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SEM: a shallow energy method for finite deformation hyperelasticity problems

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Abstract

In existing works, the deep energy method (DEM) is developed based on deep neural networks to handle partial differential equation problems of finite deformation hyperelasticity. Inspired by the DEM, we propose a shallow energy method (SEM) which utilizes simple and efficient radial basis function neural networks (RBFNNs) with only one hidden layer as the approximator of the continuous displacement fields of the investigated structures. In our method, the energy items, including the internal energy and the external energy, are calculated through numerical integration techniques and form the overall potential energy functional with respect to the parameters of the RBFNNs. Then, the minimization problem of the potential energy is carried out by the L-BFGS algorithm embedded in Python optimizer packages. Finally, the numerical approximation of the true displacement field is yielded and compared to those obtained by the DEM and the FEM, to illustrate the validity, the accuracy and other good performance of the proposed SEM in dealing with problems of finite deformation hyperelasticity.

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References

  1. Weiss, J.A., Maker, B.N., Govindjee, S.: Finite element implementation of incompressible, transversely isotropic hyperelasticity. Comput. Methods Appl. Mech. Eng. 135(1), 107–128 (1996)

    Article  Google Scholar 

  2. Chen, J.-S., Pan, C.: A pressure projection method for nearly incompressible rubber hyperelasticity, part I: theory. J. Appl. Mech. Trans. 63(4), 862–868 (1996). https://doi.org/10.1115/1.2787240

    Article  MATH  Google Scholar 

  3. Schröder, J., Neff, P., Balzani, D.: A variational approach for materially stable anisotropic hyperelasticity. Int. J. Solids Struct. 42, 4352–4371 (2005)

    Article  MathSciNet  Google Scholar 

  4. Fachinotti, V., Cardona, A., Jetteur, P.: Finite element modelling of inverse design problems in large deformation anisotropic hyperelasticity. Int. J. Numer. Methods Eng. 74, 894–910 (2008). https://doi.org/10.1002/nme.2193

    Article  MathSciNet  MATH  Google Scholar 

  5. Silber, G., Alizadeh, M., Salimi, M.: Large deformation analysis for soft foams based on hyperelasticity. J. Mech. 26, 327–334 (2010). https://doi.org/10.1017/S1727719100003889

    Article  Google Scholar 

  6. Duddu, R., Lavier, L., Hughes, T., Calo, V.: A finite strain Eulerian formulation for compressible and nearly incompressible hyperelasticity using high-order b-spline finite elements. Int. J. Numer. Methods Eng. 89, 762–785 (2012). https://doi.org/10.1002/nme.3262

    Article  MathSciNet  MATH  Google Scholar 

  7. Hu, D., Sun, Z., Liang, C., Han, X.: A mesh-free algorithm for dynamic impact analysis of hyperelasticity. Acta Mech. Solida Sin. 26, 362–372 (2013). https://doi.org/10.1016/S0894-9166(13)60033-6

    Article  Google Scholar 

  8. Le Pense, S.: Mean stress dependent nonlinear hyperelasticity coupled with damage stiffness degradation. A thermodynamical approach. Mech. Res. Commun. 60, 85–89 (2014). https://doi.org/10.1016/j.mechrescom.2014.06.007

    Article  Google Scholar 

  9. Zdunek, A., Rachowicz, W., Eriksson, T.: A five-field finite element formulation for nearly inextensible and nearly incompressible finite hyperelasticity. Comput. Math. Appl. (2016). https://doi.org/10.1016/j.camwa.2016.04.022

    Article  MathSciNet  MATH  Google Scholar 

  10. Hornik, K., Stinchcombe, M., White, H.: Multilayer feedforward networks are universal approximators. Neural Netw. 2(5), 359–366 (1989)

    Article  Google Scholar 

  11. Krizhevsky, A., Sutskever, I., Hinto, G.: Imagenet classification with deep convolutional neural networks. Arxiv 1097–1105 (2012)

  12. Goodfellow, I.J., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., Bengio, Y.: Generative adversarial networks. Adv. Neural. Inf. Process. Syst. 3, 2672–2680 (2014)

    Google Scholar 

  13. Dinh, L., Krueger, D., Bengio, Y.: Nice: non-linear independent components estimation. arXiv:1410.8516v6 (2014)

  14. Dinh, L., Sohl-Dickstein, J., Bengio, S.: Density estimation using real NVP. arXiv:1605.08803v3 (2016)

  15. Gers, F.A., Schmidhuber, J., Cummins, F.: Learning to forget: continual prediction with lstm. In: 1999 Ninth International Conference on Artificial Neural Networks ICANN 99. (Conf. Publ. No. 470), vol. 2, pp. 850–8552 (1999). https://doi.org/10.1049/cp:19991218

  16. Mnih, V., Kavukcuoglu, K., Silver, D., Rusu, A., Veness, J., Bellemare, M., Graves, A., Riedmiller, M., Fidjeland, A., Ostrovski, G., Petersen, S., Beattie, C., Sadik, A., Antonoglou, I., King, H., Kumaran, D., Wierstra, D., Legg, S., Hassabis, D.: Human-level control through deep reinforcement learning. Nature 518, 529–33 (2015). https://doi.org/10.1038/nature14236

    Article  Google Scholar 

  17. Bishop, C.: Improving the generalization properties of radial basis function neural networks. Neural Comput. 3(4), 579–588 (1991). https://doi.org/10.1162/neco.1991.3.4.579

