Similarity Conversion Based on Renormalization Group Theory: A Case Study of Typical Hull Grillage Model Tests Under Underwater Explosion

Abstract

The dynamic response of hull grillage structures subjected to the initial shock wave loading of an underwater explosion is influenced by factors such as initial conditions, boundary conditions, and system parameters. The system dynamics exhibit strong nonlinearity and non-stationary characteristics. Model test results guided by classical similarity theory struggle to accurately predict prototype behavior. This paper addresses the similarity conversion problem for under underwater explosion hull grillage structure model tests. Utilizing phase space reconstruction and parabolic mapping theory, we characterize the evolutionary behavior of the system’s dynamic response in phase space. By introducing renormalization group theory and self-similarity theory, the scaling relationship between the scale ratio and the amplitude of the superstable periodic orbit is revealed. Two similarity conversion methods for model tests guided by renormalization group theory are established. Taking the local grillage structures of the main deck, transverse bulkhead, and double bottom of a 10 000-ton class hull as the prototype, a series of model tests were designed based on period-doubling bifurcation characteristics and the scaling relationship with superstable periodic orbit amplitude. The effectiveness and universality of the two similarity conversion methods derived from the renormalization group framework were thoroughly validated. The results demonstrate that the similarity conversion methods established within the renormalization group theory framework can effectively solve the problem of converting model test results with distortion effects to prototype predictions.

Keywords

Model test / Similarity conversion method / Phase space reconstruction / Renormalization group theory / Self-similarity theory

References

[1] Abdullah N, Iqbal S (2022) The Fractional Hilbert Transform of Generalized Functions. Symmetry-Basel 14(10). https://doi.org/10.3390/sym14102096

[2] Barras G, Souli M, Aquelet N, Couty N. Numerical simulation of underwater explosions using an ALE method. The pulsating bubble phenomena. Ocean Engineering, 2012, 41: 53-66

[3] Dhadphale J, Kraemer K, Gelbrecht M, Kurths J, Marwan N, Sujith R (2024) Model adaptive phase space reconstruction. Chaos 34(7). https://doi.org/10.1063/5.0194330

[4] Emamzadeh S. Nonlinear Dynamic Response of a Fixed Offshore Platform Subjected to Underwater Explosion at Different Distances. Journal of Marine Science and Application, 2022, 21(4): 168-176

[5] Gao H, Tian H, Gao X (2023) Damage characteristics of cabin in navigational state subjected to near-field underwater explosion. Ocean Engineering 277. https://doi.org/10.1016/j.oceaneng.2023.114256

[6] Glazek W. Perturbative renormalization group for Hamiltonians. Physical review. D, Particles and field, 1994, 49(8): 4214-4218

[7] Goufo E, Ravichandran C, Birajdar G. Self-Similarity Techniques for Chaotic Attractors with Many Scrolls Using Step Series Switching. Mathematical Modelling and Analysis, 2021, 26(4): 591-611

[8] Hao BStarting from the Parabola, 2013Beijing, ChinaPeking University Press15-62

[9] He Z, Du Z, Zhang L, Li Y (2023) Damage mechanisms of full-scale ship under near-field underwater explosion. Thin-Walled Structures 189. https://doi.org/10.1016/j.tws.2023.110872

[10] Huang X, Kong X, Hu J, Fang Q (2022) Scaling the failure of concrete gravity dam subjected to underwater explosion shock loading. Ocean Engineering 261. https://doi.org/10.1016/j.oceaneng.2022.112176

[11] Kan R, Nie J, Liu Z, Jiao Q, Sun X, Guo X, Yan S, Zhu Y (2023) Non-ideal explosive underwater explosion shockwave model. Physics Of Fluids 35(8). https://doi.org/10.1063/5.0157589

[12] Kye W H, Choi M, Rim S, Kurdoglyan M, Kim C, Park Y (2004) Characteristics of a delayed system with time-dependent delay time. Physical Review E 69(5). https://doi.org/10.1103/PhysRevE.69.055202

[13] Lang S, Zhu H, Sun G, Jiang Y, Wei C (2022) A Study on Methods for Determining Phase Space Reconstruction Parameters. Journal of Computational and Nonlinear Dynamics 17(1). https://doi.org/10.1115/1.4052721

[14] Li F, Fu J, Chen J, Hu D. Two-way coupling fluid-structure interaction analysis on dynamic response of offshore wind turbine. Marine Georesources & Geotechnology, 2024, 42(11): 1677-1686

