An Introduction to the Finite Element Method with the Variational Approach

Prakash Mahadeo Dixit, Sachin Singh Gautam

• Publisher: Academic Press
• Year: 2026
• Topic: Technology & Engineering
• eBook ISBN:
  9780443333903
• Subtopic: Fluid Mechanics
• Index: Technology & Engineering

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

• Front Cover
• An Introduction to the Finite Element Method with the Variational Approach
• An Introduction to the Finite Element Method with the Variational Approach
• Copyright
• Dedication
• Contents
• Preface
• 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 for Problems 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: 2-D 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 - Galerkin FEM for Nonlinear Elastic Problems
• A - Function Space
• B - Strain Energy Expression for Rod
• C - 1-D Variational Functionals of Rod Extension and Beam Bending Problems with Interior Point Force
• D - Derivation of Shape Functions of Three-Noded 1-D Lagrangian Element
• E - Three Special Cases of 1-D Weighted Residual Method
• F - Gauss Point Coordinates and Weights for 1-D Gauss-Legendre Numerical Integration Scheme
• G - Appendix G: Shape Functions of Some 1-D Lagrangian and Hermitian Elements
• H - 1-D Finite Difference Scheme for First Derivative
• I - Complete Polynomials in Two Coordinates
• J - Shape Functions of Some C0 Continuity Triangular and Rectangular Elements
• K - Appendix K: Gauss Point Coordinates and Weights for Gauss Numerical Integration Scheme over Master Triangle
• L - Appendix L: Shape Functions of Simplest C1 Continuity Rectangular Element
• M - Appendix M: Some Commonly Misunderstood Concepts and Terms
• N - Practical Finite Element Analysis
• Index
• Back Cover