Multidimensional Analysis, of Quenching: Comparison of Inverse Techniques/67531/metadc667667/... · Multidimensional Analysis, of Quenching: Comparison of Inverse Techniques Kevin

Multidimensional Analysis, of Quenching: Comparison of Inverse Techniques

Kevin J. Dowding Sandia National Laboratories

Albuquerque, NM 871 85-0835

Abstract Understanding the surface heat transfer during quenching can be benefi-

cial. Analysis to estimate the surface heat transfer from intern a1 temperature measurements is referred to as the inverse heat conduction prsblem (IHCP). Function specification and gradiendadjoint methods, which use a gradient search method coupled with an adjoint operator, are widely used methods to solve the IHCP. In this paper the two methods are presented for the multidimensional case. The focus is not a rigorous comparison of numerical results. Instead after formulating the multidimensional solutions, issues associated with the numerical implementation and practical application of the methods are discussed. In addition, an experiment that measured the surface heat flux and temperatures for a transient experiment is analyzed. Transient temperatures are used to estimate the surface heat flux, which is compared to the measured values. The estimated surface fluxes are comparable for the two methods, but computational requirements d Iffer.

Nomenclature specific heat [J lkgC] function specification regularization method location for temperature sensorj [m]

temperature residual sensor j [ " C] nonhomogeneous term for boundary surface Ti (whole domain) gradienvadjoint method convection coefficient [ ~ / ( m 2 " C ) ] boundary condition coefficient

Tikhonov regularization matrices

number of temperature sensors

objective function 1°C sec]

sum-of-squares term in objective function [ "C sec]

regularization (Tikhonov) term in objective function ["c sec] thermal conductivity [ W/m°C]

boundary condition coefficient

function space of all "square integrable" functions

number of temporal components for estimated heat flux

2

2

2

outward-pointing unit normal vector

search direction ["C2/( W/m2)m] number of spatial components for heat flux on [r,] heat flux [ W/m2]

prior information for heat flux [ W/m2]

two-dimensional coordinate vector

number of future time steps objective function sequential gradiendadjoint method time, initial time, final time [sec]

temperature, initial temperature [ "C]

WLi' W weighting constant, matrix ("C)-'

sensitivity coefficient, matrix[ O C / ( W/m2) 1 measured temperature [ C] zeroth and first order Tikhonov regularization 1 1/( W/m2)2]

Tikhonov regularization parameter [("C/( W/m2)) l lm]

boundary surface i for domain R

update to heat flux [ W/m2]

time step [ sec]

convergence tolerance

sensitivity function [ ( "c /~ ) ( "c / ( w / ~ * ) ) ~ J density [kg/rn3]

Rh4S error in estimated heat flux [ W/m2]

basis function, basis function vector

adjoint function I l /m( "C/( W/m2))] two-dimensional domain scalar product, Eq. (12)

norm, Eq. (13)

2

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I

Introduction Estimating the conditions at the surface of a conducting body from internal

measurements is typically called the inverse heat condiiction problem (IHCP). The terminology inverse is used for this type of conduction problem because conditions at the boundary or surface of a body are estimated using internal measurements. Whereas a direct conduction probl :m uses conditions specified on the boundary to compute the internal temperature. While the direct problem is generally well-posed, the inverse problem tends to be ill-posed and very sensitive to measurement errors.

Several methods are applied to solve the IHCP. Function specification (Beck et al., 1985), Tikhonov regularization (Tikhonov and Arsenin, 1977), gradiedadjoint methods (Alifanov, 1994, Alifanov et al., 1596, and Ozisik, 1993), and mollification (Murio, 1993) are more frequently cited methods. However, other approaches are also applied: dynamic progrmming (Busby and Truijillo, 1985), Kalman filter (Tuan et al., 1996), Monte Carlo method (Haji-Sheikh and Buckingham, 1993). In addition, combining methods is useful. Beck and Murio (1986) formulated a combined functisn specification and Tikhonov regularization method; similar concepts are used in Osman et al. (1997). Jarny et al. (1991) combined a gradienthdjoint method with Tikhonov regularization. Since this paper focuses on multidimensional problems, the literature review narrows to the two-dimensional problem. Refer to books by Alifanov (1994), Beck et al. (1985), Hensel (1991), and Kurpisz and Nowak (1995) for a comprehensive survey of the literatu-e.

