1. Introduction to Finite Element Methods

2. Need for Computational Methods
Solutions Using Either Strength of Materials or Theory of Elasticity Are Normally Accomplished for Regions and Loadings With Relatively Simple Geometry
Many Applications Involve Cases with Complex Shape, Boundary Conditions and Material Behavior
Therefore a Gap Exists Between What Is Needed in Applications and What Can Be Solved by Analytical Closed-form Methods
This Has Lead to the Development of Several Numerical/Computational Schemes Including: Finite Difference, Finite Element and Boundary Element Methods

3. Introduction to Finite Element Analysis
The finite element method is a computational scheme to solve field problems in engineering and science.
The fundamental concept involves dividing the body under study into a finite number of pieces (subdomains) called elements
Particular assumptions are then made on the variation of the unknown dependent variable(s) across each element using so-called interpolation or approximation functions.
This approximated variation is quantified in terms of solution values at special element locations called nodes.
Through this discretization process, the method sets up an algebraic system of equations for unknown nodal values which approximate the continuous solution.

4. Advantages of Finite Element Analysis
- Models Bodies of Complex Shape
- Can Handle General Loading/Boundary Conditions
- Models Bodies Composed of Composite and Multiphase Materials
- Model is Easily Refined for Improved Accuracy by Varying Element Size and Type (Approximation Scheme)
- Time Dependent and Dynamic Effects Can Be Included
- Can Handle a Variety Nonlinear Effects Including Material Behavior, Large Deformations, Boundary Conditions, Etc.

5. Basic Concept of the Finite Element Method
Any continuous solution field such as stress, displacement, temperature, pressure, etc. can be approximated by a discrete model composed of a set of piecewise continuous functions defined over a finite number of subdomains.
One-Dimensional Temperature Distribution
Exact Analytical Solution x T
Approximate Piecewise Linear Solution

6. Two-Dimensional Discretization
u(x,y)
Approximate Piecewise Linear Representation

7. Discretization Concepts

8. Common Types of Elements
Two-Dimensional Elements Triangular, Quadrilateral Plates, Shells, 2-D Continua
One-Dimensional Elements Line Rods, Beams, Trusses, Frames
Three-Dimensional Elements Tetrahedral, Rectangular Prism (Brick) 3-D Continua

9. Discretization Examples
Three-Dimensional Brick Elements
One-Dimensional Frame Elements
Two-Dimensional Triangular Elements

10. Basic Steps in the Finite Element Method Time Independent Problems
- Domain Discretization
- Select Element Type (Shape and Approximation)
- Derive Element Equations (Variational and Energy Methods)
- Assemble Element Equations to Form Global System
[K]{U} = {F}
[K] = Stiffness or Property Matrix {U} = Nodal Displacement Vector
{F} = Nodal Force Vector
  - Incorporate Boundary and Initial Conditions
 - Solve Assembled System of Equations for Unknown Nodal Displacements and Secondary Unknowns of Stress and Strain Values

11. Common Sources of Error in FEA
Domain Approximation
Element Interpolation/Approximation
Numerical Integration Errors
(Including Spatial and Time Integration)
Computer Errors (Round-Off, Etc., )

12. Measures of Accuracy in FEA
Error = |(Exact Solution)-(FEM Solution)|
Convergence
Limit of Error as:
Number of Elements (h-convergence) or Approximation Order (p-convergence)
Increases
Ideally, Error  0 as Number of Elements or Approximation Order  

13. Two Dimensional Examples
Triangular Element
Scalar-Valued, Two-Dimensional Field Problems
Triangular Element
Vector/Tensor-Valued, Two-Dimensional Field Problems u1 u2 1 2 3 u3 v1 v2 v3 1 2 3 f1 f2 f3

14. Two-Dimensional Discretization Refinement
(Discretization with 228 Elements)
(Discretization with 912 Elements)
(Triangular Element)
(Node) 

15. Development of Finite Element Equation
The Finite Element Equation Must Incorporate the Appropriate Physics of the Problem
For Problems in Structural Solid Mechanics, the Appropriate Physics Comes from Either Strength of Materials or Theory of Elasticity
FEM Equations are Commonly Developed Using Direct, Variational-Virtual Work or Weighted Residual Methods
Direct Method
Based on physical reasoning and limited to simple cases, this method is worth studying because it enhances physical understanding of the process
Variational-Virtual Work Method
Based on the concept of virtual displacements, leads to relations between internal and external virtual work and to minimization of system potential energy for equilibrium
Weighted Residual Method
Starting with the governing differential equation, special mathematical operations develop the “weak form” that can be incorporated into a FEM equation. This method is particularly suited for problems that have no variational statement. For stress analysis problems, a Ritz-Galerkin WRM will yield a result identical to that found by variational methods.

