Boundary element method for an improperly posed problem in unsteady heat conduction D. Lesnic, L. Elliott & D.B. Ingham Department of Applied Mathematical Studies, University of Leeds, Leeds LS2 9JT, UK Abstract In this one-dimensional, unsteady, inverse heat conduction problem (IHCP) the boundary conditions are enforced at time t = 0, at x = d (0 < d 3 1) and x = 1 and the solution is sought using the Boundary Element Method (BEM). Because this is an improperly posed problem the direct method cannot be applied and special corrective procedures, such as least squares and minimal energy methods, have been introduced. From all the numerical results performed it has been found that both the least squares and the minimal energy methods give good agreement with the analytical solutions provided that the known boundary condition is extended over a suitable additional time interval such that accurate results can be obtained over the complete earlier time domain. However, when a sufficient amount of noise is included in the data then the least squares method produced substantially less accurate and oscillatory results. In these cases, the minimal energy technique overcomes these problems by minimizing the kinetic energy functional. 1 Introduction Inverse heat conduction problems (IHCP) arise in many heat transfer situations when experimental difficulties are uncounted in measuring or producing the appropriate boundary conditions. In practice it is often difficult to measure the surface temperature or the heat flux at the surface of a solid as a function of time. Sometimes it is easier to record the temperature at one or more interior locations within the solid. This is only one of the many examples of inverse heat conduction problems discussed in the excellent review by Beck et al. [11. In this study the solution of the one-dimensional, unsteady linear heat conduction equation in a slab geometry with constant physical properties, namely
78 Boundary Element Technology is analysed. The governing eqn (1) has to be solved subject to the boundary conditions T(x,0) = T (x) for x [0,1] (2) o a q(l,t) + p T(l,t) = f(t) for t e (0,oo) (3) where T is the temperature, q = dt/dx is the heat flux, T and f are prescribed functions and a and /3 are known constants which are not simultaneously zero. In addition, temperature readings are provided at an arbitrary space location x = d (0,1], namely T(x,t) = g(t) for x = d (4) where g is a known function. Based on the formulation in eqns (l)-(4) the IHCP considered here requires the determination only of the temperature T(0,t) and the heat flux q(0,t) on the remaining part of the boundary, i.e. x = 0. In practice only a finite time domain is analysed such that the time t (0,t ], where t is an arbitrary specified final time of interest. 2 The boundary element method The BEM for the numerical solution of this class of improperly posed problems is applied since this choice of method does not require any domain discretisation. In addition, the existence of a fundamental solution for the governing differential equation enables the problem to be reformulated in an integral representation form involving only the boundary integrals. The one-dimensional fundamental solution of the heat conduction eqn (1) is of the form, see Brebbia et al. [2], ^\2 1/7 exp - 4- I J^L_ t T 1 H(t-%) (5) [47l(t-T)]^ I 4lt TJ where H is the Heaviside function and this enables the differential eqn (1) to be transformed into the following boundary integral equation i 7)(x,t) T(x,t) = f T'(,T) F(x,t;g,%) dx - f T(,T) F' dt Jo o ^0 0! T(y,0) F(x,t;y,0) dy (6) r. 'o where (x,t) [0,l]x[0,oo), primes denote differentiation with respect to the outward normal, TJ = 1 for (x,t) <E (0,l)x(0,oo), and 17 = 0.5 for x = 0, 1 or t = 0. In eqn (6), T' = q on = 1 and T' = -q on = 0. For simplicity, in the formulation of the BEM applied here constant boundary elements, i.e. the temperature and the heat flux are assumed constant over each element, are used when the integral eqn (6) is discretised. The time interval [0,t ] is divided into N elements on each boundary x = 0 and x = 1 and the space interval [0,1] into N elements. If one takes x on the boundary, at x = 0 and x = 1, then the integral eqn (6) results in a system of linear algebraic equations, namely, (1)
