The solution of a nonlinear inverse problem in heat transfer

Transcription

1 IMA Journal of Applied Mathematics (1993) 50, The solution of a nonlinear inverse problem in heat transfer D. B. INGHAM AND Y. YUAN Department of Applied Mathematical Studies, University of Leeds, Leeds LS2 9JT, UK [Received 20 November 1991 and in revised form 22 October 1992] In this paper the authors investigate the numerical solution of a class of nonlinear inverse elliptic equations. The boundary element method, combined with a minimal energy constraint, is used and it is found that this technique gives a good stable approximation to the solution. Polynomial and piecewise quadratic approximations for the nonlinear function which occurs in the governing equation have been considered and the effects of small perturbations to the boundary condition have also been investigated. Examples in which there are known analytical or numerical solutions are presented and the agreement with the solutions obtained using the present method are excellent. 1. Introduction If the temperature is specified at the surface of a solid body, whose thermal properties are assumed to be constant, then the steady-state temperature distribution within the body is governed by the Laplace equation for which there is a unique solution. Such problems may be mathematically recast into an integral equation form and in recent years the solution of such formulations have been gaining in popularity (see Jaswon & Symm, 1977; Brebbia, 1989). These integral equations are usually intractable by analytical methods and thus much attention has been given to their numerical solution using a variety of approximating techniques (see e.g. Ingham & Kelmanson, 1984). One immediate advantage of the reformulation is that the equations apply only on the boundary of the solution domain, whereas space discretization techniques, such as finite difference (FD) or finite element (FE) methods, evaluate information at many interior points. The boundary element method (BEM) uses only the boundary data to compute the solution at any interior point and it is found that a high degree of resolution may be obtained. An immediate consequence of this is that the system of algebraic equations generated by a BEM is considerably smaller than that generated by an equivalent FD or FE approximation. Although when using the BEM the matrix associated with the system of algebraic equations is full, in many cases where the geometry of the solution domain is complex the BEM may use less CPU time, and be easier to program, than the corresponding FD and FE methods. The solution of nonlinear problems in heat transfer using the BEM technique is well established (see e.g. the investigations of Khader, 1980; Bialecki & Nowak, 1981; Ingham et al, 1981; Khader & Hanna, 1981; and Ingham & Kelmanson, 1984). All these studies deal with nonlinear boundary conditions, whilst Khader (1980), Bialecki & Nowak (1981), Khader & Hanna (1981), and Ingham & 113 Oxford University Press 1993

2 114 D. B. INGHAM AND Y. YUAN Kelmason (1984) deal with temperature-dependent thermal conductivity by first employing the Kirchhoff transformation. In this paper we consider the steady-state solution of the nonlinear heat conduction equation V [/(r) Vr] = 0 in Q (1.1) where Q is the region enclosed by the boundary dq of the solid body, T is the temperature of the solid, and f{t) is the thermal conductivity of the body which is temperature dependent. We will assume that at every point on the surface of the body that the temperature is prescribed, although it is easy to extend the analysis to include linear combinations of the temperature and the heat flux or even nonlinear boundary conditions, e.g. radiative conditions. If f{t) is known, then the techniques as described in Ingham & Kelmanson (1984) may be applied. However, frequently in practice the detailed variation of the thermal conductivity with temperature is unknown but extra information in the interior of the body is known, e.g. the temperature may be measured at a number of points within the body. This phenomenon falls into the general class of problems known as an inverse heat conduction problem since extra conditions to those normally given are specified, but there is an unknown function within the governing equation. There are numerous examples of inverse problems that arise in heat transfer and other physical problems, e.g. in solving the Maxwell equation V B = 0 for the magnetic field when B = (ih and (j. is the magnetic permeability which in general will depend in some unknown way on the strength of the magnetic field. Numerous authors have studied the identification of parameters which appear in the governing partial differential equations when extra conditions are enforced. Cannon (1967) studied the partial differential equation (1.1) subject to the boundary condition/(t) dt/dn = <f>(x, y) on d 2and T = g(x, v) on a part of the boundary dq, where <j>(x, y) and g(x, y) were specified functions, whilst Cannon & Duchateau (1973) have investigated the time-dependent heat conduction problem with additionally specified 'flux' conditions at the boundary. Chen & Seinfeld (1972) have presented two methods to estimate the parameters that occur in the governing partial differential equations with extra noisy observations and Carrera & Neuman (1986) investigated a diffusion-type partial differential equation in groundwater flow. An excellent review of inverse heat conduction is given in the recent book of Beck et al. (1985). The inverse heat conduction problem is much more difficult to solve either analytically or numerically than the direct problem, i.e. when f(t) is given, but inverse problems occur much more frequently in practice than do direct problems. In this paper we consider equation (1.1) subject to the conditions T{x,y) = 4>(x,y) on dq, (1.2) T(x,y) = y>(x,y) on T, (1.3) where <p and V ar e given functions and F is a set of points interior to the boundary dq. It is clear that if the functions/(t) and T(x, y) are solutions of the problem ( ) then the functions cf{t) and T(x, v) are also a solution of the

