Heterogeneous Peridynamic Neural Operators: Discover Biotissue Constitutive Law and Microstructure From Digital Image Correlation Measurements

TL;DR

The heterogeneous peridynamic neural operator (HeteroPNO) approach, for data-driven constitutive modeling of heterogeneous anisotropic materials, can learn a constitutive model for a biological tissue with anisotropic heterogeneous response undergoing large deformation regime.

Abstract

Human tissues are highly organized structures with collagen fiber arrangements varying from point to point. Anisotropy of the tissue arises from the natural orientation of the fibers, resulting in location-dependent anisotropy. Heterogeneity also plays an important role in tissue function. It is therefore critical to discover and understand the distribution of fiber orientations from experimental mechanical measurements such as digital image correlation (DIC) data. To this end, we introduce the Heterogeneous Peridynamic Neural Operator (HeteroPNO) approach for data-driven constitutive modeling of heterogeneous anisotropic materials. Our goal is to learn a nonlocal constitutive law together with the material microstructure, in the form of a heterogeneous fiber orientation field, from load-displacement field measurements. We propose a two-phase learning approach. Firstly, we learn a homogeneous constitutive law in the form of a neural network-based kernel function and a nonlocal bond force, to capture complex homogeneous material responses from data. Then, in the second phase we reinitialize the learnt bond force and the kernel function, and training them together with a fiber orientation field for each material point. Owing to the state-based peridynamic skeleton, our HeteroPNO-learned material models are objective and have the balance of linear and angular momentum guaranteed. Moreover, the effects from heterogeneity and nonlinear constitutive relationship are captured by the kernel function and the bond force respectively, enabling physical interpretability. As a result, our HeteroPNO architecture can learn a constitutive model for a biological tissue with anisotropic heterogeneous response undergoing large deformation regime. Moreover, the framework is capable to provide displacement and stress field predictions for new and unseen loading instances.

Figures (22)

• Figure 1: Schematic of the hetero-PNO mechanism, with a fundamental influence (kernel) function $\omega$ corresponding to a horizontal fiber orientation, which is rotated for each point independently, according to the local fiber angle.
• Figure 2: Demonstration of the heterogeneous synthetic data generation, under HGO constitutive law and heterogeneous fiber orientation field. Top row: FEM boundary value problem set up (right) and one instance of the random applied external forces (left). Middle and bottom rows: prescribed collagen fiber orientation in the two examples (right), and the resulting displacement fields (left) associated with the external forces shown on the top.
• Figure 3: Synthetic dataset, example 1: HomoPNO and HeteroPNO predictions of displacement field (given external load and boundary conditions), and external forces (given displacement field) against the ground truth.
• Figure 4: Synthetic dataset, example 1: HomoPNO and HeteroPNO predictions of the first Piola-Kirchhoff stress tensor against the ground truth.
• Figure 5: Synthetic dataset, example 1: Discovered hidden fiber orientation by Hetero-PNO on the synthetic data set and comparison against the ground truth. The averaged absolute error for the learned orientation map is 6.55$^{\text{o}}$.