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MULTI-DISCIPLINARY ANALYSIS,
INVERSE DESIGN AND OPTIMIZATION
George S. DulikravichProfessor and Director, MAIDO Institute
Department of Mechanical and Aerospace EngineeringThe University of Texas at Arlington
(Thanks to my students, postdocs and visiting scientists)
Professor Dulikravich has authored and co-authored over 300 technical publications in diverse fields involving computational and analytical fluid
mechanics, subsonic, transonic and hypersonic aerodynamics, theoretical and computational electro-magneto-hydrodynamics, conjugate heat transfer including
solidification, computational cryobiology, acceleration of iterative algorithms, computational grid generation, multi-disciplinary aero-thermo-structural inverse
problems, design and constrained optimization in turbomachinery, and multi-objective optimization of chemical compositions of alloys. He is the founder and Editor-in-Chief of the international journal on Inverse Problems in Engineering
and an Associate Editor of three additional journals. He is also the founder, chairman and editor of the sequence of International Conferences on Inverse
Design Concepts and Optimization in Engineering Sciences (ICIDES). Professor Dulikravich is a Fellow of the American Society of Mechanical Engineers, an
Associate Fellow of the American Institute of Aeronautics and Astronautics, and a member of the American Academy of Mechanics. Professor Dulikravich is also
the founder and Director of Multidisciplinary Analysis, Inverse Design and Optimization (MAIDO) Institute and Aerospace Program Graduate Student
Advisor at UTA.
A sketch of my current research interests• Geometry Parameterization• Computational Grid Generation• Flow-Field Analysis• Thermal Field Analysis• Stress-Deformation Field Analysis• Electric Field Analysis• Magnetic Field Analysis• Conjugate (Concurrent) Analysis• Multi-Disciplinary Inverse Problems• Multi-Disciplinary Optimization & Design
Multi-Disciplinary Analysis, Inverse Design and Optimization (MAIDO)
Aerodynamics
Heat Conduction Structure
s
Conjugate Heat Transfer
Aero-Elasticity
Thermo-Elasticity
Aero-Thermo-Elasticity
T Q
T
T
Q
U
U?
U?
P
P
Parallel Computer of a “Beowulf” type• Based on commodity hardware and public domain software• 16 dual Pentium II 400 MHz and 11 dual Pentium 500 MHz based PC’s• Total of 54 processors and 10.75 GB of main memory • 100 Megabits/second switched Ethernet using MPI and Linux • Compressible NSE solved at 1.55 Gflop/sec with a LU SSOR solver on a
100x100x100 structured grid on 32 processors (like a Cray-C90)• GA optimization of a MHD diffuser completed in 30 hours. Same
problem would take 14 days on a single CPU
Conjugate Simulation of Internally Cooled Gas Turbine Blade
Static temperature contours and grid in the leading edge region
Head Cooling Simulation
Animated view of outer surface mesh
Electro-Magneto-Fluid-Dynamics (EMFD):• active control of large-scale single crystal growth,• enhanced performance of compact heat exchangers,• control of spray atomization in combustion processes, • reduction of drag of marine vehicles, • flow control in hypersonics, • fast response shock absorbers, • hydraulic transmission in automotive industry, • free-flow electrophoretic separation in pharmaceutics,• large scale liquid based food processing, • biological transport under the influence of EM fields, • fuel cells and batteries, • electro-polymers and other smart materials, etc.
EMHD Conservation of Linear Momentum
30 i]ΤΤαρg[1Dt
vDρ me ppp
)vv(μ t
v )EE(σ 2
TTT
κ 2
sTEσT
κ5
5
Eq e BEσ1 BEdσ 2 BTσ 4
BTdσ 5 B)BE( σ 7 B)BT( T
κ10
EEε p BBμ1m
BEvε p
)BE(εDt
Dp
EMHD Conservation of Energy
EκΤdκΤκQDt
TDρC 421hp
BEκBΤκEdκ 1075
ΤdΤΤ
κΤEσEEσ 2
41
BΤEΤ
κΤdE
Τ
κ 105
Dt
BDEVε
μ
B
Dt
EεDE p
m
p
.
Conservation of Mass
0v
EMHD Maxwell’s Equations
ep qBvεEε ,
0B ,
t
BE
,
vqBvεEεt
)Evε(μ
Bepp
ΤdσΤσEdσEσ 5421
BΤκTBEσ 10-1
7 .
Multi-Disciplinary Analysis(Well-defined or Direct Problems)
Multi-disciplinary engineering field problems are fully defined and can be solved when the following set of information is given:
1. governing partial differential or integral equation(s),
2. shape(s) and size(s) of the domain(s),
3. boundary and initial conditions,
4. material properties of the media contained in the field, and
5. internal sources and external forces or inputs.
