5 research outputs found
Learning solution of nonlinear constitutive material models using physics-informed neural networks: COMM-PINN
We applied physics-informed neural networks to solve the constitutive
relations for nonlinear, path-dependent material behavior. As a result, the
trained network not only satisfies all thermodynamic constraints but also
instantly provides information about the current material state (i.e., free
energy, stress, and the evolution of internal variables) under any given
loading scenario without requiring initial data. One advantage of this work is
that it bypasses the repetitive Newton iterations needed to solve nonlinear
equations in complex material models. Additionally, strategies are provided to
reduce the required order of derivation for obtaining the tangent operator. The
trained model can be directly used in any finite element package (or other
numerical methods) as a user-defined material model. However, challenges remain
in the proper definition of collocation points and in integrating several
non-equality constraints that become active or non-active simultaneously. We
tested this methodology on rate-independent processes such as the classical von
Mises plasticity model with a nonlinear hardening law, as well as local damage
models for interface cracking behavior with a nonlinear softening law. Finally,
we discuss the potential and remaining challenges for future developments of
this new approach
Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domains
Physics-informed neural networks (PINNs) are a new tool for solving boundary
value problems by defining loss functions of neural networks based on governing
equations, boundary conditions, and initial conditions. Recent investigations
have shown that when designing loss functions for many engineering problems,
using first-order derivatives and combining equations from both strong and weak
forms can lead to much better accuracy, especially when there are heterogeneity
and variable jumps in the domain. This new approach is called the mixed
formulation for PINNs, which takes ideas from the mixed finite element method.
In this method, the PDE is reformulated as a system of equations where the
primary unknowns are the fluxes or gradients of the solution, and the secondary
unknowns are the solution itself. In this work, we propose applying the mixed
formulation to solve multi-physical problems, specifically a stationary
thermo-mechanically coupled system of equations. Additionally, we discuss both
sequential and fully coupled unsupervised training and compare their accuracy
and computational cost. To improve the accuracy of the network, we incorporate
hard boundary constraints to ensure valid predictions. We then investigate how
different optimizers and architectures affect accuracy and efficiency. Finally,
we introduce a simple approach for parametric learning that is similar to
transfer learning. This approach combines data and physics to address the
limitations of PINNs regarding computational cost and improves the network's
ability to predict the response of the system for unseen cases. The outcomes of
this work will be useful for many other engineering applications where deep
learning is employed on multiple coupled systems of equations for fast and
reliable computations
A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method
Physics-informed neural networks (PINNs) are capable of finding the solution
for a given boundary value problem. We employ several ideas from the finite
element method (FEM) to enhance the performance of existing PINNs in
engineering problems. The main contribution of the current work is to promote
using the spatial gradient of the primary variable as an output from separated
neural networks. Later on, the strong form which has a higher order of
derivatives is applied to the spatial gradients of the primary variable as the
physical constraint. In addition, the so-called energy form of the problem is
applied to the primary variable as an additional constraint for training. The
proposed approach only required up to first-order derivatives to construct the
physical loss functions. We discuss why this point is beneficial through
various comparisons between different models. The mixed formulation-based PINNs
and FE methods share some similarities. While the former minimizes the PDE and
its energy form at given collocation points utilizing a complex nonlinear
interpolation through a neural network, the latter does the same at element
nodes with the help of shape functions. We focus on heterogeneous solids to
show the capability of deep learning for predicting the solution in a complex
environment under different boundary conditions. The performance of the
proposed PINN model is checked against the solution from FEM on two prototype
problems: elasticity and the Poisson equation (steady-state diffusion problem).
We concluded that by properly designing the network architecture in PINN, the
deep learning model has the potential to solve the unknowns in a heterogeneous
domain without any available initial data from other sources. Finally,
discussions are provided on the combination of PINN and FEM for a fast and
accurate design of composite materials in future developments