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

dulikra@mae.uta.edu

(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