Finite Element Modeling and Analysis with a Biomechanical Application Alexandra Schönning, Ph.D. Mechanical Engineering University of North Florida ASME Southeast Regional XI Jacksonville, FL April 8, 2005
Feb 06, 2016
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