HiDeNN-FEM: A seamless machine learning approach to nonlinear finite element analysis

Area of Science:

• Computational Mechanics
• Artificial Intelligence in Engineering
• Numerical Analysis

Background:

• Traditional finite element methods (FEM) face challenges with accuracy and element distortion in complex simulations.
• Hierarchical deep-learning neural networks (HiDeNN) offer a systematic approach for numerical approximations in differential equations.

Purpose of the Study:

• To present a framework for nonlinear finite element analysis based on HiDeNN approximation (nonlinear HiDeNN-FEM).
• To integrate deep neural network building blocks for enhanced numerical computations.

Main Methods:

• Developed a nonlinear HiDeNN-FEM framework utilizing structured deep neural networks.
• Implemented three core building blocks: partial derivative operator, r-adaptivity, and material derivative.
• Employed two-step optimization schemes for r-adaptivity and integration with existing FE solvers.

Main Results:

• Demonstrated significantly higher accuracy for nonlinear HiDeNN-FEM with r-adaptivity compared to regular FEM in 2D and 3D examples.
• Showcased a substantial reduction in element distortion.
• Effectively suppressed the hourglass mode, a common artifact in FEM.

Conclusions:

• The proposed nonlinear HiDeNN-FEM framework offers a powerful and accurate alternative for solving complex engineering problems.
• The integration of deep learning enhances the capabilities of traditional finite element analysis, improving simulation reliability and efficiency.