Data-driven tissue mechanics with polyconvex neural ordinary differential equations

• Published in: Computer methods in applied mechanics and engineering 2022-08, Vol.398, p.115248, Article 115248
• Main Authors: Tac, Vahidullah, Sahli Costabal, Francisco, Tepole, Adrian B.
• Format: Article
• Language: English
• Subjects: Artificial neural networks; Boundary value problems; Closed form solutions; Computational mechanics; Constitutive modeling; Constraint modelling; Convexity; Derivatives; Differential equations; Exact solutions; Finite element method; Machine learning; Mathematical analysis; Mechanics; Neural networks; Nonlinear finite elements; Ordinary differential equations; Plastic surgery; Skin mechanics; Soft tissues
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2022.115248

Data-driven methods are becoming an essential part of computational mechanics due to their advantages over traditional material modeling. Deep neural networks are able to learn complex material response without the constraints of closed-form models. However, data-driven approaches do not a priori satisfy physics-based mathematical requirements such as polyconvexity, a condition needed for the existence of minimizers for boundary value problems in elasticity. In this study, we use a recent class of neural networks, neural ordinary differential equations (N-ODEs), to develop data-driven material models that automatically satisfy polyconvexity of the strain energy. We take advantage of the properties of ordinary differential equations to create monotonic functions that approximate the derivatives of the strain energy with respect to deformation invariants. The monotonicity of the derivatives guarantees the convexity of the energy. The N-ODE material model is able to capture synthetic data generated from closed-form material models, and it outperforms conventional models when tested against experimental data on skin, a highly nonlinear and anisotropic material. We also showcase the use of the N-ODE material model in finite element simulations of reconstructive surgery. The framework is general and can be used to model a large class of materials, especially biological soft tissues. We therefore expect our methodology to further enable data-driven methods in computational mechanics. [Display omitted] •Data-driven modeling of nonlinear, anisotropic materials with automatically polyconvex energy functions.•Invariant inputs, energy outputs enable finite element implementation for large scale simulations.•Neural ODE model can interpolate analytical models and surpasses analytical models against experimental data.•Neural ODE model finds the closest convex function even when fed non-convex training data.•Demonstration of complex static equilibrium problems, e.g. reconstructive surgery.