A mechanics‐informed artificial neural network approach in data‐driven constitutive modeling

Abstract

A mechanics‐informed artificial neural network approach for learning constitutive laws governing complex, nonlinear, elastic materials from strain–stress data is proposed. The approach features a robust and accurate method for training a regression‐based model capable of capturing highly nonlinear strain–stress mappings, while preserving some fundamental principles of solid mechanics. In this sense, it is a structure‐preserving approach for constructing a data‐driven model featuring both the form‐agnostic advantage of purely phenomenological data‐driven regressions and the physical soundness of mechanistic models. The proposed methodology enforces desirable mathematical properties on the network architecture to guarantee the satisfaction of physical constraints such as objectivity, consistency (preservation of rigid body modes), dynamic stability, and material stability, which are important for successfully exploiting the resulting model in numerical simulations. Indeed, embedding such notions in a learning approach reduces a model's sensitivity to noise and promotes its robustness to inputs outside the training domain. The merits of the proposed learning approach are highlighted using several finite element analysis examples. Its potential for ensuring the computational tractability of multi‐scale applications is demonstrated with the acceleration of the nonlinear, dynamic, multi‐scale, fluid‐structure simulation of the supersonic inflation dynamics of a parachute system with a canopy made of a woven fabric.

Finite electro-elasticity with physics-augmented neural networks

“…With the material stability condition fulfilled by construction, it can be expected to have a favorable numerical behavior, c.f. [3]. Thus, in future work we aim to apply the physics-augmented machine learning models for macroscopic finite element simulations of nonlinear EAP composites and metamaterials.…”

“…For the data generation, the following 10 load cases for the macroscopic fields F and d 0 are considered: (1) purely mechanical uniaxial tension, (2) purely mechanical pure shear load, (3)(4) the minimum mechanical value for uniaxial tension, and then two datasets with an electric displacement field applied in X 1 and X 3 direction, respectively, (5-6) the maximum mechanical value for uniaxial tension, and then two datasets with an electric displacement field applied in X 1 and X 3 direction, respectively, (7-8) the minimum mechanical value for shear load, and then two datasets with an electric displacement field applied in X 1 and X 3 direction, respectively, and (9-10) the maximum mechanical value for shear load, and then two datasets with an electric displacement field applied in X 1 and X 3 direction, respectively. With each single load case consisting of 100 data points, this results in an overall calibration dataset size of 1, 000 datapoints.…”

“…For this, a special choice of network architecture is required. Using input convex neural networks (ICNNs) [2], some of the present authors were the first to propose a model which combines the flexibility of ANNs with the polyconvexity condition, see Klein et al [44], later also implemented by [3], and followed by Tac, Costabal, and Tepole [82], who exploited special properties of neural ordinary differential equations [14]. Furthermore, ANNs can be embedded in a modeling framework, by using them only as a part of the overall model and using additional analytical terms, e.g., which fulfill coercivity conditions [44].…”

Neural networks meet hyperelasticity: A guide to enforcing physics

“…To remedy this weakness, a fairly new trend in NN-based constitutive modeling, and in scientific machine learning in general [41], is to include essential underlying physics in a strong form, e.g., by using adapted network architectures, or in a weak form, e.g., by modifying the loss term for the training [34]. These types of approaches, coined as physics-informed [22], mechanics-informed [4], physics-augmented [25], physics-constrained [19], or thermodynamics-based [37], enable an improvement of the extrapolation capability and the usage of sparse training data [22,27], which is particularly important when constitutive models are to be fitted to experimental data.…”

Data-Driven Anisotropic Finite Viscoelasticity Using Neural Ordinary Differential Equations

“…All of these studies enforced the desired constraint in the weak sense using specially designed loss functions that penalize deviations from convexity. Later, methods were developed using input convex neural networks (ICNNs) [9] and neural ordinary differential equations (NODEs) [10] to satisfy convexity conditions a priori [11,3,12,13]. A parallel approach to constitutive modeling of hyperelasticity has been the automated discovery of constitutive laws from a large catalog of existing closed-form expressions [14,15,16,17].…”