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.

Table of Contents:
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 22. 2-D Galerkin FEM for Time-Dependent Problems 23. 2-D Galerkin FEM for Nonlinear Elastic Problems