Basic Finite Element Method as Applied to Injury Biomechanics - 1st Edition - ISBN: 9780128098318, 9780128098325

Basic Finite Element Method as Applied to Injury Biomechanics

1st Edition

Authors: King-Hay Yang
eBook ISBN: 9780128098325
Paperback ISBN: 9780128098318
Imprint: Academic Press
Published Date: 20th September 2017
Page Count: 748
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Basic Finite Element Method as Applied to Injury Biomechanics provides a unique introduction to finite element methods. Unlike other books on the topic, this comprehensive reference teaches readers to develop a finite element model from the beginning, including all the appropriate theories that are needed throughout the model development process.

In addition, the book focuses on how to apply material properties and loading conditions to the model, how to arrange the information in the order of head, neck, upper torso and upper extremity, lower torso and pelvis and lower extremity. The book covers scaling from one body size to the other, parametric modeling and joint positioning, and is an ideal text for teaching, further reading and for its unique application to injury biomechanics.

With over 25 years of experience of developing finite element models, the author's experience with tissue level injury threshold instead of external loading conditions provides a guide to the "do’s and dont's" of using finite element method to study injury biomechanics.

Key Features

  • Covers the fundamentals and applications of the finite element method in injury biomechanics
  • Teaches readers model development through a hands-on approach that is ideal for students and researchers
  • Includes different modeling schemes used to model different parts of the body, including related constitutive laws and associated material properties


Biomedical Engineers, Biomechanical Engineers, Graduate Students, Clinicians, R&D Professionals, Mechanical Engineers, Materials Scientists

Table of Contents


Chapter 1 Introduction

1.1 Finite element method and analysis

1.2 Calculation of strain and stress from the FE model

1.2.1 Average strain and point strain

1.2.2 Normal and shear strain

1.2.3 Calculation of stress

1.3 Sample matrix structural analysis

1.3.1 Element stiffness matrix of a linear spring

1.3.2 Element stiffness matrix of a linear spring not in line with the x-axis

1.3.3 Element stiffness matrix of a homogeneous linear elastic bar

1.3.4 Global stiffness matrix of multiple inline linear springs or bars

1.3.5 Global stiffness matrix of a simple biomechanics problem

1.3.6 Global stiffness matrix of a simple truss bridge 

1.3.7 Gaussian or Gauss elimination

1.4 From MSA to a finite element model

1.5 Exercises

Chapter 2 Meshing, Element Types, and Element Shape Functions

2.1 Structure idealization and discretization

2.2 Node

2.3 Element

2.3.1 Simplest element types

2.3.2 1D element type 

2.3.3 2D element type

2.3.4 3D element type

2.4 Formation of finite element mesh 

2.5 Element shape functions and [B] matrix

2.5.1 1D 2-node element shape functions

2.5.2 2D 3-node linear triangular and 4-node bilinear plane stress quadrilateral elements 

