Download - Fitting a Round Peg to a Square Hole
Fitting a Round Peg Fitting a Round Peg to a Square Holeto a Square Hole
Janna Balling Janna Balling
and Andrew Andersonand Andrew Anderson
A Method for Physical Validation of Finite Element Pressure ModelsA Method for Physical Validation of Finite Element Pressure Models
IntroductionIntroduction
Validate subject-Validate subject-specific FE models specific FE models of the hip using of the hip using experimental dataexperimental data
Single-leg-stance Single-leg-stance and stair-climbingand stair-climbing
Cartilage contact Cartilage contact stress measured stress measured using pressure using pressure sensitive filmsensitive film
IntroductionIntroduction
Pressure sensitive filmPressure sensitive film– Cut into rosette patternCut into rosette pattern– Fit to femoral headFit to femoral head– Film scanned in 2DFilm scanned in 2D– CalibratedCalibrated– Color intensity is Color intensity is
proportional to applied proportional to applied pressurepressure
ObjectiveObjective
Validate FE model predictions of Validate FE model predictions of contact stress with pressure film contact stress with pressure film measurementsmeasurements– Convert 3D FE model pressure plot into a Convert 3D FE model pressure plot into a
2D synthetic image2D synthetic image– Compare synthetic image with pressure film Compare synthetic image with pressure film
imageimage
?
Fit A SphereFit A Sphere
GivenGivend={x,y,z} for n pointsd={x,y,z} for n points
Least-Squares FitLeast-Squares Fit
WhereWhere
LoopLoop
r
(a,b,c)
a= -0.000422081b= 0.000270402c= 0.146251r= 20.5007 only 317 iterations
n
i i
ion
ii
n
ii L
xa
nL
nx
na
111
)(1*
11
n
i i
ion
ii
n
ii L
yb
nL
ny
nb
111
)(1*
11
n
i i
ion
ii
n
ii L
zc
nL
nz
nc
111
)(1*
11
n
iiLn
r1
1
aaoo-a ≈ 0 b-a ≈ 0 bo-b ≈-b ≈ 0 c0 co-c≈-c≈ 00
else aelse ao=a b=a bo=b c=b co=c=c
0 approximation = 2.2204460492503131e-0160 approximation = 2.2204460492503131e-016
222 )()()( oioioii czbyaxL
origin:
radius:
Transform to Sphere Transform to Sphere CoordinatesCoordinates
GivenGivend={x,y,z} origin=(a,b,c) d={x,y,z} origin=(a,b,c)
radius=rradius=r Surface Point along vector to Surface Point along vector to
originorigin(x,y,z)
(a,b,c)
(x’,y’,z’)
(0,0,0)x
y
r
x’-ax-a
L
x’
r
top
centerorigin
Transform to Femur Transform to Femur CoordinatesCoordinates
GivenGivend={x,y,z} origin=(a,b,c) radius=rd={x,y,z} origin=(a,b,c) radius=r
center=(cx,cy,cz) top=(tx,ty,tz)center=(cx,cy,cz) top=(tx,ty,tz) Find Coordinate Transform to Find Coordinate Transform to
originorigin
Apply To Each PointApply To Each Point
vxvzvy vzvyvx origincentervz
topcentervy
)](,,[][][]'[ rcbadRd
y
-z
origin top
]0,0,1[][][ vxR
]1,0,0[][][ vzR]0,1,0[][][ vyR
Transform to Plane Transform to Plane CoordinatesCoordinates
GivenGivend={x,y,z}d={x,y,z}
origin=(a,b,c)origin=(a,b,c)
radius=rradius=r
center=(0,0,0)center=(0,0,0)
top=(tx,ty,tz)top=(tx,ty,tz) Preserve arc length and x-y orientationPreserve arc length and x-y orientation
..
*rd
d(0,0,0)
(x,y,z)r
Lσ
x
z
y
(0,0,0)
(x,y,z)
xβ
(tx,ty,tz)
(0,0,0)
(x’,y’)d
x’
y’
(tx,ty,tz)
β)/(tan 1 yx
)cos(*' dx )sin(*' dy
)/2(tan*2 1 rL
Map Plane to JPEGMap Plane to JPEG
GivenGivend={x,y} pressure={p(x,y)}d={x,y} pressure={p(x,y)}
resolution=(xres,yres)resolution=(xres,yres)
Determine Bins and Determine Bins and SampleSample
bwx = dx/xres bwy= dy/yresbwx = dx/xres bwy= dy/yres
Pressure = average of 3 nearest nodesPressure = average of 3 nearest nodes
bwx
bwy
dy
dx
Future WorkFuture Work
JPEG Bin assignmentJPEG Bin assignment– 3 node average3 node average– all nodes averageall nodes average– average of 4 cornersaverage of 4 corners– transfer transfer
function/smoothingfunction/smoothing
Comparison of JPEG values Comparison of JPEG values to film JPEGto film JPEG– bitwisebitwise– regionregion
Real Femur FE MeshReal Femur FE Mesh– increased complexityincreased complexity– sphere fittingsphere fitting