DEDEM: Discontinuity Embedded Deep Energy Method for solving fracture mechanics problems

TL;DR

Results demonstrate that DEDEM can accurately model the mechanical behaviors of cracks on a large variety of fracture problems, and it is found that DEDEM achieves significantly higher computational efficiency and accuracy than the existing methods based on domain decomposition techniques.

Abstract

Physics-Informed Neural Networks (PINNs) have aroused great attention for its ability to address forward and inverse problems of partial differential equations. However, approximating discontinuous functions by neural networks poses a considerable challenge, which results in high computational demands and low accuracy to solve fracture mechanics problems within standard PINNs framework. In this paper, we present a novel method called Discontinuity Embedded Deep Energy Method (DEDEM) for modeling fracture mechanics problems. In this method, interfaces and internal boundaries with weak/strong discontinuities are represented by discontinuous functions constructed by signed distance functions, then the representations are embedded to the input of the neural network so that specific discontinuous features can be imposed to the neural network solution. Results demonstrate that DEDEM can accurately model the mechanical behaviors of cracks on a large variety of fracture problems. Besides, it is also found that DEDEM achieves significantly higher computational efficiency and accuracy than the existing methods based on domain decomposition techniques.

Figures (8)

• Figure 1: The schematic of DEDEM: (a) The schematic of a problem with a crack; (b) strong and (c) weak discontinuous functions constructed from SDF; (d) The whole training workflow.
• Figure 2: An example of the embedding of a crack: (a) A domain with a crack; (b) the sign function constructed from $\varphi$ to separate the crack surface and (c) continuous functions constructed from $\psi$ to guarantee the continuity outside the crack; (d) The constructed embedding $\gamma$ for describing the crack.
• Figure 3: A center crack in a homogeneous plate: (a) the geometry of the symmetric part for computation; (b) Comparison of $K_1$ with the reference solution Tada2000; (c) Sampling points for numerical integration and (d) the crack embedding for the case $a/b=0.5$; Predicted solutions and absolute error compared with FEM for the case $a=0.5$m: displacement $u_2$ (e,f) and stress $\sigma_{22}$ (g,h).
• Figure 4: A bi-material interface crack: (a) the geometry of the symmetric part for computation; (b) Comparison of SIFs with the reference solution miyazaki1993stress; (c-j): Predicted displacement and stress and absolute error compared with FEM for the case $E_1/E_2=10$.
• Figure 5: A plate with multiple cracks: (a) The geometry; (b) The domain decomposition strategy for CENN and CPINN; (c-d) The embeddings to describe cracks for DEDEM; The evolution history of rRMSE on (e) displacement and (f) von-Mises stress predicted by CENN, CPINN and DEDEM.