determination of unsteady container temperatures during freezing of three-dimensional organs with...
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DETERMINATION OF UNSTEADY CONTAINER TEMPERATURES DURING FREEZING OF THREE-DIMENSIONAL
ORGANS WITH CONSTRAINED THERMAL STRESSES
Brian DennisAerospace EngineeringPenn State University
G.S. DULIKRAVICHMultidisciplinary Analysis, Inverse Design, and Optimization (MAIDO)
Department of Mechanical and Aerospace EngineeringThe University of Texas at Arlington
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Objectives
• Use finite element method(FEM) model of transient heat conduction and thermal stress analysis together with a Genetic Algorithm(GA) to determine the time varying temperature distribution that will cool the organ at the maximum cooling rate allowed without exceeding allowed stresses
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Mathematical Model
0Xu2Gzx
w2
yx
v2
2x
u2 G
Thermoelasticity
0Yv2Gzy
w2
2y
v2
yx
u2 G
0Zw2G2z
w2
zy
v2
zx
u2 G
12
EG ,
211
E
x
TG23X
y
TG23Y
z
TG23Z
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Mathematical ModelHeat Conduction
Tk t
TCeffective
dT
dHCeffective
2
1
222
222
dzdT
dydT
dxdT
dzdH
dydH
dxdH
dT
dH
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Comparison with Analytic Solution
0 1 2 3 4Time
25
30
35
40
45T
AnalyticNumerical
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Geometry and Material Properties
• Actual Geometry from MRI or other sources was not available so a kidney like object was constructed from analytical functions
• Some material properties are available in various publications but some had to be estimated for this test case
• The procedure can be easily repeated when accurate geometry and material data are available
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Algorithm
Ti = Tinitial
t = 0
run GA with analysisusing t=tI to tf and T=Ti
tf=tI+t
Ti=Tbest
is kidneyTmax < Tfrozen
N
Y
END
START
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Outer Boundary
-1
-0.5
0
0.5
1
Z
-1
-0.5
0
0.5
1
X
-1
-0.5
0
0.5
1
Y
X Y
Z
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Kidney Cortex
X
Y
Z
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Kidney Medula
X Y
Z
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Kidney Cortex and Medula
X
Y
Z
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Parameterization
• Outer sphere (cooling container) divided into six equally spaced patches.
• Each patch is covered by biquadratic Lagrange polynomials with nine nodes
• 26 design variables describing container surface temperature
• Allowed to vary from +20 C to -30 C
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Objective Function
2
yield
maxPt
TF
If( max stress > yield stress)
P = 100
else
P = 0
Minimize average cooling rate with max. stress as a constraint
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Numerical Results• Time interval for outer loop was 5
minutes
• Time step for analysis was 15 seconds
• Was run for 30 minutes (6 surface temperature distribution optimizations)
• Grid with 5184 parabolic tetrahedral elements was used for heat conduction
• Grid with 5184 linear elements was used for stress analysis
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GA Parameters Used• Each design variable represented with 4-bit
binary string• Uniform crossover operator was used• 50% probability of crossover, 2% probability
of mutation• 15 generations were run for each optimization• Used a population size of 31• Was run on a 32 processor parallel computer
composed of commodity components• One optimization takes ~45 minutes
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Numerical Results
10 20 30time(min)
2
4
6
8
10
12
14
16
18
maxstress
(MPa)
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Numerical Results
10 20 30time(min)
-2.25
-2
-1.75
-1.5
-1.25
-1
-0.75
-0.5
averagedT/dt(degreesC/sec)
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Numerical Results
5 10 15generation
-0.007
-0.006
-0.005
-0.004
-0.003
-0.002
-0.001
0
0.001
0.002
0.003
0.004
0.005
averagedT/dt
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Numerical Results
-1 -0 .8 -0 .6 -0 .4 -0 .2 0 0 .2 0 .4 0 .6 0 .8 1Y
-2 0
-10
0
10
T
T a t 1 0 m inT a t 1 5 m inT a t 2 0 m inT a t 2 5 m inT a t 3 0 m in
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Conclusion• An algorithm was developed for the inverse
determination of surface temperature distributions in the freezing of a kidney organ
• Current results show that after 30 minutes the kidney is not entirely frozen and more time intervals are needed
• This simulation process was made practical through the use of parallel optimization algorithms(GA) and cheap parallel computer composed of PC components
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Results(movie)
Animation of temperature contours along slice at z=0 for every 5 minute interval
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Results( movie)
Animation of container surface temperature contours for every 5 minute interval