Solution to Shape Identification Problem of UnsteadyHeat-Conduction Fields

Eiji Katamine,1 Hideyuki Azegami,2 and Yasuhiro Matsuura2

1Department of Mechanical Engineering, Gifu National College of Technology, 501-0495 Japan2Department of Mechanical Engineering, Toyohashi University of Technology, 441-8580 Japan

This paper presents a numerical analysis method for shape determinationproblems of unsteady heat-conduction fields in which time histories of temperaturedistributions on prescribed subboundaries or time histories of gradient distributions oftemperature in prescribed subdomains have prescribed distributions. The square errorintegrals between the actual distributions and the prescribed distributions on theprescribed subboundaries or in the prescribed subdomains during the specified periodof time are used as objective functionals. Reshaping is accomplished by the tractionmethod that was proposed as a solution to shape optimization problems of domains inwhich boundary value problems are defined. The shape gradient functions of theseshape determination problems are derived theoretically using the Lagrange multipliermethod and the formulation of material derivative. The time histories of temperaturedistributions are evaluated using the finite-element method for a space integral and theCrank–Nicolson method for a time integral. Numerical analyses of nozzle and coolantflow passage in a wing are demonstrated to confirm the validity of this method.© 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(3): 212–226, 2003; Publishedonline in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ htj.10086

Key words: inverse problem, optimum design, numerical analysis, unsteadyheat-conduction, finite-element method, domain optimization, shape identification,traction method

1. Introduction

Cases in which the heat transfer phenomenon influences the performance of machines andstructures are abundant. For example, the technique of applying a special clay to a sword blade invarying amounts of thickness depending on the position on the blade ensured the control of the heattransfer phenomenon in the hardening process of the traditional Japanese sword. The method is a veryrational one devised in order to produce a metal form which has the appropriate structure at any givenposition as a result of controlling the thickness of the special clay at each position. Based on modernengineering which considers mathematical models, it is possible to consider a numerical solution tothe problem which determines the boundary shape of the special clay so that this technique mayinclude the temperature history for the heat transfer field produced at the boundary where the specialclay came in contact with the sword blade, for each point of the boundary. In this paper, a new practical

© 2003 Wiley Periodicals, Inc.

Heat Transfer—Asian Research, 32 (3), 2003

212

method of numerical analysis for the shape identification problem of unsteady heat-conduction fieldsto control temperature distribution or temperature gradient distribution is presented. The shapeidentification problem in this unsteady heat-conduction field is an inverse problem.

There has been much research on the numerical analysis of inverse problems for unsteadyheat-conduction fields focused on the identification of time histories of temperature and heat flux [1,2], the intensity and position of the heat source [3, 4], and characteristic values such as the heatconductivity coefficient and the heat transfer coefficient [5]. However, the research on boundary shapeidentification has been far from satisfactory. A study limiting the number of design variables to twoor three has been made [6, 7]. Even for the steady-state problem, there have not been many researchreports [8–12]. In previous studies of numerical analysis for shape identification, the numericaloptimization procedure involves first providing a substitute model with a finite number of degrees offreedom as a continuum model and then formulating the optimization problems. Based on this model,a shape optimization (or shape identification) problem, in which the design variables are defined ina finite dimensional vector space, can be analyzed numerically using mathematical programmingtechniques. This method, however, was not effective in locating an optimal solution to problems witha large number of design variables because of the large number of dimensions in the design space.Thus, the study of the boundary shape identification has not been active.

On the other hand, the present authors have focused on the solution of the shape determinationproblem based on the distributed sensitivity function which uses adjoint variables. In previous papers,we presented a numerical analysis method for these shape inverse problems of steady heat-conductionfields and potential flow fields in which the distributions of state functions (temperature [13], velocity[14], and pressure [15]) on subboundaries or in subdomains were controlled to the prescribeddistributions. Reshaping was accomplished by the traction method [16, 17] which was proposed byone of the authors as a solution to shape optimization problems in which boundary value problemswere defined.

