Analytical methods and recent advances in computation for the dynamics of continuous systems
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structural dynamics,
finite element method,
model-order reduction,
scientific machine learning
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Abstract
Continuous systems such as beams, plates, shells, and membranes are distributed-parameter systems whose dynamic behavior is governed by partial differential equations. Their analysis is essential for predicting vibration, resonance, stability, and transient structural responses, yet exact solutions are often difficult to obtain because of complex geometries, material nonlinearities, and boundary conditions. This study reviews analytical methods and recent computational advances in continuous-system dynamics using the PRISMA 2020 framework. A total of 58 studies published between 2018 and 2026 were analyzed, covering analytical and semi-analytical techniques, conventional numerical discretization, spectral element methods, isogeometric analysis, model-order reduction, physics-informed machine learning, and neural operators. The findings show that classical analytical solutions and perturbation methods remain essential as verification benchmarks and for nonlinear resonance analysis. Finite element and high-order structure-preserving formulations effectively handle complex geometries, while nonlinear model reduction using spectral submanifolds and quadratic manifolds substantially reduces computational cost. Scientific machine learning and neural operators enable rapid prediction for parametric differential-equation families, although their reliability strongly depends on training-data representativeness and physical consistency outside the training regime.
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References
Agathos, K., Vlachas, K., Garland, A., & Chatzi, E. (2024). Accelerating structural dynamics simulations with localised phenomena through matrix compression and projection-based model order reduction. International Journal for Numerical Methods in Engineering, 125(10), e7445. https://doi.org/10.1002/nme.7445
Andersson, L., Persson, P., & Persson, K. (2023). Efficient nonlinear reduced order modeling for dynamic analysis of flat structures. Mechanical Systems and Signal Processing, 191, 110143. https://doi.org/10.1016/j.ymssp.2023.110143
Azizzadenesheli, K., Kovachki, N., Li, Z., Liu-Schiaffini, M., Kossaifi, J., & Anandkumar, A. (2024). Neural operators for accelerating scientific simulations and design. Nature Reviews Physics, 6(5), 320–328. https://doi.org/10.1038/s42254-024-00712-5
Badrkhani, V., Hiemstra, R. R., Mika, M., & Schillinger, D. (2023). A matrix-free macro-element variant of the hybridized discontinuous Galerkin method. International Journal for Numerical Methods in Engineering, 124(20), 4427–4452. https://doi.org/10.1002/nme.7320
Bettini, L., Cenedese, M., & Haller, G. (2024). Model reduction to spectral submanifolds in piecewise smooth dynamical systems. International Journal of Non-Linear Mechanics, 163, 104753. https://doi.org/10.1016/j.ijnonlinmec.2024.104753
Bondsman, B., Ahn, C.-U., & Kim, J.-G. (2024). Interface reduction technique for Enhanced Craig–Bampton method. Mechanical Systems and Signal Processing, 208, 111074. https://doi.org/10.1016/j.ymssp.2023.111074
Brugnoli, A., Matignon, D., & Morlier, J. (2025). A linearly-implicit energy-momentum preserving scheme for geometrically nonlinear mechanics based on non-canonical Hamiltonian formulations. Nonlinear Dynamics, 113, 27539–27566. https://doi.org/10.1007/s11071-025-11601-6
Buffa, A., Gantner, G., Giannelli, C., Praetorius, D., & Vázquez, R. (2022). Mathematical foundations of adaptive isogeometric analysis. Archives of Computational Methods in Engineering, 29(7), 4479–4555. https://doi.org/10.1007/s11831-022-09752-5