Several methods applied to the one-dimensional problem have been extended for the two-dimensional problem. Function specific ition and gradient/adjoint methods have received the most attention. Functilm specification specifies a functional form for the heat flux over a future interval (Beck et al., 1985). Specifying a functional form over the future interval is a form of regularization to stabilize the ill-conditioned problem (Lamm, 1995). In conjunction with specifying a function form, function specification solves the problem in an efficient sequential manner.

Several researchers have investigated the two-dimensional application of the function specification method. It was applied to estima.e spatially and time varying convective heat transfer coefficients, Osman and Beck (1989a, 1990), and surface heat flux, Osman et al. (1997). A boundary element method was coupled with function specification to investigate multidimensional problems by Zabaras and Liu (1988). Hsu et al. (1992) applied a finite element method to solve the general two-dimensional problcm with inverse methods similar to function specification.

Gradienthdjoint methods, which typically apply a conjugate gradient iterative scheme and use an adjoint operator, employ iterative or Tikhonov regularization to stabilize the solution and solve the multidimeniional problem. Iterative regularization depends on the slowness or “viscosi .y” of the solution and uses the iteration index as the regularization parmeter. Several papers use iterative regularization; see Alifanov and Kerov (1981), Kerov (1983), and Alifanov and Egorov (1985). Additional inve:,tigations using gradient/ adjoint methods, but not iterative regularization, are given in Zabaras and Yang (1996), Reinhardt and Hao (1996a,b) and Jarny et al.

Others have studied the multidimensional inverse probleni with a variety of approaches. Tuan et al. (1996) uses a Kalman filter to delelop an on-line algorithm. A new method, called “direct sensitivity coefficicnt,” is claimed by Tseng and Zhao (1996) and Tseng et al. (1996). Pasquetii and Le Niliot (1991) employ Tikhonov regularization with the boundary element method. An adjoint approach to compute sensitivities and relate me: sured temperature to unknown surface conditions is used by Hensel and Hills (1989). A Monte-Carlo method is given by Haji-Sheikh and Buckingham (1993). Mol- lification with a space marching technique is used by Murio (1993b) and Guo and Murio (1991). Busby and Trujillo (1985) use dynamic program- ming. Transform methods are studied by Imber (1974, 1975).

While there are several methods and approaches to solve th: IHCP, distinct advantages or disadvantages of individual methods are cloudy at best. There

(1991).

have been a limited number of studies to directly compare methods. Raynaud and Beck (1988) suggested basic test cases to compare IHCP solutions and compared four methods. The basic test cases quantify the performance of a method for several important criteria, suggested by Beck (1979), for comparing IHCP methods. A comparison of several methods using an experimental case is given by Beck et al. (1996) and Beck (1993). A fortunate outcome of past comparisons, though they are limited and for one-dimensional problems, is that the estimated heat flux from different methods does not differ significantly.

Function specification and gradientladjoint methods are the most widely used approaches. These two methods are discussed in this paper. Though a two dimensional numerical comparison using experimental data is shown, the focus is on application and implementation issues. In particular I will discuss the numerical solution and computational requirements, spatial representation of the surface heat flux, handling of the nonlinear problems, and regularization issues. With the previously demonstrated agreement between different methods, important issues, especially for the multidimensional case, are those related to getting a solution for practical problems. By practical I mean a problem that requires numerical solution, has temperature- dependent properties, and measurements at a limited number of locations.

Knowledge of the surface heat transfer during quenching can aid in understanding the quenching process. In many cases a one-dimensional solution may not be sufficient to model the process. Generalizing the methods for the multidimensional case will allow them to be applied to more practical cases. However, the multidimensional IHCP is significantly more difficult than the one-dimensional case. The multidimensional case is more computationally intensive and ill-posed. The objective of this paper is to describe and contrast two main methods for solving the multidimensional IHCP. Several issues that concern the practical application are discussed. This paper is the first known comparison of methods for the two-dimensional IHCP.

In the subsequent section a description of the multidimensional IHCP is given. The following two sections describe solution methodologies for function specification and gradientladjoint methods, respectively. A discussion contrasting several aspects of the two methods is given next. In addition, an experimental case is considered. The paper concludes with a brief summary.

Problem Description A schematic of the general multidimensional IHCP is shown in Figure 1.

