[Paper Review] Meshless physics-informed deep learning method for three-dimensional solid mechanics

Diab Abueidda, Qiyue Lu+1 more|arXiv (Cornell University)|Dec 2, 2020

Model Reduction and Neural NetworksPhysics and Astronomy83 references154 citations

TL;DR

This paper proposes a meshfree physics-informed deep learning method, termed the Deep Collocation Method (DCM), that solves three-dimensional partial differential equations in solid mechanics without requiring data generation or meshing. By training a deep neural network to satisfy the strong form of PDEs and boundary conditions via collocation, the method achieves accurate, real-time solutions for linear elasticity, hyperelasticity, and elastoplasticity with isotropic and kinematic hardening.

ABSTRACT

Deep learning and the collocation method are merged and used to solve partial differential equations 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 deep learning 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 deep collocation method is potentially a promising standalone technique to solve partial differential equations involved in the deformation of materials and structural systems as well as other physical phenomena.

Motivation & Objective

• To develop a meshfree alternative to the finite element method for solving PDEs in solid mechanics.
• To eliminate the need for data generation via traditional numerical methods such as FEM, which is often the bottleneck in data-driven models.
• To enable fast, real-time inference of displacement and stress fields using a single trained deep neural network.
• To extend physics-informed deep learning to complex material behaviors, including large deformation and elastoplasticity with hardening.
• To demonstrate the feasibility and accuracy of the deep collocation method across diverse constitutive models in 3D.

Proposed method

• The method uses a deep neural network (DNN) to approximate the displacement field in the strong form of the PDEs.
• The loss function is constructed from the residual of the strong form PDEs, boundary conditions (essential and natural), and is minimized using optimizers like Adam and L-BFGS.
• Automatic differentiation computes gradients of the displacement field, enabling evaluation of strain, stress, and internal variables.
• The method is applied to three constitutive models: linear elasticity, neo-Hookean hyperelasticity, and von Mises plasticity with isotropic and kinematic hardening.
• A pseudo-time stepping scheme is used for path-dependent plasticity, with state variables carried over between steps.
• Sampling of collocation points is performed via Monte Carlo, with separate sets for interior, displacement, and traction boundaries.

Experimental results

Research questions

• RQ1 Can a physics-informed deep learning method solve 3D solid mechanics problems without requiring data generated by FEM or other numerical methods?
• RQ2 How accurately can a deep neural network approximate solutions for linear elasticity, hyperelasticity, and elastoplasticity using only the strong form of the PDEs and boundary conditions?
• RQ3 Does the deep collocation method maintain accuracy and efficiency across different material models, including those with complex hardening behavior?
• RQ4 How do sampling strategies and optimization techniques affect the convergence and accuracy of the DCM in 3D problems?
• RQ5 Can the method handle path-dependent problems like elastoplasticity without retraining or re-sampling at each step?

Key findings

• The DCM achieves qualitative and quantitative agreement with FEM solutions for 3D cantilever beam problems under linear elasticity, hyperelasticity, and elastoplasticity.
• For the elastoplastic case with C = 0.2 m, the DCM predicted vertical displacement, von Mises stress, and plastic strain components that closely matched FEM results.
• The method enables near-instantaneous inference of solutions at any spatial point after training, bypassing expensive re-simulation.
• The use of automatic differentiation allows accurate computation of gradients and stress fields without mesh-based integration.
• The method avoids meshing and tangent matrix assembly, reducing computational overhead typical in FEM.
• The study demonstrates that physics-informed deep learning can serve as a standalone solver for complex solid mechanics problems without data-driven pretraining.