Bui, T. A., Park, J., & Kim, J.-S. (2024). A review of combining component mode synthesis and model order reductions for geometrically nonlinear analysis. Journal of Mechanical Science and Technology, 38(9), 4699–4711. https://doi.org/10.1007/s12206-024-0807-4
Cao, Q., Goswami, S., & Karniadakis, G. E. (2024). Laplace neural operator for solving differential equations. Nature Machine Intelligence, 6(6), 631–640. https://doi.org/10.1038/s42256-024-00844-4
Castro, R., Bergonzi, M., Pecker Marcosig, E., Fernández, J., & Kofman, E. (2024). Discrete-event simulation of continuous-time systems: Evolution and state of the art of quantized state system methods. SIMULATION, 100(6), 613–638. https://doi.org/10.1177/00375497241230985
Cenedese, M., Axås, J., Bäuerlein, B., Avila, K., & Haller, G. (2022). Data-driven modeling and prediction of non-linearizable dynamics via spectral submanifolds. Nature Communications, 13(1), 872. https://doi.org/10.1038/s41467-022-28518-y
Chen, H.-Y., Ding, H., Li, S.-H., & Chen, L.-Q. (2020). Convergent term of the Galerkin truncation for dynamic response of sandwich beams on nonlinear foundations. Journal of Sound and Vibration, 483, 115514. https://doi.org/10.1016/j.jsv.2020.115514
Chen, Z., Liu, Y., & Sun, H. (2021). Physics-informed learning of governing equations from scarce data. Nature Communications, 12(1), 6136. https://doi.org/10.1038/s41467-021-26434-1
Dong, X., Bai, Y., & Wan, W.-D. (2024). Kernel functions embed into the autoencoder to identify the sparse models of nonlinear dynamics. Communications in Nonlinear Science and Numerical Simulation, 131, 107869. https://doi.org/10.1016/j.cnsns.2024.107869
Eisenträger, S., Kapuria, S., Jain, M., & Zhang, J. (2023). On the numerical properties of high-order spectral (Euler–Bernoulli) beam elements. ZAMM – Journal of Applied Mathematics and Mechanics, 103(9), e202200422. https://doi.org/10.1002/zamm.202200422
Fu, Z.-J., Xi, Q., Gu, Y., Li, J., Qu, W., Sun, L., Wei, X., Wang, F., Lin, J., Li, W., Xu, W., & Zhang, C.-Z. (2023). Singular boundary method: A review and computer implementation aspects. Engineering Analysis with Boundary Elements, 147, 231–266. https://doi.org/10.1016/j.enganabound.2022.12.004
Geelen, R., Wright, S. J., & Willcox, K. (2023). Operator inference for non-intrusive model reduction with quadratic manifolds. Computer Methods in Applied Mechanics and Engineering, 403, 115717. https://doi.org/10.1016/j.cma.2022.115717
Gervasio, P., Dedè, L., Chanon, O., & Quarteroni, A. (2020). A computational comparison between isogeometric analysis and spectral element methods: Accuracy and spectral properties. Journal of Scientific Computing, 83(1), 18. https://doi.org/10.1007/s10915-020-01204-1
Goswami, S., Yin, M., Yu, Y., & Karniadakis, G. E. (2022). A physics-informed variational DeepONet for predicting crack path in quasi-brittle materials. Computer Methods in Applied Mechanics and Engineering, 391, 114587. https://doi.org/10.1016/j.cma.2022.114587
Hafeez, M. B., & Krawczuk, M. (2023). A review: Applications of the spectral finite element method. Archives of Computational Methods in Engineering, 30(5), 3453–3465. https://doi.org/10.1007/s11831-023-09911-2
Harik, I. E., Liu, X., & Balakrishnan, N. (1992). Analytic solution to free vibration of rectangular plates. Journal of Sound and Vibration, 153(1), 51–62. https://doi.org/10.1016/0022-460X(92)90626-9
Jacinto Portela, C. C., Tadeu, A., & de Lacerda, L. A. (2022). A new approach for solving heat conduction under zero and non-zero initial conditions. Engineering Analysis with Boundary Elements, 144, 185–198. https://doi.org/10.1016/j.enganabound.2022.08.015
Jain, S., & Haller, G. (2022). How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models. Nonlinear Dynamics, 107(2), 1417–1450. https://doi.org/10.1007/s11071-021-06957-4
Kaheman, K., Kutz, J. N., & Brunton, S. L. (2020). SINDy-PI: A robust algorithm for parallel implicit sparse identification of nonlinear dynamics. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 476(2242), 20200279. https://doi.org/10.1098/rspa.2020.0279
