A numerical method for the solution of two-dimensional inverse

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 32, 363-383 (1991)

A NUMERICAL METHOD FOR THE SOLUTION OF TWO- DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEMS

H.-J. REINHARDT

Fachhereich Mathematik, Unio.-GH-Siegen, 5900 Siegen, F.R. Germany

SUMMARY A numerical method for the solution of inverse heat conduction problems in two-dimensional rectangular domains is established and its performance is demonstrated by computational results. The present method extends Beck's' method to two spatial dimensions and also utilizes future times in order to stabilize the ill-posedness of the underlying problems. The approach relies on a line approximation of the elliptic part of the parabolic differential equation leading to a system of one-dimensional problems which can be decoupled.

1. INTRODUCTION

The classical (direct) problem in heat conduction consists in determining the interior temperature distribution of a body from data given on its surface. However, in many physical problems data are not available over the entire surface, or are available only at certain locations in the interior. In such cases, the aim is to determine surface temperatures and fluxes from a (measured) temperature history at fixed locations inside the body or at certain parts of the surface. This problem is called the inverse heat conduction problem (IHCP).

There are many practical problems in physics and in the engineering sciences which essentially rely on the solution of an IHCP. Fields of applications comprise the development of new materials, the casting and welding in steel or polymer processing, the development of a sophisti- cated temperature measurement related to lifetime analysis of plans, the development of transient calorimeters, ill-posed problems in rheometry, etc. A small selection of related papers is collected in the references (see Brockel and Graf,' Friedrich and H~f rnann , '~ Taler,39 Engl and Lan- gthaler," Kaiser and T r o l t ~ s c h , ~ ~ Weir4').

The underlying mathematical problem turns out to be an ill-posed one. This essentially means that small errors in the data can lead to large errors in the solution. The analysis of ill-posed problems is a growing field in the mathematical and engineering literature during the last years. An extensive reference list going back to the last century has been collected by Dinh Nho Hao and Gorenflo.I6 Numerical methods have been developed and used for thirty years. However, most of the methods used are more or less heuristic. There is a wide field for a numerical analyst to provide rigorous statements concerning numerical stability and error estimates. Works in the latter direction are provided by Cannon and Ewing,'* car ass^,'^ Manselli and Miller,2g Monk3' and M ~ r i o . ~ '

The purpose of this paper is to present a numerical method solving for IHCPs in two- dimensional rectangular domains. The use of the method is restricted to rectangular domains which, however, is not a serious limitation, since many irregular domains of complex shape can be transformed into this simple shape without altering the form of the heat equation. The method is

0029-5981/91/100363-21$10.50 @ 1991 by John Wiley & Sons, Ltd.

Received 26 April 1990 Revised 7 September 1990

364 H.-J. RETNHARDT

based on a well established algorithm for IHCPs in one spatial dimension which is extended to two-dimensional domains via a method-of-line approximation. A suitable modification of an earlier numerical procedure (see Reference 4) allows the stabilization by means of arbitrary future times also in two-dimensional calculations. Moreover, the present numerical scheme allows (inhomogeneous) source terms in the heat equation. The term ‘future times’ was invented by Beck5* ’, * in order to make the calculations of one-dimensional IHCPs more stable.

By means of the numerical method presented in this note we are able to compute surface temperatures of rectangular regions in a stable and efficient way. We are, however, far from presenting a rigorous numerical analysis of our method, and our numerical analysis is more on the experimental side. The essential question remains still open how the number of future times should balance the smallness of the time increment. The question is open even for the Beck method in one spatial dimension; the answer should be given in a quantitative manner including error estimates. Our approach may lead toward the answer to this crucial question.

Numerical methods for IHCPs in two spatial dimensions were developed and computational results were published by Bass and Ott,’ Busby and Trujillo,” Macqueene et Yoshimura and I k ~ t a ~ ~ and in our earlier papers Baumeister and Reinhardt? Reinhardt and V a l e n ~ i a . ~ ~ Our approach is similar to that of Yoshimura and I k ~ t a ~ ~ who also invented a certain generalization of the one-dimensional Beck method. Despite the great need there seems to be only a few computer programs available for solving practical problems in more than one dimension.

Following this introduction, in the second section we state the problem of inverse heat conduction and discuss the ill-posedness of such problems. In the third section, Beck’s method for one-dimensional IHCPs is presented. Hints are given how such problems can be treated in the framework of optimal control. In the fourth section, the extension of Beck’s method to two-dimensional rectangular regions is established. Our approach relies on a method-of-line approximation leading to a system of one-dimensional IHCPs which can be decoupled. The decoupling process is based on the complete solution of an eigenvalue problem. A selection of computational results is presented in the final section. We concentrate on mathematical test cases, among which the benchmark problems can be viewed as extremely difficult from the practical point of view. The section on computational results is devoted to a stability analysis of the numerical method by taking perturbations into account as well as toward a verification of the computer program.