2.5.3 2D, 4-node plate element shape functions

2.5.4 3D, 4-node shell element 

2.5.5 3D, 8-node trilinear element shape functions

2.6 Exercises

Chapter 3 Isoparametric Formulation and Mesh Quality 

3.1 Introduction 

3.2 Natural coordinate system

3.3 Isoparametric formulation of 1D elements

3.3.1 1D linear bar element isoparametric shape functions

3.3.2 1D beam element isoparametric shape functions 

3.4 Isoparametric formulation of 2D element

3.4.1 Isoparametric formulation of 2D triangular element

3.4.2 Isoparametric formulation of 2D bilinear element

3.5 Isoparametric formulation of 3D element 

3.5.1 Constant strain tetrahedral element 

3.5.2 Trilinear hexahedral element

3.6 Transfer mapping function

3.7 Jacobian matrix and determinant of Jacobian matrix 

3.8 Element Quality (Jacobian, warpage, aspect ratio, etc.) 

3.8.1 Normalized Jacobian 

3.8.2 Internal and skew angles

3.8.3 Warpage

3.8.4 Aspect ratio

3.8.5 Distortion 

3.8.6 Stretch

3.8.7 Generation of high quality mesh

3.8.8 Patch test

3.9 Exercises

Chapter 4 Element Stiffness Matrix

4.1 Introduction

4.2 Direct method 

4.2.1 Direct formation of structure stiffness matrix

4.2.2 Direct method for a 2-node beam element 

4.3 Strong formulation

4.4 Weak formulation 

4.4.1 Variational method

4.4.2 Weight residual method 

4.5 Derive element stiffness matrix from shape functions 

4.5.1 Gauss Quadrature 

4.5.2 1D element stiffness matrix using Gauss quadrature

4.5.3 Gauss integration points for 2D and 3D elements 

4.5.4 2D element stiffness matrix using Gauss quadrature 

4.5.5 Full and reduced integration 

4.5.6 Zero-energy mode 

4.6 Method of superposition

4.6.1 Stiffness matrix of a 2D frame element

4.6.2 Stiffness matrix of a 3D frame element 

4.7 Coordinate transformation

4.7.1 2D rotation of a vector

4.7.2 2D rotation of the stiffness matrix

4.7.3 3D rotation

4.8 Exercises

Chapter 5 Material Laws and Properties

5.1 Material Laws 

5.1.1 Linear elastic material 

5.1.2 Elastic plastic material 

5.1.3 Hyperelastic material 

5.1.4 Viscoelastic material 

5.1.5 Orthotropic material 

5.1.6 Foam material 

5.1.7 Material defined by equation of state 

5.2 Material Test Strategy and Associated Property 

5.2.1 Experimental types for biological tissue testing

5.2.2 Reverse engineering methodology 

5.2.3 List of common material properties of biological tissues

Chapter 6 Boundary and loading conditions 

6.1 Essential and natural boundary conditions 

6.2 Nodal constraint and prescribed displacement

6.2.1 Nodal constraint 

6.2.2 Prescribed displacement

6.2.3 Penalty method

6.2.4 Symmetrical FE modeling through nodal constraint 

6.3 Natural boundary/loading conditions 

6.3.1 Concentrated loads 

6.3.2 Distributed load 

6.3.3 Initial velocity and acceleration

6.4 Exercises

Chapter 7 Stepping through finite element analysis 

7.1 Introduction 

7.2 Gaussian elimination and iterative procedures

7.2.1 Jacobi or simultaneous displacement method 

7.2.2 Gauss-Seidel or successive displacement method

7.3 Verification and validation

7.3.1 Historical aspect 

7.3.2 Verification

7.3.3 Validation

7.3.4 Quantifying the extent of validation 

7.3.5 Uncertainty Qualification

7.4 Response Variables

7.5 Parametric studies 

7.6 Design of computer experiments

7.7 Stochastic simulation versus Monte Carlo simulation 

Chapter 8 Modal and Transient Dynamic Solutions 

8.1 Mass calculation 

8.2 Central difference time integration method 

8.3 Implicit modal analysis and explicit transient dynamic FE solutions 

8.4 Common problems encountered

8.4.1 Over and under constraints

8.4.2 Negative volume 

8.4.3 Hourglass energy and prevention

8.4.4 Fail to detect contact 

8.5 Failure simulation

8.5.1 Strain criteria based failure simulations

8.5.2 Spot weld force based failure simulations 

Chapter 9 Biological Components Modeling 

9.1 Convert biomedical images to finite element mesh

9.1.1 Anatomical components 

9.1.2 Biomedical Imaging 

9.1.3 Registration and Segmentation 

Chapter 10 Parametric Modeling

10.1 Methods to morph subject-specific FE mesh for other anthropometries 

10.2 Modeling the obese subject 

Chapter 11 Modeling passive and active muscle

11.1 Introduction

11.2 Methods for modeling passive muscle