    Article  Google Scholar 

  18. Schilling, R., Al-Ajlouni, A.: Approximation of nonlinear systems with radial basis function neural networks. IEEE Trans. Neural Netw. 12, 1–15 (2001). https://doi.org/10.1109/72.896792

    Article  Google Scholar 

  19. Sarimveis, H., Alexandridis, A., Tsekouras, G., Bafas, G.: A fast and efficient algorithm for training radial basis function neural networks based on a fuzzy partition of the input space. Ind. Eng. Chem. Res. (2002). https://doi.org/10.1021/ie010263h

    Article  Google Scholar 

  20. Reddy, R., Ganguli, R.: Structural damage detection in a helicopter rotor blade using radial basis function neural networks. Smart Mater. Struct. (2003). https://doi.org/10.1088/0964-1726/12/2/311

    Article  Google Scholar 

  21. Ng, W., Dorado, A., Yeung, D., Pedrycz, W., Izquierdo, E.: Image classification with the use of radial basis function neural networks and the minimization of the localized generalization error. Pattern Recogn. 40, 19–32 (2007). https://doi.org/10.1016/j.patcog.2006.07.002

    Article  MATH  Google Scholar 

  22. Sideratos, G., Hatziargyriou, N.D.: Probabilistic wind power forecasting using radial basis function neural networks. IEEE Trans. Power Syst. 27(4), 1788–1796 (2012). https://doi.org/10.1109/TPWRS.2012.2187803

    Article  Google Scholar 

  23. Samaniego, E., Anitescu, C., Goswami, S., Nguyen-Thanh, V.M., Guo, H., Hamdia, K., Zhuang, X., Rabczuk, T.: An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Comput. Methods Appl. Mech. Eng. 362(15), 112790–111279029 (2020)

    Article  MathSciNet  Google Scholar 

  24. Nguyen-Thanh, V.M., Zhuang, X., Rabczuk, T.: A deep energy method for finite deformation hyperelasticity. Eur. J. Mech. A Solids (2020). https://doi.org/10.1016/j.euromechsol.2019.103874

    Article  MathSciNet  MATH  Google Scholar 

  25. Nguyen-Thanh, V.M., Anitescu, C., Alajlan, N., Rabczuk, T., Zhuang, X.: Parametric deep energy approach for elasticity accounting for strain gradient effects. Comput. Methods Appl. Mech. Eng. 386, 114096 (2021). https://doi.org/10.1016/j.cma.2021.114096

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, W., Bazant, M.Z., Zhu, J.: A physics-guided neural network framework for elastic plates: comparison of governing equations-based and energy-based approaches. Comput. Methods Appl. Mech. Eng. (2021). https://doi.org/10.1016/j.cma.2021.113933

    Article  MathSciNet  MATH  Google Scholar 

  27. Abueidda, D., Koric, S., Sobh, N., Sehitoglu, H.: Deep learning for plasticity and thermo-viscoplasticity. Int. J. Plast 136, 102852 (2021). https://doi.org/10.1016/j.ijplas.2020.102852

    Article  Google Scholar 

  28. Dehghani, H., Zilian, A.: Ann-aided incremental multiscale-remodelling-based finite strain poroelasticity. Comput. Mech. 68, 131–154 (2021). https://doi.org/10.1007/s00466-021-02023-3

    Article  MathSciNet  MATH  Google Scholar 

  29. Gärtner, T., Fernández, M., Weeger, O.: Nonlinear multiscale simulation of elastic beam lattices with anisotropic homogenized constitutive models based on artificial neural networks. Comput. Mech. (2021). https://doi.org/10.13140/RG.2.2.18450.17604

  30. Li, H.: Gaussian process regression a machine learning approach to derivative pricing. Thesis of MSc. Imperial College London (2018)

  31. Abramowitz, M., Stegun, I.A., Romer, R.H.: Handbook of mathematical functions with formulas, graphs, and mathematical tables. American Association of Physics Teachers (1988)

  32. Sebah, P., Gourdon, X.: Introduction to the gamma function. Am. J. Sci. Res. 2–18 (2002)

  33. Rasmussen, C.E.: Gaussian processes in machine learning. In: Summer School on Machine Learning, pp. 63–71. Springer (2003)

  34. Heikkinen, V., Mirhashemi, A., Alho, J.: Link functions and matérn kernel in the estimation of reflectance spectra from RGB responses. J. Opt. Soc. Am. A 30, 2444–2454 (2013). https://doi.org/10.1364/JOSAA.30.002444

    Article  Google Scholar 

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Acknowledgements

This research is supported by the Fundamental Research Funds for the Central Universities, JLU (93K172020K28). We would like to express my sincere thanks to the Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University.

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Authors and Affiliations

  1. School of Mechanical and Aerospace Engineering (SMAE), Jilin University, No. 5988 Renmin Street, Changchun, 13022, Jilin, China

    Zhangyong Liang & Huanhuan Gao

  2. Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, No. 2699 Qianjin Street, Changchun, 130015, Jilin, China

    Huanhuan Gao & Tingting Li

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  1. Zhangyong Liang
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  2. Huanhuan Gao
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Correspondence to Huanhuan Gao.

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Liang, Z., Gao, H. & Li, T. SEM: a shallow energy method for finite deformation hyperelasticity problems. Acta Mech 233, 1739–1755 (2022). https://doi.org/10.1007/s00707-022-03174-x

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  • DOI: https://doi.org/10.1007/s00707-022-03174-x

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