2. ILL-POSEDNESS OF INVERSE HEAT CONDUCTION PROBLEMS

In this section, we pose the problem of inverse heat conduction in one and two dimensions. Moreover, the ill-posedness of such problems is studied.

The governing equation for transient heat flow in one spatial dimension (and Cartesian co-ordinates) is given by

p c z = z a- +cD(x,t), O < x < L , t > O , au a ( ;:)

where p, c denote the density and specific heat, resp., 0 is the heat source function, u denotes the temperature itself. The heat conduction coefficient A may be temperature dependent.

Dividing by pc leads to

au a au - =-a-++(x,t), O < x < L , t > O at ax ax

with the positive thermal conduction coefficient (or thermal diffusion coefficient) a = a/( pc) and

2D INVERSE HEAT CONDUCTION PROBLEMS 365

F = @/pc. As boundary conditions, one can prescribe the temperature itself (i.e. Dirichlet boundary condition), or the heat flux q = - Mu/ax (i.e. Neumann boundary condition), or a boundary condition of mixed type. The heat conduction problem is said to be of direct type if one boundary condition is prescribed at each end of the spatial interval. It is of inverse type if two boundary conditions are given at one end of the interval and none at the other end. In any case, an initial condition for the temperature is prescribed.

The direct problem can be solved by standard methods. Since we are mainly interested in the inverse problem, let us first consider the solution of

aZ a Z 2 -_. a-=o, O < x < l , t > O at ax2

(3)

(4) dZ

z(1, t) = g(t), -(l, t ) = h(t), ax

z(x, 0) = ZO(X), 0 < x < 1 (5 )

t > 0

with a constant thermal diffusivity a > 0. Under suitable differentiability assumptions one knows that z is completely determined by g and h for all x, t,

1 (1 - x)Zn g'"'(t)

" = O (2n)! a"

O0 1 (1 - X)2" - ( 1 -x ) c ~

" = O (2n + l)! a"

z(x, t ) = c - h(")(t)

This formula was first given by Stefan37 in 1889 and was rediscovered by Holmgren2'*22 in 1904-1908; it was used in many, more recent works (see e.g. References 10, 14, 16,23,40 and 42). As a consequence of (6), it should be noted that a compatibility condition at t = 0 has to be satisfied.

On the basis of (6), a well-known example with boundary values g1 , g2 at x = 1 depending on a parameter p > 0 such that llgl - g2 /I --+ 0 ( B --+ GO) has corresponding solutions zl, z2, resp., such that I/zl - z2 / / --+ 00 ( B --+ GO) (see, e.g. Beck et u Z . , ~ Section 4.2.1). The underlying norm is the maximum norm over a finite time interval. This shows that the problem (3)-(5) is not 'well posed' in the sense of Hadamard and is called 'ill-posed' (or 'improperly posed').

Another approach leads to Volterra convolution integral equations of the first kind which are known to constitute ill-posed problems. The following relation between the desired temperature f ( t ) = v(0, t) at x = 0 and the given temperature g(t) = u(1, t ) holds in case zo = 0.

1; ic(t - s)f(s)ds = g(t), t > 0 (7)

where m

x(r) = 2 c (-l)kpkexp(-pir), pk = (2k + 1)7@ k = O

This is a Volterra integral equation of the first kind. The associated convolution operator

A : f€L,[O, T] - -+g€Lz[O, 7-1

is known to be compact, and problem (7) is ill-posed; T > 0 denotes a finite time. The problem is, moreover, severely ill-posed since all derivatives of the kernel IC vanish to zero.

366 H.-J. REINHARDT

We shall now establish a two-dimensional analogue of the above inverse heat conduction problem. The heat conduction on a two-dimensional domain R is modelled by the following parabolic differential equation:

(8) a u at pc __ = vmu + @(x, y, t), (x, Y)€R, t > 0

For simplicity, we again restrict ourselves to a constant thermal diffusivity u > 0. Moreover? we let R be a rectangular domain R = (0, 1) x (0, d). Then an inverse heat conduction problem is given by the following equations:

au

a Y -(x,O,t)=:O, u(x,d,t)=O, O < X < l , t > O

4x9 y, 0) = 0, (x, Y) E fJ (12)

The situation is depicted graphically in Figure 1. The problem consists in determining the temperaturef( y, t ) and the heat flux q( y, t ) at x = 0

from the known temperature g at x = 1. The first condition in (11) is a symmetry requirement. Without restriction of generality, the temperatures in the second condition of (1 1) and in the initial condition (12) are set to zero.