[15] Li G, Shi D, Wang L, Zhao K (2022) Measurement technology of underwater explosion load: A review. Ocean Engineering 254. https://doi.org/10.1016/j.oceaneng.2022.111383

[16] Li H, Zhu Y, Liu K, Zou H, Chen X, Wang H (2024b) Experimental study of the coupled damage characteristics of a large-scale hull girder subjected to an underwater near-field explosion. Thin-Walled Structures 196. https://doi.org/10.1016/j.tws.2023.111547

[17] Liu B, Peng JNonlinear Dynamics, 2003Beijing, ChinaHigher Education Press74-389

[18] Lohmann A, Tepichin E, Ramirez J. Optical implementation of the fractional Hilbert transform for two-dimensional objects. Applied Optics, 1997, 36(26): 6620-6626

[19] Ma S, Wang Z, Li S, Chen Y, Zhu Q, Min B (2022) Similarity Research on the Action of a Centrifuge Scaled Near-Field Underwater Explosion on a Concrete Gravity Dam. International Journal of Structural Stability and Dynamics 22(8). https://doi.org/10.1142/S0219455422500845

[20] Marchesi G, Bardiani J, Lomazzi L, Manes A (2025) Dimensional analysis and scalability of a simplified hull girder subjected to underwater explosion shock loading. International Journal of Impact Engineering 203. https://doi.org/10.1016/j.ijimpeng.2025.105332

[21] Peng Z, Li J, Hao H (2024) Finite Linear Representation of Nonlinear Structural Dynamics Using Phase Space Embedding Coordinate. Journal of Engineering Mechanics 150(5). https://doi.org/10.1061/JENMDT.EMENG-7349

[22] Phuong N, Duc V, Giang N, Ly L, Xuan N, Nam V (2024) 24 Nonlinear Vibration and Dynamic Buckling of Complex Curved Functionally Graded Graphene Panels Reinforced with Inclined Stiffeners. International Journal of Structural Stability and Dynamics 24(19). https://doi.org/10.1142/S0219455424502237

[23] Pishevar A, Amirifar R. An adaptive ALE method for underwater explosion simulations including cavitation. Shock Waves, 2010, 20(5): 425-439

[24] Rossopoulos G, Papadopoulos C (2024) A π-Theorem-Based Advanced Scaling Methodology for Similarity Assessment of Marine Shafting Systems. Journal of Marine Science and Engineering 12(6). https://doi.org/10.3390/jmse12060894

[25] Shahriar A, Shoele K (2024) Nonlinear shock-induced flutter of a compliant panel using a fully coupled fluid-thermal-structure interaction model. Journal of Fluids and Structures 124. https://doi.org/10.1016/j.jfluidstructs.2023.104047

[26] Verma M, Priyadarshi A, Verma S. Vector-valued fractal functions: Fractal dimension and fractional calculus. Indagationes Mathematicae-New Series, 2023, 34(4): 830-853

[27] Xu TSimilarity Theory and Model Testing, 1982Beijing, ChinaChina Agricultural Machinery Press14-72

[28] Yao X, Yin Y, Zhao K, Huang R (2025a) A study of diffeomorphism characterization in similar transformation of underwater explosion ship model tests. Nonlinear Dynamics 22(113). https://doi.org/10.1007/s11071-025-11676-1

[29] Yao X, Zhao K, Huang R, Yin Y (2025b) Research on the similarity of ship structure’s shock response under underwater explosion based on scaling theory. International Journal of Impact Engineering 203. https://doi.org/10.1016/j.ijimpeng.2025.105333

[30] Yao X, Zhao K, Shi D (2024) Study on the effectiveness of impact factor in underwater explosion model test. International Journal of Impact Engineering 185. https://doi.org/10.1016/j.ijimpeng.2023.104839

[31] Zhang S, Jia J, Gao M, Han X. Study on the parameters determination for reconstructing phase-space in chaos time series. Acta Physica Sinica, 2010, 59(3): 1576-1582

[32] Zhang T, Liang X, Zou Y (2021) Modeling of Robot’s Low-Speed Motion Nonlinear Dynamics Based on Phase Space Reconstruction Neural Network. Journal of Computational and Nonlinear Dynamics 16(9). https://doi.org/10.1115/1.4051337

[33] Zheng W, Hao BApplied Symbolic Dynamics, 1994ShanghaiShanghai Science and Technology Education Press11-39

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