[Paper Review] Physics informed deep learning for computational elastodynamics without labeled data

Chengping Rao, Hao Sun+1 more|arXiv (Cornell University)|Jun 10, 2020

Model Reduction and Neural Networks39 references33 citations

TL;DR

The paper develops a physics-informed neural network (PINN) for elastodynamics that trains without labeled data, using mixed displacement-stress outputs and a hard enforcement scheme for initial/boundary conditions via a composite of three neural networks, and demonstrates static, dynamic, and wave-propagation problems.

ABSTRACT

Numerical methods such as finite element have been flourishing in the past decades for modeling solid mechanics problems via solving governing partial differential equations (PDEs). A salient aspect that distinguishes these numerical methods is how they approximate the physical fields of interest. Physics-informed deep learning is a novel approach recently developed for modeling PDE solutions and shows promise to solve computational mechanics problems without using any labeled data. The philosophy behind it is to approximate the quantity of interest (e.g., PDE solution variables) by a deep neural network (DNN) and embed the physical law to regularize the network. To this end, training the network is equivalent to minimization of a well-designed loss function that contains the PDE residuals and initial/boundary conditions (I/BCs). In this paper, we present a physics-informed neural network (PINN) with mixed-variable output to model elastodynamics problems without resort to labeled data, in which the I/BCs are hardly imposed. In particular, both the displacement and stress components are taken as the DNN output, inspired by the hybrid finite element analysis, which largely improves the accuracy and trainability of the network. Since the conventional PINN framework augments all the residual loss components in a "soft" manner with Lagrange multipliers, the weakly imposed I/BCs cannot not be well satisfied especially when complex I/BCs are present. To overcome this issue, a composite scheme of DNNs is established based on multiple single DNNs such that the I/BCs can be satisfied forcibly in a "hard" manner. The propose PINN framework is demonstrated on several numerical elasticity examples with different I/BCs, including both static and dynamic problems as well as wave propagation in truncated domains. Results show the promise of PINN in the context of computational mechanics applications.

Motivation & Objective

• Motivate and model elastodynamics problems without labeled data using PINNs.
• Incorporate both displacement and stress fields as DNN outputs to improve accuracy and trainability.
• Develop a hard enforcement scheme for initial/boundary conditions to ensure unique solutions.
• Demonstrate the framework on static and dynamic elastodynamics problems and wave propagation in bounded/truncated domains.

Proposed method

• Use a mixed-variable PINN output (u, sigma) to approximate displacement and stress fields.
• Formulate elastodynamics via strong form with governing equations and constitutive relations.
• Introduce a composite hard-BC enforcement scheme using three interconnected networks (particular, distance, general) to satisfy I/BCs exactly.
• Pre-train the distance and particular solution networks, then train the general solution network to satisfy the PDE residuals.
• Evaluate residuals at collocation points and employ automatic differentiation for exact derivatives.

Experimental results

Research questions

• RQ1 Can a PINN model solve elastodynamics without labeled data while accurately capturing displacement and stress fields?
• RQ2 Does hard enforcement of initial/boundary conditions via a composite DNN improve accuracy over soft enforcement in elastodynamics problems?
• RQ3 How does the mixed displacement-stress output affect training stability and prediction accuracy in elastodynamics PINNs?
• RQ4 Can the framework handle static, dynamic, and wave-propagation scenarios in bounded and truncated domains?

Key findings

• The mixed u, sigma PINN output improves training and accuracy compared to displacement-only formulations.
• Hard enforcement of I/BCs via the composite DNN scheme yields more accurate boundary and initial condition satisfaction than soft enforcement.
• The approach demonstrates convergence of stress and displacement fields toward FE solutions as network capacity increases.
• Dynamic and cyclic loading cases show the PINN can capture time-evolving stress distributions and wave propagation in bounded domains.
• Predicted stress fields and displacement fields show good agreement with reference FE solutions in presented examples.