In this study, we applied the traction method to the prescribed problem of time histories oftemperature distributions or time histories of temperature gradient distributions on unsteady heat-con-duction fields. The traction method is a method of applying the gradient method of the distributionsystem which directly uses the sensitivity functions (shape gradient functions) of the domain variationwhich are theoretically derived from the optimization problem. In the traction method, the domainvariations that minimize the objective functions are obtained as solutions of pseudolinear elasticproblems of continua defined on the design domains and loaded with pseudodistributed traction inproportion to the shape gradient functions on the design domains. The numerical solutions of boththe shape gradient function and the pseudolinear elastic problems, used for evaluation of the domainvariation, can be obtained using the finite-element method or the boundary-element method. There-fore, the traction method can be applied to complex shape determination problems with a large numberof design variables. Additionally, this method is advantageous in that the oscillating phenomenondoes not occur during the iterative process of shape optimization because the design boundaries arenot moved by the movement of finite-element nodal points but by the traction in proportion to theshape gradient functions [17].

First, we formulate a temperature square error integrals minimization problem and theoreti-cally derive a sensitivity function for this problem. The shape gradient functions are derived

213

theoretically using the Lagrange multiplier method and the formulation of the material derivative. Ina similar manner, we can derive the shape gradient function for a prescribed problem where the timehistories of temperature gradient distributions are specified in the prescribed subdomains. Then, thenumerical method based on the traction method for these prescribed problems is presented. Lastly,the successful results for the two-dimensional problems demonstrate the validity of the presentedmethod.

2. Temperature Square Error Integrals Minimization Problem

2.1 Formulation of problem

We consider a domain Ω ∈ Rn, (n = 2, 3), with boundaries Γ = Γ0 ∪ Γ1. The temperature isrepresented by φ = φ(x, t) in space x ∈ Ω and time t ∈ [0, T]. R is the set of real numbers. Thetemperature φ(Γ0, [0, T]) distributed on Γ0, the heat flux q(Γ1, [0, T]) distributed on Γ1, the heatconductivity coefficient k, density ρ, capacity c, and heat source f(x, t) distributed in Ω, and the initialtemperature distribution φ(x, 0) = φ0(x) are given as known functions or values.

A domain variation problem where the temperature distribution φ(ΓD, [0, T]) is specified tobe φD(ΓD, [0, T]) on subboundaries ΓD ⊆ Γ1 can be regarded as a shape determination problem, asshown in Fig. 1. The domain variation of the heat-conduction field from domain Ω to Ωs = Ts(Ω) canbe described using a one-to-one mapping Ts. The index s denotes the history of domain variation.Ts(Ω) is assumed to be the elements of a set D of admissible functions which satisfy the restrictionson domain variation. For simplicity, we assume that the subboundaries Γ1 are invariable, that is,Ts(Γ1) = Γ1. This problem is formulated as

given Ω and

k, ρ, c, f, q, φ, φ0: fixed in space (1)

find Ωs or Ts(Ω) ∈ D (2)

Fig. 1. Determination problem of domain with prescribed temperature on prescribed subboundaryΓD ⊂ Γ1.

214

that minimize ∫EΓD

0

T

dt (3)

subject to ∫0

T

a(φ, w) + b(φ,t, w) − c(w) − d(w)dt = 0 φ ∈ Ψt Ww ∈ Wt (4)

where the square error integrals EΓD are defined by

EΓD = EΓD

(φ − φD, φ − φD)dt = ∫(ΓD

φ − φD)2dΓ (5)

Equation (4) is the variational form, or the weak form using the adjoint temperature w(x, t), of thestate equation. The bilinear forms a(φ, w) and b(φ,t, w) and the linear forms c(w) and d(w) are definedby

a(φ, w) = ∫ Ωs

kφ,jw,jdx, b(φ,t, w) = ∫ Ωs

ρcφ,twdx,

c(w) = ∫ Ωs

fwdx, d(w) = ∫ Γ1

qwdΓ

The sets Ψt and Wt of the admissible temperature φ and adjoint temperature w are given by

Ψt = φ(x, t) ∈ H1(Ωs × [0, T])|φ(x, t) = φ(x, t), t ∈ [0, T], x ∈ Ts(Γ0),

kφ(x, t),jnj = q(x, t), t ∈ [0, T], x ∈ Ts(Γ1), φ(x, 0) = φ0(x), x ∈Ωs (6)

