This book is intended for students, engineers, and researchers interested in both computational mechanics and deep learning. It presents the mathematical and computational foundations of Deep Learning with detailed mathematical formulas in an easy-to-understand manner. It also discusses various applications of Deep Learning in Computational Mechanics, with detailed explanations of the Computational Mechanics fundamentals selected there. Sample programs are included for the reader to try out in practice. This book is therefore useful for a wide range of readers interested in computational mechanics and deep learning.
Author(s): Genki Yagawa, Atsuya Oishi
Series: Lecture Notes on Numerical Methods in Engineering and Sciences
Publisher: Springer
Year: 2022
Language: English
Pages: 407
City: Cham
Preface
Computational Mechanics
Deep Learning
Computational Mechanics with Deep Learning
Readership
Structure of This book
Acknowledgements
Contents
Part I Fundamentals
1 Overview
1.1 Deep Learning: New Way for Problems Unsolvable by Conventional Methods
1.2 Progress of Deep Learning: From McCulloch–Pitts Model to Deep Learning
1.3 New Techniques for Deep Learning
1.3.1 Numerical Precision
1.3.2 Adversarial Examples
1.3.3 Dataset Augmentation
1.3.4 Dropout
1.3.5 Batch Normalization
1.3.6 Generative Adversarial Networks
1.3.7 Variational Autoencoder
1.3.8 Automatic Differentiation
References
2 Mathematical Background for Deep Learning
2.1 Feedforward Neural Network
2.2 Convolutional Neural Network
2.3 Training Acceleration
2.3.1 Momentum Method
2.3.2 AdaGrad and RMSProp
2.3.3 Adam
2.4 Regularization
2.4.1 What Is Regularization?
2.4.2 Weight Decay
2.4.3 Physics-Informed Network
References
3 Computational Mechanics with Deep Learning
3.1 Overview
3.2 Recent Papers on Computational Mechanics with Deep Learning
References
Part II Case Study
4 Numerical Quadrature with Deep Learning
4.1 Summary of Numerical Quadrature
4.1.1 Legendre Polynomials
4.1.2 Lagrange Polynomials
4.1.3 Formulation of Gauss–Legendre Quadrature
4.1.4 Improvement of Gauss–Legendre Quadrature
4.2 Summary of Stiffness Matrix for Finite Element Method
4.3 Accuracy Dependency of Stiffness Matrix on Numerical Quadrature
4.4 Search for Optimal Quadrature Parameters
4.5 Search for Optimal Number of Quadrature Points
4.6 Deep Learning for Optimal Quadrature of Element Stiffness Matrix
4.6.1 Estimation of Optimal Quadrature Parameters by Deep Learning
4.6.2 Estimation of Optimal Number of Quadrature Points by Deep Learning
4.7 Numerical Example A
4.7.1 Data Preparation Phase
4.7.2 Training Phase
4.7.3 Application Phase
4.8 Numerical Example B
4.8.1 Data Preparation Phase
4.8.2 Training Phase
4.8.3 Application Phase
References
5 Improvement of Finite Element Solutions with Deep Learning
5.1 Accuracy Versus Element Size
5.2 Computation Time versus Element Size
5.3 Error Estimation of Finite Element Solutions
5.3.1 Error Estimation Based on Smoothing of Stresses
5.3.2 Error Estimation Using Solutions Obtained by Various Meshes
5.4 Improvement of Finite Element Solutions Using Error Information and Deep Learning
5.5 Numerical Example
5.5.1 Data Preparation Phase
5.5.2 Training Phase
5.5.3 Application Phase
References
6 Contact Mechanics with Deep Learning
6.1 Basics of Contact Mechanics
6.2 NURBS Basis Functions
6.3 NURBS Objects Based on NURBS Basis Functions
6.4 Local Contact Search for Surface-to-Surface Contact
6.5 Local Contact Search with Deep Learning
6.6 Numerical Example
6.6.1 Data Preparation Phase
6.6.2 Training Phase
6.6.3 Application Phase
References
7 Flow Simulation with Deep Learning
7.1 Equations for Flow Simulation
7.2 Finite Difference Approximation
7.3 Flow Simulation of Incompressible Fluid with Finite Difference Method
7.3.1 Non-dimensional Navier–Stokes Equations
7.3.2 Solution Method
7.3.3 Example: 2D Flow Simulation of Incompressible Fluid Around a Circular Cylinder
7.4 Flow Simulation with Deep Learning
7.5 Neural Networks for Time-Dependent Data
7.5.1 Recurrent Neural Network
7.5.2 Long Short-Term Memory
7.6 Numerical Example
7.6.1 Data Preparation Phase
7.6.2 Training Phase
7.6.3 Application Phase
References
8 Further Applications with Deep Learning
8.1 Deep Learned Finite Elements
8.1.1 Two-Dimensional Quadratic Quadrilateral Element
8.1.2 Improvement of Accuracy of [ B ] Matrix Using Deep Learning
8.2 FEA-Net
8.2.1 Finite Element Analysis (FEA) With Convolution
8.2.2 FEA-Net Based on FEA-Convolution
8.2.3 Numerical Example
8.3 DiscretizationNet
8.3.1 DiscretizationNet Based on Conditional Variational Autoencoder
8.3.2 Numerical Example
8.4 Zooming Method for Finite Element Analysis
8.4.1 Zooming Method for FEA Using Neural Network
8.4.2 Numerical Example
8.5 Physics-Informed Neural Network
8.5.1 Application of Physics-Informed Neural Network to Solid Mechanics
8.5.2 Numerical Example
References
Part III Computational Procedures
9 Bases for Computer Programming
9.1 Computer Programming for Data Preparation Phase
9.1.1 Element Stiffness Matrix
9.1.2 Mesh Quality
9.1.3 B-Spline and NURBS
9.2 Computer Programming for Training Phase
9.2.1 Sample Code for Feedforward Neural Networks in C Language
9.2.2 Sample Code for Feedforward Neural Networks in C with OpenBLAS
9.2.3 Sample Code for Feedforward Neural Networks in Python Language
9.2.4 Sample Code for Convolutional Neural Networks in Python Language
References
10 Computer Programming for a Representative Problem
10.1 Problem Definition
10.2 Data Preparation Phase
10.2.1 Generation of Elements
10.2.2 Calculation of Shape Parameters
10.2.3 Calculation of Optimal Numbers of Quadrature Points
10.3 Training Phase
10.4 Application Phase
References
Index
