Solving Inverse Design Problems Involving Radiant Enclosures through

Gradient-Based Optimization

K. J. Daun, J. R. Howell, and D. P. MortonDepartment of Mechanical Engineering

The University of Texas at Austin

Radiant Enclosure Design Problems

• Radiant enclosure design problems are encountered in diverse industrial settings

• Examples:– Paint drying in the automotive industry– Baking ovens in the food industry– RTP of semiconductor wafers– Optimal geometry of solar collector-concentrators

General Form of the Design Problem

• Enclosure contains a heater surface and a design surface

• Goal is to find the enclosure geometry and heater settings that produces the desired qs and T over the design surface

• In order for the problem to be well-posed, only one BC can be specified on each surface!

• Can be solved using forward, inverse, or optimization design methodologies

Design SurfaceqsDS, TDS are

known

Heater Surface

qs = ?

qs = ?

qs = ? qs = ?

qs = ?

qs = ?

Forward Design Methodology

• Either TDS or qsDS is specified over each surface

• Assume TDS is specified over the design surface

• The designer

1.Guesses heater settings and enclosure geometry

2.Calculates qs over the design surface

3.Adjusts design and repeats analysis if necessary

• Requires many iterations, and the final solution quality is limited

Inverse Design Methodology

• Both TDS and qsDS are specified over the design surface, while the heater settings are unspecified

• The inverse design problem is in its explicit form, which is ill-posed

• The resulting set of ill-conditioned equations, Ax = b, must be solved using regularization methods

• Finds a solution in few iterations, but it is often non-physical• This method is only applicable to certain types of enclosure

design problems

Optimization Design Methodology

• TDS is specified over the design surface

• Define an objective function, F,

and design parameters, , that control enclosure geometry and heater settings

• Optimal design is found by minimizing F, usually through gradient-based minimization

• Much more efficient than forward “trial-and-error” method

• Easy to implement design constraints by restricting the domain of

DSN

j

targetsjDSsjDS

DS

qqN

F1

21

Gradient-Based Minimization

• Objective function local minimum is found iteratively

• At the kth step:

1. Check if k = , usually by checking if Fk < crit

2. Choose search direction, pk

3. Choose step size, k, usually by minimizing fk = Fk+kpk

4. Take a step, k+1 = k + kpk

Gradient-Based Minimization• Search direction is chosen based on 1st and 2nd-order

curvature at Fk:

• Search direction choices:

T

N

kkkk FFF

F

,,,21

112

12

212

122

122

112

2

FF

F

FFF

F

n

k

Steepest Descent pk = Fk

Newton 2Fkpk = Fk

Quasi-Newton Bkpk = Fk

Classes of Enclosure Design Problems

Design SurfaceT = Tspec, q = qspec

Heater Surface

q1 = ?

q2 = ?q3 = ?

q4 = ?q5 = ?

Design SurfaceT = Tspec, qs = qspec

q1 = ?Heater Surface

q2 = ? q3 = ? q4 = ? q5 = ?

Design SurfaceT = Tspec, q = qspec

q1 = ? q2 = ?

q3 = ? q4 = ?

Design SurfaceTt = Tspec, qt = qspec

V

q1t = ?Heater Surface

q2t = ? q3t = ? q4t = ? q5t = ?

Inverse Boundary Condition Design

• Goal is to find heat flux distribution over the heater surface

• Enclosure geometry is fixed

• Solved using both inverse and optimization design methodologies

• Inverse design methodologies by Oguma and Howell (1995), Harutunian et al. (1995)

• Optimization design methodology by Daun et al. (2003)

Design SurfaceT = Tspec, qs = qspec

qs = ?

Heater Surface

qs = ? qs = ? qs = ? qs = ?

Governing Radiosity Equations

,uqo b

a

o duuukuqu '',,' ,4 uTu

Radiosity Emitted Radiation Reflected Incident Radiation

= +

b

a

o duuukuq '',,' ,uqs ,uqo

Outgoing Energy Incoming Energy

= +

Solution of Governing Equations• Calculating F and 2F requires 1st- and 2nd-

order heat flux sensitivity, e.g.