Wt = w(x, t) ∈ H1(Ωs × [0, T])|w(x, t) = 0, t ∈ [0, T], x ∈ Ts(Γ0),

w(x, T) = wT(x) = 0, x ∈ Ωs (7)

In the tensor notation of this paper, the Einstein summation convention and partial differentialnotations (⋅),i = ∂(⋅) / ∂xi and (⋅),t = ∂(⋅) / ∂t, respectively, are used. Hm(⋅) and n denote the Sobolev spaceand the outward unit normal vector, respectively.

2.2 Shape gradient function

Applying the Lagrange multiplier method, this problem can be rendered into a stationaryproblem of the Lagrange functional

L = ∫0

T

EΓD(φ − φD, φ − φD)dt − ∫

0

T

a(φ, w) + b(φ,t, w) − c(w) − d(w)dt (8)

215

The derivative L. with respect to s is derived using the velocity function V(Ωs) = ∂Ts(Ω) / ∂s =

∂Ts(Ts−1(Ωs)) / ∂s [16, 17] as

L. = −∫

0

T

a(φ, w′) + b(φ,t, w′) − c(w′) − d(w′)dt

−∫0

T

a(φ′, w) + b(φ,tg , w) − 2EΓD

(φ′, φ − φD)dt + lG(V) (9)

where (⋅) is the material derivative, and (⋅)′ is the shape derivative for domain variation of thedistributed function under a spatially fixed condition. The linear form lG(V) of the velocity functionV is given by

lG(V) = ∫ Γ0

Gn ⋅ VdΓ (10)

G = ∫0

T

−kφ,jw,jdt (11)

From Eq. (9), φ(x, t) and w(x, t) are obtained by

∫ 0

T

a(φ, w′) + b(φ,t, w′) − c(w′) − d(w′)dt = 0 Ww′ ∈ Wt (12)

∫ 0

T

a(φ′, w) + b(φ,tg , w′) − 2EΓD

(φ′, φ − φD)dt = 0 Wφ′ ∈ Ψt (13)

which indicate the variational form of the original state equation for temperature φ(x, t) and thevariational form for w(x, t), which we call an adjoint equation. Under the conditions satisfying Eqs.(12) and (13), the derivative of the Lagrange functional agrees with that of the objective functionaland the linear form lG(V) with respect to V:

L.|φ,w = (∫

0

T

E.

ΓDdt)|φ,w = lG(V) (14)

The coefficient vector function Gn in Eq. (10) represents a sensitivity function relative to domainvariation and is the so-called shape gradient function. The scalar function G is called the shape gradientdensity function. The shape gradient function can be derived theoretically, thus, domain variation canbe analyzed by the traction method [16, 17].

216

3. Temperature Gradient Square Error Integrals Minimization Problem

In the previous section, we formulated a prescribed problem for time histories of temperaturedistributions φ(ΓD, [0, T]) on the prescribed subboundaries ΓD ⊆ Γ1 in unsteady heat-conductionfields and derived the shape gradient function for the problem. In a similar manner, we can derive theshape gradient function for a prescribed problem where the time histories of temperature gradientdistributions ∇

→φ(ΩD, [0, T]) are specified by gD(ΩD, [0, T]) in the prescribed subdomains ΩD ⊂ Ω,

as shown in Fig. 2.

3.1 Formulation of problem

This problem is formulated as

given Ω and

k, ρ, c, f, q, φ, φ0: fixed in space (15)

find Ωs or Ts(Ω) ∈ D (16)

that minimize ∫ 0

T

EΩDdt

(17)

subject to ∫0

T

a(φ, w) + b(φ,t, w) − c(w) − d(w)dt = 0 φ ∈ Ψt Ww ∈ Wt (18)

where the square error integrals EΩD are defined by

EΩD = EΩD

(∇→

φ − gD, ∇→

φ − gD) = ∫(ΩD

∇→

φ − gD) ⋅ (∇→

φ − gD)dx (19)

Fig. 2. Determination problem of domain with prescribed temperature gradient in prescribedsubdomain ΩD ⊂ Ω.

