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