Description
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.- Explains the fundamentals of the Finite Element Method (FEM) with a focus on linear and time-independent aspects, employing a variational approach- Covers variational FEM formulations for 1-D and 2-D scenarios in solid mechanics, fluid mechanics, and heat conduction problems- Explores the application of 1-D Galerkin FEM to address challenges presented by material nonlinearity and time-dependent problems- Delves into the intricacies of FEM algorithms and provides a comprehensive overview of coding implementation- Offers insights into Machine Learning and includes a section on Isogeometric analysis
Table of Contents
1. Introduction2. 1-D Variational Functional3. 1-D Ritz's Method4. 1-D Variational FEM: Rod Extension Problem5. 1-D Variational FEM: Rod Extension Problem with Point Forces in the Interior6. Elements and Shape Functions for 1-D Variational FEM7. 1-D Weighted Residual Integral and Galerkin FEM8. 1-D Numerical Integration9. Coding for 1-D Variational and Galerkin FEM10. 1-D Galerkin FEM for Nonlinear Problems11. 1-D Galerkin FEM for Time-Dependent Problems12. 2-D Variational Functional13. Straight-Sided Elements with C0 Continuity for 2-D Variational FEM14. 2-D Variational FEM: 2D Steady-State Heat Conduction Problem15. Straight-Sided Elements with C1 Continuity for 2-D Variational FEM16. Variational FEM for 2-D Solid Mechanics Problems17. Variational FEM for 2-D Fluid Mechanics Problems18. Curved-Sided Elements with C0 Continuity for 2-D Variational FEM19. 2-D Codes for Solid Mechanics and Heat Transfer Problems20. Overview of Some Recent Developments21. Machine Learning and Isogeometric Analysis22. 2-D Galerkin FEM for Time-Dependent Problems23. 2-D Galerkin FEM for Nonlinear Elastic Problems