217

For simplicity, we assumed that the subboundaries Γ0 and the prescribed subdomains ΩD areinvariable, that is, Ts(Γ0) = Γ0 and Ts(ΩD) = ΩD.

3.2 Shape gradient function

In a manner similar to the previous section, the derivative L. is derived as [16, 17]

L. = ∫

0

T

a(φ, w′) + b(φ,t, w′) − c(w′) − d(w′)dt

− ∫ 0

T

a(φ′, w) + b(φ,tg , w) − 2EΩD

(∇→

φ′, ∇→

φ − gD)dt + lG(V) (20)

The linear form lG(V) of the velocity function V and the shape gradient function for this problem areobtained in the same manner as in the previous section by

lG(V) = ∫ Γ1

Gn ⋅ VdΓ (21)

G = ∫ 0

T

−kφ,jw,j − ρcφ,tw + fw + ∇n(qw) + (qw)κdt (22)

where ∇n(⋅) ≡ ∇→

(⋅) ⋅ n and κ denote the mean curvature. The distribution of temperature φ(x, t) andthe adjoint temperature w(x, t) are obtained by

∫ 0

T

a(φ, w′) + b(φ,t, w′) − c(w′) − d(w′)dt = 0 Ww′ ∈ Wt (23)

∫ 0

T

a(φ′, w) + b(φ,tg , w) − 2EΩD

(∇→

φ′, ∇→

φ − gD)dt = 0 Wφ′ ∈ Ψt (24)

Under the condition satisfying Eqs. (23) and (24), the derivative of the Lagrange functional agreeswith that of the objective functional and the linear form lG(V) with respect to V:

L.|φ,w = (∫

0

T

EΩDdt)|φ,w = lG(V) (25)

218

4. Numerical Solution Technique

4.1 Traction method

The traction method is a procedure for determining the amount of domain variation (velocityfunction V) that reduces the objective functional, based on the governing equation (26). This methoduses the gradient method in a Hilbert space, a technique that is also employed in distributed parameteroptimal control problems.

aE(V, y) = −lG(y), Wy ∈ D (26)

where the bilinear form aE(V, y) gives the strain energy of the pseudolinear elastic body defined inthe domain Ωs as

aE(u, v) = ∫ Ωs

Aijkluk,lvi,jdx (27)

where u, v, and Aijkl are displacement, variational displacement, and an elastic tensor, respectively.

The governing equation (26) indicates that the velocity field V is a displacement field whennegative shape gradient functions −Gn act on the boundaries as external forces. In other words, usingthe traction method, domain variation is a displacement field when the shape gradient functions actas external forces in a pseudoelastic problem. Accordingly, Eq. (26) can be solved using a solutionto ordinary linear-elastic problems, thus confirming the general applicability of the traction method.In this study, the finite-element method is used. The elastic tensor is given for simplicity as

Aijkl = δikδjl (28)

4.2 Numerical procedure

A flowchart of the shape optimization system is shown in Fig. 3. The main elements of thesystem include two unsteady heat conduction analyses, calculation of the shape gradient function, avelocity analysis based on the traction method, and shape updating. The shape gradient function isevaluated using the two heat conduction analyses in which the distributions of temperature φ(x, t) andadjoint temperature w(x, t) are analyzed. The time histories of temperature distributions are evaluatedusing the finite-element method for a space integral and the Crank–Nicolson method for a timeintegral. In calculating the solution to the original state equation (12) or (23), the temperatureφ(x, t) is analyzed using the initial condition φ(x, 0) = φ0(x) for time from t = 0 to t = T. On the otherhand, in calculating the solution to the adjoint equation (13) or (24), the adjoins temperature w(x, t)is analyzed using the initial condition w(x, T) = wT(x) = 0 for time from t = T to t = 0. Then, the shapegradient function is calculated using the results. The domain variation V of Eq. (26) is analyzed usingthe finite-element method. Shape optimization analysis is performed by repeatedly executing theseelements sequentially. To perform an analysis, a domain variation coefficient ∆s is set which adjuststhe magnitude of the domain variation ∆sV per iteration.