3 A NONLINEAR INVERSE PROBLEM IN HEAT TRANSFER 115 problem ( ) for any constant c. Physically, the constant c is determined by specifying the thermal conductivity of the material at a given temperature. Hence the mathematical solution of the problem ( ). Further, the existence and stability of the problem ( ) is also questionable, for example, because of the maximum principle, namely if max \ip(x, y)\ > max 0(x, y)\ then there is no solution to the problem ( ). Although there are some mathematical difficulties relating to the solution of the problem ( ), the problem is of much physical interest. In this paper we will show how the BEM may be modified in order to obtain a physically correct approximate solution to the problem ( ). A transformation of equation (1.1) is employed such that all the nonlinear aspects of the problem are transferred to the boundary of the solution domain. The function/(t) is then represented by a simple function/(t), where F(T) e & and 3P is a set of polynomials or piecewise quadratic functions, and a modified BEM is developed. In this paper all the calculations have been performed in a square region with T given on the boundary dq, but the extension to boundaries of arbitrary given shape is trivial. 2. Formulation In order to solve the problem ( ) by using the BEM the first step is the introduction of the transformed variable A which satisfies VA=f(T)VT. (2.1) Thus the governing equation becomes the Laplace equation Vl4 = 0, (2.2) and then all the nonlinear aspects of the problem are transferred to the boundary of the solution domain. The application of the BEM to the solution of equation (2.2) is well documented. The formula for A may be expressed as (see Ingham & Kelmanson, 1984) = j A(q) In' \p -q\dq- J^'ta) In \p - q\ dq. (2.3) Here, p e Q and q e dq, where Q = Q + dq; the prime denotes differentiation with respect to the outward normal to dq at q; and {2JI if p e Q, 9 if p e Q, where 8 is the angle included between the tangents to dq on either side of p. If either A or A' is prescribed at each point q e dq, then the solution of the boundary integral equation obtained by letting p e dq in equation (2.3) determines the boundary distribution of both A and A'. Equation (2.3) may now be used to generate the solution A(p) at any point p e Q. Let ' -(T)dTr, (2.4)

4 116 D. B. INGHAM AND Y. YUAN and, employing equation (2.1), we may write the transformation in the form Combining equations (2.3) and (2.5), we obtain f g(t(q))w\p-q\dq-l Jan A=g(T), A'=f(T)T'. (2.5) J aa T'(q)f(T(q))\n\p - q\dq = r,(p)g(t(p)) peq, qed 2 (2.6) as the nonlinear integral equation on 9Q. In order to obtain the numerical solution of the integral equation (2.6), a constant-element BEM has been employed. Linear and higher-order elements could have been used, but since the accuracy achieved by using the simple constant-element approach is very good the more general discretizations were thought to be unnecessary. Further, the use of higher-order elements is not without its difficulties near corners (see e.g. Ingham et al., 1991). Thus the boundary d 2 is first subdivided into JV segments d 2j (/ = 1,...,N), and on each segment 3Q t we take constant <pj and (f>j to represent <j> and <f>', where (j), and (p take the values of (/> and <j>', respectively, at the midpoint of the segment dq t. The integral formula then becomes f [ In \p - q\ dq = r){p)g(t(p)). (2.7) j=\ Jail/ j=\ J Taking p, = q, e 9Q h we obtain Let -2 #/(*>)[ \n\p~q\dq = 0 (i = l,...,n). (2.8) i=\ Jan, Ei,= \ in'lpi-qldq-bij-rii and Gy = In \p,,-q\ dq. Jaii/ JaQj Then, from (2.8), we obtain the linear system of equations Similarly, if p t e F cz Q, let Then, we have 2 EijgWj) ~ 2 GutjWj) =0 (i = 1,..., TV). (2.9) EIij=\ \n'\pi-q\dq and O/, v = I \n\pi~q\dq. Jaoj Jai2j EI iig (<t>,) - 2 GWJf {</>;) = 2ng(T( Pi )) (i = 1,..., 1). (2.10) l