In the homogeneous case F = 0, analogous considerations as above imply the explicit repres- entation of the solution of (9)-(12) as a function of the unknown temperaturef(y, t),

't

T d L X

0 1

Figure 1. Model of a two-dimensional IHCP

2D INVERSE HEAT CONDUCTION PROBLEMS

where

367

$ k , l = cos(vk(l - x))cos(ply)

f;W = p ( Y , s)cos(pLly)dy 0

This, again, leads to Volterra integral equations of the first kind for determining&,

j: x j ( t - s)f j (s) ds = gj( t ) , j = 0, 1, .

where

2 " x j ( r ) = - C (- l)kvkexp(- (v: + p f ) r ) y/;i k = O

g j ( t ) = g ( Y , t)cos(pjY)dY 1: At this point it would be appropriate to discuss the possibilities of stabilizing and regularizing

the above Volterra integral equations on finite intervals. This can be achieved by Laplace transformations in suitable function spaces. An optimal stability bound can be derived which behaves as O(E(ln(E/E)))-21, where E / E represents the signal-to-noise ratio and E denotes a certain degree of differentiability. Several regularizations, like truncation of series, Tikhonov regularizations, the mollification method and hyperbolic regularization, can be considered in this context and error estimates can be provided. For results in this direction, we refer to the works of K n a b ~ ~ e r , ' ~ - ' ~ M ~ r i o , ~ ~ , ~ ~ Weber4' and to a recent paper of Dinh Nho Hao and Gorenflo16 and the references therein.

3. THE NUMERICAL SOLUTION OF ONE-DIMENSIONAL IHCPs

In this section, we recall the method of Beck' for one-dimensional IHCPs and give a description of this method in the framework of optimal control which enables us to understand it in a broader context.

Let us consider the one-dimensional inhomogeneous inverse heat conduction problem (cf. equation (2))

au a au at ax ax - =;-a-+F(x , t ) , X E C O , ~ ] , ~ E [ O , T ]

uI,=l = g(t), = 0, ~ E [ O , T ] ax x = l

with initial condition u(x, 0) = uo(x), x E [0, 11, and finite terminal time T > 0. The desired quantities are the temperature and flux at x = 0,

For simplicity, let a( .) be a function of the spatial variable only.

368 H.-J. REINHARDT

Let us assume that u = U" and q = q" have already been computed for the discrete time t = t , on a not necessarily equidistant grid 0 = xo < x1 < . . . < XI = 1 in [0, 11. Measured temperatures will be denoted by Uf ; at the boundary this means that U; = g(tn). Beck's method uses data from several 'future times'--from t,+ to t,+,-and proceeds as follows:

(i) Compute u " + ~ from

a w + m aun+m -a- + F ( x , t), - Iz - 1 -- __

a u n + m

at ax ax ax x = o

f o r r time steps with initial conditions (Numerical solutions: t$+m, i = 0, . . . , I , m = 1, . . . , r).

(ii) Compute 'Sensitivity Coeficients' s n f m =

= u", m = 1 , . . . , r

aq(qn) from a as"+, - a-

asn+m

at ax ax

for r time steps with initial conditions s"+'"lt,, = 0, m = 1, . , . , r

(iii)

(iv)

(Numerical solutions: $ + m , i = 0, . . . , I , m = 1, . . . , r). Determine new f lux by

l r L An m = l

q"+' = q" + - c 1 ( u f z + m - Gf[+m):f[+m

r L

where An = 1 1 L, for t = tn+,,,, m = 1, . . . , r. Update un+l by the solution of

and Ufz+m are the measured temperatures at zz = x i I , 1 = 1 , . . . , m = l I = 1

with initial condition u"+' I t = f n = u". (Numerical solution: uf + ', i = 0, . . . , I).

(v) Go back to (i) with t = t n+l , uf", qnf1.

If the heat conduction coefficient is independent of the temperature itself-which turns out to be a linear problem-then the sensitivity coefficients in Step (ii) do not depend on time and can be computed beforehand for all times.

In Step (iii), qn+l is determined by the problem

Minimize I I .

Since one can consider minimization in (16) is equivalent to the determination of the zeros of the derivative of

at zI as a function of one variable only, namely the flux q, the

One step of Newton's method for calculating the zeros of Z ( q ) then gives the formula for qn+l in Step (iii) (cf. Reference 8, equations (10a, b)).

In the case of one future time, i.e. r = 1, and a single transducer position at x = XI, exact


matching of the measured temperature data at x = 0 is possible by using Taylor's formula

Inserting the numerical approximations 6;" and du""/dq(q") at x = xz = 1, respectively, an estimated flux can be obtained in this case by

for u"+'(q") = u(x, t,+ 1 ; q") and

= q" + ( q + l __ $+1)/$;+1

This is the basic equation for Stolz's method38 to which the formula for qn+l in Step (iii) reduces in the case r = 1, L = 1, x = x f = z l . Contrary to Step (i), the numerical approximations in Stolz's method are calculated by a discrete form of a convolution integral equation for the heat flux; the latter is known as Duhamel's theorem (cf. e.g. Beck et ~ l . , ~ Section 3.2).