219

5. Numerical Results

We present the results of two numerical analyses for 2D shape determination problems usingthe traction method and shape gradient function derived in previous sections.

5.1 Coolant flow passage in a wing

We analyzed a shape determination problem of coolant flow passage in a wing for atemperature prescribed problem, as shown in Fig. 4. The outer surface boundary of the wing wasassumed as the prescribed subboundary ΓD = Γ1. The shape shown in Fig. 5 was chosen as theobjective shape, and the temperature distribution over the outer surface boundary was assumed to bethe prescribed temperature distribution φD(x, t). The design boundary is the left-side boundary ofcoolant flow passage. For simplicity, the case we considered had the following conditions: length ofwing l = 0.25 m, specified temperature φ = 20 deg, heat-conductivity coefficient k = 0.204 kW/m deg,heat flux q = 150 kW/m2, density ρ = 2710 kg/m3, capacity c = 0.896 kJ/kg deg, initial temperature

Fig. 3. Numerical procedure.

Fig. 4. Shape determination problem of coolant flow passage in a wing.

220

distribution φ0(x) = φ0 = 20 deg, and specified period of time T = 0.05 s. By dividing the time intervalinto 100 steps, the Crank–Nicolson method was applied. The heat source f was disregarded.

Numerical results are shown in Figs. 6 to 9. Figure 6 shows a comparison of the shapes andtemperature distributions between the initial domain and the converged domain. Figure 7 shows thefinite-element meshes. Figure 8 shows the time histories of temperature at point A in Fig. 4. Figure9 shows the iterative history ratio of the objective functional normalized with the initial value. Fromthese results, we observe a slight difference between the objective domain and converged domain. Itis possible to regard the set shape identification problem as a problem with strong multicrestedness.Therefore, it is dependent on the chosen shape of the initial domain, and the analyzed convergeddomain seems to be one localized solution. However, it was confirmed that the time histories oftemperature in the converged domain agreed with those in the objective domain, and the value of theobjective functional approached zero. According to the numerical results of this basic problem, we

Fig. 5. Objective temperature distributions of coolant flow passage problem.

Fig. 6. Results of temperature distributions in the coolant flow passage problem.

221

confirmed the validity of the present method. This numerical calculation required about 1 hour ofCPU time on a single Alpha21164A/500 MHz processor.

5.2 Nozzle

In the prescribed problem for temperature gradient distribution, we analyzed a nozzlethickness shape problem as shown in Fig. 10. Considering that the two halves are symmetrical, theupper half of domain A–B–C–D was analyzed. The subdomain in the neighborhood of the inner wallof the nozzle was assumed as the prescribed subdomain ΩD. The shape and temperature gradientdistributions shown in Fig. 11 were chosen as the objective shape and the temperature gradient

Fig. 7. Finite-element meshes in the coolant flow passage problem.

Fig. 9. Iterative history of objective functionalin the coolant flow passage problem.

Fig. 8. Time histories of temperature in thecoolant flow passage problem at the

left-hand-side point in Fig. 4.

222

Fig. 10. Shape determination problem of a nozzle.

Fig. 11. Objective temperature gradient distributions in the nozzle problem.

Fig. 12. Results of temperature gradient distributions in the nozzle problem.

223

distributions, respectively. The design boundary is the outside boundary A–D. The case we consideredhad the following conditions: length of wing l = 0.06 m, specified temperature φ = 100 deg,heat-conductivity coefficient k = 0.204 kW/m deg, heat flux q = 0 kW/m2, density ρ = 2710 kg/m3,capacity c = 0.896 kJ/kg deg, initial temperature distribution φ0(x) = φ0 = 20 deg, and specified periodof time T = 0.001 s. In the time integral, by dividing the time period into 100 steps, the Crank–Nicolsonmethod was applied. The heat source f was disregarded.

Numerical results of this problem are shown in Figs. 12 to 15 in a similar manner to theabove-mentioned results. We confirmed that the value of the objective functional approached zero,

Fig. 13. Finite-element meshes in the nozzle problem.

Fig. 15. Iterative history of objective functionalin the nozzle problem.

