An Introduction to the Finite Element Method with the Variational Approach offers a comprehensive solution to the gaps often found in introductory texts on the Finite Element Method (FEM). The book provides a thorough introduction to the fundamental principles of linear and time-independent FEM within the variational framework. It meticulously covers the derivation of 1-D FEM equations based on variational functionals, encompassing both linear and higher-order elements, and shape functions driven by convergence criteria. Furthermore, it explores 1-D numerical integration, outlines coding procedures, and provides insights into handling material nonlinearity and time-dependent scenarios.
Expanding into 2-D problems, the book offers derivations of 2-D FEM equations tailored to diverse engineering disciplines, including Steady-State Heat Conduction, Solid Mechanics (covering torsion, plane strain/axisymmetric cases, and the bending, stability, and vibrations of thin plates), as well as Fluid Mechanics (addressing incompressible inviscid and viscous fluids). It includes detailed discussions on element continuity, numerical integration techniques, and even includes 2-D codes for selected problems. The book concludes by delving into recent advancements in FEM, with a specific focus on applications in machine learning and isogeometric analysis.

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 in 1980 and ended as a Professor in Mechanical Engineering at IIT Kanpur in 2018. His research work is in the areas of metal forming processes, ductile fracture and damage mechanics, contact-impact problems and dynamic, large deformation, damage-coupled, thermo-elasto-plastic, contact finite element formulation.

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
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