It has been observed that the method becomes more stable as r increases but it is still unstable for small time steps (cf. Reference 7). One has to achieve a balance between the smallness of the time increment and the number of future times. Such a balancing is typical for ill-posed problems. However, for this specific method there exists no rigorous statement concerning balancing.

It is obvious that the underlying ill-posed problem (15) can be written in the form of a control problem,

Minimize

subject to

IluCL .) - 911 u, - (au,), = F

u(0, t ) = f ( t ) , u x ( l , t ) = 0, u(x, 0) = uO(x)

which, moreover, in the homogeneous case F = 0, uo = 0 is equivalent to

Minimize / I *f - 9 112 f e L2 r.0,TI

The associated operator A = ic *f maps L2 [O, T ] into itself and is compact. Beck's method can now be considered as a regularization method by discretization or as

a regularization in finite dimensional spaces; the basis functions are piecewise constant. To be more precise, let At = TIN, At' = MAt and

V, = {v : (nAt, nAt + At') -+ [w I v is piecewise constant on ( (n + i)At, (n + i + l)At), i = 0, . . . , M - 1)

Then Beck's method can be described as follows:

(i) Set wo = u'for n = 0, t = 0. (ii) Minimize Ilw(1, -1 - Q I / [ t , t + A t ' ]

subject to w, - (a%), = m, t )

w(0, t ) = f ( t ) , WJ1, t ) = 0, w(x, t ) = wo, f € V,

(Solution: w") (iii) S e t wo = w"(x, t + At), replace t by t + At, n by n + 1, and go to (i).

We close this section by briefly discussing numerical schemes for the solution of the one- dimensional heat conduction problem. Up to now we are free in the choice of a specific numerical method for the initial-boundary-value problems appearing in Steps (i), (ii) and (iv) of Beck's method. We propose the Crank-Nicolson method because it is second order in time. For the

370 H.-J. REINHARDT

discretization in space, one can use a difference approximation or a Galerkin method. The usual difference scheme, however, has been observed to produce the greatest errors at boundary points of the spatial interval where a Neumann boundary condition is prescribed (cf. Reference 34, Estimate 12.(13)). In all the present problems to be solved, at x = 0 and x = 1 only fluxes are prescribed, i.e. Neumann boundary conditions are present. We thus propose a Galerkin approximation where the local errors are of the same magnitude over the whole interval.

4. A NUMERICAL METHOD FOR TWO-DIMENSIONAL PROBLEMS

A two-dimensional inverse heat conduction problem has already been presented in Section 2 (cf. equations (9)-(12) and Figure 1). We now consider a more general situation and present a numerical scheme which takes Beck's one-dimensional method as a basis and where, initially, an approximation by a method-of-lines is performed. The line approximation leads to a system of coupled initial-boundary-value problems which can be decoupled owing to the special structure of the underlying matrix.

Let us consider a situation which is an extension of the situation of Figure 1 in so far as two materials can be present and where the temperature itself or, alternatively, a flux boundary condition is prescribed at y = D. Let the heat conduction coefficients A(i), i = 1, 2-and the thermal diffusivity di)-be constant in the respective materials and, for the time being, independent of the temperature itself. Figure 2 displays this problem.

Let 0 = xo < x1 < . . . < xI = 1 be a not necessarily equidistant grid in [0, 11 and let y j = jAy, j = 0, 1, . . . , & J , J A Y = d, be an equidistant discretization of the y-interval [0, d]. Addition- ally, we set y i f J + = & D. Discretization of d2u/dy2 by central differences in (9) leads to the

Y

- D

ulx=,

= g(y,t)

given

and -d

uxlx=, = 0

( I n s u l a t i o n )

Y -0

Figure 2. Inverse heat conduction problem at the boundary of a two-dimensional plate


following system for u j approximating u(x, y j , t):

For reasons to be explained later, we consider the whole region from y = D to y = - D, set y j = y - j for the negative indices, and the system (18) is considered from j = - ( J - l), . . . , 0, . . . , J - 1.

We now pose the physical assumption, that no heat is lost by the flow from Material 1 to Material 2, which means that the heat flux is continuous at y = d, i.e. A(') au/ d y = A ( 2 ) au/ ay. An approximation of first order derivatives by difference quotients thus leads to the relation

Such a coarse approximation of the flux in Material 2 is justified if this material is nearly an insulator properties and no steep gradients occur.

We are thus able to determine the temperature at y = d from those at y = d - A y and y = D,

Here, p = (L(')/Ac2))(D - d) /Ay . Defining b = 1/(1 + p ) , we have p/(l + p ) = 1 - 1. Note, that the cases A ( 2 ) = 0 and D = d are allowed and lead to 6 = 0 and 6 = 1, respectively.