5 A NONLINEAR INVERSE PROBLEM IN HEAT TRANSFER 117 P "7-1 FIG. 1. Notation for the analytic evaluation of integrals on straight-line segment geometry. If the segment dqj is a straight line, then the integrals occurring in the equations (2.7) may be written J go. ln\x-y\dy = J, and then we may obtain (see Manzoor, 1984) 1 = Y, J = a cos i3(ln a - In b) + h(\n b-l) + ay sin 6. (2. 11) (2-12) (2-13) (2-14) If we denote q t _ x and q, as the endpoints of dqj (see Fig. 1), then a, b, and h are the lengths of the lines joining p to q,-\, p to q jt and <7 y _, to q r respectively. Further, /3 and y are the angles qjqj^xp and qj_ x pqj, respectively. 3. Minimal energy scheme Han (1982) and Falk & Monk (1986) studied the Cauchy problem for elliptic equations by using the minimal energy technique and Ingham et al. (1991b) investigated an inverse heat transfer problem by using this technique. However, all these studies were restricted to linear problems. Here, we use this technique to deal with the nonlinear elliptic equation (1.1). Instead of considering the problem ( ) we investigate the solution of the problem v-[/(r)vr] = o in Q, ] T = (j) on afl, (3.1) -v *e onf, J where >0 is a preassigned small quantity. In general, there are many solutions

6 118 D. B. INGHAM AND Y. YUAN of the problem (3.1), and clearly the solution of problem ( ) is one of the solutions of problem (3.1). Inserting transformation (2.4) into (3.1), we obtain We now let in Q, on dq, on F. H[m,M] = ig = j /(T) dt:/eh'[m,m] and /(r) (3.2) (3.3) (3.4) and then, clearly, H[m, M] is a subspace of the Sobolev space H 2 [m, M] and for any given g e H there is a unique function T which satisfies the problem (3.2, 3.3) (see Gilbert & Trudinger, 1983). Therefore we now consider the constrained minimal problem j(g(t)) = inf J(g(f)) = inf f f \Vg\ 2 dx dy] ie'h gek } J (3-5) J where f is the solution of the problem (3.2, 3.3) for a given function g. The expression of the functional J(g) is similar to the energy functional for the Laplace equation, so we call J(T) the energy functional of equation (3.2). It is clear that the solution of the problem (3.5) is a solution of the problem ( ). Since the function g(f(x, y)) satisfies the Laplace equation V 2 g = 0, employing Green's formula to the minimal problem we then have where n is the outward normal to dq, which on discretization becomes = f g(<t>)-^-g(<t>) ds = 1(0/) -H-g&iW, Jan on,=i an \ x (36) (3.7) where <5, denotes the length of dq, and equation (2.9) has been used. Then the problem reduces to finding the function g which satisfies j(g(t)) = inf i 2 g(<pdg- k >E ki g(<pj), (i = l,...,/), (3.8) / + c) where W tj = Ely GIi,Gi^E mj, i/;, = T(p,), and p t F. Solving the constrained minimal problem (3.8), we obtain the functions g(t) and f(t). Then we may