The transient temperature distribution inside a multidimensional region is described by the heat conduction equation:

9 (1) aT(r, t ) ( r ) in R v . ( k ( T ) V T ( r , t ) ) = pcp(T)-

at ’ ( t , < t < r , ) with the boundary conditions

Figure 1. Schematic of multidimensional general IHCP

( r ) o n r i , ( i = l , 2 , 3 ) 9 ( 2 4 a -k . -T(r, t ) + h i T ( r , t ) = f i ( r , t ) ,

I afi ( to < t I t 1 )

and the initial condition T ( r , to) = To('), ( r ) in (Q u r). (2c)

The thermal conductivity k(T) and volumetric heat capacity pc,(T) are assumed to be temperature dependent. The spatial domain R has boundaries represented with symbols ri, (i =1, 2, 3, 4). The boundary conditions in Eq. (2a) are, for generality, the first, second, and third kind (i=I, 2, and 3, respectively). The boundary coefficients k; and hi are specified to form the correct boundary condition -- e.g., kl = 0 and hl = 1 specifies a temperature boundary condition (first kind); k2 = k(T) and h2 = 0, a flux condition (second kind); and k3 = k(T) and h3 = h(r; T), a convective condition (third kind). The outward-pointing normal vector is denoted li ; the heat flux leaving a surface is positive.

The thermal properties (kpc,), boundary conditions f&, arid initial condition (TO) are assumed to be known. The heat flux q(

mon choice. The estimated vector is retained only for the first time interval tm Then time index rn is increased by one time step, and the solution process is repeated by marching in time until the last sequential interval. Retaining more than one component is possible, but has not been extensively studied.

In Eq. (7a) the temperature and sensitivity are needed. The original nonlinear temperature problem in Eqs. (1) and (2) is linearized in the sequential implementation. Temperature-dependent parameters are evaluated at the initial temperature. Parameter variation on account of transient temperature change during the sequential interval is neglected; parameters can vary spatially with temperature. Evaluating the parameters at the initial temperature is referred to as quasi-linearization. After quasi-linearizing, T(q*) is assembled from the solution of the following problem.

with the boundary conditions

In Eq. (9c) p,,, - l ( r ) is the computed temperature at time t , - when qm-] is estimated (the previous sequential interval).

Sensitivity coefficients are the first derivative of temperature with respect to the unknown heat flux vector qm- There are P spatial components in this vector, and sensitivity to each component is solved for independently. The sensitivity equations can be derived by differentiating Eqs. (I) and (2) with respect to qkm after substituting the approximation for the heat flux in Eq. (4) and evaluating the thermal properties at the initial temperature. The sensitivity equations for spatial component k are given next; the kth column in X is assembled by solving these equations.

Sensitivity Coefficient Equations.

with the boundary conditions

and the initial condition Xk(r,tm-l) = 0, ( r ) i n ( R u r ) . (1 IC)

There are P sensitivity equations, k = 1, 2, ..., P. The kth sensitivity problem in Eqs. (IO) and (1 1) is exactly the same as the

temperature problem in Eqs. (8) and (9) with three simplifications. First, the

initial condition for all sensitivity equations is zero. Second, the heat flux distribution q* in Eq. (9b) is replaced by the kth basis function $,Js) in Eq. (1 1 b). Third, the known boundary conditions in Eq. (9a) are homogeneous for the sensitivity equation, Eq. (1 la).

An alternative to deriving sensitivity equations, as done here, is to use a finite difference approximation of the derivative. The sensitivity equation method is typically more accurate than the finite difference approximation of the sensitivity coefficients, Beck and Arnold (Chapter 7, 1977) and Black- well et al. (1998). The sensitivity equation approach removes the dependence on the numerical step size of the finite difference approximation of the derivative. In situations that require significant effort to solve the sensitivity equations the finite difference approximation may be adequate and much less work to implement. Care must be used to guard against numerical errors when using a finite difference approximation, however. Blackwell et al. provide a detailed discussion of deriving sensitivity equations for heat conduction problems and their solution.

GradienVAdjoint Method Gradientkidjoint methods, which couple a gradient search method with an

adjoint equation approach, are growing in popularity. These methods are more mathematically based. I will try not to get too involved in the mathematics while describing the methods. Completely avoiding the mathematics is not possible or appropriate, however.

