When subjected to large deformations, hyperelastic materials, such as polymers and soft tissues, present a highly nonlinear behaviour. This often makes their constitutive description complex and computationally expensive. Surrogate models can replace these traditional and costly models and overcome some computational limitations by learning directly from acquired data. Currently, the usage of surrogate models in commercial finite element (FE) software, such as Abaqus, is non-existent. Therefore, this work advances the current state of the art by developing a methodology to use surrogate models to describe the constitutive behaviour of hyperelastic materials. These models were then incorporated into the FE commercial software, Abaqus, using a user defined material – UMAT, programmed in Fortran Language. Artificial neural networks were trained to predict the isochoric part of the Cauchy stress tensor and the spatial elasticity tensor from the existing data. With the parameters of the trained neural networks, a user defined material was developed. The present method was validated using classical benchmark problems, and the results obtained using the developed constitutive models were compared with the ones obtained with the conventional approach. The correctness of the obtained results highlights the possibility of using data-driven constitutive models to describe the behaviour of incompressible hyperelastic materials. The developed method can be a viable alternative to avoid the need to directly express a given material constitutive equation that is not implemented in the software and show the potential to use this to model materials with experimental data, without the need to derive a material model.
KalinaKALindenLBrummundJ, et al.Automated constitutive modeling of isotropic hyperelasticity based on artificial neural networks. Comput Mech2022; 69: 213–232.
2.
LiuMLiangLSunW. A generic physics-informed neural network-based constitutive model for soft biological tissues. Comput Methods Appl Mech Eng2020; 372: 113402.
3.
SajjadiniaSSCarpentieriBShriramD, et al.Multi-fidelity surrogate modeling through hybrid machine learning for biomechanical and finite element analysis of soft tissues. Comput Biol Med2022; 148: 105699.
4.
HumphreyJO’RourkeSL. An introduction to biomechanics: solids and fluids, analysis and design. New York, NY: Springer. 2015. ISBN https://doi.org/10.1007/978-1-4939-2623-7. https://doi.org/10.1007/978-1-4939-2623-7.
5.
HannunACaseCCasperJ, et al.Deep speech: scaling up end-to-end speech recognition. 2014, DOI: 10.48550/arXiv.1412.5567.
6.
HeKZhangXRenS, et al.Delving deep into rectifiers: surpassing human-level performance on imagenet classification. In: 2015 IEEE international conference on computer vision (ICCV), 2015, pp.1026–1034. ISBN 2380-7504, DOI: 10.1109/ICCV.2015.123.
7.
HeKZhangXRenS, et al.Deep residual learning for image recognition. In: 2016 IEEE conference on computer vision and pattern recognition (CVPR), 2016, pp.770–778. ISBN 1063-6919, DOI: 10.1109/CVPR.2016.90.
8.
HeXAvrilSLuJ. Machine learning prediction of tissue strength and local rupture risk in ascending thoracic aortic aneurysms. Mol Cell Biomech2019; 16: 50–52.
9.
KokkinosI. Pushing the boundaries of boundary detection using deep learning. 2015, DOI: 10.48550/arXiv.1511.07386.
10.
KrizhevskyASutskeverIHintonGE. Imagenet classification with deep convolutional neural networks. Commun ACM2017; 60: 84–90.
11.
LeCunYBengioYHintonG. Deep learning. Nature2015; 521: 436–444.
12.
TaigmanYYangMRanzatoM, et al.Deepface: closing the gap to human-level performance in face verification. In: 2014 IEEE conference on computer vision and pattern recognition, 2014, pp.1701–1708. ISBN 1063-6919, DOI: 10.1109/CVPR.2014.220.
13.
WuYSchusterMChenZ, et al.Google’s neural machine translation system: bridging the gap between human and machine translation. 2016, DOI: 10.48550/arXiv.1609.08144.
14.
CillaMPérez-ReyIMartínezMA, et al.On the use of machine learning techniques for the mechanical characterization of soft biological tissues. Int J Numer Method Biomed Eng2018; 34: e3121.
15.
DabiriYVan der VeldenASackKL, et al.Prediction of left ventricular mechanics using machine learning. Front Phys2019; 7. DOI: https://doi.org/10.3389/fphy.2019.00117.
16.
DoHNIjazAGharahiH, et al.Prediction of abdominal aortic aneurysm growth using dynamical Gaussian process implicit surface. IEEE Trans Biomed Eng2019; 66: 609–622.
17.