7 A NONLINEAR INVERSE PROBLEM IN HEAT TRANSFER 119 obtain an approximate solution of the problem ( ) by the use of the BEM (see Ingham & Kelmanson, 1984). In order to solve the problem (3.8), we define a set of functions, 9 say, such that and we need to find the best approximation, /(T) say, to the unknown function f{t) in 3F. Inserting the functions / and g into the linear system of equations (2.9), we obtain E^^-G^f (<!>,) = 0 (i = l,...,a0. (3.9) Since (3.9) is a homogeneous system of equations, which is because the solution of the problem ( ) is not unique, we need one extra condition in order to be able to obtain a unique solution, and hence/(t) has to be fixed at one point, T o say. Therefore the minimal problem (3.8) may be rewritten in the form where s =f(t 0 ) is given. = inf \ 2 AJ geh ' /=' f(t Q )=s [/ =!,...,/), (3.10) N 1 8(1>i - e) «2 Wfc) «g( V, + e) 2n i=l 4. Polynomial approximate solutions In order to solve the problem (3.10), the simplest choice of & is a set of linear functions or more generally polynomial functions. Initially we assume that & is the set of linear functions, and that the function f(t) = at + b, and therefore g(t) = \at 2 + bt, such that f(t) and g(t) satisfy (3.10), where a and b are constants to be determined. Hence the problem reduces to the determination of the two constants a and b, which may be found using the NAG routine E04UCF. The routine E04UCF is designed to minimize an arbitrary smooth function subject to certain constraints which may include simple bounds on the variables, linear constraints, and smooth nonlinear constraints. The method of E04UCF is a sequential quadratic programming method (see e.g. Fletcher, 1981; Gill et al., 1981). Solutions have also been obtained assuming that 9 is a set of quadratic functions, namely f(t) = at 2 + bt + c and g(t) = $at 3 + ibt 2 + c. Then the problem reduces to the determination of the constants a, b, and c, and the NAG routine E04UCF has again been used. Example 4.1. We consider the problem in which '0 (0= c= l; v=0), (0«v= l;;t=0),

8 120 D. B. INGHAM AND Y. YUAN and V(x,y) = {\ + 2xy) m for (x,y)er, where T contains the five points (, {), (3, \), (I, 3), (3, 2), (I, 2)- It is clear that T = (1 + 2xy) in - 1 and f(t) = l + T is the analytical solution of the problem ( ). Numerical solutions have been obtained with N = 40, 80, 160, i.e. 10, 20, 40 segments, respectively, on each side of the boundary 9Q, and & is assumed to be a set of linear functions and/(t) is fixed at r o = 0. The results for N = 40, 80, 120 and e = 0001 give a = and 6 = 1-0, a = 1008 and 6 = 10, and a = 1003 and 6 = 1-0, respectively. These results illustrate the excellent agreement between the numerical and the analytical solutions. The choice of the parameter e has also been investigated. When N = 160 and = 001, 0001, 00001, we get a = and b = 10, a = and b = 10, and a = and 6 = 1-0, respectively. These results show that the accuracy improves as e decreases but that the choice of e does not significantly change the results. However, if we let e tend to zero, then we find that no solution can be obtained if < 10~ 8. This is because we have taken six extra conditions (five extra interior points and one condition is required to fix/(r)) but we have introduced only two extra unknowns, a and 6. Therefore we cannot find any function f{t) which can satisfy the problem (3.10) as e tends to zero. In all the further results presented in this paper, we have taken e = f(t) FIG. 2. The solutions f(t) for Example 4.2, where 1 is the analytical solution, 2 is the linear approximate solution, and 3 denotes the quadratic approximate solution.

9 A NONLINEAR INVERSE PROBLEM IN HEAT TRANSFER 121 Example 4.2 Let 0 and In (1 + 2*) (O= ;t= l;y = l), _0 (O=Sy= l;;t=o), K, y) = In (1 + 2xy) for {x, y) e F, where F contains the five points as used in Example 4.1. The solution to this problem is f(t) = exp T and T = In (1 + 2xy). The numerical solutions are obtained with the set & being linear and quadratic functions, respectively, and f(t) is fixed at T o = 0. Figure 2 shows the analytical solution f{t) and the numerical solutions f(t) with N = 160. It shows that the accuracy of the numerical solutions are reasonable, bearing in mind that we have restricted/(t) to be either a linear or quadratic function. We have also investigated the effect of the choice of the fixed point T o. In the case of the linear approximation for f(t) we have taken T 0 = ln3 and T o = \ and the variation of f(t) as a function T is shown in Fig. 3. Further, Table 1 shows the values of T at some interior points of f(t) FIG. 3. The solutions/(t) for Example 4.2, where 1 is the analytical solution and 2, 3, 4 are linear approximate solutions with 7J, = 0, \, In 3, respectively.