Kan, Z., Dong, K., Chen, B., Peng, H., & Song, X. (2021). The direct force correction based framework for general co-rotational analysis. Computer Methods in Applied Mechanics and Engineering, 385, 114018. https://doi.org/10.1016/j.cma.2021.114018
Khaniki, H. B., Ghayesh, M. H., Chin, R., & Amabili, M. (2022). A review on the nonlinear dynamics of hyperelastic structures. Nonlinear Dynamics, 110(2), 963–994. https://doi.org/10.1007/s11071-022-07700-3
Kim, T. H., & Lee, U. (2021). Temporal and spatial-domain DFT-based spectral element model for the dynamic analysis of a rectangular Mindlin plate. Journal of Sound and Vibration, 509, 116220. https://doi.org/10.1016/j.jsv.2021.116220
Li, M., & Haller, G. (2022). Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, Part II: Bifurcation and quasi-periodic response. Nonlinear Dynamics, 110(2), 1045–1080. https://doi.org/10.1007/s11071-022-07476-6
Li, M., Jain, S., & Haller, G. (2022). Nonlinear analysis of forced mechanical systems with internal resonance using spectral submanifolds, Part I: Periodic response and forced response curve. Nonlinear Dynamics, 110(2), 1005–1043. https://doi.org/10.1007/s11071-022-07714-x
Li, Z., Kovachki, N. B., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., & Anandkumar, A. (2021). Fourier neural operator for parametric partial differential equations. International Conference on Learning Representations. https://openreview.net/forum?id=c8P9NQVtmnO
Lu, K., Jin, Y., Chen, Y., Yang, Y., Hou, L., Zhang, Z., Li, Z., & Fu, C. (2019). Review for order reduction based on proper orthogonal decomposition and outlooks of applications in mechanical systems. Mechanical Systems and Signal Processing, 123, 264–297. https://doi.org/10.1016/j.ymssp.2019.01.018
Lu, L., Jin, P., Pang, G., Zhang, Z., & Karniadakis, G. E. (2021). Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nature Machine Intelligence, 3(3), 218–229. https://doi.org/10.1038/s42256-021-00302-5
Lu, L., Meng, X., Cai, S., Mao, Z., Goswami, S., Zhang, Z., & Karniadakis, G. E. (2022). A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data. Computer Methods in Applied Mechanics and Engineering, 393, 114778. https://doi.org/10.1016/j.cma.2022.114778
Mazzilli, C. E. N., Gonçalves, P. B., & Franzini, G. R. (2022). Reduced-order modelling based on non-linear modes. International Journal of Mechanical Sciences, 214, 106915. https://doi.org/10.1016/j.ijmecsci.2021.106915
Meesala, V. C., & Hajj, M. R. (2019). Response variations of a cantilever beam–tip mass system with nonlinear and linearized boundary conditions. Journal of Vibration and Control, 25(3), 485–496. https://doi.org/10.1177/1077546318809853
Messenger, D. A., & Bortz, D. M. (2021). Weak SINDy for partial differential equations. Journal of Computational Physics, 443, 110525. https://doi.org/10.1016/j.jcp.2021.110525
Müller, A., & Bischoff, M. (2022). A consistent finite element formulation of the geometrically non-linear Reissner–Mindlin shell model. Archives of Computational Methods in Engineering, 29(5), 3387–3434. https://doi.org/10.1007/s11831-021-09702-7
Page, M. J., McKenzie, J. E., Bossuyt, P. M., Boutron, I., Hoffmann, T. C., Mulrow, C. D., Shamseer, L., Tetzlaff, J. M., Akl, E. A., Brennan, S. E., Chou, R., Glanville, J., Grimshaw, J. M., Hróbjartsson, A., Lalu, M. M., Li, T., Loder, E. W., Mayo-Wilson, E., McDonald, S., . . . Moher, D. (2021). The PRISMA 2020 statement: An updated guideline for reporting systematic reviews. PLOS Medicine, 18(3), e1003583. https://doi.org/10.1371/journal.pmed.1003583
Patuelli, C., Cestino, E., & Frulla, G. (2024). A nonlinear beam finite element with bending–torsion coupling formulation for dynamic analysis with geometric nonlinearities. Aerospace, 11(4), 255. https://doi.org/10.3390/aerospace11040255
Pellicano, F., Strozzi, M., & Avramov, K. V. (2019). Nonlinear vibration of continuous systems. Shock and Vibration, 2019, 6870697. https://doi.org/10.1155/2019/6870697
