Finite Element Modeling and Analysis with a
Biomechanical Application
Alexandra Schönning, Ph.D.Mechanical Engineering
University of North Florida
ASME Southeast Regional XIJacksonville, FL
April 8, 2005
Presentation overview
Finite Element Modeling The process Elements and meshing Materials Boundary conditions and
loads Solution process Analyzing results
Biomechanical Application Objective Need for modeling the human
femur Data acquisition Development of a 3-
Dimensional model Data smoothing NURBS Finite element modeling Initial analysis Discussion and future efforts
Finite Element Modeling (FEM)
What is finite element modeling? It involves taking a continuous structure and “cutting” it into
several smaller elements and describing each of these small elements by simple algebraic equations. These equations are then assembled for the structure and the field quantity (displacement) is solved.
In which fields can it be used? Stresses Heat transfer Fluid flow Electromagnetics
FEM: The process
Determine the displacement at the material interfaces
Simplify by modeling the material as springs.
Co
F3 = 30kNF2 = 20kN
St
k1 k2
F3 = 30kNF2 = 20kN
n1 n2 n3
FEM: The process Draw a FBD for each node, sum
the forces, and equate to zero k1 k2
F3 = 30kNF2 = 20kN
n1 n2 n3
n3
F3Spring force2 = k2(x3-x2)
ΣF = 0:
-k2(x3-x2)+F3 = 0
k2*x2-k2*x3+F3 = 0
-k2*x2+k2*x3 = F3
Spring force1 = k1(x2-x1)
n2F2
Spring force2 = k2(x3-x2)
ΣF = 0:
-k1(x2-x1)+k2(x3-x2)+F2 = 0
-k1*x1+(k1+k2)*x2-k2*x3 = F2
n1
Spring force1 = k1(x2-x1)
R
ΣF = 0:
R+k1(x2-x1)= 0
k1*x1-k1*x2 = R
FEM: The process
Re-write equations in matrix form
k1*x1-k1*x2 = R (node 1)-k1*x1+(k1+k2)*x2-k2*x3 = F2 (node 2)-k2*x2+k2*x3 = F3 (node 3)
k1
k1
0
k1
k1 k2
k2
0
k2
k2
x1
x2
x3
R
F2
F3
Stiffness matrix [K] Displacement vector {δ} Load vector {F}
k1 k2
F3 = 30kNF2 = 20kN
n1 n2 n3
FEM: The process
Apply boundary conditions and solve
At left boundary Zero displacement
(x1=0)
Simplify matrix equation
Plug in values and solve
k1 k2
k2
k2
k2
x2
x1
F2
F3
k1 k2
F3 = 30kNF2 = 20kN
n1 n2 n3
k1=40 MN/m
k2 = 60 MN/m
x2
x1
40 60
60
60
60
120
30
x2
x1
1.25
1.75
x3
x3
x3
FEM: The process
The continuous model was cut into 2 smaller elements
An algebraic stiffness equation was developed at each node
The algebraic equations were assembled and solved
This process can be applied for complicated system with the help of a finite element software
FEM: Element types
1-dimensional Rod elements Beam elements
2-dimensional Shell elements
3-dimensional Tetrahedral elements Hexahedral elements
Special Elements Springs Dampers Contact elements Rigid elements
Each of the elements have an associated stiffness matrix
Different degrees of freedom (DOF) in each of the elements Spring developed has 1 DOF Beam has 6 DOF
Linear, quadratic, and cubic approximations for the displacement fields.
FEM: Materials
Properties Modulus of elasticity (E) Poisson’s ratio () Shear modulus (G) Density Damping Thermal expansion (α) Thermal conductivity Latent heat Specific heat Electrical conductivity
Isotropic, orthotropic, anisotropic
Homogeneous, composite Elastic, plastic, viscoelastic
Strain (%)
FEM: Boundary Conditions (constraints and loads) Boundary conditions are used to mimic the surrounding
environment (what is not included in your model) Simple example: Cantilever beam
Beam is bolted to a wall and displacements and rotations are hindered. More complex example: Tire of a car
Is the bottom of the tire fixed to the ground? Is there friction involved? How is the force transferred into the tire?
Are the transfer characteristics of the bearings considered? Are breaking loads considered? Interface between components?
Garbage in – garbage out… …but not in FEM
Garbage in – beautiful, colorful, and believable… …garbage out
k1 k2
F3 = 30kNF2 = 20kN
n1 n2 n3
FEM: Solution process Today’s computer speeds have made FEM computationally affordable. What
before may have required a couple of days to solve may now take only an hour. Inverse of the stiffness matrix
K*δ = F δ = K-1*F
Displacements strains stress
k1
k1
0
k1
k1 k2
k2
0
k2
k2
x1
x2
x3
R
F2
F3
FEM: Analyzing results
Interpreting results Consider the results wrong until you have convinced
your self differently. Sanity checks
Does the shape of the deformation make sense? Check boundary condition configurations
Are the deformation magnitudes reasonable? Check load magnitudes and unit consistency
Is the quality of the stress fringes OK? Smoothness of unaveraged and noncontinuous reslts Review mesh density and quality of elements
Are the results converging? Is a finer mesh needed? Verification of results
Local unexpected results may be OK FBD, simplified analysis, relate to similar studies. Check reaction forces and moments
Pedestal assembly
FEM: summary
Use of FEM Predict failure Optimize design
The process Elements and meshing Materials Boundary conditions and loads Solution process Analyzing results