11.3 Methods for modeling muscular activation

11.4 Application of muscle models 

Chapter 12 Modeling the Head 

12.1 Introduction of corresponding anatomy 

12.2 Injury mechanism 

12.3 Material models 

12.4 Material properties

12.5 Test data available for model validation

12.6 Concluding remarks 

Chapter 13 Modeling the Neck 

13.1 Introduction of corresponding anatomy

13.2 Injury mechanism 

13.3 Material models

13.4 Material properties

13.5 Test data available for model validation

13.6 Concluding remarks

Chapter 14 Modeling the Upper Torso and Upper Extremity

14.1 Introduction of corresponding anatomy

14.2 Injury mechanism

14.3 Material models

14.4 Material properties

14.5 Test data available for model validation

14.6 Concluding remarks

Chapter 15 Modeling the Lower Torso

15.1 Introduction of corresponding anatomy

15.2 Injury mechanism

15.3 Material models 

15.4 Material properties

15.5 Test data available for model validation 

15.6 Concluding remarks 

Chapter 16 Modeling the Lower Extremity 

16.1 Introduction of corresponding anatomy 

16.2 Injury mechanism 

16.3 Material models 

16.4 Material properties 

16.5 Test data available for model validation

16.6 Concluding remarks 

Chapter 17 Modeling Vulnerable subjects 

17.1 Modeling pediatric subjects 

17.2 Modeling elderly female subjects 

Chapter 18 Fundamental of Blast Modeling

18.1 Friedlander blast wave 

18.2 Equation of state 

18.3 Coupling fluid and solid interfaces 

18.4 Primary blast induced head injuries


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About the Author

King-Hay Yang

King-Hay Yang

Dr. King-Hay Yang received his B.S. degree from the National Taiwan University in 1976 and Ph.D. degree from Wayne State University in 1985. He served as an Assistant Professor at the Department of Orthopedic Surgery, West Virginia University from 1985 to 1988. Presently, Dr. Yang is a Professor of the Department of Biomedical Engineering and the Director of the Bioengineering Center at Wayne State University. Since 2014, he has been serving as a member of the Human Factors Panel of the National Academies. Dr. Yang’s research interests include injury biomechanics, contact impact biomechanics, and bone fracture biomechanics. His most recent research involves detailed modeling of the human body from head to toe in efforts to investigate mechanisms of injury related to motor vehicle collisions, contact sports, and blast-induced injuries. His team developed a human brain model that is now on a permanent display at the Computer World of the Smithsonian Institution. Dr. Yang received the 1984 Volvo award in Biomechanics, The Creators: Metro Detroit’s Inventors award by Crane’s Detroit Business; the Best Biomechanics Paper Award of the 2001 International Congress on Whiplash Associated Disorders; the 2001, 2007, and 2008 John Paul Stapp Best Paper Awards; and the 2009 Ralph H. Isbrandt Award for the best paper in automotive safety at the Society of Automotive Engineers (SAE) World Congress. He is a Fellow of the American Institute for Medicine and Biological Engineering (AIMBE) and a Fellow of SAE. He has served as a Visiting Professors at the Swiss Federal Institute of Technology in Zurich, Switzerland, in 1994; Monash University in Melbourne, Australia from 2001 to 2007; the National Cheng-Kung University in Tainan, Taiwan in 2003; the Honk Kong Polytech University in Kowloon, Hong Kong, in 2005; and the Taipei Medical University in 2011. He has also been a Chang-Jiang (Yangtze River) Professor of Hunan University in Changsha, Hunan, China since 2005. Dr. Yang is a member of the Stapp Advisory Committee of the Stapp Car Crash Conference, the premier forum for presentation of research in impact biomechanics, human injury tolerance, and related fields that advance the knowledge of land-vehicle crash-injury protection. Additionally, he is a council member of the International Research Council on Biomechanics of Injury (IRCOBI), which aims at improving the scientific basis on which safety and crashworthiness standards are formulated. Lastly, he has been serving on the Advisory Committee of the Life Science Advisory Board, Lawrence Technical University, since 2007.

Affiliations and Expertise

Professor, Wayne State University, Michigan, USA