An Introduction to the Finite Element Method with the Variational Approach
- 1st Edition - April 24, 2026
- Latest edition
- Authors: Prakash Mahadeo Dixit, Sachin Singh Gautam
- Language: English
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 Elemen… Read more
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Description
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.
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.
Key features
Key features
- 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
Readership
Readership
Graduate, and undergraduate students, practitioners
Table of contents
Table of contents
1: Introduction
1.1. Steps of the FEM
1.2. Historical Background
1.3. Developments after 1960
1.4. Different Approaches for Numerical Solution of Differential Equations
1.5. References on FEM Learning Using Commercial FE Packages
1.6. Expected Learning Objectives and Outcomes
1.7. Plan of the Book
2: 1-D Variational Functional
2.1. Rod Extension Problem (1-D Boundary Value Problem)
2.2. Integral Form Corresponding to the Variational Formulation
2.3. Differential and Extremum of a Function
2.4. Calculus of Variations: Variation and Extremum of a Functional
2.5. Derivation of the Variational Functional
2.6. Derivation of the Euler Equation and Natural Boundary Condition of the Variational Functional
2.7. Physical Interpretations of the Variational Formulation
2.8. Existence of Variational Functional
2.9. Variational Functional for Other BCs and Interior Point Forces
2.10. Variational Functional for Other 1-D Problems
2.11. Summary Exercise Problems
3: 1-D Ritz’s Method
3.1. Ritz Method for the Rod Extension Problem
3.2. Ritz Method for Other 1-D Problems
3.3. Example on Rod Extension Problem
3.4. Convergence of the Ritz Method [1]
3.5. Need for FEM
3.6. Summary Exercise Problems
4: 1-D Variational FEM: Rod Extension Problem
4.1. Discretization and Approximation
4.2. Functional in Terms of Global DOF
4.3. Extremization of the Functional
4.4. Application of the Essential Boundary Condition
4.5. Simplification of Global Assembly
4.6. Development of FE Equations for Other 1-D Problems
4.7. Example on Rod Extension Problem
4.8. Summary Exercise Problems
5: 1-D Variational FEM: Rod Extension Problem with Point Forces in the Interior
5.1. Boundary Value Problem Corresponding to Rod Extension with Interior Point Force
5.2. Variational Functional Corresponding to BV Problem of Section 5.1
5.3. Finite Element Formulation of BV Problem of Section 5.1
5.4. Solvability of FE Equations of Rod Extension Problem
5.5. Physical Interpretation of the Solvability Condition for Rod Extension Problem
5.6. Solvability of FE Equations of Other Problems
5.7. Summary Exercise Problems
6: Elements and Shape Functions for 1-D Variational FEM
6.1. Guidelines for Choosing the Mesh
6.2. Choice of the Approximation
6.3. Simplest Approximation/Element for the Rod Extension and Beam Bending Problems
6.4. Higher Order Approximations for the Rod Extension Problem
6.5. Finite Element Formulations for Beam Bending, Transverse Beam Vibrations and Column Stability Problems
6.6. Summary Exercise Problems
7: 1-D Weighted Residual Integral and Galerkin FEM
7.2. Different Forms of Weighted Residual Method
7.3. Galerkin Finite Element Method
7.4. Weighted Residual Integral for Other BCs and Interior Point Forces
7.5. Weighted Residual Integral for Other 1-D Problems
7.6. Weighted Residual Method for Other 1-D Problems
7.7. Galerkin FEM for Other 1-D Problems
7.8. Summary Exercise Problems
8: 1-D Numerical Integration
8.1. Gauss-Legendre Numerical Integration Scheme
8.2. Mapping of Typical Element e onto Master Interval
8.3. Shape Functions in Natural Coordinate