A sequential implementation of the gradientkidjoint method is not the usual approach. It is a relatively recent implementation (Dowding, 1997; Dowding and Beck, 1998; Reinhardt and Hao, 1996a,b; and Arytukhin and Gedzhadze, 1994). A more common implementation considers the entire time range simultaneously, referred to as whole domain. Dowding (1997) and Dowding and Beck (1998) compare a whole domain and sequential implementation for a linear problem (constant thermal properties). Computa- tion requirements for a sequential implementation are shown to be greater than whole domain in many cases for the linear problem; estimated heat flux is comparable. The nonlinear problem has not been studied, but a sequential solution might have potential.

A sequential implementation, including quasi-linearization, as discussed in the previous section, is discussed in this section for the gradientkidjoint method. The method described is referred to as a sequential gradienthdjoint method (SGAM). Comments on the whole domain solution are given near the end of this section. In future discussions I refer to this method as a whole domain gradientkidjoint method (GAM).

Spatial-dependent heat flux approximation. In contrast to the FSRM, no assumptions are required about the specific functional form of the unknown heat flux to formulate an inverse solution using gradientladjoint methods. Schemes to minimize the objective function J ( q ) (as defined and discussed next), which use iterative methods, such as steepest descent or conjugate gradient, require the gradient of J ( q ) . Methods to compute this gradient depend on the function space where q(r, t ) is assumed to reside. Two possibilities are a finite-dimensional space and an infinite-dimensional space. The approximation of heat flux for the FSRM in the previous section is finite-dimensional. Characteristics are assumed (a priori) about the heat flux. For the infinite-dimensional problem a priori information is not required concerning the (unknown) function q(r, t) . However, two additional problems, which are the adjoint and sensitivity problems are required. For the special case when a priori information is available concerning the function -- or assumed, as with FSRM - the problem is considered finite- dimensional and standard differential calculus can be used to compute the gradient. Lamm (1990) provides a detailed description of the differences in the formulations when heat flux is finite versus infinite-dimensional.

The analysis considered here is the more general infinite-dimensional problem. The heat flux is assumed to be in the function space Lz, all square

integrable functions on r4 x ( I , , t , + - I ) . In this case the scalar product is defined by

for arbitrary functions Zl(r,t) and Z*(r,f). The associated ncrm of function Z(

(VJ(q"), VJ(q"- ' ) - VJ(q"))L* II VJ(q" - ,nZ,

p"=

See Eq. (12) and Eq. (13) for the definition of ( , )L2 and I l l L 2 . 49. Solve the sensitivity problem for e(r, t )

e(r, 0) = 0, ( r ) in R . ( 2 6 ~ ) 4ii). Evaluate the optimal step size 0"

j = 1

5.Compute the improved value for q

( r ) on I y 4 q"+l(r , t ) = q"(r , t ) -p"p"(r , t ) , ,(28)

( t , - 1 < f 5 f,, + - 1 1 6. Check for convergence of the estimated heat flux

(29) 2

If convergence has not been obtained, let n = n + 1 and return to Step 2 with the updated heat flux. If convergence has been met, m w e to the next sequential interval.

The describing equations for the direct, sensitivity, and adjoint problem have a similar form. The differential equations and boundary conditions have the identical forms. Differences between the three problems a-e: (1) The sensitivity and adjoint problem have homogeneous known boundary conditions (Eq. (224 and Eq. (26a)). 2) The sensitivity problem's driving term is the search directionp"(r,f) on surface r4 (Eq. (26b)). (3) The adjoint problem's driving term is the residual (Eq. (21)). (4) The sensitivity has ;I specified zero initial condition (Eq. (26c)). (5) The adjoint problem has a specified zero final condition (Eq. (22c)).

A whole domain solution is mentioned previously. If the IHCP in Eqs. (1) and (2) is linear (constant properties), the whole domain gradient method is obtained by lengthening the time interval for the sequential formulation presented. Apply the sequentid solution over (to < f 5 lf) , instead of ( tm-] < t I f m + r - l ) , for the whole domain method. Jamy et al. (1991) also give the formulation for the linear problem using the whde domain gradient method. Extending the whole domain gradient methods for nonlinear problems is discussed in Artyukhin (1996) and Loulou et al. (1996). Though the equations are similar to those for a linear problem, the solution of a nonlinear partial differential equation (PDE) is required. All PDEs are linear for a sequential implementation using quasi-linearization.

