Nonlinear electro-elastic finite element analysis with neural network constitutive models

TL;DR

This work tackles the challenge of simulating electro-elastic composites under very large deformations and instabilities by developing invariant-based physics-augmented neural network (PANN) constitutive models. The authors couple a neural-network potential with volumetric growth and stress-normalisation terms to enforce thermodynamic consistency, objectivity, symmetry, and polyconvexity when required. They calibrate isotropic and transversely isotropic PANNs to analytically and numerically homogenised microstructures (including RVE with spherical inclusions and rank-one laminates) and demonstrate that PANNs can accurately reproduce first- and second-gradient responses, ensuring ellipticity and stability in nonlinear FE analyses. The results show that non-polyconvex PANNs can learn ellipticity from data and still deliver highly accurate and robust FE predictions for actuation and wrinkling, highlighting the method’s potential for efficient multiscale design and optimization of electro-active composites.

Abstract

In the present work, the applicability of physics-augmented neural network (PANN) constitutive models for complex electro-elastic finite element analysis is demonstrated. For the investigations, PANN models for electro-elastic material behavior at finite deformations are calibrated to different synthetically generated datasets, including an analytical isotropic potential, a homogenised rank-one laminate, and a homogenised metamaterial with a spherical inclusion. Subsequently, boundary value problems inspired by engineering applications of composite electro-elastic materials are considered. Scenarios with large electrically induced deformations and instabilities are particularly challenging and thus necessitate extensive investigations of the PANN constitutive models in the context of finite element analyses. First of all, an excellent prediction quality of the model is required for very general load cases occurring in the simulation. Furthermore, simulation of large deformations and instabilities poses challenges on the stability of the numerical solver, which is closely related to the constitutive model. In all cases studied, the PANN models yield excellent prediction qualities and a stable numerical behavior even in highly nonlinear scenarios. This can be traced back to the PANN models excellent performance in learning both the first and second derivatives of the ground truth electro-elastic potentials, even though it is only calibrated on the first derivatives. Overall, this work demonstrates the applicability of PANN constitutive models for the efficient and robust simulation of engineering applications of composite electro-elastic materials.

Figures (17)

• Figure 1: Illustration of the physics-augmented neural network (PANN) constitutive model. Note that the hidden-layer (yellow) of the NN may be multilayered.
• Figure 2: Microstructures considered in this work. (a) RVE with spherical inclusion. (b) Rank-one laminate composite material (particular case with $\boldsymbol{N}=[1\,0\,0]^T$).
• Figure 3: Uniform deformation induced in a prism of dimensions $[a_0,b_0,c_0]$ after applying a voltage gradient $\Delta V=-\frac{e_0}{c_0}$, yielding a uniformly deformed prism with dimensions $[a,b,c]$.
• Figure 4: Equilibrium path for the PANN model and the RVE with a spherical inclusion. Contour plot distribution of $\alpha_2$ (mechanical micro-fluctuation) and $\beta$ (micro-fluctuation of the electric potential). The PANN model shows an excellent prediction quality.
• Figure 5: Stress correspondence plots for the (a) polyconvex $\mathcal{SP}^+(64)$ and the (b) non-polyconvex $\mathcal{SP}(8)$ PANN models calibrated to rank-one laminate data, evaluated for the $P_{31}$ stress of the test dataset. The polyconvex model shows some visible deviations from the ground truth model, while the non-polyconvex model shows an excellent performance.

Theorems & Definitions (5)

• Theorem A.1
• proof
• Corollary A.2
• Theorem A.3
• proof