Inserting u - ~ and uJ in (18), we obtain for the first and last equation

resp., where CI = ,I - 1, P = 2 - p/(l + p) .

order is perturbed in the above system with perturbations in the first and last row, One observes that the tridiagonal matrix belonging to the central difference quotient of second

- P 1 0 1 - 2 1 0 0 1 - 2 1 0 ;;i . . . 0 1 - 2 1

0 1 - p

The system (18) can be written in matrix form as

372

where

H.-J. REINHARDT

9(X,t)=(F(x,y-(J- l ) , t ) , . * * >F(x,yJ- l , t ) ) ' The matrix M has distinct eigenvalues and, thus, there exists an orthonormal matrix B such that D = BTMB is a diagonal matrix consisting of the eigenvalues d j of M. The columns of B are the corresponding (orthogonal) eigenvectors. Since B is orthonormal, B- = BT; moreover d j < 0 for all j .

A decoupling of (23) can now be performed, if one considers V = BTU instead of U . The following system for V = ( u - ( J - '), . . . , uo, . . . , uJ- l )T has to be solved,

au j a Z v . a(') _ - - a ( ' ) d + 7 d j u j + F j , x ~ [ O , l ] , t ~ ( O , T ] , j = 0 , k l , . . . , k(J-1) (24) at ax2 (AY) 1

where R = (F-( J - '), . . . , Yo, . . . , F J - l )T = BTR. The associated boundary conditions are given by

uj1,=1 = g j , 91 =o, t€[O,T] , j=O, Jc l ) . . . ) f ( J -1) ax -

T where G = BTG with G = (g-( J - l ) , . . . ,go, . . . , g J - 1 ) . For an initial-boundary-value problem with given temperature at y = D and y = - D, the

decoupled system (24), (25) can be solved by Beck's method on every jth line using a suitable numerical scheme and the solution U itself is obtained by 'coupling' the v i s , U = BV.

The complete numerical solution procedure may be described as follows. Assuming that numerical approximations uf, j , i = 0, . . . , I , j = 0, f 1 , . . . , +( J + l ) , for the temperature field and 47, j = 0, Jc 1, . . . , Jc ( J + l), for the heat flux at x = 0 at time t = t , have already been computed, then for the next discrete time t = tn+l:

(i) Compute the eigenvalues and eigenuectors of the matrix M. (ii) Solve the ill-posed problems (24, (25) for one time step and every j th line, j = 0, f 1, . . . ,

( J - l), with initial conditions vj"+'(xi, t,) = uf, j , i = 0,. . . , I , andflux Gj" at x = 0, where

vf = ( U ? , - ( J - ~ ) , . . . , Vy, 0, . . . , Uf, ~ - 1 ) ~ = BTUf

Uf = (uf,-.(J-l), . . . , u ~ O , . . . , u Z J - ~ ) ~ , Q" = ($-( J - ~ ) , . . . , qo , . "n . . ,$-') = BTQ"

i = 0,. . . ,I - P=(@-(J-l),* - - , 4 ~ , * . - , & - 1 )

(Numerical solution: V?+' , i = 0,. . . , I, pin+').

u;+' = B V ~ + ' , i = 0,. . . ,I, p+l= B@+'

(iii) 'Couple' temperature and Jux,

(iv) Compute temperature and flux on the Jth line by formula (20). (v) Go buck to (i) with t =

We should like to make several remarks concerning this method. First, in the case of heat conduction coefficients independent of the temperature, Step (i) need not to be performed for

$,;', 4;".


every time step, but only once in advance. Note that our method solves non-linear problems with temperature dependent coefficients; the above procedure may be viewed as a linearization where, in every time step, the heat conduction coefficients are ‘frozen’ during the associated future time intervals. In the case where no Material 2 is present, i.e. D = d, p = 2, the eigenvalues and eigenvectors are explicitly known.

Secondly, the determination of the reference temperatures for the minimization process on every line (see Step (ii)) gives a hint how to design experiments and how to construct sensors. For the discretization we assume that all transducer positions coincide with mesh points. The ‘decoupling rule’ then requires that, for every fixed x-grid point, the vector of temperature values for yj, j = 0, & 1, . . . , & (J - l), has to be ‘decoupled’ via multiplying it by BT. Hence it is advantageous that at every discrete y-position a transducer is located whenever at least one measurement is possible for this x-position. Otherwise, if there is no measurement available, the numerical approximations at the previous time t have to be taken as reference temperatures, which will cause additional computational errors.

