An Inverse Solution for Determining Arbitrary

冤

An Inverse Solution for Determining Arbitrary Boundary-Conditions using a Least-Squares Approach A. E. Segall Associate Professor Mem. ASME Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA 16802

⳵⌬Tn ⳵⌬Tn+1 ⳵⌬TN ⴱ ⌽共␰,t1兲 ⴱ ⌽共␰,t1兲 . . . ⴱ ⌽共␰,t1兲 ⳵t ⳵t ⳵t ⳵⌬Tn ⳵⌬Tn+1 ⳵⌬TN ⴱ ⌽共␰,t2兲 ⴱ ⌽共␰,t2兲 . . . ⴱ ⌽共␰,t2兲 ⳵t ⳵t ⳵t ] ] ] ⳵⌬Tn ⳵⌬Tn+1 ⳵⌬TN ⴱ ⌽共␰,tN兲 ⴱ ⌽共␰,tN兲 . . . ⴱ ⌽共␰,tN兲 ⳵t ⳵t ⳵t

冤冥冤 冥 v共␰,t1兲

a1

⫻

a2

=

]

Despite the versatility of numerical approaches to the inverse problem 共Beck et al. 关1兴 and Xue et al. 关2兴兲, there is still a strong need for analytical solutions. In fact, many numerical simulations require a starting point and must be verified and bounded to help ensure the validity of the solution. In addition to bounding the problem, there is always a need for closed-form solutions or firstorder approximations that can be quickly used to highlight the significance of various parameters and their often complicated interrelationships. Even with this enduring importance, significant limitations remain including a reliance on higher-order derivatives that magnify data errors, restrictions to small time frames, or the inability to handle arbitrary boundary-conditions. Fortunately, many of these limitations can be avoided and the inverse-solution found for a variety of geometries by using a generalized directsolution combined with a least-squares approach.

Provided the unit-response, ⌽共␰ , t兲 can be derived analytically or numerically, the inverse problem for any combination of geometry and boundary conditions can be solved with the determination of the coefficients and the use of Eq. 共2兲. Once the coefficients are determined, the direct solution obtained via Eq. 共1兲 can then be used to determine the temperature distribution throughout the solid.

Generalized Response A generalized inverse-solution for both slabs 共␰ = x兲 and cylinders 共␰ = r兲 is possible by recognizing a common form to their unit response: ⬁

⌽共␰,t兲 = ␺ + ␧

冕

k

v共␰,t兲 =

兺a j=1

j

冋

0

⳵⌬T共␶兲 ⳵⌬T ⌽共␰,t − ␶兲d␶ = ⴱ ⌽共␰,t兲 ⳵␶ ⳵t

共4兲

j ␺t j/2 + ␧ ␦ j共␰,t兲 2

⬁

共1兲

兺冋

兺

2

k=1

b2j k ⬁

兺b

共− 1兲it j−i−1␹共i, j − 1兲

i=0

⌬T共t兲 = a1t

1/2

+ a 2t + a 3t

3/2

+ ¯ =

兺a t n

共2兲

n=1

A generalized inverse solution is then possible by fitting temperatures obtained at location, ␰ to Eq. 共1兲 with the polynomial coefficients now satisfying the following linear equations in a leastsquares sense, Contributed by the Heat Transfer Division of ASME for publication in the JOURHEAT TRANSFER. Manuscript received June 30, 2004; final manuscript received May 24, 2005. Review conducted by A. F. Emery.

NAL OF

Journal of Heat Transfer

册

共6a兲

⬁

− 1兲兴 n/2

k=1

⌮k 2共i+1兲 k

2 共− 1兲 j+1 ⌮ k⍀ k ␨共j − 1,2j − 3兲 tan−1关e10共j ␦2j−1共␰,t兲 = 2j−2 + ¯ + 2 j−1 ␲ k=1 bk j−2

N

共5兲

⌮ke−bk t

j−1

+

While Eq. 共1兲 is clearly a direct solution of the problem, its form can also be used for the corresponding inverse problem by employing a polynomial with coefficients an to represent the unknown temperature boundary-condition,

册

where the response functions for even 共2j兲 and odd 共2j − 1兲 terms are defined as follows and all other terms have been previously defined 共Segall 关3,4兴, Austin 关5兴, and Vedula et al. 关6兴兲:

␦2j共␰,t兲 = 共− 1兲 j␹共j − 1, j − 1兲 t

−bk2t

where ␧, ␺, and Z reflect geometry, thermal diffusivity 共␬兲, and known boundary-conditions. Given this generalized form, the direct response to a system can be shown to be 共Segall 关3兴兲:

Inverse Solution

v共␰,t兲 =

兺⌮ e k=1

N

It has long been established that the response v共␰ , t兲 of a structure of coordinate, ␰ subjected to an arbitrary temperature-history, ⌬T共t兲 can be expressed by Duhamel’s form of the convolution integral using the unit response, ⌽共␨ , t兲:

共3兲

]

v共␰,tN兲

aN

Introduction

v共␰,t2兲

冥

兺 i=0

冋

兺

共− 1兲

it

2共 j−i兲−3 2 ␨共i,2j i

2

⬁

− 3兲

兺b k=1

⌮k 2共i+1兲 k

册

共6b兲

For a semi-infinite solid, a relationship has already been expressed in a form suitable for the least-squares approach by Carslaw and Jaeger 关7兴: N

v共␰,t兲 =

兺 a 2 ⌫共j + 1兲t n

n=1

n

n/2 n

冉冑 冊

i erfc

␰

2 ␬t

共7兲

Hence, when using Eq. 共5兲 for slabs and cylinders or Eq. 共7兲 for a semi-infinite solid, the coefficients, an are determined by Eq. 共3兲 in a least-squares sense with Eq. 共2兲 then providing the inverse solution.

Copyright © 2005 by ASME

DECEMBER 2005, Vol. 127 / 1403

Fig. 1 Comparison of inverse predictions and asymptotic boundary-condition of a semi-infinite slab as a function of nondimensional time

Results Confirmatory calculations were conducted for a semi-infinite slab 共thickness L = 0.05 m and ␬ = 0.085 cm2 / s兲 under an asymptotic boundary-condition, ⌬T共t兲 = 1 · 共1 − e−0.5t兲 at x / L = 0 and adiabatic at x / L = 1. A solution derived by Austin 关5兴 was then used to generate the response at the adiabatic boundary with a random number generator used to simulate experimental errors as high as ±10%. The error-laden data was then fit to a polynomial of the form of Eq. 共5兲 and the unknown polynomial coefficients determined via a least-squares algorithm. As shown in Fig. 1, the resulting inverse solution via Eq. 共5兲 gives excellent agreement with the applied boundary-condition. As shown by Fig. 2, excellent agreement was also seen with a triangular temperature-history except at the apex. The flattening of the apex by the curve-fit does highlight a shortcoming of continuous functions including polynomials, as well as some numerical approaches 共Frankel and Keyhani 关8兴兲 in that they cannot reflect abrupt changes; the use of more sophisticated piecewise relationships such as B-splines or radial basis functions might solve this problem. Similar calculations were also performed for an infinitely long cylinder 共Ri / Ro = 0.667兲 subjected to the same asymptotic temperature-history on

Fig. 3 Comparison of inverse predictions and asymptotic boundary-condition of an infinitely long and hollow cylinder as a function of nondimensional time

the i.d. with convection on the o.d. 共Bi= 1200兲. Again using existing relationships 共Vedula et al. 关6兴兲 with artificial error to generate a response, the least-squares approach determined the inverse solution as shown in Fig. 3. For both the slab and cylinder, the late upswing in the results reflects the typical breakdown of the polynomial near the end of the defining time-interval. In order to further validate the proposed method, as well as highlight two interrelated advantages with respect to generalized polynomials, a comparison was also made to the seminal solution derived by Burggraf 关9兴 for a plate:

兺 2n! 冉 ␬ 冊 ⬁

⌬T共t兲 = Ts共t, ␰兲 +

␰2

1

n=1

d Ts共t, ␰兲 dtn

n n

共8兲

where Ts共␰ , t兲 represents the known temperature history and all other terms are as previously defined. As discussed in Burggraf 关9兴, there is a limiting requirement of Eq. 共8兲 in that the function Ts共␰ , t兲 must be infinitely-differentiable to allow sufficient terms for convergence and a uniform temperature-distributions at t = 0. Since a uniform initial-temperature was desired and a truncated polynomial in the form of Eq. 共2兲 is not infinitely-differentiable, a new polynomial capable of matching the remote responses was required. Therefore, instead of using Eq. 共2兲 with its limited number of derivatives, a new polynomial containing integral orders of time and the complimentary error function was derived for this study: N

Ts共␰,t兲 =

兺t

m

erfc共␻兲

共9兲

m=1

where

␻=

m␰

共10兲

2L冑␬t

By using a polynomial in this form, the nth order derivative for each term in Eq. 共9兲 can be determined by dn m t erfc共␻兲 = dtn Fig. 2 Comparison of inverse predictions and triangular boundary-condition of a semi-infinite slab as a function of nondimensional time

1404 / Vol. 127, DECEMBER 2005

n if n⬍m m if n⬎m

兺 i=0

冉冊

⌫共m + 1兲 m−1 t i ⌫共m + 1 − i兲

n

再兺 n

p=1

U p 共 p兲 F 共␻兲 p!