Fig. 14. Time histories in the nozzle problem oftemperature gradient in the x2 direction at point

P in Fig. 10.

224

and the converged domains analyzed by the proposed method exhibited good agreement with theobjective domain. Based on these results, we confirmed the validity of the present method. Thisnumerical calculation required about 20 minutes of CPU time on a single Alpha21164A/500MHzprocessor.

6. Conclusions

In this paper, we derived the shape gradient functions with respect to the shape identificationproblems of unsteady heat-conduction fields to control temperature distributions and temperaturegradient distributions to prescribed distributions. The validity of the traction method using the derivedshape gradient functions was confirmed by the numerical results.

Literature Cited

1. Nakamura M, Tanaka M, Ishikawa H. Inverse analysis using BEM to estimate unknownboundary values in transient heat conduction problems. Trans JSME Ser A 1992;58:2206–2221. (in Japanese)

2. Bai Q, Fujita Y. A finite element analysis for inverse heat conduction problems. Trans JSMESer B 1997;63:1320–1326. (in Japanese)

3. Kubo S, Ohnaka K, Ohji K. Identification of heat-source and force using boundary integrals.Trans JSME Ser A 1988;54:1329–1334. (in Japanese)

4. Momose K, Kimoto H. Green’s function approach to optimal arrangement of heat sources.Trans JSME Ser B 1995;61:1762–1767. (in Japanese)

5. Tanaka M, Nakamura M, Shiozaki A. A boundary-element inverse analysis procedure forestimation of thermal properties in transient heat conduction. J Soc Mat Jpn 1993;42:708–713.(in Japanese)

6. Matsuzaki K, Nakamura H, Kubo S, Ohji K. Identification of a crack from transient temperaturemeasurements. Proc 7th Computational Mechanics Conference, 1995, No. 950-54, p 370–371.(in Japanese)

7. Nakamura H, Kubo S, Sakagami T, Ohji K. Identification of an edge crack form transienttemperature measurement. Proc 8th Computational Mechanics Conference, 1995, No. 95-4, p343–344. (in Japanese)

8. Barone MR, Caulk DA. Optimal arrangement of hole in a two-dimensional heat conductor bya special boundary integral method. Int J Num Methods Eng 1982;18:675–685.

9. Kennon SR, Dulikravich GS. Inverse design of multiholed internally cooled turbine blades.Int J Num Methods Eng 1986;22:363–375.

10. Yoshikawa F, Nigo S, Kiyohara S, Taguchi S, Takahashi H, Ichimiya M. Estimation ofrefractory wear and solidified layer distribution in the blast furnace hearth and its applicationto the operation. Tetsu-to-Hagane 1987;73:2068–2075. (in Japanese)

11. Shau R, Batista J, Carey GF. An improved algorithm for inverse design of thermal problemswith multiple materials. Trans ASME J Heat Transfer 1990;112:274–279.

12. Meric RA. Differential and integral sensitivity formulations and shape optimization by BEM.Eng Anal Boundary Elements 1995;15:181–188.

13. Katamine E, Azegami H, Kojima M. Boundary shape determination on steady-state heatconduction fields. Heat Transfer Asian Res to appear.

14. Katamine E, Azegami H. Domain optimization analysis of potential flow field. Trans JSMESer B 1995;61:103–108. (in Japanese)

225

15. Katamine E, Azegami H, Yamaguchi S. Shape identification analysis of flow field (prescribedproblems of pressure distribution and solution by the traction method). Trans JSME Ser B1998;64:1063–1070. (in Japanese)

16. Azegami H. Solution to domain optimization problems. Trans JSME Ser A 1994;60:1479–1486. (in Japanese)

17. Azegami H, Kaizu S, Shimoda M, Katamine E. Irregularity of shape optimization problemsand an improvement technique. In: Hernandez S, Brebbia CA (editors). Computer aidedoptimum design of structures V. Computational Mechanics Publications; 1997. p 309–326.

"F F F"

Originally published in Trans JSME Ser B 66, 2000, 227–234.Translated by Eiji Katamine, Department of Mechanical Engineering, Gifu National College of

Technology, Shinsei-cho, Motosu-gun, Gifu, 501-0495 Japan.

226