8.4. Expressions for the Element Coefficient Matrix and Right Side Vector using the Gauss-Legendre Numerical Integration Scheme for the Rod Extension Problem
8.5. Selection of the Number of Gauss Points
8.6. Summary Exercise Problems
9: Coding for 1-D Variational and Galerkin FEM
9.1. 1-D Pre-Processor
9.2. 1-D Processor
9.3. 1-D Post-Processor
9.4. Code for 1-D Rod Extension Problem
9.5. Summary Exercise Problems
10: 1-D Galerkin FEM for Nonlinear Problems
10.1. 1-D Force-Displacement Relation for the Rod with Material Nonlinearity
10.2. Boundary Value Problem for the Rod with Material Nonlinearity and Corresponding Weighted Residual Integral
10.3. 1-D Galerkin FEM for the Rod with Material Nonlinearity
10.4. Iterative Solution Procedure to Solve the Nonlinear FE Equations
10.5. Example on Rod with Material Nonlinearity
10.6. Summary Exercise Problems
11: 1-D Galerkin FEM for Time-Dependent Problems
11.1. Boundary-Initial Value (BIV) Problem Governing the Time-Dependent Rod Extension Problem
11.2. Weighted Residual Integral of the BIV Problem
11.3. 1-D Galerkin FEM for the BIV Problem
11.4. Example on Time-Dependent Rod Extension Problem
11.5. Summary Exercise Problems
12:2-D Variational Functional
12.1. 2-D Steady-State Heat Conduction Problem (2-D Boundary Value Problem)
12.2. Derivation of the Variational Functional of the BV Problem of Section 12.1
12.3. Derivation of the Euler Equation and Natural Boundary Condition of the Variational Functional
12.4. Summary Exercise Problems
13: Straight-Sided Elements with C0 Continuity for 2-D Variational FEM
13.1. Guidelines for Choosing the Mesh
13.2. Choice of the Approximation
13.3. Requirements of Convergence Conditions for Problems with First Derivative of the Variable as the Highest Derivative in the Functional
13.4. Simplest Triangular Element for Problems with C0 Continuity Requirement
13.5. Higher Order Triangular Elements for Problems with C0 Continuity Requirement
13.6. Simplest Rectangular Element for Problems with C0 Continuity Requirement
13.7. Higher Order Rectangular Elements for Problems with C0 Continuity Requirement
13.8. Summary Exercise Problems
14: 2-D Variational FEM: 2D Steady-State Heat Conduction Problem
14.1. Variational Functional in Array Form
14.2.Domain Discretization and Approximation
14.3. Boundary Discretization and Approximation
14.4. Functional in Terms of Global DOF
14.5. Extremization of the Functional
14.6. Application of the Essential Boundary Condition
14.7. Evaluation of Element Quantities
14.8. Global Assembly
14.9. FE Equations when There is Point Heat Source
14.10. FE Equations when There is Second Natural BC (i.e., the Convective BC)
14.11. Symmetry of Geometry, Boundary Conditions and Other Parameters
14.12. Axisymmetric Steady-State Heat Conduction Problem
14.13. Summary Exercise Problems
15: Straight-Sided Elements with C1 Continuity for 2-D Variational FEM
15.1. Convergence Conditions for Plate Bending Problem
15.2. Simplest C1 Continuity Triangular Element
15.3. Simplest C1 Continuity Rectangular Element
15.4. Summary Exercise Problems
16: Variational FEM for 2-D Solid Mechanics Problems
16.1. Torsion of a Shaft of Non-Circular Cross-Section
16.2. Plane Strain Solid Mechanics Problem
16.3. Axisymmetric Solid Mechanics Problem
16.4. Bending of Thin Plates with Negligible Shear Deformation
16.5. Stability of Thin Plates with Negligible Shear Deformation
16.6. Vibrations of Thin Plates with Negligible Shear Deformation
16.7. Summary
17: Variational FEM for 2-D Fluid Mechanics Problems
17.1. 2-D Flow of an Incompressible Inviscid Fluid
17.2. 2-D Flow of an Incompressible Viscous Fluid with Negligible Inertia (Stream Function Formulation)
17.3. Axisymmetric Flow of an Incompressible Viscous Fluid with Negligible Inertia (Stream Function Formulation)