Ilq.n+ 1 - 4 n llL2 < E .

Results and Discussion Several issues related to the implementation and practical application of

inverse methods are discussed. I will discuss the function specification regu-

larization method (FSRM) and sequential gradientladjoint method (SGAM). Both methods are formulated in the previous two sections. (In addition, discussion of a whole domain gradientladjoint method (GAM) is given to contrast it with the SGAM.) The goal is to highlight aspects of the methods and indicate differences. This information may help indicate when it is advanta- geous to choose one method over the other. Many of the issues may be in a grey area. Nevertheless, it is important when applying the methods to understand their limitations and the underlying assumptions.

Numerical Solution and Computational issues. The inverse solutions formulated for the FSRM and SGAM require numerical solution for practical cases, including irregular geometry, multiple materials, temperature dependent properties, etc. The PDEs shown for the solution methodologies presented in the previous sections can be solved using any of several numerical techniques, such as finite element, finite difference, control volume finite element, or boundary element.

Two approaches have been used to take an existing numerical solver -- for example a general finite element code -- and combine it with inverse techniques to create a general inverse code. The first approach takes the existing code and modifies it to be a subroutine (Osman and Beck, 1989b). The subroutine becomes an integral aspect of the inverse code and is called to compute direct solutions as needed. An alternate approach combines an existing numerical code with an inverse code but external to the codes. A general driver (Eldred et al., 1996) that provides such a code-to-code communication calls it Reusable Interface Technology (RIT). RIT has been successfully applied for parameter estimation (Blackwell and Eldred, 1997). It allows both codes (inverse and numerical solver) to be developed independently. In the former approach when the numerical code is converted to a subroutine, it usually requires significant modification. The modifications typically sever the link with the original numerical solver code. Hence future updates in the numerical solver cannot be implemented without effort to modify the code according to the subroutine structure. A main advantage of RIT is that it does not have this problem.

Both inverse methods require the solution of several PDEs. However, as previously discussed, there are similarities between the PDEs. The similarity allows the same numerical solver to be used to solve all PDEs. It also permits possible computational savings by storing certain matrices and reusing them to solve other PDES. To store these matrices requires access to the details of the solver. If RIT is used, it may not be possible to access these matrices. Issues of computational savings are not discussed further here. See Osman et al. (1997) and Dowding (1997) for a discussion of computational savings when solving with FSRM and SGAM.

Of possibly more significance to the computational requirements is the number of PDEs that must be solved. (Minor differences in the PDEs are ignored, and all PDEs are assumed to require similar effort to solve.) In the FSRM, a PDE for temperature and P PDEs for sensitivity are solved, where P is the number of spatial parameters approximating the heat flux on r4. Hence, the computational requirements increase with the spatial discretization of the heat flux for FSRM. In contrast the SGAM solves three PDEs (direct, adjoint, and sensitivity) regardless of the spatial discretization of the heat flux. The three problems are solved iteratively, however. The trade-off is solving P i 1 problems one time for the FSRM or three problems iteratively for the SGAM. As more spatial components are estimated, the SGAM may gain a computational edge over FSRM. Because conjugate gradient search methods are efficient and converge in a few iterations, SGAM is more computationally efficient than FSRM. The required iterative solution for SGAM does not increase computational expense as much as the large number of sensitivity problems required for FSRM.

Solving several PDEs for each method requires care to ascertain the effect of discretization errors. Although the solution of one PDE -- for example temperature -- may be insensitive to the discretization, the solution for other PDEs may not be insensitive. The discretization required for one PDE may be different from the discretization required for other PDEs. This is because the solutions for different problems vary in the locations where gradients are

large. Consequently, the PDEs may require different discretization to capture the gradients and accurately predict the solution. Users shoidd investigate this point.

Sequential and Quasi-Linearization vs. Whole Domain. A sequential implementation of the two inverse methods is used. In conjunction with a sequential method, quasi-linearization can be used. Quasi-linearization refers to the process of temporarily linearizing the problem by evaluating temperature-dependent quantities at the initial temperature of a sequ :ntial interval. An alternative to the sequential method is a whole domain apl~roach. Whole domain means the entire time domain (to, ff) is considered. Heat flux is estimated for all time and spatial locations simultaneously.

