An Introduction to the Finite Element Method with the Variational Approach

Prof. Prakash Mahadeo Dixit earned a BTech in Aeronautical Engineering from the Indian Institute of Technology Kharagpur in 1974 and a PhD in Mechanics from the University of Minnesota, USA, in 1979. His teaching journey began as a Lecturer in Aerospace Engineering at IIT Kharagpur from 1980 to 1984. From 1984 to 2018, he was a Professor in Mechanical Engineering at IIT Kanpur. Prof. Dixit has over 38 years experience in teaching and research, notably in Metal Forming, Elasto-Plastic Impact, Ductile Fracture, and Finite Element Method, and has mentored 10 PhD and 80 master's students. He organized 8 short-term courses, benefitting academia and industry professionals. His contributions include 55 journal papers, 31 conference publications, a research monogram, and two textbooks. He also created a web-based Finite Element Method course. Prof. Dixit also has a lifelong membership in the Indian Association of Computational Mechanics. Dr. Sachin Singh Gautam is an Associate Professor in Mechanical Engineering at the Indian Institute of Technology Guwahati, specializing in computational mechanics. His research encompasses isogeometric analysis, contact-impact issues, GPU computing, and machine learning's application in finite elements. Dr. Gautam has an extensive publication record, including 32 journal papers, 17 book chapters, 7 referred conference proceedings, and 83 conference publications. He has guided 3 PhD students and 20 postgraduate dissertations and is currently overseeing 8 PhD candidates and 3 master's students. Dr. Gautam maintains membership in professional organizations, including the Institution of Engineers (India), Indian Association of Computational Mechanics, Indian Society for Theoretical and Applied Mechanics, and Indian Society for Applied Mechanics (ISAM). He also reviews manuscripts for journals such as the International Journal for Numerical Methods in Engineering. Furthermore, he's engaged in developing the contact and isogeometric module for the FEAST® software, a project by VSSC, ISRO.

1. Introduction
2. 1-D Variational Functional
3. 1-D Ritz’s Method
4. 1-D Variational FEM: Rod Extension Problem
5. 1-D Variational FEM: Rod Extension Problem with Point Forces in the Interior
6. Elements and Shape Functions for 1-D Variational FEM
7. 1-D Weighted Residual Integral and Galerkin FEM 7.1. Different Forms of Weighted Residual Integral
8. 1-D Numerical Integration
9. Coding for 1-D Variational and Galerkin FEM
10. 1-D Galerkin FEM for Nonlinear Problems
11. 1-D Galerkin FEM for Time-Dependent Problems
12. 2-D Variational Functional
13. Straight-Sided Elements with C0 Continuity for 2-D Variational FEM
14. 2-D Variational FEM: 2D Steady-State Heat Conduction Problem
15. Straight-Sided Elements with C1 Continuity for 2-D Variational FEM
16. Variational FEM for 2-D Solid Mechanics Problems
17. Variational FEM for 2-D Fluid Mechanics Problems
18. Curved-Sided Elements with C0 Continuity for 2-D Variational FEM
19. 2-D Codes for Solid Mechanics and Heat Transfer Problems
20. Overview of Some Recent Developments
21. Machine Learning and Isogeometric Analysis