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DOI: 10.14489/vkit.2026.11.pp.018-025
Вяткин С. И., Долговесов Б. С.
МЕТОД ПРОЕКТИРОВАНИЯ ОБРАТНОЙ ФОРМЫ ДЛЯ ЭЛАСТИЧНЫХ ФУНКЦИОНАЛЬНО ЗАДАННЫХ ОБЪЕКТОВ
(c. 18-25)
Аннотация. Рассмотрена задача проектирования обратной формы для эластичных функционально заданных объектов, что позволяет пользователям уделить больше внимания желаемым формам, не рассматривая упругие деформации. Решение этой задачи с использованием классических итерационных методов имеет медленную сходимость к желаемому результату. Предложено асимптотическое локальное разложение для обратной силы упругости нелинейной модели материала. Это позволило построить быстрый и надежный численный метод для решения обратной задачи статического равновесия. Разработан интерактивный инструмент для задач проектирования обратных форм.
Ключевые слова: функционально заданные объекты; интерактивное проектирование; метод конечных элементов; нелинейная оптимизация.
Vyatkin S. I., Dolgovesov B. S.
A METHOD OF DESIGNING AN INVERSE SHAPE FOR ELASTIC FUNCTIONALLY DEFINED OBJECTS
(pp. 18-25)
Abstract. Elastic objects are used in biomechanics, computer animation, robotics, and 3D printing, the most popular additive manufacturing technology, finds many applications in various industries. When designing the shapes of elastic objects, the focus is on the desired target shapes. To do this, it is necessary to develop an inverse shape design tool that automatically calculates the resting shape. Then the resting shape is deformed into the desired target shape under the action of specified external forces. The article considers the problem of designing an inverse shape for elastic functionally defined objects. This allows users to pay more attention to the desired shapes without considering elastic deformations. Solving this problem using classical iterative methods has a slow convergence to the desired result. The aim of the work is to develop a reliable method for designing an inverse elastic shape using an asymptotic numerical approach. An asymptotic numerical method is proposed using the basic mathematical structure of specific nonlinear models of materials. As a result, it works orders of magnitude faster than traditional Newton methods. The main results of the work are the following algorithms, methods and tools. An asymptotic local decomposition for the inverse elastic force of a nonlinear material model is proposed. This allowed us to build a fast and reliable numerical method for solving the inverse problem of static equilibrium. An interactive tool has been developed for reverse shape design tasks. Thanks to the method, interactive force adjustment, multi-purpose reverse shape design, as well as solutions for direct static equilibrium can be supported. The practical significance is as follows. As a result, thanks to the method, interactive force adjustment, multi-purpose reverse shape design, as well as solutions for direct static equilibrium can be supported.
Keywords: Functionally defined objects; Interactive design; Finite element method; Nonlinear optimization.
С. И. Вяткин, Б. С. Долговесов (Институт автоматики и электрометрии Сибирского отделения Российской академии наук, Новосибирск, Россия) E-mail:
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S. I. Vyatkin, B. S. Dolgovesov (Institute of Automation and Electrometry of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia) E-mail:
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1. Bern, J. M., Zargarbashi, F., Zhang, A., et al. (2022, May 23–27). Simulation and fabrication of soft robots with embedded skeletons. In IEEE International Conference on Robotics and Automation (ICRA) (pp. 5205–5211). Philadelphia, PA, USA. https://doi.org/10.1109/ICRA46639.2022.9811844
2. Vyatkin, S. I., & Dolgovesov, B. S. (2022). Functionally defined models for additive manufacturing. Issledovaniya. Innovatsii. Praktika, 4(4), 16–25. [in Russian language]. https://doi.org/10.18411/iip-08-2022-04
3. Schegg, P., & Duriez, C. (2022). Review on generic methods for mechanical modeling, simulation and control of soft robots. PLoS ONE, 17(1), 1–14. https://doi.org/10.1371/journal.pone.0251059