Finally we should like to mention that, up to now, we have not taken the symmetry condition at y = 0 into account. Our method is thus applicable also to non-symmetric problems. In case of symmetry, one can clearly save some storage during the computation, but note that the system from - ( J - 1) up to J - 1 is to be considered-and has to be decoupled. The principle reason for taking the whole system (1 8) from - ( J - 1) to J - 1 into account lies in the fact that a symmetry condition at y = 0 and the corresponding approximation of that boundary condition have restricted our method in Reference 4 to the case of only one future time. One way to overcome this disadvantage lies in the present approach, which allows an arbitrary number of future times.

5. COMPUTATIONAL RESULTS

The computer program used works in a similar manner to the SINCO program by which the computations presented in References 4 and 35 were performed. The present new program extends SINCO in several aspects. In particular, an arbitrary number of future times is now allowed, and an inhomogeneous term in the differential equation representing sources can be taken into account. The latter is also useful for designing mathematical test examples where the solution is known. The computer program has more features and is more flexible than demonstrated in this paper. Boundary conditions can be prescribed also in the form of fluxes at part of the boundary, the thermal diffusivity may be temperature dependent and two materials as described in Figure 2 may be present. An extension to spherical domains is under development.

In Reinhardt and Valencia3’ it was demonstrated that the method is applicable to and works well for measured data obtained from specifically designed Heat Transfer Measuring Blocks during a fire experiment. In the present paper it is our aim to test our method and verify the computer program by applying it to several mathematical examples.

For one-dimensional IHCPs, a widely studied benchmark problem is the ‘step heat flux’ case, i.e. q(t) = 0, for t < 0, q(t) = 1, for t > 0. The solution is explicitly known to be (with a = 1, F = 0, u = u in (15))

1 1 2 “ l 3 2 n2 n2 u(x, t ) = t +- - x + - x 2 -- C -exp(-n2n2t)cos(nnx), O 6 x < 1, t o

u(x, t ) = 0, t 6 0

At t = 0, the sum with the preceding factor represents the Fourier series of 3 - x + +x2. There is also a close relation of u(x, t ) to the Theta function.

314 H.-J. REINHARDT

Example 1

With the function v+(x, t) = t - x + i x 2 we set for the solution

u(x,y, t )=cos -y v+(x,t), O < X < l , - l < y < l , t > O c 1 We then have an inhomogeneous differential equation, F(x, y, t ) = 4 n2 cos(4 n y)v+ (x, t), with exact flux 4( y, t ) = cos(4 n y), t 3 0. The function v c fulfills the one-dimensional heat equation

v: - v : x = o , O < X < l , t > O

with boundary fluxes q(t) = - v: I x = o = 1, v: I x T l = 0, t 3 0. The problem is similar to but simpler than the 'step heat flux' case since there is no jump in the flux at t = 0; the one- dimensional example with solution v + may be called the 'constant heat flux' case.

Example 2

y-direction by a cosine factor, With the solution v of the step heat flux case we construct a solution which again varies in the

u(x, y, t ) = cos -y v(x, t ) (1 ) For the source term we obtain F(x, y, t ) = & n2 cos(3n y) v(x, t); the flux turns out to be q( y, t ) = 0, t Q 0, q( y, t ) = coS(+zy), t > 0.

Example 3 (see also Weber?' p . 1786)

The solution is set to be

u(x, y, t ) = exp(pt)cos(l - x), 0 Q x Q 1, - 1 6 y 6 1, t > 0

i.e. we test a one-dimensional example which is constantly extended in the y-direction. Following Weber?' the parameter p is set to be - 0.5 and the nearest measurement position to the surface is at x = 0.5. The source term and flux, resp., are given by

F (x ,y , t )= ( l +B)exp(pt)cos(l - x ) , q(y , t )= -exp(Jt)sin(l.)

Table I shows a selection of test cases for which computations have been performed. In every case, we prescribe the solution at y = 1 and consider the case d = D = 1 (see Figure 2). For the thermal diffusivity, we always choose a = 1. To test the stabilizing effect of the method with more

Table I. Examples tested ~~ ~~~

Number Initial Transducer CPU-time (s) time time for positions Exact Inacc.

r At steps comput. in x-direction data data Figures

0.0 0.125 s .o 65.21 65.29 3, 4, 5, 6 EX. 1 3 0.01 100 EX. 2 3 0.01 105 - 005 0.125 1 .o 158.9 1591 7, 8, 9, SO

EX. 3 4 0.05 20 0.0 0.5 1 .o 21.15 14 EX. 3 3 0.05 20 00 0.5 s .o 16.48 16.50 11, 12,s3


than one future time involved, every example is additionally computed with perturbed data by adding csin(ot)-always with E = 5E - 3 and o = 20. Following Beck et al.,' we call this the case with 'inaccurate temperature data' or, as in Beck et al.,' the case of 'simulated measurement errors'. We have always chosen eight equidistant subintervals in the x-direction and five in the y-direction. In the last column of the table, references to the associated figures collected at the end of this section are given.