Boundary Element Technology 79 2N 2 Mr 1 y G(x,t; ;t,t ) T' -y E(x,t ;,t.,t.) + 7). 5 I T. L I' l'^' j-l' J J L[ I. j j-ij i uj j N 0 H(x,t;y,y ) T = 0, i = 1,2N (7) j=i ' l J-' J ^ where 77 = 0.5, 6 is the Kronecker symbol, t _^ and t. are the endpoints of an element on the boundaries x = 0 and x = 1 and t^ its midpoint, y t = 0, T, T' =-q and y, are the endpoints of an element on the boundary for j = T7N and T' = q. for j = N+1,2N are the values of the temperature T and heat flux ±q at the midpoint of the element [t,t ], T takes the value of T at the midpoint of the j-l j Oj 0 element [y._,y.], x. = 0 for i < N, x = 1 for i > N,?. = 0 for j < N,, = 1 for j > N and the influence matrices are defined as ^b ~(x.t:.t) H(t-x) dx (8) G(x,t;g;a,b) = [ r^ E(x,t; ;a,b) = F' (x,t;?,x) H(t-x) dx (9) H(x,t;c,d) = [ F(x,t;y,0) H(t) dy (10) The discretised form of the boundary condition (3) is a q. + 3 T. = f., i = N + 1.2N (11) where f = f(t ), and in order to complete the system of equations one must add further information represented by the internal measurements (4). The time interval [0,t ] on the space location x = d is further discretised into N elements with t' denoting the midpoint of the i-th element on this interval. Applying the discretised form of equation (4) into the boundary integral (6} results in another N linear equations, namely 2N 2N G(d,t';g;t,t)T' -y E(d,t';^,t,t)T. i J J-l J J L i J J-l J J J=l J=l N 0 V H(d,t'.;y.,y.) T. = 7)(d,t'. ) g., i = 1,N (12) Lt i j-l j oj i i * j-i where g = g(t'). At this stage one recalls that the application of BEM for the IHCP produced an algebraic system (7), (11) and (12) of (3N + NM linear equations with 4N unknowns. In particular, eqns (7) and (11) enable some of the unknowns, preferably the discretised temperatures T. for j = N+1,2N, i.e. on the boundary x = 1, and the heat fluxes q. for j = 1,2N, i.e. on the boundaries x = 0 and x = 1, to be expressed
80 Boundary Element Technology as a function of the remaining discretised temperatures T for j = i,n, i.e. on the boundary x = 0. These substitutions, when introduced into eqns (12), lead to a lower order system of N linear equations with N unknowns, which in a generic form can be written in the form A T = b (13) where A = (A ) for i = 1,N, j = 1,N is a matrix depending on the geometrical matrices introduced in expressions (8)-(10), b = (by) for i = IN is a known vector and T = (T ) for j = 1,N is the unknown T - j vector of the temperature on the boundary x = 0. So far, the application of the BEM and boundary conditions (2)-(4) has reduced the IHCP to the linear system of algebraic eqns (13). A necessary condition for a solution is that N ^ N. Taking N = N, then the system of eqns (13) contains N linear equations with N unknowns and, if the matrix A has an invers, then one simply has T = A~* b. However, because the IHCP (l)-(4) is improperly posed the system of eqns (13) is ill-conditioned, i.e. the matrix A is singular, and hence no direct solution is possible. Since the system of eqns (13) is ill-conditioned more information is necessary and this is achieved from the known data (4) by taking N > N and the system becomes overdetermined. Using the least-squares method, based on the minimization of the Euclidian norm inf A T - b I ^, the solution of this system is given by, see Pasquetti & Le Niliot [3] T = (A^ A)~* A^ b (14) However, it has been found that some of the results are not sufficiently accurate nor stable and convergent. Therefore we replace the exact internal condition (4) with the inequality T(d,t) - g(t) * c for t e (0,oo) (15) where e > 0 is a preassigned small quantity and then the system of eqns (11) becomes a system of N linear inequalities with N unknowns, namely, A T - b * c (16) where the value of c > 0 should be chosen as small as possible. The system of inequalities (26) will form the set of constraints which should be satisfied by an energy functional introduced in the next section. 3 Minimal Energy Method In order to improve the stability of the solution with respect to large experimental errors, the minimal energy technique has been considered. This method was developed by Han et al. [4] for the backward heat conduction problem and it has been extended to the present IHCP in this paper. A physical argument is now adopted, in an analogy with the