Multi-Disciplinary Inverse Problems(Ill-posed or ill-defined)
If any of this information is unknown or unavailable, the field problem becomes an indirect (or inverse) problem and is generally considered to be ill posed and unsolvable. Specifically, inverse problems can be classified as:
1. Shape determination inverse problems,
2. Boundary/initial value determination inverse problems,
3. Sources and forces determination inverse problems,
4. Material properties determination inverse problems, and
5. Governing equation(s) determination inverse problems.
The inverse problems are solvable if additional information is provided and if appropriate
numerical algorithms are used.
Inverse prediction of temperature-dependent thermal conductivity of an arbitrarily shaped object
Inverse determination of boundary conditions
Inverse Determination of Convective Boundary Condition on a Rectangular Plate
1n
conv
a
0
ambO
abn
coshakn
abn
sinh
axn
sina
ybnsinh
dxa
xnsinT
a
h2y,xT
Hybrid Constrained Optimization
• Minimize one or more objective functions of a set of design variables subject to a set of equality and inequality constraint functions.
ALGORITHMS• Gradient Search (DFP, SQP, P&D)• Genetic Algorithm(s)• Differential Evolution• Simulated Annealing• Simplex (Nelder-Mead)• Stochastic Self-adaptive Response Surface
(IOSO)
DETERMINATION OF UNSTEADY CONTAINER TEMPERATURES DURING FREEZING OF THREE-
DIMENSIONAL ORGANS WITH CONSTRAINED THERMAL STRESSES
• 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
Diffuser flow separation with no applied magnetic field
Significantly reduced diffuser flow separation with optimized distribution of magnets located in the geometric expansion only
Two-stage axial gas turbine entropy fields and total efficiencies before and after optimization of hub and shroud shapes
using a hybrid constrained optimizer
Results
Comparison of 3 optimized airfoil cascades against the original VKI airfoil cascade.
Multi-objective Constrained Design Optimization
Comparison of total pressure loss versus total lift for optimized airfoil cascades and the inversely designed original VKI airfoil cascade.
Internally cooled blade example and its triangular surface mesh
Passage shape in x-z plane for initial design and for IOSO optimized design
Principal stress contours for initial design andfor IOSO optimized design of cooling passage
Temperature contours on pressure side for initial design and for IOSO optimized design
Temperature contours on suction sidefor initial design and for IOSO optimized design
Objective function convergence history andTemperature constraint function convergence history
ITERATION
OB
JEC
TIV
E
20 40 60 80
0.003
0.004
0.005
0.006
0.007
0.008
0.009
IOSOPGA
ITERATION
TE
MP
ER
AT
UR
EC
ON
ST
RA
INT
20 40 60 8010-6
10-5
10-4
10-3
10-2
10-1
100
IOSOPGA
Extremum search dynamic
Extremum search dynamic
Extremum search dynamic
Extremum search dynamic
Extremum search dynamic
Extremum search dynamic
Extremum search dynamic
Extremum search dynamic
Extremum search dynamic
Extremum search dynamic
Optimization of chemical composition of an alloy Purpose: To determine optimal properties of an alloy havingdifferent chemical compositions by using an existing database Problem features:
variable parameters: chemical composition of an alloy
C, S, P, Cr, Ni, Mn, Si, Cu, Mo, Pb, Co, Cb, W, Sn, Al, Zn, Ti
( 8…17 variables).
criterion: •Stress (PSI – maximize);•Operating temperature (T – maximize);•Time to "survive" until rupture (Hours – maximize).
mathematical model: have none; use an existing database
Optimization of chemical composition of an alloy(8…17 chemical components in a steel alloy )
C, S, P, Cr, Ni, Mn, Si, Cu, Mo, Pb, Co, Cb, W, Sn, Al, Zn, Ti
temperature
1980
2000
2020
2040
2060
2080
2100
2120
1 2 3 4 5
number of chemical composition of an alloy
Max
imu
m o
f T
emp
erat
ure
Stress
8000
9000
10000
11000
1 2 3 4 5
number of chemical composition of an alloy
Max
imu
m o
f st
ress
Hours
1000
3000
5000
7000
9000
11000
1 2 3 4 5
number of chemical composition of an alloy
Max
imu
m o
f h
ou
rs
0.00E+00
2.00E+03
4.00E+03
6.00E+03
8.00E+03
1.00E+04
1.20E+041
2 3 4 56
78
910111213141516
1718
1920
2122232425
2627282930
3132
3334
353637383940414243
4445
464748 4950
Optimization of chemical composition of an alloy Problem No. 1.
Optimization of chemical composition of an alloy Problem No. 1.