JiangZDoHChoiJ, et al.A deep learning approach to predict abdominal aortic aneurysm expansion using longitudinal data. Front Phys2020; 7: 235.
18.
LuoYFanZBaekS, et al.Machine learning–aided exploration of relationship between strength and elastic properties in ascending thoracic aneurysm. Int J Numer Method Biomed Eng2018; 34: e2977.
19.
LinkaKHillgärtnerMAbdolaziziKP, et al.Constitutive artificial neural networks: a fast and general approach to predictive data-driven constitutive modeling by deep learning. J Comput Phys2021; 429: 110010.
20.
LiuZWuC. Exploring the 3D architectures of deep material network in data-driven multiscale mechanics. J Mech Phys Solids2019; 127: 20–46.
21.
LiuZWuCKoishiM. A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials. Comput Methods Appl Mech Eng2019; 345: 1138–1168.
22.
GhaboussiJWuXPhDGK. Neural network material modelling. Statyba1999; 5: 250–257.
23.
HoerigCGhaboussiJWangY, et al.Machine learning in model-free mechanical property imaging: novel integration of physics with the constrained optimization process. In: Frontiers of physics. 2021, https://api.semanticscholar.org/CorpusID:236503901.
24.
ZhangWRossiniGKamenskyD, et al.Isogeometric finite element-based simulation of the aortic heart valve: integration of neural network structural material model and structural tensor fiber architecture representations. Int J Numer Method Biomed Eng2021; 37: e3438.
25.
KudelaJMatousekR. Recent advances and applications of surrogate models for finite element method computations: a review. Soft comput2022; 26: 13709–13733.
26.
GhaboussiJGarrettJHWuX. Knowledge-based modeling of material behavior with neural networks. J Eng Mech1991; 117: 132–153.
27.
WuXGarrett JrJHGhaboussiJ. Representation of material behavior: neural network-based models. In: 1990 IJCNN international joint conference on neural networks, vol. 1, 1990, pp.229–234. DOI: 10.1109/IJCNN.1990.137574.
28.
HashashYMAJungSGhaboussiJ. Numerical implementation of a neural network based material model in finite element analysis. Int J Numer Methods Eng2004; 59: 989–1005.
29.
As’adFAveryPFarhatC. A mechanics-informed artificial neural network approach in data-driven constitutive modeling. Int J Numer Methods Eng2022; 123: 2738–2759.
30.
DalHDenliFAAçanAK, et al.Data-driven hyperelasticity–part I: a canonical isotropic formulation for rubberlike materials. Available at SSRN 4386964, 2023, DOI: 10.2139/ssrn.4386964.
31.
TacVLinkaKSahli-CostabalF, et al.Benchmarks for physics-informed data-driven hyperelasticity. 2023, DOI: 10.48550/arXiv.2301.10714.
32.
FuhgJNBouklasNJonesRE. Learning hyperelastic anisotropy from data via a tensor basis neural network. J Mech Phys Solids2022; 168: 105022.
33.
LinkaKKuhlE. A new family of constitutive artificial neural networks towards automated model discovery. Comput Methods Appl Mech Eng2023; 403: 115731.
34.
TacVSreeVDRauschMK, et al.Data-driven modeling of the mechanical behavior of anisotropic soft biological tissue. Eng Comput2022; 38: 4167–4182.
35.
HolzapfelGA. (2000) Nonlinear solid mechanics: a continuum approach for engineering. New York: John Wiley and Sons.
36.
TreloarLRG. The elasticity of a network of long-chain molecules. II. Rubber Chem Technol1944; 17: 296–302.
37.
HolzapfelGGasserTOgdenR. A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast2000; 61: 1–48.
38.
GasserTCOgdenRWHolzapfelGA. Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J R Soc Interface2006; 3: 15–35.
39.
RosenblattF. The perceptron: a probabilistic model for information storage and organization in the brain. Psychol Rev1958; 65: 386.
40.
KelleherJD. Deep learning. Cambridge, MA: MIT press. 2019.
41.
KingmaDPBaJ. Adam: a method for stochastic optimization. 2017.
42.
AkibaTSanoSYanaseT, et al.Optuna: a next-generation hyperparameter optimization framework. In: Proceedings of the ACM SIGKDD international conference on knowledge discovery and data mining, 2019, pp.2623–2631. DOI: 10.1145/3292500.3330701. Cited by: 970.
43.
BergstraJBardenetRBengioY, et al.Algorithms for hyper-parameter optimization. Adv Neural Inf Process Syst2011; 24: 2546–2554.