16. Simple Element Equation Example Direct Stiffness Derivation1 2 k u1 u2 F1 F2
Stiffness Matrix
Nodal Force Vector

17. Common Approximation Schemes One-Dimensional Examples
Polynomial Approximation
Most often polynomials are used to construct approximation functions for each element. Depending on the order of approximation, different numbers of element parameters are needed to construct the appropriate function.
Linear
Quadratic
Cubic
Special Approximation
For some cases (e.g. infinite elements, crack or other singular elements) the approximation function is chosen to have special properties as determined from theoretical considerations

18. Theoretical Basis: Formulating Element Equations
Several approaches can be used to transform the physical formulation of a problem to its finite element discrete analogue.
If the physical formulation of the problem is described as a differential equation, then the most popular solution method is the Method of Weighted Residuals.
If the physical problem can be formulated as the minimization of a functional, then the Variational Formulation is usually used.

19. Theoretical Basis: MWR
One family of methods used to numerically solve differential equations are called the methods of weighted residuals (MWR).
In the MWR, an approximate solution is substituted into the differential equation. Since the approximate solution does not identically satisfy the equation, a residual, or error term, results.
Consider a differential equation
Dy’’(x) + Q = 0 (1)
Suppose that y = h(x) is an approximate solution to (1). Substitution then gives Dh’’(x) + Q = R, where R is a nonzero residual. The MWR then requires that
ò Wi(x)R(x) = 0 (2)
where Wi(x) are the weighting functions. The number of weighting functions equals the number of unknown coefficients in the approximate solution.

20. Sources of Error in the FEM
The three main sources of error in a typical FEM solution are discretization errors, formulation errors and numerical errors.
Discretization error results from transforming the physical system (continuum) into a finite element model, and can be related to modeling the boundary shape, the boundary conditions, etc.
Discretization error due to poor geometry
representation.
Discretization error effectively eliminated.

21. Sources of Error in the FEM (cont.)
Formulation error results from the use of elements that don't precisely describe the behavior of the physical problem.
Elements which are used to model physical problems for which they are not suited are sometimes referred to as ill-conditioned or mathematically unsuitable elements.
For example a particular finite element might be formulated on the assumption that displacements vary in a linear manner over the domain. Such an element will produce no formulation error when it is used to model a linearly varying physical problem (linear varying displacement field in this example), but would create a significant formulation error if it used to represent a quadratic or cubic varying displacement field.

22. Sources of Error in the FEM (cont.)
Numerical error occurs as a result of numerical calculation procedures, and includes truncation errors and round off errors.
Numerical error is therefore a problem mainly concerning the FEM vendors and developers.
The user can also contribute to the numerical accuracy, for example, by specifying a physical quantity, say Young’s modulus, E, to an inadequate number of decimal places.

23. Advantages of the Finite Element Method
Can readily handle complex geometry:
The heart and power of the FEM.
Can handle complex analysis types:
Vibration
Transients
Nonlinear
Heat transfer
Fluids
Can handle complex loading:
Node-based loading (point loads).
Element-based loading (pressure, thermal, inertial forces).
Time or frequency dependent loading.
Can handle complex restraints:
Indeterminate structures can be analyzed.

24. Advantages of the Finite Element Method (cont.)
Can handle bodies comprised of nonhomogeneous materials:
Every element in the model could be assigned a different set of material properties.
Can handle bodies comprised of nonisotropic materials:
Orthotropic
Anisotropic
Special material effects are handled:
Temperature dependent properties.
Plasticity
Creep
Swelling
Special geometric effects can be modeled:
Large displacements.
Large rotations.
Contact (gap) condition.

25. Disadvantages of the Finite Element Method
A specific numerical result is obtained for a specific problem. A general closed-form solution, which would permit one to examine system response to changes in various parameters, is not produced.
The FEM is applied to an approximation of the mathematical model of a system (the source of so-called inherited errors.)
Experience and judgment are needed in order to construct a good finite element model.
A powerful computer and reliable FEM software are essential.
Input and output data may be large and tedious to prepare and interpret.