4. Chen, F., & Wang, M. Y. (2020). Design optimization of soft robots: A review of the state of the art. IEEE Robotics & Automation Magazine, 27(4), 27–43. https://doi.org/10.1109/MRA.2020.3024280
5. Ren, Y., Kusupati, U., Panetta, J., et al. (2022). Umbrella meshes: Elastic mechanisms for freeform shape deployment. ACM Transactions on Graphics, 41(4), 1–15. https://doi.org/10.1145/3528223.3530089
6. Zehnder, J., Coros, S., & Thomaszewski, B. (2022). SGN: Sparse Gauss-Newton for accelerated sensitivity analysis. ACM Transactions on Graphics, 41(1), 1–10. https://doi.org/10.1145/3470005
7. Hsu, J., Truong, N., Yuksel, C., & Wu, K. (2022). A general two-stage initialization for sag-free deformable simulations. ACM Transactions on Graphics, 41(4), 1–13. https://doi.org/10.1145/3528223.3530165
8. Liu, M., Zhang, X., Yang, B., et al. (2023). Three-dimensional modeling of heart soft tissue motion. Applied Sciences, 13(4), Article 2493. https://doi.org/10.3390/app13042493
9. Fu, H., Bian, S., Li, O., et al. (2022). 3D modelling with C2 continuous PDE surface patches. Mathematics, 10(3), 1–20. https://doi.org/10.3390/math10030319
10. Fang, J., Chaudhry, E., Iglesias, A., et al. (2022). Reconstructing dynamic 3D models with small data by integrating position-based dynamics and PDE-based modelling. Mathematics, 10(5), 1–24. https://doi.org/10.3390/math10050821
11. Bahrami, H., Sichetti, F., Puppo, E., et al. (2023). Physically-based simulation of elastic-plastic fusion of 3D bioprinted spheroids. Biofabrication, 15(4), 1–25. https://doi.org/10.1088/1758-5090/acf2cb
12. Chen, Q., Cao, J., Lin, W., et al. (2024). Predicting dynamic responses of continuous deformable bodies: A graph-based learning approach. Computer Methods in Applied Mechanics and Engineering, 420(4), Article 116669. https://doi.org/10.1016/j.cma.2023.116669
13. Shao, H., Kugelstadt, T., Hadrich, T., et al. (2021). Accurately solving rod dynamics with graph learning. In Advances in Neural Information Processing Systems 34 (NeurIPS) (Vol. 35, pp. 4829–4842). https://doi.org/10.48550/arXiv.2006.03897
14. Borges, E. I. A., Rieder, J., Aschenbrenner, D., et al. (2022). Framework for armature-based 3D shape reconstruction of sensorized soft robots in eXtended reality. Frontiers in Robotics and AI, 9, 1–12. https://doi.org/10.3389/frobt.2022.810328
15. Bacher, M., Knoop, E., & Schumacher, C. (2021). Design and control of soft robots using differentiable simulation. Current Robotics Reports, 2(51), 211–221. https://doi.org/10.1007/s43154-021-00052-7
16. Maloisel, G., Schumacher, C., Knoop, E., et al. (2023). Optimal design of robotic character kinematics. ACM Transactions on Graphics, 42(6), 1–15. https://doi.org/10.1145/3618404
17. Fang, G., Matte, C.-D., Scharff, R., et al. (2020). Kinematics of soft robots by geometric computing. IEEE Transactions on Robotics, 36, 1272–1286. https://doi.org/10.1109/TRO.2020.2985583
18. Arnavaz, K., Nielsen, M. K., Kry, P. G., et al. (2023). Differentiable depth for Real2Sim calibration of soft body simulations. Computer Graphics Forum, 42(1), 277–289. https://doi.org/10.1111/cgf.14720
19. Pinskier, J. (2021). From bioinspiration to computer generation: Developments in autonomous soft robot design. Advanced Intelligent Systems, 4(1), 10–20. https://doi.org/10.1002/aisy.202100086
20. Zheng, A., Zhu, Z., Bian, S., et al. (2021). Optimized unidirectional and bidirectional stiffened objects for minimum material consumption of 3D printing. Mathematics, 9(21), Article 2835. https://doi.org/10.3390/math9212835
21. Vyatkin, S. I., & Dolgovesov, B. S. (2023). Method of anisotropic deformation of elastic materials based on free-form patches. BOHR International Journal of Biocomputing and Nano Technology, 2(1), 8–13. https://doi.org/10.54646/bijbnt.2023.12
22. Galadima, Y. K., Xia, W., Oterkus, E., & Oterkus, S. (2021). Multiscale modeling with peridynamics. In Peridynamic modeling, numerical techniques, and applications (Chapter 17). Elsevier. https://doi.org/10.1016/B978-0-12-820069-8.00018-4
23. Kossa, A., Valentine, M. T., & McMeeking, R. M. (2023). Analysis of the compressible, isotropic, neo-Hookean hyperelastic model. Meccanica, 58(1), 217–232. https://doi.org/10.1007/s11012-022-01633-2
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