Department of BioEngineering College of Engineering

University of Illinois at Chicago

BioE 594 – COMPUTATIONAL METHODS IN BIOMECHANICS Fall 2011

Time & Place: Tuesdays 1:00 P.M. to 3:00 P.M.; ROOM 520 at

New Orthopedic Building, Rush University Medical Center Goals: Computer models are being increasingly used for the solution of many complex problems in biomechanics. This course will give the students an insight on how computer models based on numerical methods are applied in orthopedic biomechanics. Students will be required to complete mini projects in each of the applications that will be discussed in this course. Instructor: Raghu N. Natarajan Ph.D. Professor, Department of Orthopedics, Rush-Presbyterian-St.Luke’s Medical Center and Department of BioEngineering, University of Illinois at Chicago. (e-mail : raghu_natarajan@rush.edu) Topics (1) Introduction to Orthopedics (2) Numerical methods (3) Introduction to Finite Element Method (4) Contact Analysis: Total Knee Replacement & Total Hip Replacement (5) Stress shielding in Total Hip Replacement (6) Micromotion in Orthopedics (7) Effect of surgical interventions on lumbar motion segment (8) Projects Grading: Homework 30% Projects 40% Final comprehensive examination 30%

NUMERICAL METHODS THAT CAN BE USED IN BIOMECHANICS

1) Mechanics of Materials Approach

(A) Complex Beam Theory

(i) Straight Beam

(ii) Curved Beam

(iii) Composite Beam

From:Daviddarling.info

NUMERICAL METHODS THAT CAN BE USED IN BIOMECHANICS

Mechanics of Material Approach (Cont)

NUMERICAL METHODS THAT CAN BE USED IN BIOMECHANICS

(2) Finite Difference Method

NUMERICAL METHODS THAT CAN BE USED IN BIOMECHANICS

(2) Finite Difference Method (Contd)

Consider an ordinary differential equation

One of the difference equation method is using:

To approximate the differential equation.

Solution is:

APPLICATION OF FINITE ELEMENT METHOD TO BIOMECHANICS

Introduction

Re-invented around 1963Initially applied to engineering structures

Concrete dams Aircraft structures

(Civil engineers) (Aeronautical

engineers)

Introduction

FEM is based on

EnergyMethod

Methodof

Residuals

Introduction

Energy method

Total potential energy must be stationary

δ (U + W) = δ ( П ) = 0

Introduction

Residual method

Differential equation governing the problem is given by A ( ø ) = 0

Minimise R = A ( ø* ) - A ( ø )

ø is actual solution

ø* is assumed solution

Introduction

Both methods give us a set of equations

[ K ] { a } = { f }

Stiffness Matrix

Displacement Matrix

Force Matrix

Introduction - FEM Procedure

Continuum is separated by imaginary lines or surfaces into a number of “finite elements”

Finite Elements

Introduction - FEM Procedure Elements are assumed to be interconnected at a

discrete number of “nodal points” situated on their boundaries

Finite ElementsNodal Points

Displacements at these nodal points will be the basic unknown

Introduction - FEM Procedure A set of functions is chosen to define uniquely the

state of displacement within each finite element ( U ) in terms of nodal displacements ( a1 , a2 , a3 )

U = Σ Ni ai i= 1, 3

x

y

a1

a2

a3

Finite Element

Nodal Point

Introduction - FEM Procedure This displacement function is input into either

“energy equations” or “residual equations” to give us element equilibrium equation

[ K ] { a } = { f }

x

y

a1

a2

a3

Finite Element

Nodal Point

ElementDisplacementMatrix

ElementForceMatrix

ElementStiffnessMatrix

Introduction - FEM ProcedureElement equilibrium equations are assembled

taking care of displacement compatibility at the connecting nodes to give a set of equations that represents equilibrium of the entire continuum

Finite ElementsNodal Points

Introduction - FEM ProcedureSolution for displacements are obtained after

substituting boundary conditions in the continuum equilibrium equations

Finite ElementsNodal Points

Support PointsSupport Points

Introduction Finite element method used to solve:

Elastic continuum Heat conduction Electric & Magnetic potential Non-linear (Material & Geometric) -plasticity, creep Vibration Transient problems Flow of fluids Combination of above problems Fracture mechanics

Introduction Finite elements:

Truss , Cable and Beam elements Two & Three solid elements Axi-symmetric elements Plate & Shell elements Spring, Damper & Mass elements Fluid elements

Application to Spine Biomechanics

Finite Element Mesh of C4-C7

Intact With Graft at C5-C6 Level

C4

C5

C6

C7

C5-C6Graft

Facet Joints

von Mises Stress in C4-C5 Annulus (Flexion)

Neutral Graft Kyphotic Graft

5 MPa 6 MPaAnterior Anterior

Finite Element Mesh of L1-S1

Vertical Displacement Distribution in L1-S1

Finite Element Mesh of L2-L5With 25% Translational

Spondylolisthesis

Vertical Displacement Distribution in L2-L5 Under Flexion Moment

(25% translational spondylolisthesis)

Application to Knee Implant Biomechanics

Finite Element Mesh to Represent Tibial Insert & Femoral Component

Contact Compressive Stress

Motion of Femoral Implant with respect to UHMWPE Knee Insert

Application to Femoral Implant Biomechanics

Finite Element Mesh of an Intact Femur

Distribution of SIGMA-ZZ in an intact femur

Finite Element Mesh of a Femur with Implant

SIGMA-ZZ in a Femur With Implant

Implant fixed with cement layer in a femur

Von Misses stress in cement layer

SIGMA-ZZ in cortical bone in a femur with implant attached using cement

Advantage of using FEMIrregular complex geometry can be modeledEffect of large number of variables in a problem

can be easily analysedMultiple phase problems can be modeledEffect of various surgical techniques can be

compared using appropriate FE modelsBoth static and time dependent problems can be

modeledSolution to certain problems that cannot be (or

difficult) obtained otherwise can be solved by FEM