10 122 D. B. INGHAM AND Y. YUAN TABLE 1 1ABLE 1 The values of T at several interior points with T o = 0, \, In 3 for Example 4.2 (x.y) (0-25, 0-25) (0-25, 0-5) (0-25, 0-75) (0-5, 0-25) (0-5, 0-5) (0-5, 0-75) (0-75,0-25) (0-75,0-5) (0-75,0-75) r 0 = o Numerical solutions * 0 =ln ftllulj Ll^tll solutions the solution domain with different fixed points T o. Again the results show that, bearing in mind that f(t) is not a linear function over the range of values of T considered, the approximate solutions are in reasonably good agreement with the analytical solution. 5. Piecewise quadratic solutions In Section 4 we obtained an approximate solution to the problem ( ) when we approximated the unknown thermal conductivity as either a linear or a quadratic function of the temperature and we had only a few extra interior data points. This is the situation which normally occurs in practice, but if more interior information is known then we may assume that & is a set of &-piecewise quadratic functions, i.e. we use a A:-piecewise quadratic function/(t) to approximate the unknown function f{t). In view of the maximal principle, we know that (see e.g. Gilbarg & Trudinger, 1983), there are points p\,p 2 e 9Q such that T{p x ) = min T(p) = m and T(p 2 ) = max T(p) = M. pei2 pe(2 Hence, the defining region of f{t) is a closed interval [m, M], and this is also true for the function f(t). Subdividing [m, M] into k equal intervals with the mesh points atm = t 0, t it..., t k _ u t k = M, then f(t) may be written in the form { re[f o,r,], where a,, b it and c, (i = l,...,k) are constants to be determined. In the transformation (2.4) we let f{t) represent f(t), and then we obtain a piecewise

11 A NONLINEAR INVERSE PROBLEM IN HEAT TRANSFER 123 third-order polynomial, say g(t), which may be written in the form c k T + d k, Te[t k -,,t k ], where d t (i = 1,...,&) are integral constants of the transformation (2.4), which are to be determined. If we insert the functions/and g into the linear system (3.9), then it includes N equations and N + 4k unknown variables. In order to solve the problem (3.9), we need 4k extra equations. The continuity of f(t), f'(t), and g{t) at the mesh points, except the two end points T = m and T = M, may offer 3fc 3 equations, and clearly d,=0 is another equation. As we noted earlier the homogeneous system of equations (3.9) is such that/(r) is fixed at one point. However, if only one point is fixed, then the accuracy of the solution is not good (see Example 5.2), so we suppose that f{t) is fixed at two mesh points, for example at the two end points m and M. If the set F contains / = k points, then the equation (1.3) gives rise to the remaining k equations. Combining all the conditions above, and (3.9), we obtain a new solvable system of equations: f = 0 (/ = 1,..., N), jb Wto) = f (V.) (i = 1,.., k), afi + biu + c, = a i+l tf + b i+1 tj + c, +1 " 6, = 2a /+,/, ffc 1 m + c 1 =5, a k M 2 + b k M + c k = S, (i = l,...,*-!), (5.1) where s=f{m) and S=f(M). Unfortunately, it is not possible to obtain an accurate solution of (5.1) by solving this problem directly, as we shall demonstrate later. Further, if l>k, then the system of equations (5.1) becomes overdetermined and this suggests that a least-squares technique should be employed. However, as in the case when I = k, we found that it is impossible to obtain an accurate solution by using this technique (see Example 5.1), and this conclusion is consistent with that reported by Ingham et al. (1991b) when studying a linear inverse heat conduction problem. Hence, as in Section 4, we use the minimal energy technique to solve the problem (5.1). In Section 3 we described the minimal energy scheme and we now give examples to verify the accuracy of this technique. In all the examples presented in this section the calculations have been performed in a square region Q, and the set F contains n x n points which are evenly distributed in a checkerboard formation in the solution domain Q. The extension of the method to deal with a region of any shape and irregular placed points in Fis trivial. The results obtained using the method described in this paper are compared with the analytical solutions where they exist.

12 124 D. B. INGHAM AND Y. YUAN FIG. 4. The solutions f(t) for Example 5.1, where 1 is the analytical solution, 2, 3, 4 are the numerical solutions using minimal energy method with k = 4, 8, 12, respectively, and 5 and 6 denote the solution by solving the system of equations (5.1) directly and using the least-squares method. Example 5.1 Let us first recalculate Example 4.2 using piecewise quadratic functions. Here the numerical solutions are obtained with N = 160 and k = 4, 8, 12, where k denotes the number of pieces of the piecewise quadratic function f{t). The set Tis taken to contain 25 and 64 points for k = 4 and 8, and 100 points for k = 12, respectively. The function f(t) was fixed at the two end points, T = 0 and 7 = In 3. In Fig. 4 we present the analytical solution f(t) and the numerical solutions FIG. 5. The lines of constant T for Example 5.1, where (a) illustrates the analytical solution and (b) shows the numerical solution with k = 4.

