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Request a sales quote*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.### Prakash Mahadeo Dixit

### Sachin Singh Gautam

- 1st Edition - April 1, 2025
- Authors: Prakash Mahadeo Dixit, Sachin Singh Gautam
- Language: English
- Paperback ISBN:9 7 8 - 0 - 4 4 3 - 3 3 3 8 9 - 7
- eBook ISBN:9 7 8 - 0 - 4 4 3 - 3 3 3 9 0 - 3

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

LIMITED OFFER

Immediately download your ebook while waiting for your print delivery. No promo code needed.

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

Graduate, and undergraduate students, practitioners

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

- No. of pages: 850
- Language: English
- Edition: 1
- Published: April 1, 2025
- Imprint: Academic Press
- Paperback ISBN: 9780443333897
- eBook ISBN: 9780443333903

PD

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

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