17.4. Summary
18: Curved-Sided Elements with C0 Continuity for 2-D Variational FEM
18.1. Example of Construction Procedure
18.2. Choice of the Parent Element, Mapping Function and Images of the Geometric Nodes
18.3. Approximation for the Primary Variable
18.4 . Types of Curved-Sided Elements
18.4. Summary Exercise Problems
19: 2-D Codes for Solid Mechanics and Heat Transfer Problems
19.1. 2-D Pre-Processor
19.2. 2-D Processor 19.3. 2.D Post-Processor
19.4. Code for 2-D Cantilever Beam Problem
19.5. Code for 2-D Cook’s Beam Problem
19.6. Code for 2-D Plate with Hole Problem
19.7. Code for 2-D Wrench Problem
19.8. Code for 2-D Heat Transfer Problem
19.9. Summary
20: Overview of Some Recent Developments
20.1. New Developments
20.2. Virtual Element Method 20.3. Enriched Finite Elements
20.4 Summary
21: Machine Learning and Isogeometric Analysis
21.1. Review of Machine Learning in Finite Element Analysis
21.2. Application of Machine Learning in Finding Optimal Number of Gauss Quadrature Points
21.3. Application of Machine Learning in Stress Recovery
21.4. Isogeometric Analysis of 1-D Rod Extension Problem
21.5. Isogeometric Analysis of 2-D Solid Mechanics Problem
21.6 Summary
Appendix
A. Function Space
B. Strain Energy Expression for Rod
C. 1-D Variational Functional of Rod Extension Problem 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. 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. Gauss Point Coordinates and Weights for Gauss Numerical Integration Scheme over Master Triangle
L. Shape Functions of Simplest C1 Continuity Rectangular Element
M. Some Commonly Misunderstood Concepts and Terms
N. Practical Finite Element Analysis
1.1. Steps of the FEM
1.2. Historical Background
1.3. Developments after 1960
1.4. Different Approaches for Numerical Solution of Differential Equations
1.5. References on FEM Learning Using Commercial FE Packages
1.6. Expected Learning Objectives and Outcomes
1.7. Plan of the Book
2: 1-D Variational Functional
2.1. Rod Extension Problem (1-D Boundary Value Problem)
2.2. Integral Form Corresponding to the Variational Formulation
2.3. Differential and Extremum of a Function
2.4. Calculus of Variations: Variation and Extremum of a Functional
2.5. Derivation of the Variational Functional
2.6. Derivation of the Euler Equation and Natural Boundary Condition of the Variational Functional
2.7. Physical Interpretations of the Variational Formulation
2.8. Existence of Variational Functional
2.9. Variational Functional for Other BCs and Interior Point Forces
2.10. Variational Functional for Other 1-D Problems
2.11. Summary Exercise Problems
3: 1-D Ritz’s Method
3.1. Ritz Method for the Rod Extension Problem
3.2. Ritz Method for Other 1-D Problems
3.3. Example on Rod Extension Problem
3.4. Convergence of the Ritz Method [1]
3.5. Need for FEM
3.6. Summary Exercise Problems
4: 1-D Variational FEM: Rod Extension Problem
4.1. Discretization and Approximation
4.2. Functional in Terms of Global DOF
4.3. Extremization of the Functional
4.4. Application of the Essential Boundary Condition
4.5. Simplification of Global Assembly
4.6. Development of FE Equations for Other 1-D Problems
4.7. Example on Rod Extension Problem
4.8. Summary Exercise Problems
5: 1-D Variational FEM: Rod Extension Problem with Point Forces in the Interior
5.1. Boundary Value Problem Corresponding to Rod Extension with Interior Point Force
5.2. Variational Functional Corresponding to BV Problem of Section 5.1
5.3. Finite Element Formulation of BV Problem of Section 5.1
5.4. Solvability of FE Equations of Rod Extension Problem
5.5. Physical Interpretation of the Solvability Condition for Rod Extension Problem