13 A NONLINEAR INVERSE PROBLEM IN HEAT TRANSFER 125 f{t) which are obtained by using the minimal energy method, the least-squares technique, and solving the system of equations (5.1) directly. It shows that the least-squares and the direct methods of solution are not suitable for solving this problem. For k = 4, 8, 12, we obtain the values of the minimal energy, Eng, to be , , , respectively. The correct value is Eng = 8/3, and this, like Fig. 4, shows that as A: increases the solution becomes more accurate. Figure 5 shows the lines of constant T for the analytical solution and the numerical solution with k = 4. The differences between these solutions are graphically indistiguishable and as k increases it is found that the numerical solution is convergent and stable, and the accuracy improves. Example 5.2 Here we take another simple test function, namely, f(t) = sin T, with T = arccos \(x 2 - y 2 ), and impose the following boundary conditions: arccos \x 2 \\y = 0), <t>( x arccos - \y 2 ) (0 «y = 1; x = 1), > y) = arccos {{x 2 -{) (0 «x «1; y = 1), ^arccos (\ - y 2 ) (O^y «1; JC = 0). The purpose of choosing this test problem was to investigate how the technique FIG. 6. The solutions /(T) for Example 5.2, where 1 is the analytical solution, 2 is the numerical solution with two points fixed, and 3 denotes the solution with one point fixed.

14 126 D. B. INGHAM AND Y. YUAN TABLE 2 The values of f(t) at various mesh points with respectively, for Example 5.2 = 40, 80, 160, Mesh points (T) N = Numerical solutions N = W = Analytical solution deals with functions f(t) not being monotonic. Solutions were obtained with N = 40, 80, 160, k - 10, and T containing 100 points. The function/(t) was fixed at two points, namely T = IJI and T = \n and results have been obtained with N = 40 and 80. In the case of N = 160, the solution was obtained with j{t) being fixed at one point, T = JI, and at the two points T = 5JI and T = \n. Figure 6 shows the variation of f(t) as a function of T and the numerical solution f(t) with N = 160 with both one and two points being fixed. It is clear that in order to obtain an accurate solution at least two points must be fixed and this observation is typical of numerous other numerical results we have investigated. Table 2 shows the values of the analytical solution and numerical solution with N = 40, 80, 160, respectively, at mesh points of f(t). It is seen from this table that, as expected, as the value of N increases the closer does the numerical solution approach the analytical solution. Table 3 illustrates the TABLE 3 The values of T at various interior points with N = 160 for Example 5.2 (0-25,0-25) (0-25,0-5) (0-25, 0-75) (0-5,0-25) (0-5,0-5) (0-5,0-75) (0-75,0-25) (0-75,0-5) (0-75,0-75) Numerical solution Analytical solution

15 A NONLINEAR INVERSE PROBLEM IN HEAT TRANSFER 127 temperature at some interior points, using the analytical solution and the numerical solution with N = 160. It is estimated that the relative errors are about 01% for/(t) and about 0005% for all values of T. All the results show that the accuracy of the numerical solution is very good and it becomes more accurate as the value of N increases. Example 5.3 We now consider a problem for which there is no simple analytical solution for T. However, in order to test the numerical technique developed here, we assume that f(t) is given, and for this case we take f(t) = 1 + 2T + 3T 2, and evaluate the solution using the method described by Ingham & Kelmanson (1984). Having obtained the solution, we then specify T at the interior mesh points and attempt to determine f(t) and T using the minimal energy method. In order to illustrate the method we take the boundary condition as r x 2 Solutions were obtained with N = 40, 80, 160, and k = 4, 8, 12, respectively, and F containing 100 points. Table 4 shows the minimal energy of the problem in each case. Although the exact value for the minimal energy is not known in this case, it is observed that results obtained appear to be consistent with an estimate of about In Table 5 we present the values of T at some interior points for N = 160, and k = 4, 8, 12, respectively. It is clear that as the number of pieces of }{T) increases the solution is converging and is stable. Examples show that if the temperature on the boundary is specified and some extra information is given inside the body then using the minimal energy technique piecewise quadratic function always gives an excellent approximate solution to the problem. However, in practice the temperature on the boundary is obtained by measurement and is therefore always subject to some error. It may be estimated that all temperature measurements are correct to within e,, say. In order to verify that the minimal energy technique works well in TABLE 4 The variation of J(T) as a function of k and N for Example 5.3 k N

