character embedding of non-movable elements are combined to expand the number of watermark carriers, and achieve multiple embedding of digital watermark information. In addition, a mapping mechanism was used to further increase the watermark robustness. The experimental results show that the algorithm has good imperceptibility, and the embedding of invisible characters adds little redundancy. The proposed algorithm is robust to translation, element deletion, element addition, level-of-detail change attacks, as well as number-mode separation attacks. Furthermore, the watermark detection process is blind detection, which does not require the participation of the original BIM data, making the algorithm highly practical. In summary, the proposed algorithm solves the previous problem that 3D robust watermarking cannot be applied to BIM data, and provides a feasible solution for protecting the copyright of BIM data. Conceptualization, Q.Z., C.Z., and N.R. ; methodology, Q.Z. ; validation, Q.Z., C.Z., and N.R. ; formal analysis, Q.Z. and N.R. ; writing--original draft preparation, Q.Z. ; writing--review and editing, Q.Z., C.Z., and N.R. ; visualization, Q.Z. ; supervision, C.Z. and N.R. ; funding acquisition, C.Z. and N.R. All authors have read and agreed to the published version of the manuscript. This work was supported by the Natural Science Foundation of China under grant 42071362, the National Key Research and Development Program of China under grant2022 YFC3803600 and 2023 YFB3907100. Not applicable. **Data Availability Statement:** BIM data pertaining to this study can be obtained from Autodesk Revit 2017 ([[https://help.autodesk.com/view/RVT/2017/CHS/?guid=GUID-61](https://help.autodesk.com/view/RVT/2017/CHS/?guid=GUID-61) EF2F22-3A1F-4317-B925-1E85F138 BE88]([https://help.autodesk.com/view/RVT/2017/CHS/?guid=GUID-61](https://help.autodesk.com/view/RVT/2017/CHS/?guid=GUID-61) EF2F22-3A1F-4317-B925-1E85F138 BE88), accessed on 1 April 2023). The other data and the code that support the findings of this study are available on request from the corresponding author. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**BIM Data**} & \multirow{2}{*}{**Attacks**} & \multicolumn{4}{c}{**NC with Different Watermark Lengths**} \\ \cline{3-6}  & & **32** & **64** & **128** & **256** & **512** \\ \hline \multirow{6}{*}{Office building} & Translate 100 m & 0.99 & 0.99 & 0.99 & 0.99 & 0.99 \\  & Change level of detail & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\  & Delete 20\% of elements & 1.00 & 1.00 & 1.00 & 1.00 & 0.93 \\  & Add 20\% of elements & 1.00 & 1.00 & 1.00 & 1.00 & 0.91 \\  & Exporting FBX format data & 1.00 & 1.00 & 1.00 & 1.00 & 0.96 \\  & Exporting JSON format data & 1.00 & 1.00 & 1.00 & 1.00 & 0.95 \\ \hline \multirow{6}{*}{Gymnasium} & Translate 100 m & 0.98 & 0.98 & 0.98 & 0.98 & 0.98 \\  & Change level of detail & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\  & Delete 20\% of elements & 1.00 & 1.00 & 1.00 & 0.99 & 0.88 \\  & Add 20\% of elements & 1.00 & 1.00 & 0.99 & 0.98 & 0.87 \\  & Exporting FBX format data & 1.00 & 1.00 & 1.00 & 0.99 & 0.86 \\  & Exporting JSON format data & 1.00 & 1.00 & 1.00 & 0.98 & 084 \\ \hline \multirow{6}{*}{Structural model} & Translate 100 m & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\  & Change level of detail & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\ \cline{1-1}  & Delete 20\% of elements & 1.00 & 1.00 & 1.00 & 0.97 & 0.83 \\ \cline{1-1}  & Exporting FBX format data & 1.00 & 1.00 & 1.00 & 0.98 & 0.81 \\ \cline{1-1}  & Exporting JSON format data & 1.00 & 1.00 & 1.00 & 1 & 0.86 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Results of attacks under different watermark lengths.