In Table I, for every example there are also given the CPU-times (in seconds) on a SUN SPARCstation 1; times for the Input/Output are included which, for our program, amount to 3-9 per cent. We believe that on the basis of this machine it is not difficult to make comparisons with other computers and thus compare the performance of our computer code with others.

Each of the following figures displays five curves related to the positions (0, yj) at the surface for equidistant y j = jAy, j = 0, 1,2, 3,4. As far as absolute errors are shown, they are multiplied by lo2; relative errors are given in per cent. All curves for errors in the following figures are thus related to pointwise errors at the surface x = 0. The x-axis is always related to the (dimensionless) time and reaches to 1.0. For Example 2, computations are started at a negative initial time

It had been observed by numerical experiments that, in certain cases, especially when the flux is very small, an additional update of the flux yields better results. For this purpose, after having calculated q"+l and u"+' by Steps (iii) and (iv) of Beck's method, resp., a first order difference quotient of higher accuracy may be used.

( t o = - 0.05).

N

a, L 3

0 c a, 0. e 0)

r

Y

m 0

0

0.0 0.2 0.4 0. 6 0. 8 1 . 0

I : Pos. [ i . j I - l O $ O I 4 : P O S . I i ' I - 1 0, 31 2 : POS. I i 1 - 1 0 . 1 1 5 : ~ o s . I ~ : ~ I - [ D . ~ I 3 : ~ o s . t i : 1 1 - 1 0 . 2 1

Figure 3. Surface temperatures at x = 0 for Example 1 (constant heat flux, 2-d)

T i me

316 H.4. REINHARDT

Example 1

Figure 3 shows the five curves for the surface temperatures varying by the factor cos($ n: y) in the y-direction; the exact temperature at x = 0 is tcos(*ny). The absolute errors of the temperatures remain below 7E - 4 (see Figure 4) in the case of exact temperature data. With simulated measurement error of magnitude 5E - 3 the absolute errors in the surface temperatures vary with maximal amplitude of 7E - 3 (see Figure 5). One also observes very well the periodicity with period n/20 caused by the added perturbation. The absolute errors in the flux are one magnitude larger, having a maximal amplitude of 4E - 2 where again the periodicity is obvious (see Figure 6).

Example 2

Due to the number (three) of future times used and the initial time to = - 005, the computed temperatures and fluxes start to become positive before t = 0 when the jump in the heat flux occurs. Figure 7 shows the fluxes at (0, yj),j = 0,. . . ,4; the exact fluxes are known to be cos($.nyj) for t > 0. The relative errors are large in the first three time steps following t = 0 and reach 20 per cent at t = 001; they stay below 1.1 per cent during the rest of the time interval (see Figure 8). The absolute errors in the temperatures and fluxes, resp., remain below 2E - 3 and 1E - 2, resp., after t = 003 (not displayed in a figure). Figure 9 shows a considerable perturbation of the fluxes in the case of inaccurate data. The relative errors in the temperatures are below 2 to 6 per cent,

m 0

- t\l

I - I I CI

N

W L 3

0

Q Q E W

-

-u

L

c.

L 0 L L

w

ln

U n

0

W 0

0

w 0

0

N 0

0

0.0 0. 2 0 . 4 0. 6 0.8 1 . 0 Ti me

1 : P O S . l i . jl-10.01 4 : P o s . l i , 1 1 - 1 0 . 3 1

3: POS. ( i . J I - 1 0. 21 2: POS. r i . i l - t D . ’ l 5: p O S * 1 ~ e J l - f o ~ ~ l

Figure 4. Absolute errors of temperatures ( x 10’) for Example 1

- N

I

I

I R 0

R

0,

3

0

0, Q E cu

..-.

L

Y

c

r-

L 0 L L

W

v)

U n

- N

I

L

I I 0

R

X 3

- - LL

Y

0 (D

r L 0 L

c W

v)

U n

0

W

0

.c

0

cu 0

I

I I I

0.0 0.2 0.4 0. 6 0. B 1 . 0 T i me

1 : POS. [ i . ! l - [ O . O l 4 : Pos. I i I j l - 1 0,31 2: Pos. [ i 1 - [ 0 . 1 1 5: P O S . I i , j l -I 0. 4 1 3 : ~ o s . [ ~ : ; I - ~ O . Z I

Figure 5. Absolute errors of temperatures ( x 10') for Example 1 with inaccurate data

0.0 0.2 0.4 0. 6 0. B 1 . 0

1 : POS. ( i . ! l - ( O . O l 4 : Pos. I i ' 1 - 1 0.31 2 : P o s . I i 1 -[O. 1 1 5: ~ o s . I ~ : ~ I - I O . ~ I 3 : ~ o s . I ~ : ~ I - ~ o . z I

T i me

Figure 6. Absolute errors of heat fluxes ( x 10') for Example 1 with inaccurate data