冎

共11兲

The following terms used in Eq. 共11兲 are defined as Transactions of the ASME

Discussion The results shown in Figs. 1–3 indicate the versatility and reasonable accuracy of the proposed method without the need for derivatives and the potential errors contained within. Additional or combinations of functions including piecewise polynomials could also be used with Eq. 共1兲 as long as they are reasonably capable of describing the excitation over the known time interval and not used beyond. Provided the thermophysical properties are independent of temperature, the least-squares method can be used for any geometry and boundary conditions as long as a unit response is known in closed-form or determined numerically. When temperature dependent properties are warranted, the iterated-polynomials advocated by Imber 关11兴 could be used in Eq. 共2兲 instead of the convolution terms with the resulting coefficients again determining the inverse solution. However, it is probably more prudent to simply use the current least-squares approach with minimum and maximum values of diffusivity to help bound the inverse solution. Fig. 4 Comparison of the dominant first-derivatives of the asymptotic and triangular boundary-conditions as fit by Eq. „9… and used in the Burggraf †9‡ solution

F 共 p兲共 ␻ 兲 =

2

n −共 y 兲2

冑␲ 共− 1兲 e

Hn−1共␻兲

共12兲

with Hn representing a Hermite polynomial of nth order Hn共␻兲 = n!

冋

integer共n/2兲

兺 j=0

and p−1

Up =

兺 j=0

共− 1兲 j

共− 1兲−1

⌫共p + 1兲 ␻ j ⌫共p + 1 − j兲 j!

冋再 兿 冉 n−1

i=0

共2␻兲n−2m m!共n − 2m兲!

j−p −i 2

冊冎

册

共13兲

a p−jt关共 j−p兲/2兴−n

册

共14兲

As shown in Fig. 1, the polynomial given by Eq. 共8兲 does a relatively poor job of predicting the asymptotic boundarycondition when the data with artificial errors was introduced. In contrast, a relatively smooth estimation of the triangular boundary-condition was made with the same polynomial as shown in Fig. 2. This seemingly contradictory performance can be ascribed to the dependence on derivatives and their tendency to magnify errors in the underlying data 共Rowlands et al. 关10兴兲 as shown by the oscillations of temperature and first derivative in Figs. 1 and 4, respectively. In terms of the asymptotic boundarycondition, the first derivative is clearly dominating the solution with the oscillations reflecting a greater polynomial “wiggle” about the original data points relative to the triangular input; the observed oscillations probably originate in the term U p since it contains various powers of time. Higher-order derivatives also displayed similar tendencies, but were always at least one orderof-magnitude lower than the proceeding order and did not appear to influence the solution. The least-squares method advocated in this paper avoids these problems altogether.

Journal of Heat Transfer

Conclusions A least-squares methodology based on a generalized directsolution has been developed to solve the inverse problem for common geometries without the need for higher order derivatives. Good agreement was seen between the current study and earlier results for various test cases including triangular and asymptotic boundary conditions. While the mixed-order polynomials used in this study are clearly versatile, they do have limitations in that they cannot readily model abrupt temperature changes. Nevertheless, the method appears well suited for estimating boundary conditions provided the analysis is restricted to the time interval used to determine the polynomial and the thermophysical properties do not vary with temperature. Methods for handling different geometries and temperature dependent properties are suggested.

References 关1兴 Beck, J. V., Blackwell, B., and Clair, C. R., 1985, Inverse Heat Conduction-Ill Posed Problems, Wiley-Interscience, New York. 关2兴 Xue, X., Luck, R., and Berry, J. T., 2005, “Comparisons and Improvements Concerning the Accuracy and Robustness of Inverse Heat Conduction Algorithms,” Inverse Probl., 13, pp. 177–199. 关3兴 Segall, A. E., 2003, “Thermal Stresses in an Infinite Slab under an Arbitrary Thermal Shock,” ASME J. Appl. Mech., 70, pp. 779–782. 关4兴 Segall, A. E., 2001, “Relationships for the Approximation of Direct and Inverse Problems With Asymptotic Kernels,” Inverse Probl., 9, pp. 127–140. 关5兴 Austin, J. B., 1932, “Temperature Distribution in Solid Bodies during Heating or Cooling,” J. Appl. Phys., 3, pp. 179–184. 关6兴 Vedula, V. R., Segall, A. E., and Rangarazan, S. K., 1998, “Transient Analysis of Internally Heated Tubular Components With Exponential Thermal Loading and External Convection,” Int. J. Heat Mass Transfer, 41, pp. 3675–3678. 关7兴 Carslaw, H. S., and Jaeger, J. C., 1959, Conduction of Heat in Solids, Oxford University Press, Great Britain. 关8兴 Frankel, J. I., and Keyhani, M., 1997, “A Global Time Treatment for Inverse Heat Conduction Problems,” J. Heat Transfer, 119, pp. 673–383. 关9兴 Burggraf, O. R., 1964, “An Exact Solution of the Inverse Problem in Heat Conduction Theory and Applications,” J. Heat Transfer, 86C, pp. 373–382. 关10兴 Rowlands, R. E., Libor, T., Daniel, I. M., and Rose, P. G., 1973, “Higher Order Numerical Differentiations of Experimental Information,” Exp. Mech., 14, pp. 105–112. 关11兴 Imber, M., 1979, “Nonlinear Heat Transfer in Planar Solids: Direct and Inverse Applications,” AIAA J., 17, pp. 204–213.

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