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

FEM: A biomechanical application

Objective: Develop a high fidelity finite element model of a standard femurHexahedral elementsOrthotropic material propertiesMacro-inhomogeneous Realistic loads Make model available to other researchers

Need for Modeling the Human Femur and Adjacent Bones Improved treatment options for

patients with different types of diseases Legg-Calve-Perthes disease Osteoporosis

Implant design Improve implant life and understand

failure mechanism Basic research: Understand the

stress distribution in the bone (lower limb) to learn what effect it has on disease and how we can stop or reduce effect of disease or deformities

Image Processing Computed Tomography (CT) data

acquisition Scanning device completes a 360o revolution Slices are 1 to 5 mm apart (generally). 1mm for male. Result: Matrix with gray scaled pixels based on tissue

density

Scanning the objectScanning the object

Computed Tomography Data

Slice distance

Resulting Image SetResulting Image Set

Computed Tomography Data

Select the desired regionSelect the desired region

… … and Growand Grow

Development of a 3-dimensional model in Mimics Computed tomography

data Density threshold in

Hounsfield units Cortical : 2000-3200 Cancellous: (1100-

2000) Bone Marrow: <1100

Manual editing Region growing

Development of a 3-Dimensional model

3-D models created by interpolation of 2D slices in Mimics Data smoothing NURBS (Non-Uniform Rational B-Splines) *.igs files

Data Smoothing

Why smoothen the data? Data lost through scanning (Interpolation between 1 mm slices) Estimate the threshold values Manual editing Result: Model has rough surfaces

Goal: Want a model that can easily be meshed yet properly represent the object

For meshing to be performed and in order to solve the model it is necessary to remove some detail (partially created from inaccuracies)

Data smoothing through Geomagic by Raindrop Geomagic

Problem geometry

Surfaces are not properly closed

Portion of surfaces are inside out

Preparing the geometry for NURBS NURBS = Non-Uniform Rational B-Splines What is NURBS? Curves that approximate

a surface. From a rough surface of “random” points to a surface that can be expressed as polynomials. It creates an analytical surface that the mesher will better understand.

Develops a rectangular grid structure in place of the triangular grid structure. The imported geometry consists of triangular

surfaces Hexahedral mesh more easily developed

with rectangular surfaces Why NURBS?

Consecrated method of going from random to analytical surfaces.

Initial try vs. final grid structure Need to prepare for the mesh block structure

Creation of the NURBS surface

NURBS surfaces Subdivided

Grid Need to plan for

the mesh 90O angles is

optimal

Development of the mesh: The TrueGrid Interface Why hexahedral mesh?

Simplicity of mesh Regularity Angle distribution Higher control of mesh

Why TrueGrid? Specializes in hexahedral

meshing of complicated geometry

Allows for easy modification of the final mesh

Physical window Geometry Elements and nodes

Computational window Block structure

Command window Environment window Mesh input file is created

Development of a hexahedral mesh in TrueGrid

Create block structure Remove blocks not needed (4) Define block boundary surfaces (5, 6) Attach the corners of the block to the

geometry (7) Map blocks’ surfaces to the geometry’s

“combined” surfaces (7) Create the mesh by specifying the node

increase in each direction. BB surfaces need to be consistent Coincident nodes on BB surfaces must be

removed

Problem elements

Butterfly mesh

1

2 3

48765

a

b

Surface 1 projected

Development of the Femoral Hexahedral Mesh in TrueGrid More complicated block structure

Three separate volumes: cortical, cancellous, marrow Long section – easy Condyles, trochanter and femoral head causes problems

Visualization difficulty: how to create the block boundaries in 3 dimensions

In some cases it is impossible to ensure the 3D connectivity Ensuring that blocks will allow for different material properties.

Geometry is free-formed 90 degree corners don’t exist Try to map the blocks’ vertices to convex geometry

Must carefully plan (from the beginning) how to map the blocks.

Preparing the geometry in TG

*.iges file contains more surfaces than block surfaces needed

Surfaces are combined Vertices are attached Edges are attached Curves are generated to

steer the mesh Faces are projected

Developing the block structure

Blocks missing

Blocks missing

6 edgesNot possible

Creating the block structure Need to ensure that

separate blocks are created for cortical, cancelleous, and marrow materials

Connectivity between blocks may not be possible in some regions. Create separate files and

then merge the files

Solution

Meshing difficulty due to geometry Lowest level of blocks

Excluding cortical shell

Block face follows geometry that is partially concave

Elements intersect themselves Zero and negative

volume

Meshing difficulty due to geometry Angle 1 is acceptable Angle 2 is negative Only edges attached to

geometry are initially controlled Curves and internal surfaces are

created to control the mesh

2

1

Non controlled edge

Meshing difficulty due to geometry

Resolving geometry difficulties Go back to Geomagics and recreate the *iges files.

Move the edges of the surfaces to a location less likely to create problems (less concave)

Create internal edges, surfaces and points to steer the internal mesh

Use bias commands to increase/decrease the mesh density in a local region

Resolving block structure difficulties May need to build separate

Resulting mesh

1 2 3 4 5 6

How can mesh be modified?

Element size Element material Insert partitions in the mesh

Insert geometry of an implant Attach the internal block surfaces/edges/vertices to

the implant Change the material properties of the implant

Merge different mesh files

F1

F2

z

xy

Analysis

Initial analysis performed to verify mesh solvability

Material Linearly elastic Isotropic macroinhomogeneous

Boundary conditions Removed all DOF from the distal

end of the condyles

Load One legged stance Distributed load on femoral head Distributed load on greater

trochanter Results

Max deflections: 3 mm Peak von Mises stress: 37 MPaMarco Viceconti, Mario Davinelli, Fulvia Taddei, Angelo Cappello, “Automatic generation of

accurate subject-specific bone finite element models to be used in clinical studies”, Journal of Biomechanics 37 (2004) 1597 - 1605

Cortical bone Cancellous bone

Bone marrow

E 17,000 MPa 750 MPa 300 MPa

0.33 0.33 0.45

x y z

Joint reaction force (F1) -616 N 171 N 2800 N

Abductor muscle force (F2)

430 N 0 N 1160 N

Future Work and Discussion Model improvements

Femur Orthotropic material properties Improved loading conditions Compare to tetrahedral mesh

Analyze different stances Complete a model of the lower limb

Model use Optimize implant designs Improved treatment options for patients with different types of

diseases Make available to the public so research can more easily be

advanced

Summary

Garbage in – garbage out! Even though you obtain pretty pictures. Anyone can run a FE analysis… Pay close attention to boundary conditions, degrees

of freedom, mesh quality and validity of results Applications

Failure analysis, optimization, heat transfer, fluid flow, electromagnetic analysis

Biomechanical application