W

0

X 3

LL - Y

0 0) 5

0.0 0 . 2 0.4 0. 6 0 . 8 1 T i m e

1 : Pos . [ i , j l - [ O . O l 4 : Pos. [ i , j l - [ 0 , 3 1 2: Pos. t i . j l - ~ O ~ I l 5: Pos. I i j l - [ 0. 4 1 3 : Pos . I i . j l - 1 0 . 2 1

Figure 7. Heat flux curves for Example 2 (step heat flux, 2-d)

. o

m

N

0.0 0 . 2 0.4 0. 6 0 . 8 I T i m e

1 : P o s . [ i . j l - [ O , O l 4 r Pos. [ i . j I - [ O . 3 ) 2: Pos. [ i . j l - ( O . l ) 5: P o s . t i . j l - I O . 4 1 3 : POS. [ i . j I - [ O . 2 1

Figure 8. Relative errors of heat fluxes (in per cent) for Example 2

. o


m

0

0.0 0 . 2 0 . 4 0. 6 0. fl 1 . 0 T i m e

1 : P o s . [ i . j l - 1 0 . 0 1 4 : P o s . l i , j ) - I O ~ 3 1 2 : POS. t i 1 - [ 0 . 1 1 5: P o s . I i . j l - t O . 4 1 3: Po$. ( i : j 1 - [ 0 * 2 1

Figure 9. Heat flux curves for Example 2 with inaccurate data

depending on the y-position (see Figure 10) whereas the relative errors in the fluxes are between 4 and 10 per cent (no figure for this). Comparing the absolute errors, these computations can be considered as stable ones.

Example 3

Due to the fact that the nearest transducer position to the surface is relatively far away, computations with At = 0.01, as in the other examples, exhibit unstable behaviour even in the unperturbed case. Stabilization can be achieved by increasing the size of the time steps At or by increasing the number of future times (or both). The first remedy is applied with At = 0.05 and the resultant flux for the perturbed case is shown in Figure 1 1 where the exact flux is drawn as a dotted line. The relative errors in the fluxes for the case of exact temperature data remain below 0.6 per cent (see Figure 12). The relative errors of the temperatures in the case of simulated measurement errors are mostly below 5 per cent (see Figure 13) whereas the relative errors in the fluxes still reach 40 per cent (no figure for the latter). Applying the second remedy and increasing the number of future times from 3 to 4 results in a decrease of the temperature errors (see Figure 14) as well as of the errors in the fluxes-mostly below 10 per cent (no figure for the fluxes).

To conclude this section we can state that the test examples-and many others-can be computed by our two-dimensional extension of Beck's method in a stable and efficient way. Moreover, the last example demonstrates the ill-posed character of the underlying IHCP which can be stabilized by a numerical solution procedure in the indicated manner.

0.0 0.2 0.4 0. 6 0. 8 1 . 0 T i me

I : P O S . [ i * I - [ O . O 1 4 : POS. I i * 1 - 1 0 , 3 1 2 : or. I i : i l - [ 0 . 1 1 5: ~ o s . l i : ~ ~ - [ o o ~ ~ 3: Pos . I i . j l - I 0. 2 1

Figure 10. Relative errors of temperatures (in per cent) for Example 2 with inaccurate data

19

0

0

0.0 0.2 0. 4 0. 6 0. 8 1

1 : P O S . [ i . ~ l - I O . O I 4 : P o s . [ i . j l - I D . 3 1 2: P O S . I i 1 -I 0. 1 1 5: P O S . I i I j l -I D , 4 1 3 : ~ o s . t i : i l * ( 0 . 2 1

T i me

Figure 11. Heat flux curves at surface for Example 3

. o

I .\- C

I- - m L f

0 L a2

iy

v

0

cu 0

Figure 12. Relative errors of heat fluxes (in per cent) for Example 3 with exact data

0.0 0. 2 0. 4 0 , 6 0.3 1 . 0

I : Pos. ( i + j J - ( O , O l I t P O S . l i , j l - 1 0 . 3 1 2: Po$. t i . j ! - f 0. I 1 5: P o s . l i , j I - i 0 , 4 1 3 : P o s - I i . j l - t 0 . 2 1

T i me


382 H.-J. REINHARDT

- h.

C .- - W L 3

0 L

01 (1 E W

Y

c-

L 0 L L

W

- 01 U

, 1 I I

0 . 0 0. 2 0. 4 0. 6 0.8 1 . 0

1 : Pos. 1 i . j ) - 1 0 . 0 1 4 : P O S . I i ‘ 1 - 1 0 , 3 1 2 : Pos. l i . ~ l - ~ O . l l 5: ~ o s . l i : ~ l - l o . ~ l 3 : Pos. 1 i . j l - 1 0 . 2 1

T i me


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A numerical method for the solution of two-dimensional inverse