It is commonly accepted that a sequential method is more c:fficient than a whole domain approach for function specification methods. It 1 s far less clear for the gradiendadjoint methods whether a sequential method is more efficient; further investigation is needed. There are several supporing reasons to consider a sequential implementation.

1. The physics indicate a sequential process; the describing equations for the IHCP are parabolic in time. Consequently, temperature/heat flux at a much later time should not influence temperatureheat flux at 5 n earlier time. Simultaneously solving for all time, 1s done in a whole domain solution, allows late time estimates to influence the early time estimates

2. The final time is arbitrary. The time selected to end the analysis should not be influential on the estimated heat flux. An estimation in “real time” requires a sequential implementation.

3. Less computer memory is needed in a sequential implementation. Cases with temperature-dependent properties require saving the temperature for the entire time and space domain in a whole domain implementati m.

4. Computational requirements increase linearly as the time domain is increased for a sequential implementation. Analysis for 100 time steps requires approximately twice as much time as 50 time steps. A whole domain implementation increases the number of components simultaneously estimated as the time domain is increased. Conceptually, an increase in the time domain for a whole domain method results in solving ,I larger set of simultaneous equations. As this set gets large, the computational increase for the whole domain is unlikely to be linear.

A separate issue from the whole versus sequential solution is the concept of quasi-linearization; it is only appropriate for a sequential solution. The idea is to evaluate the properties at the initial temperature, thereby linearizing the problem. Then a nonlinear solution is not required for b e sequential interval. Though there may be special cases that require a nonlinear solution on a sequential interval, in most cases it is not necessary. Sevei a1 reasons are cited. First, thermal properties are typically not known to an accuracy sufficient to warrant iteration. Second, the estimated surface heat flux depends on regularization parameters (discussed below). It is well known that the estimated heat flux varies depending on these parameters. Criteri2 to select the parameters are based on generally unknown measurement o rrors. Conse- quently, with all the uncertainty due to other factors, a non1inc:ar solution is unlikely to improve the true accuracy of the estimated heat flux. Simulated cases with highly nonlinear thermal properties have demonsirated that the quasi-linearization process does not adversely affect the estimated heat flux (Beck et al., 1982 and Osman et al., 1997).

Quasi-linearization is recommended for the SGAM as well. Though SGAM requires iteration already, for the reasons previously specified, the problem should not be treated as nonlinear. The computational requirements to solve nonlinear PDEs are greater than those for the linear PDEs, which result from quasi-linearization.

Infinite- vs. Finite-Dimensional. In some publications the iiifinite-dimensional representation of the heat flux for gradient methods is touted as a major advantage. Infinite-dimensional means heat flux is not Fanmetenzed. It is assumed to be a function. and no other approximations about the unknown flux are introduced. In the FSRM, an a prior[ approx mation of the

.OM

, composite specimen /

mica heater

assembly

7.62 (3.00) I- AU dimensions in cm (in.) * thermocouple ra Active heater

(NOT DRAWN TO SCALE) eB Inactive heater Figure 2. Experimental configuration with measured transient temperature and heat flux heat flux is needed. To numerically solve the IHCP with an infinite-dimensional heat flux, as presented for the SGAM, the heat flux is made finite- dimensional (discretized) during the numerical solution -- that is, a numerical approximation of the infinite-dimensional heat flux is introduced -- thus making it finite-dimensional. For example, using an element-based numerical method, the spatial heat flux may be assumed constant along an element boundary.

In many cases exactly the same numerical approximation of the heat flux can be used for the infinite-dimensional (SGAM) and finite-dimensional (FSRM). However, the infinite-dimensional inverse formulation -- i.e., the number of problems solved -- is not dependent on the spatial approximation. Regardless of the numerical approximation, the same (three) problems are solved. This is not true for the FSRM. The numerical approximation of the heat flux influences the inverse formulation and number of problems to solve. The distinction between finite and infinite-dimensional is more a computational issue than an approximation issue.

Regularization. Both FSRM and SGAM require selecting the magnitudes of one or more regularization parameters. A whole domain method requires fewer regularization parameters than a sequential solution because r is not needed. To select the proper magnitude of the regularization parameter selec- tion, criteria are used. One criteria is called the residual principle (Alifanov, 1994). It suggests the sum-of-squares function should be decreased to the expected level, which is based on errors in the measurements. Typically, these errors are unknown and must be estimated.