26. Disadvantages of the Finite Element Method (cont.)
Numerical problems:
Computers only carry a finite number of significant digits.
Round off and error accumulation.
Can help the situation by not attaching stiff (small) elements to flexible (large) elements.
Susceptible to user-introduced modeling errors:
Poor choice of element types.
Distorted elements.
Geometry not adequately modeled.
Certain effects not automatically included:
Buckling
Large deflections and rotations.
Material nonlinearities .
Other nonlinearities.

27. FEM in BIOMECHANICS
The process of developing finite element models begins with the acquisition of
data that will be used to define the three-dimensional geometry of the joint tissues.
These data can come from several imaging modalities, including CT and MRI.
Three-dimensional data sets are acquired and segmented, i.e., each tissue type of interest to the modeler is labeled within the data set.
From the segmented data, three-dimensional surfaces are calculated, and fully volumetric meshes (the geometric portion of the finite element model) are generated.
In the finite element analysis, the tissue types described by the finite element models are assigned specific material characteristics, and the simulation is completed, with externally calculated boundary conditions defining the specific joint behavior (e.g., flexion of the joint due to flexor tendon action).
Each of these steps is described in more detail below and will be demonstrated.

28. DATA ACQUISITION
Accurate geometry is one key to successfully modeling joint behavior using finite element
techniques.
Both CT and MRI data are used in model development.
Once the data are acquired, model development is independent of the imaging modality.
Typically, scanners used in the medical field have a spatial resolution that is not acceptable
for a precise definition of articular surfaces.
A sample of high resolution
CT data, in a transverse cut through a
human hand.

29. Segmentation of the imaged, three-dimensional data sets is the process of identifying tissues and their boundaries.
Edges filled in by automated segmentation and partially manually corrected.
Edges generated by automated segmentation.
Final mask resulting from semi-automated segmentation

30. SURFACE GENERATION VOLUMETRIC MESHING
Three-dimensional surfaces are generated directly from the masks that are the final product of the segmentation step
VOLUMETRIC MESHING
Mesh generation algorithms focus on volumetric tetrahedral meshes. The
methods used generally rely on a subdivision algorithm of the volume.
A mesh is then built by triangulating each of the cells of the volume.
Slightly changing the coordinates of the vertices helps smooth the mesh and improve its quality.
However, tetrahedral meshes are not suited for the dynamic simulations.
Structural engineers prefer hexahedral meshes, which help speed up the
convergence of the numerical algorithms.

31. VISUALIZATION
Joint kinematics and tissue stresses calculated by the finite element code are visualized on a workstation
Visualization image showing knee ligament stresses during flexion

32. Bone biomechanics

35. Artificial Hip joint

36. Anatomy of Hip Joint Largest weight bearing joint
SIVA FROM IITG
Anatomy of Hip Joint
Largest weight bearing joint
Composed of rounded head of the femur joining the acetabulum of pelvis in a ball and socket arrangement

37. Damaged Femoral Head Femoral head cartilage
SIVA FROM IITG
Damaged Femoral Head
Femoral head cartilage
The neck is cut-off as in figure
Marrow cavity is made inside the femur
Hip prosthesis is fitted either by PMMA cement or press fitted

38. Biomaterials Stainless Steel Alloys Cobalt-Chrome alloys (Vitallium)
SIVA FROM IITG
Biomaterials
Stainless Steel Alloys
Cobalt-Chrome alloys (Vitallium)
Titanium alloys
Composites

39. Comparison of Characteristics
SIVA FROM IITG
Comparison of Characteristics
Characteristic
S-Steel
Co-Cr alloy
Titanium Alloy
Stiffness
High
Medium
Low
Strength
Corrosion -resistance
Biocompatibility

40. SIVA FROM IITG
Composite Prosthesis
Clinical studies reported early fatigue fracture of a femoral component made from laminated fiber reinforced composites.
The new designs are Constructed of short glass fibers/epoxy resin and CF/PEEK composites.

