Meshless physics‐informed deep learning method for three‐dimensional solid mechanics

• Published in: International journal for numerical methods in engineering Vol. 122; no. 23; pp. 7182 - 7201
• Main Authors: Abueidda, Diab W., Lu, Qiyue, Koric, Seid
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 15.12.2021; Wiley Subscription Services, Inc
• Subjects: Collocation methods; computational mechanics; Deep learning; Deformation; Finite element method; machine learning; Mathematical models; meshfree method; Meshless methods; Neural networks; Numerical methods; Partial differential equations; physics‐informed learning; Solid mechanics
• DOI: 10.1002/nme.6828
• ISSN: 0029-5981

Deep learning (DL) and the collocation method are merged and used to solve partial differential equations (PDEs) describing structures' deformation. We have considered different types of materials: linear elasticity, hyperelasticity (neo‐Hookean) with large deformation, and von Mises plasticity with isotropic and kinematic hardening. The performance of this deep collocation method (DCM) depends on the architecture of the neural network and the corresponding hyperparameters. The presented DCM is meshfree and avoids any spatial discretization, which is usually needed for the finite element method (FEM). We show that the DCM can capture the response qualitatively and quantitatively, without the need for any data generation using other numerical methods such as the FEM. Data generation usually is the main bottleneck in most data‐driven models. The DL model is trained to learn the model's parameters yielding accurate approximate solutions. Once the model is properly trained, solutions can be obtained almost instantly at any point in the domain, given its spatial coordinates. Therefore, the DCM is potentially a promising standalone technique to solve PDEs involved in the deformation of materials and structural systems as well as other physical phenomena.