5.6. Solvability of FE Equations of Other Problems
5.7. Summary Exercise Problems
6: Elements and Shape Functions for 1-D Variational FEM
6.1. Guidelines for Choosing the Mesh
6.2. Choice of the Approximation
6.3. Simplest Approximation/Element for the Rod Extension and Beam Bending Problems
6.4. Higher Order Approximations for the Rod Extension Problem
6.5. Finite Element Formulations for Beam Bending, Transverse Beam Vibrations and Column Stability Problems
6.6. Summary Exercise Problems
7: 1-D Weighted Residual Integral and Galerkin FEM
7.2. Different Forms of Weighted Residual Method
7.3. Galerkin Finite Element Method
7.4. Weighted Residual Integral for Other BCs and Interior Point Forces
7.5. Weighted Residual Integral for Other 1-D Problems
7.6. Weighted Residual Method for Other 1-D Problems
7.7. Galerkin FEM for Other 1-D Problems
7.8. Summary Exercise Problems
8: 1-D Numerical Integration
8.1. Gauss-Legendre Numerical Integration Scheme
8.2. Mapping of Typical Element e onto Master Interval
8.3. Shape Functions in Natural Coordinate
8.4. Expressions for the Element Coefficient Matrix and Right Side Vector using the Gauss-Legendre Numerical Integration Scheme for the Rod Extension Problem
8.5. Selection of the Number of Gauss Points
8.6. Summary Exercise Problems
9: Coding for 1-D Variational and Galerkin FEM
9.1. 1-D Pre-Processor
9.2. 1-D Processor
9.3. 1-D Post-Processor
9.4. Code for 1-D Rod Extension Problem
9.5. Summary Exercise Problems
10: 1-D Galerkin FEM for Nonlinear Problems
10.1. 1-D Force-Displacement Relation for the Rod with Material Nonlinearity
10.2. Boundary Value Problem for the Rod with Material Nonlinearity and Corresponding Weighted Residual Integral
10.3. 1-D Galerkin FEM for the Rod with Material Nonlinearity
10.4. Iterative Solution Procedure to Solve the Nonlinear FE Equations
10.5. Example on Rod with Material Nonlinearity
10.6. Summary Exercise Problems
11: 1-D Galerkin FEM for Time-Dependent Problems
11.1. Boundary-Initial Value (BIV) Problem Governing the Time-Dependent Rod Extension Problem
11.2. Weighted Residual Integral of the BIV Problem
11.3. 1-D Galerkin FEM for the BIV Problem
11.4. Example on Time-Dependent Rod Extension Problem
11.5. Summary Exercise Problems
12:2-D Variational Functional
12.1. 2-D Steady-State Heat Conduction Problem (2-D Boundary Value Problem)
12.2. Derivation of the Variational Functional of the BV Problem of Section 12.1
12.3. Derivation of the Euler Equation and Natural Boundary Condition of the Variational Functional
12.4. Summary Exercise Problems
13: Straight-Sided Elements with C0 Continuity for 2-D Variational FEM
13.1. Guidelines for Choosing the Mesh
13.2. Choice of the Approximation
13.3. Requirements of Convergence Conditions for Problems with First Derivative of the Variable as the Highest Derivative in the Functional
13.4. Simplest Triangular Element for Problems with C0 Continuity Requirement
13.5. Higher Order Triangular Elements for Problems with C0 Continuity Requirement
13.6. Simplest Rectangular Element for Problems with C0 Continuity Requirement
13.7. Higher Order Rectangular Elements for Problems with C0 Continuity Requirement
13.8. Summary Exercise Problems
14: 2-D Variational FEM: 2D Steady-State Heat Conduction Problem
14.1. Variational Functional in Array Form
14.2.Domain Discretization and Approximation
14.3. Boundary Discretization and Approximation
14.4. Functional in Terms of Global DOF
14.5. Extremization of the Functional
14.6. Application of the Essential Boundary Condition
14.7. Evaluation of Element Quantities
14.8. Global Assembly
14.9. FE Equations when There is Point Heat Source
14.10. FE Equations when There is Second Natural BC (i.e., the Convective BC)