Experimental Case. The schematic of an experimental set-up, which was used to estimate the thermal properties of the carbon-carbon composite mate- rial, is shown in Figure2. By measuring the transient temperatures and power supplied to the mica heater assembly, the thermal properties (orthotro- pic thermal conductivity and volumetric heat capacity) of a carbon-carbon composite were measured, Dowding et al. (1995, 1996). In this paper using the same experimental data, the opposite problem is considered; the surface heat flux is recovered using the measured thermal properties and transient temperature measurements. A two-dimensional surface heat flux is estimated. Dowding et al. (1995) describes the one-dimensional case.

The thermal model of the experimental set-up is shown in Figure 3. All outer surfaces are assumed to be adiabatic, except the surface where the energy is introduced by the heater. The energy to the heater is assumed to divide equally between the two symmetric halves and emanate from the mid-

v l

y5 '4 (5.08-6.35) 43

- 1 (0-1.27) 42

1 (1.27-2.54) (2.54".81)(3.81-5.08) (6 35-7.62)

Figure 3. Thermal model of experimental configuratic'n (origin is at left face between composite specimen and mica heater ;issembly)

dle of the heater assembly (y = -0.042 cm); see Figure 3. The heater assembly contains three independently controlled heaters. Only one of the three heaters is energized (active heater in Figure 2, the other two heaters are iden- tified as inactive). The measured temperatures are averaged on opposite sides of the heater assembly to determine the temperature at each location. The sensors are located along y = 0.0 cm at x = 0.89, 1.91, 3.13,4,45, and 6.73 cm and along y = 0.91 cm at x = 1.27 and 6.35 cm.

The numerical solution for this problem uses 21 nodes along the x-direction for all materials. In the y-direction there are 2 nodes across the mica heater, 11 nodes across the composite, and 11 nodes across the insulation. A numerical time step of 0.64 seconds, which is the same as 1 he measurement time step, is used. Because effective properties have been estimated for the mica heater, the interface between the heater and composite is modeled with perfect contact. Different numerical solvers are used for the FSRM and SGAM. A finite element code is used for FSRM. The mesh is shown in Figure 3. A finite difference-based method with a similar discretization as the finite element mesh is used for the SGAM.

Although the mica heater is quite thin (0.42 mm), because its properties include the contact resistance between the heater and the c a ~ boncarbon, it is thermally significant. Based on the effective properties of the mica heater, the dimensionless time step (Fourier number) based on the depth of the temperature sensors nearest the surface is 0.23. A dimensionless time step in this range does not represent a difficult one-dimensional IHCP; nowever, in two- dimensions with a limited number of sensors, the difficulty is increased.

An important issue in the study of the multidimensional HCP is defining the spatial representation of the (unknown) heat flux. A general rule suggested for the FSRM is that the number of spatial components should not exceed the number of sensors. If more components than this are estimated, additional regularization is typically required to stabilize t ie solution. The parameterization for the FSRM is shown in Figure 3. Six equally spaced seg- ments over which the heat flux is uniform are used (the x-range of each heat flux component is shown in Figure 3). For the SGAM a general approximation is used that assumes heat flux is constant over each nodid surface; a total of 21 spatial components define the heat flux for SGAM.

The measured surface heat flux is shown in Figure 4a. It has a step change in time and with position. Thermal properties are assumed tc be constant; the experiment considered has a maximum temperature change of 25 "C . The heat flux is estimated using FSRM and SGAM. In addition, heat flux estimated with a whole domain gradienthdjoint method (GAM) is included. For constant thermal properties the GAM formulation is the satr e as the SGAM, except the time range is extended to cover the whole timi: domain. In all cases the residual principle (Alifanov, 1994) is used to select the number of future time steps and Tikhonov parameter. Only zeroth ord:r regularization

is used. An approximation of the measurement error, which is required for the residual principle, is estimated by comparing temperature measurements across the symmetry plane (Dowding, 1997).