2000 4000 6000 8000 10000
PSI
2000
4000
6000
8000
10000
HO
UR
S
If a researcher knows exactly in what temperature span the alloy being designed will work, it is more efficient that the problem of two-criteria optimization be solved with additional constraint for the third efficiency parameter.
This figure presents interdependence of the optimization criteria built onthe obtained set of Pareto optimal solutions. Analysis of this picture showsthat the increase of temperature, forinstance, leads to the decrease ofcompromise possibilities betweenPSI and HOURS.
Optimization of chemical composition of an alloy Problem No. 2-6.
This slide presents results of solution of five additional two-criteria problems in whichPSI and HOURS were regarded as criteria, and different constraints were placed on temperature:
•Problem 2. - , number of Pareto optimal solutions is 20.•Problem 3. - , number of Pareto optimal solutions is 20.•Problem 4. - , number of Pareto optimal solutions is 20.•Problem 5. - , number of Pareto optimal solutions is 15.•Problem 6. - , number of Pareto optimal solutions is 10.
2000 4000 6000 8000 10000
PS I
0
4000
8000
12000
HO
UR
S
2 - T>=1600
3 - T>=1800
4 - T>=1900
5 - T>=2000
6 - T>=2050 Maximum achievable values of PSI and HOURS, and possibilities of compromise between these parameters largelydepends on temperature. For instance, the increase of minimum temperature from 1600 to 1900 degrees leads to thedecrease of attainable PSI by more than twice. At the same time, limiting value of HOURS will not alterwith the change of temperature. Further increase of temperature leads to further decrease of other parameters, by both limiting value and compromise possibility.
The decrease of the number of optimization criteria (transition from three- to two-criteria problem with constraints) leads to the decrease of the number ofadditional experiments, at the expense of bothdecreasing the number of Pareto optimal points anddecreasing the variation range of chemical composition of alloys.
Optimization of chemical composition of an alloy Problem No. 1.
4000 6000 8000 10000
(TE M P +460)*LO G 10(H O U R S +20)
0
2000
4000
6000
8000
10000
PS
I
4000 6000 8000 10000
(TE M P +460)*LO G 10(H O U R S +20)
0
2000
4000
6000
8000
10000
PS
I
T>=1600F
T>=1800F
T>=1900F
T>=2000F
T>=2050F
Larsen-Mueller diagram for 3-criteria optimization results. Larsen-Mueller diagrams for five 2-criteria optimization problems results
Inverse problem of finding chemical composition of an alloy with specified properties
(Problem # 8 )
Purpose: To define chemical composition of an alloy for required properties of material by using an existing database
Problem features:
variable parameters: chemical composition of the alloy
C, S, P, Cr, Ni, Mn, Si, Mo, Co, Cb, W, Sn, Zn, Ti ( 14 variables).
criterion: (multi- objective statement – 10 simultaneous objectives)
•Stress (PSI) (PSI-PSI req.)**2 –> minimize
•Operating temperature (T) (T-T req.)**2 –> minimize
•Time to "survive" until rupture (Hours) (Hours-Hours req.)**2 –> minimize
Cr -> minimize; Ni->minimize; Mo->minimize; Co->minimize; Cb >minimize;
W >minimize; Sn >minimize; Zn >minimize; Ti >minimize;
constraints: none
mathematical model: have none; use an existing experimental database
Comparative Analysis of Inverse Problem Formulations for Determining Chemical Composition of an Alloy
Eps str Eps t Eps h Eps sum NConst
N Obj
N Point(Poreto)
NCalls
Score
Prob.1 .408E-19 .356E-06 .536E-06 .297E-06 0 3 50 417 0.590
Prob.2 .269E-08 .267E-07 .172E-08 .104E-07 3 1 1 703 0.246
Prob.3 .897E-10 .143E-09 .134E-12 .777E-10 3 3 50 445 0.817
Prob.4 .434E-13 .289E-12 .244E-18 .111E-12 3 1 1 1020 0.246
Prob.5 .413E-13 .139E-05 .549E-06 .646E-06 2 1 1 601 0.239
Prob.6 .954E-06 .576E-15 .980E-04 .646E-06 2 1 1 774 0.180
Prob.7 .408E-10 .515E-10 .299E-12 .309E-10 2 1 1 776 0.256
Prob.8 .714E-09 .928E-09 .127E-10 .552E-09 3 10 46 834 1.000
0.00E+00
2.00E-01
4.00E-01
6.00E-01
8.00E-01
1.00E+00
1.20E+00
1 2 3 4 5 6 7 8
Inverse Problem #
Sco
re
1
2
3
4
5
6
7
8
Pareto Set of Cr
1.00E+01
2.00E+01
3.00E+01
4.00E+01
5.00E+01
0 10 20 30 40 50 60
Number of Pareto point
Lev
el o
f C
r
My current management views