16 128 D. B. INGHAM AND Y. YUAN TABLE 5 The value of T at various interior points as a function of k for Example 5.3 Numerical solutions k=4 * = 8 k = 12 (0-25, 0-25) (0-25, 0-5) (0-25, 0-75) (0-5, 0-25) (0-5, 0-5) (0-5, 0-75) (0-75, 0-25) (0-75, 0-5) (0-75, 0-75) this situation, we again consider the test problem (3.1) in which f(t) = exp T and T = In (1 + 2xy). Example 5.4 We take the boundary conditions as '6(x,y) (0^x^l;y = 0), 6{x,y) (0«y * 1; x = 1), n(l+2x)+ 6(x,y) (0«JC«1; j> = 1), where e, = d(x, y) =s e x and the values of 6{x, y) are uncorrelated from point to point and are given stochastically. The numerical solution is obtained with TABLE 6 The values off(t) at various mesh points with k = 10 and N = 160 for Example 5.4 T , = e,= Numerical solutions,= E, = Analytical solution

17 A NONLINEAR INVERSE PROBLEM IN HEAT TRANSFER 129 TABLE 7 The values off(t) at various mesh points with k = 10 and N = 160 for Example 5.4 Numerical solutions T e 3 = e 4 = Analytical solution N= 160, k = 10, T containing 100 points, and, = 0001, 001, The function f(t) is fixed at the two end points T = 0 and In 3. Table 6 shows the numerically obtained values of f(t) at various mesh points, the analytical solution and the numerical solution without perturbation, i.e. e, = 0. It shows that only a small error is introduced when a relatively small perturbation is introduced into the boundary conditions. In some applications the values of f(t) at the fixed points may also be in error. To illustrate the stability of the minimal energy technique in this situation, we have solved the problem with f(t 2 )-e 2 <J{T 2 )<f(t 2 ) + e 2, where T, and T 2 are twofixedpoints and e 2 is a preassigned small quantity. As an example we have taken N = 160, A: = 10, F to contain 100 points,, = 0-001, and several values of/(t,) =/(T,) + 3 and /(T 2 ) =f(t 2 ) + e 4. Table 7 gives the numerically obtained results and the analytical results at various values of T. The results given in this section confirm that the numerical method described here, which uses the minimal energy technique, is stable and can deal adequately with small amounts of noise that inevitably are introduced into such problems in practice. 6. Conclusions A boundary element method with minimal energy has been presented which enables an accurate treatment of a class of nonlinear inverse elliptic equations. It has been found that the minimal energy technique always gives an accurate, convergent, and stable solution with an increasing accuracy as the number of discretizations increases.

18 130 D. B. INGHAM AND Y. YUAN If the thermal conductivity is constant, then in order to solve the heat conduction equation the value of this quantity is required, and this requirement is transmitted into this problem in that we require to fix at one point the value of /(T). Mathematically this is seen from the homogeneous system of equations (3.9), for which one condition is required in order to obtain a unique solution. If linear, quadratic, or even polynomial functions are used with this extra condition supplied, then satisfactory results may be obtained. However, it is interesting to note that when using piecewise quadratic functions the results obtained by fixing only one value of j(t) frequently does not give very accurate results, but if two points are fixed then accurate solutions are always obtained. The mathematical reasons for this are not clear and further mathematical analysis on the problem is being performed. One possible explanation is that when using piecewise quadratic functions the function/(t) is not only continuous but has continuous derivatives everywhere, and it may therefore be postulated that f(t) and df/dt should be specified at one point in space. In all the cases considered where two conditions are specified on f(t), very accurate results may be obtained (see Example 5.2). Numerous calculations have been performed, and in all cases where we know the analytical solution we have been able to obtain/and Tto within about 0 1%. We have also investigated the effects of introducing a small perturbation into the boundary conditions and again we have found that the method is always stable. It is important to observe that when the minimization problem is solved numerically a good starting guess is not important, and in all the examples presented in this paper the starting guesses for all the variables were set to be either 0 or 1. Further, since the minimization problem (3.7) is in the form of a positive-definite quadratic form, there is a unique minimum solution. Therefore a poor starting guess will not result in a local minimum solution being obtained. Further, one might expect that there would be many conductivities that would produce a good fit to a finite number of interior measurements, but the uniqueness of the minimization process gives one confidence that probably this may not be the situation. Clearly a more rigorous mathematical analysis of the problem is now required. In practice the temperature can only be measured at a few points inside the body, typically at less than of the order of 10 points. In this case the piecewise function approach will fail as there is not sufficient information inside the body. Therefore in this situation we suggest using a simple polynomial approximation for f(t). As we discussed in Example 4.2, the fixed point T o may affect the results for f{t) and T, but the accuracy is the same for any point being fixed. Frequently, the engineer is provided with only three to five temperature measurements inside the body from which he is expected to find the temperature variation of the thermal conductivity. Fortunately the thermal conductivities of most materials are a linear function of the temperature over a large temperature range (see e.g. Jacob, 1949). Therefore we may postulate that the linear approach, as given in Example 4.1, will give a good engineering solution in the majority of practical cases and this formulation will be of the most use to the practising engineer.