TABLE 1. Estimation results for two-dimensional IHCP with experimentally measured heat flux

I .5 E-06

660 FSRM(r=6) I FEh4 I 6 1 1.5E-06 I 7000 I 330 The heat flux distribution estimated with FSRM is shown in Figure 4b.

Estimated heat flux using the gradientkidjoint methods are shown in Figure 4c for SGAM (sequential) and Figure 4d for GAM (whole domain). Table 1 identifies the direct method used to solve PDEs, number of spatial heat flux components, Tikhonov parameter, computational time, and RMS error between the measured and estimated heat flux.

FSRM more accurately predicts the measured heat flux. It has a RMS error (final column of Table 1) approximately one-half of that for SGAM and GAM. It is, however, fortuitous that the heat flux parameterization (Figure 3) coincides with the spatial dependence of the measured heat flux; only six spatial components are estimated. The SGAM and GAM have similar estimates of the heat flux. In both gradient methods the estimated flux spatially smooths the sharp step more than the FSRM. But 21 spatial components are estimated for the gradient methods. Although not shown, improved accuracy in the estimated heat flux is obtained if the same six spatial components defined for FSRM are estimated with the SGAM (and GAM); an RMS m o r in the estimated heat flux is comparable to the that for the FSRM.

There is a trade-off. It is possible to estimate many spatial components with reduced accuracy or fewer components with improved accuracy, assum- ing an appropriate spatial distribution can be selected. In many situations selecting the appropriate spatial distribution will be difficult. An advantage of the gradienthdjoint methods is the flexibility with regard to the spatial distribution. This flexibility does not give more information than the measurements provide, however. The spatial distribution can be estimated in this case because sensors are located along the entire surface. (I don't believe FSRM could estimate 21 spatial components as SGAM and GAM did. Because of program limitations this could not be investigated.)

There is a big difference in the computational time. The SGAM (sequential) requires nearly double the computational time of the GAM (whole domain). For linear problems that do not require a sequential solution, GAM is often computationally superior to SGAM. A detailed explanation of this point, as well as suggested improvements for the SGAM, are given in Dowd- ing (1998). A comparison for the nonlinear case is needed; the SGAM is anticipated to be competitive for this case.

The gradient methods (SGAM and GAM) require on the order of 100 seconds, whereas FSRM requires on the order of 7000 seconds (Column 5 of Table 1). These computational times cannot be directly compared because different direct solvers are used. The gradient methods employ an alternating direction implicit (ADI) finite difference method, which typically is much faster than the finite element method ( E M ) used for FSRM. To understand the computational difference, the time required to obtain one direct solution with comparable numerical conditions is conducted. The AD1 method required 3.6 seconds, whereas the FEM required 110 seconds; a factor of 30. This difference indicates that the gradient methods are more computationally efficient because the inverse solution required a factor of 70 more computational time. Furthermore, FSRM only estimated 7 components, compared to 21 for the SGAM. Computational costs for FSRM would further increase as more spatial components are considered.

a) Measured Surface Heat Flux

e) Sequential GradienUAdjoint method (SGAM, r = 6)

... . . . . . . . . . . . . . ........... ......... . . . . . .... . . . . . . . . . . . . . . . . . . . . . . . : ’

. . -‘1 : : :.. . . . .

120\ -

Figure 4. Numerical results from the experimental case: a) experim flux, c) SGAh4 estimated heat flux with r =6, and d) Whole domain L

Summary Methods to solve the multidimensional IHCP based on fiwtion specifica-

tion and gradientladjoint methods were discussed. Tikhonciv regularization was included to stabilize both methods. Several issues associated with imple- menting the methods for practical problems, including the numerical solution, approximation of the estimate surface heat flux, and computational requirements, were discussed. Computational requirements for FSRM may be larger than SGAM and GAM for multidimensional cases, particularly when many spatial components are estimated. An experime ita1 case demonstrated that the methods provide comparable estimated heat flux. In this example computational requirements are greater for the function specification method than the gradientladjoint method.

Acknowledgments The insights and comments from Ben Blackwell and James Beck are

appreciated. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Depart- ment of Energy under Contract DE-AC04-94AL8.5000.

b) Combined Function Specifcation Regularization Method (FSM, r = 6)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... . . . . . . . . . . . .., . I . . . . . . . . . . . . . . . . . : . :

69 l(seconds)

0 -0

d)Whole Domain Gradient Adjoint Method (GAM)

0 0

tally measured surface heat flux, b) FSM estimated surface heat imated heat flux

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