14.11. Symmetry of Geometry, Boundary Conditions and Other Parameters
14.12. Axisymmetric Steady-State Heat Conduction Problem
14.13. Summary Exercise Problems
15: Straight-Sided Elements with C1 Continuity for 2-D Variational FEM
15.1. Convergence Conditions for Plate Bending Problem
15.2. Simplest C1 Continuity Triangular Element
15.3. Simplest C1 Continuity Rectangular Element
15.4. Summary Exercise Problems
16: Variational FEM for 2-D Solid Mechanics Problems
16.1. Torsion of a Shaft of Non-Circular Cross-Section
16.2. Plane Strain Solid Mechanics Problem
16.3. Axisymmetric Solid Mechanics Problem
16.4. Bending of Thin Plates with Negligible Shear Deformation
16.5. Stability of Thin Plates with Negligible Shear Deformation
16.6. Vibrations of Thin Plates with Negligible Shear Deformation
16.7. Summary
17: Variational FEM for 2-D Fluid Mechanics Problems
17.1. 2-D Flow of an Incompressible Inviscid Fluid
17.2. 2-D Flow of an Incompressible Viscous Fluid with Negligible Inertia (Stream Function Formulation)
17.3. Axisymmetric Flow of an Incompressible Viscous Fluid with Negligible Inertia (Stream Function Formulation)
17.4. Summary
18: Curved-Sided Elements with C0 Continuity for 2-D Variational FEM
18.1. Example of Construction Procedure
18.2. Choice of the Parent Element, Mapping Function and Images of the Geometric Nodes
18.3. Approximation for the Primary Variable
18.4 . Types of Curved-Sided Elements
18.4. Summary Exercise Problems
19: 2-D Codes for Solid Mechanics and Heat Transfer Problems
19.1. 2-D Pre-Processor
19.2. 2-D Processor 19.3. 2.D Post-Processor
19.4. Code for 2-D Cantilever Beam Problem
19.5. Code for 2-D Cook’s Beam Problem
19.6. Code for 2-D Plate with Hole Problem
19.7. Code for 2-D Wrench Problem
19.8. Code for 2-D Heat Transfer Problem
19.9. Summary
20: Overview of Some Recent Developments
20.1. New Developments
20.2. Virtual Element Method 20.3. Enriched Finite Elements
20.4 Summary
21: Machine Learning and Isogeometric Analysis
21.1. Review of Machine Learning in Finite Element Analysis
21.2. Application of Machine Learning in Finding Optimal Number of Gauss Quadrature Points
21.3. Application of Machine Learning in Stress Recovery
21.4. Isogeometric Analysis of 1-D Rod Extension Problem
21.5. Isogeometric Analysis of 2-D Solid Mechanics Problem
21.6 Summary
Appendix
A. Function Space
B. Strain Energy Expression for Rod
C. 1-D Variational Functional of Rod Extension Problem 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. 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. Gauss Point Coordinates and Weights for Gauss Numerical Integration Scheme over Master Triangle
L. Shape Functions of Simplest C1 Continuity Rectangular Element
M. Some Commonly Misunderstood Concepts and Terms
N. Practical Finite Element Analysis
Product details
Product details
- Edition: 1
- Latest edition
- Published: April 24, 2026
- Language: English
About the authors
About the authors
PD
Prakash Mahadeo Dixit
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.
Affiliations and expertise
Formerly IIT Kanpur, Department of Mechanical Engineering, Kanpur, IndiaSG
Sachin Singh Gautam
Dr. Sachin Singh Gautam is an Associate Professor in Mechanical Engineering at the Indian Institute of Technology Guwahati, specializing in computational mechanics. He completed his Ph.D. from IIT Kanpur in 2010 and worked as a post-doctoral fellow in AICES, RWTH Aachen University, Germany till 2013 before joining his current position. His research encompasses isogeometric analysis, contact-impact problems, GPU computing, and machine learning's application in finite elements.
Affiliations and expertise
Associate Professor, Mechanical Engineering, Indian Institute of Technology Guwahati, India