Ponsioen, S., Jain, S., & Haller, G. (2020). Model reduction to spectral submanifolds and forced-response calculation in high-dimensional mechanical systems. Journal of Sound and Vibration, 488, 115640. https://doi.org/10.1016/j.jsv.2020.115640
Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686–707. https://doi.org/10.1016/j.jcp.2018.10.045
Rao, C., Sun, H., & Liu, Y. (2021). Physics-informed deep learning for computational elastodynamics without labeled data. Journal of Engineering Mechanics, 147(8), 04021043. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001947
Ravichandran, A., Mohanty, P. K., Kareemullah, M., Smerat, A., & Khan, M. A. (2026). Dynamic and vibration analysis of beam, plates, and shells on composites materials – a literature review. Journal of Engineering and Applied Science, 73(1), 184. https://doi.org/10.1186/s44147-026-00925-8
Ren, J., & Lin, H. (2023). A survey on isogeometric collocation methods with applications. Mathematics, 11(2), 469. https://doi.org/10.3390/math11020469
Song, N., Peng, H., & Kan, Z. (2022). A hybrid data-driven model order reduction strategy for flexible multibody systems considering impact and friction. Mechanism and Machine Theory, 169, 104649. https://doi.org/10.1016/j.mechmachtheory.2021.104649
Song, N., Peng, H., Kan, Z., & Chen, B. (2020). A novel nonsmooth approach for flexible multibody systems with contact and friction in 3D space. Nonlinear Dynamics, 102(3), 1375–1408. https://doi.org/10.1007/s11071-020-05972-1
Szalai, R. (2020). Invariant spectral foliations with applications to model order reduction and synthesis. Nonlinear Dynamics, 101(4), 2645–2669. https://doi.org/10.1007/s11071-020-05891-1
Takacs, S. (2023). Discretization error estimates for discontinuous Galerkin isogeometric analysis. Applicable Analysis, 102(5), 1439–1462. https://doi.org/10.1080/00036811.2021.1986023
Tripura, T., & Chakraborty, S. (2023). Wavelet Neural Operator for solving parametric partial differential equations in computational mechanics problems. Computer Methods in Applied Mechanics and Engineering, 404, 115783. https://doi.org/10.1016/j.cma.2022.115783
Vizzaccaro, A., Opreni, A., Salles, L., Frangi, A., & Touzé, C. (2022). High order direct parametrisation of invariant manifolds for model order reduction of finite element structures: Application to large amplitude vibrations and uncovering of a folding point. Nonlinear Dynamics, 110(1), 525–571. https://doi.org/10.1007/s11071-022-07651-9
Wang, S., Teng, Y., & Perdikaris, P. (2021). Understanding and mitigating gradient flow pathologies in physics-informed neural networks. SIAM Journal on Scientific Computing, 43(5), A3055–A3081. https://doi.org/10.1137/20M1318043
Wang, S., Wang, H., & Perdikaris, P. (2021). Learning the solution operator of parametric partial differential equations with physics-informed DeepONets. Science Advances, 7(40), eabi8605. https://doi.org/10.1126/sciadv.abi8605
Wu, L., Tiso, P., & Van Keulen, F. (2018). Interface reduction with multilevel Craig–Bampton substructuring for component mode synthesis. AIAA Journal, 56(5), 2030–2044. https://doi.org/10.2514/1.J056196
Xing, Y., Li, G., & Yuan, Y. (2022). A review of the analytical solution methods for the eigenvalue problems of rectangular plates. International Journal of Mechanical Sciences, 221, 107171. https://doi.org/10.1016/j.ijmecsci.2022.107171
Yessenbayeva, G. A., Yesbayeva, D. N., Bauyrzhankyzy, D., & Nurgali, D. A. (2018). On mathematical and analytical methods for solving problems on vibrations of membranes and plates. Bulletin of the Karaganda University. Mathematics Series, 1(89), 113–120. https://doi.org/10.31489/2018m1/113-120
Zhou, L., Yao, H., Li, M., Wei, S., & Kan, Z. (2026). A geometric rotation-dispersion-guided adaptive co-rotational formulation for flexible multibody dynamics. Applied Sciences, 16(15), 7697. https://doi.org/10.3390/app16157697