• Radiosity sensitivities are solved by post-processing the radiosity solution

1. Solve A x = b, xi = qoi

2. Solve A x = b, xi = qoip

3. Solve A x = b, xi = 2qoipq

• Heat flux sensitivities are found from radiosity sensitivities

DSN

j p

sjtargetsjsj

DSp

qqq

N

F

1

2

Example Problem: Inverse BC Design

Design Surfaceqs

target = 2 W/m2

Eb = 1 W/m2

= 0.5

qs = 0 W/m2 qs = 0 W/m2

1

2

3

4

5

6

7

8

9 10 11 12 12 11 10 9

1

2

3

4

5

6

7

8

Heater Surface = 0.9

Example Problem: Inverse BC Design

u

0

2

4

6

8

10

12

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

T4(u) [W/m2]

Example Problem: Inverse BC Design

0.95-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0.65 0.7 0.75 0.8 0.85 0.9 0.95

u

T4(u) [W/m2] (specified)

qs(u, ) [W/m2]

Optimization of Transient Problems• Solved using:

– Gradient-based optimization (Fedorov et al., 1998)

– Inverse design methodology (Ertürk et al., 2001)

– Nonlinear controls with regularization (Gwak and Masada, 2002)

Design SurfaceTt = Tspect, qt = qspect

V

q1t = ?

Heater Surface

q2t = ? q3t = ? q4t = ? q5t = ?

Governing Equations

b

a

o duuukuq '',,' ,uqs ,uqo

Outgoing Energy Incoming Energy

= + t

uTuucu

,

Stored Energy

u

• Enclosure surface discretized into N elements, and time domain discretized into Nt time steps

• Results in a non-linear system of equations that must be solved at each time step

,,,4iiiii ttttt bTT BA

Transient Optimization• Objective function is defined as

• Heater settings are controlled by cubic splines,

where = ttmax.

• Objective function sensitivities found analytically, through direct differentiation

t DSN

i

N

ji

targetij

DSt

tTtTNN

F1 1

2,

1

3

32

2

123

313

131,

hh

hhshq

Example Problem: Transient Optimization

T1 = 300 K1= 1

T2 = 1000 K2 = 1

=0

=1

V

Heater surface

Design surface Refractory surface

1 2 3 4 5 6 7 8

Material Thickness [cm]

Design Surface AISI 1010 Steel 0.1 0.8

Heater Surface Refractory Brick 1 0.7

Refractory Surface Refractory Brick 1 0.3

Example Problem: Transient Optimization

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

qs

T

24

1 2 3 4

5

6

7

8

Example Problem: Transient Optimization

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

T

Ts

x = 1

x = 0

Target

x = 0 x = 1

Design Surface

Geometric Optimization

• Both heater settings and geometry can be designed

• Only optimization methods have been used

• Deterministic optimization for diffuse-gray enclosures (infinitesimal-area method)

• Stochastic optimization for enclosures with specular surfaces (Monte Carlo method)

Design SurfaceT = Tspec, q = qspec

Heater Surface

qs

qsqsqsqs

Enclosures Containing Specular Surfaces

`

Energy leaving by emission

Energy in from surrounding surfaces

Energy in from other sources

= –

ΦΦ isi Aq ΦΦ ibii AE

N

jjijbjj AE

1

ΦΦΦ F=

• The exchange factor, Fji, is estimated using Monte Carlo analysis:

• The uncertainty in Fji induces a random error in qs and F

bj

ibjijij N

N FF ~

Stochastic Optimization• Uses Kiefer-Wolfowitz scheme based on steepest-

descent, used when an unbiased estimate of F is unavailable

• At the kth iteration,

1.Check if k =

2.Set

3.Set k = 0ka, 0 a 1

4. k+1 = k + kpk

• The gradient is estimated by central finite difference,

with

kkk FF ~~ p

k

kp

kkp

kk

p h

hFhFF

2

~~~

ee ΦΦ

Φ

10,0 bk

hh

bk

Example Problem: Geometric Optimization

Specular Surfaceqs3 = 0 W/m2

3 = 1

Specular Surfaceqs2 = 0 W/m2

2 = 1

Design SurfaceEb4 = 0 W/m2, qs

target = -1 W/m2, 4 = 1

1

2

u

Heater Surfaceqs1 = 1 W/m2

1 = 1

Example Problem: Geometric Optimization

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1

2

1

2

Initial Enclosure Geometry

Optimal Enclosure Geometry

Minimization Path

Example Problem: Geometric Optimization

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

u

qs(u

,

) [W

/m2]

qs(u, 0) (initial)

qs(u, *) (optimal)

Conclusions

• Optimization is used to design many types of radiant enclosures

• Solves the inverse design problem implicitly, through iteration

• More efficient than the forward design method, and usually produces better solutions

• More straightforward than inverse design (regularization) method, and easier to implement design constraints