41. SIVA FROM IITG
Composite Model
Basic Composite Model With Elements

42. Conical Stem Cemented prosthesis model contains three main parts:
SIVA FROM IITG
Conical Stem
Cemented prosthesis model contains three main parts:
Conical Stem with head
Cement layer
Cortical bone
Basic Model

43. Model With Chopped Fiber Core
SIVA FROM IITG
Chopped Fiber Core
Model With Chopped Fiber Core

44. Material Properties Used for Analysis of Total Hip Prosthesis
Parts
Material
Young’s Modulus (MPa)
Poisson's Ratio
Geometrical Parameter (All dimensions are in mm)
Head and Stem
 Ti6Al4V
110x103
0.33
Sphere radius 25
Stem radius 10
Stem outer radius 10
Stem inner radius 7.5
Cement Layer
UHMWPE-AL2O3
1x103
0.39
Inner radius 10.5
Outer radius
Length 100
Cortical Bone
AS4/PEEK
3x103
0.30
Inner radius 20.5
Outer radius 30

45. Maximum Shear Stress Region
SIVA FROM IITG
Maximum Shear Stress Region
Enlarged View of the Deformed Stem and Cortical Bone Showing the Maximum Shear Stress Region (Path Aa)

46. Variation of Shear Stresses
Variation of Maximum Shear Stress
With System Parameters
Stem Length
(in mm)
Maximum Shear Stress
(in MPa)
145
17.314
145.5
15.522
147.5
21.033
150
20.919
152.5
17.144
155
20.262
Neck Length
(in mm)
Maximum Shear Stress
(in MPa) 45
13.337
47.5
17.376 50
17.314
52.5
25.363
Neck Inclination
(in degree)
Maximum Shear Stress
(in MPa) 45
17.314
47.5
20.383 50
22.964
Stem Inner Radius
(in mm)
Maximum Shear Stress
(in MPa)
7.5
17.314 8
20.655
8.5
19.443

47. Continued...
The variation in the above parameters do not show a particular trend
Hence the design optimization has been carried out to minimize the magnitude of maximum shear stress

48. Dimensions of Hip Prosthesis Before Optimization
Dimensions of Hip Prosthesis Before Optimization
Parts
State Variables
Design Variables
Femur
Sphere Radius
25 mm
Stem Outer Radius
10 mm
Stem Inner Radius
7.5 mm
Neck Inclination
450
Stem Length
145.5 mm
Neck length
50 mm

49. Design Variables of Femoral Components
After Optimisation
Design Variables
Dimension (mm)
Stem outer radius
9.9301
Stem inner radius
8.0405
Stem length
153.22
Neck length
50.975

50. SIVA FROM IITG
Shear Stresses
SXY
x-y component
SYZ
y-z component
SXZ
z-x component
Shear Stresses in the Interface of Stem and Cortical Bone

51. Shear Stresses - Continued…
SIVA FROM IITG
Shear Stresses - Continued…
SXY
x-y component
SYZ
y-z component
SXZ
z-x component
Shear Stresses in the Interface of Stem and Cortical Bone

52. Dentistry

54. Lumbar disc degeneration

57. Neck Injury Prevention
Experimental Research and Development

58. Neck Injury Prevention
Existing numerical models of the cervical musculature
Eindhoven (MADYMO)
(Van der Horst 2002)
Duke (LS-DYNA)
(Chancey et al. 2003)
KTH (LS-DYNA)
(Brolin et al. 2005)
France (RADIOSS)
(Frechede et al. 2006)
JAMA (LS-DYNA)
(Ejima et al. 2005)

59. Numerical Modeling The KTH FE Neck Model Intervertebral Disks
and Ligaments
Vertebrae
Muscles

60. Geometry of the Cervical Musculature
The FE Muscle Model Geometry created from MRI
Segmented from MRI (50th percentile)
Interpolated into 3D surfaces
Anatomical guide books /morphometric literature
Neurosurgical expertise and dissection

61. Geometry of the Cervical Musculature
The FE Muscle Model Geometry created from MRI
Segmented from MRI (50th percentile)
Interpolated into 3D surfaces
Positioned relative the KTH neck model in line with the literature

62. Geometry of the Cervical Musculature
The FE Muscle Model Geometry created from MRI
25 individual muscle pairs
Rigid body insertions to the vertebrae
One muscle can have multiple origins/insertions

63. Geometry of the Cervical Musculature
Anterior: Hyoid, SCM
Lateral: SCM, TZ
Create the geometry of the solid element muscle model and compare the kinematic response with a spring muscle model, using the same active spring elements
Posterior: TZ, SplCap
Posterior: Suboccipital

64. Evaluation of Muscle Models
Discrete Muscle Model
DMM
Continuum Muscle Model
CMM
SMFE Muscle Model
SMFEMM

65. Evaluation of Muscle Models
Rear end ~4G
[Ono et al 1999 and Davidsson et al 1999]

66. Evaluation of Muscle Models
Frontal~15G [Ewing et al 1977]
Lateral ~7G [Ewing et al 1977]

67. Heart and FEM

74. Finite Element Modeling of the
Human Foot and Footwear