19 A NONLINEAR INVERSE PROBLEM IN HEAT TRANSFER 131 In the light of the numerical results obtained, we conclude that this inverse problem is not well-posed since the direct method of solution does not give accurate results. Further, the governing matrix is not nearly singular and hence the problem is not ill-conditioned. Clearly a detailed mathematical investigation into this problem would now be very appropriate. Acknowledgement The authors would like to thank the Sino-British Friendship Scholarship Scheme for the financial support of Y. Yuan. REFERENCES BECK, J. V., BLACKWELL, B., & ST. CLAIR, C. R Inverse Heat Conduction. New York: Wiley-Interscience. BIALECKI, R., & NOWAK, A. J Boundary value problems in heat conduction with nonlinear material and boundary conditions. Appl. Math. Modelling 5, BREBBIA, C. A. (ed.) 1989 Proc. 12th Int. Conf. Boundary Element Methods in Engineering. Berlin: Springer. CANNON, J. R Determination of the unknown coefficient k(u) in the equation V k(u) VM = 0 from overspecified boundary data. /. Math. Anal. Appl. 18, CANNON, J. R., & DUCHATEAU, P Determining unknown coefficients in a nonlinear conduction problem. SIAM J. Appl. Math. 24, CARRERA, J., & NEUMAN, S. P Estimation of aquifer parameters under transient and steady state conditions, Parts 1, 2, and 3. Water Resources Res. 22, CHEN, W. H., & SEINFELD, J. H Estimation of spatially varying parameters in partial differential equations. Int. J. Control 15, FALK, R. S., & MONK, P. B Logarithmic convexity for discrete harmonic functions and the approximation of the Cauchy problem for Poisson's equations. Math. Comp. 47, FLETCHER, R Practical Methods of Optimization, Vol. 2, Contrained Optimization. New York: Wiley. GILBARG, D., & TRUDINGER, N. S Elliptic Partial Differential Equations of Second Order, 2nd edn., Berlin: Springer. GILL, P. E., MURRAY, W., & WRIGHT, M. H Practical Optimization. London: Academic Press. HAN, H The finite element method in a family of improperly posed problems. Math. Comp. 38, INGHAM, D. B., HEGGS, P. J., & MANZOOR, M Boundary integral equation solution of nonlinear plane potential problems. IMA J. Numer. Anal. 1, INGHAM, D. B., & KELMANSON, M. A Boundary Integral Equation Analysis of Singular, Potential and Biharmonic Problems, Lecture Notes in Engineering 7. Berlin: Springer. INGHAM, D. B., RITCHIE, J. A., & TAYLOR, C. M. 1991a The linear boundary element solution of Lalpace's equation with Dirichlet boundary conditions. Comput. Math. Applic. IS, INGHAM, D. B., YUAN, Y., & HAN, H. 1991b The boundary element method for an improperly posed problem. IMA. J. Appl. Math. 47, JACOB, M Heat Trasfer, Vol. 1. New York: Wiley. JASWON, M. A., & SYMM, G. T Integral Equation Methods in Potential Theory and Elastostatics. London: Academic Press.

20 132 D. B. INGHAM AND Y. YUAN KHADER, M. S Heat conduction with temperature dependent thermal conductivity. Paper 80-HT-4, National Heat Transfer Conf., ASME, Orlando, Florida. KHADER, M. S., & HANNA, M. C An iterative boundary numerical solution for general steady heat conduction problems. Trans. ASME: J. Heat. Transfer 103, MANZOOR, M Heat Flow Through Extended Surface Heat Exchanges, Lecture Notes in Engineering 5. Berlin: Springer.