Boundary Element Technology 81 steady case, and consequently one minimizes the kinetic energy - f' f' I %^1'dxdT (17) ^ ^ In order to put the relation (17) into a boundary integral form the energy equation is derived by multiplying the governing eqn (1) by u(x,t) and integrating over the domain [0,l]x[0,t^], namely, _ r'f T* 9T(x,%),2 ^ ^ ^ 1 r ^(^ )dx - 1 [ f (x,0)dx (18) Jo Jo ^^ ^o * ^o where the left hand side denotes the absorbed (or released) thermal energy through both ends x = 0 and x = 1 and the kinetic energy KE(T), whilst the right hand side denotes the inner energy of the system at times t = 0 and t = t. Introducing the identity (18) into relation (17) after discretisation yields N 2N, 0 _. _ KE(T) = U - t^j T. r + I[ <y, - y^ - T <y,t,» (19) where y. is the midpoint of the i-th element on t = t^. the boundary 2N integral (6) the terms T(y.,t ), can be expressed as 2N G(y,t;^;t,t)T' -Y E(y,t;g.,t,t)T. i f j j-l j j L i f J j-l J J According to N 0 _ + V L H(y,t ;y,y ) T = n(y,t ) T(y.,t ), i = 1,N I f j-l j oj if 11 u (20) The process of substitution mentioned in section 2, when introduced into expression (20), results in a quadratic form for the kinetic energy as a function of the unknowns temperatures T = (T.) for j = Ij\J which can be written in the following generic form KE(T) = T'" Q T + q'" T + q^ (21) where Q = (QJ for i, j = T^N, q^ = (q^) for i = TIN and q^ are known matrix, vector and constant, respectively, depending on the geometry and the known boundary conditions (2) and (3). The minimal energy method now reduces to finding the temperature vector T which minimizes the kinetic energy (21), subject to the linear constraints of the type (16). This problem is solved using the NAG routine E04UCF based on the minimization of an arbitrary smooth function subject to certain constraints which may include simple bounds of variables and both linear and nonlinear constraints. 4 Numerical results and discussion As expected, the IHCP is ill-posed and therefore the direct method was found to be inappropriate since the system of eqns (13) is ill-conditioned. For illustrating various features of the numerical
82 Boundary Element Technology schemes a simple the test function is considered, namely, T(x,t) = 2t + x* (22) The function T satisfies equation (1), together with the general boundary condition at x = 1, namely a = = 1, and the internal condition is taken on the boundary, i.e. d = 1. In all the numerical calculations presented herein the mesh size was fixed at N = N = 40, N =80 and the small control quantity c was chosen to be O(10 ), so as to ensure that the results were accurate. In the many examples performed the numerical results for the surface temperature were found to agree well with the analytical solution and therefore, in this study the results are presented only for the surface heat flux which is more difficult to accurately calculate. The effect of the choice of the value of t is illustrated by solving the problem in the domain [0,l]x[0,l] and in the domain [0,l]x[0,2] with the numerical results for the heat flux on the boundary x = 0 shown in tables la and Ib, respectively. From table la it can be seen that the numerical results for the surface heat flux obtained by using the minimal energy methods are very similar to the analytical solution up to a value of approximately 0.9 t. However, inaccurate and oscillatory results in predicting the surface heat flux are strongly present when using the least squares method. However, from table Ib, where t = 2, it can be observed that the numerical results for the surface heat flux obtained by using both the least squares and minimal energy methods are sufficiently close to the analytical solution for t ^ t /2 = 1. Table la. Surface heat flux for Table Ib. Surface heat flux for t 125 0.2125 0.4125 0.6125 0.8125 0.9875 Leas t squares 7837 2096-0.24720-0.20005-0729 24566. 0 Mini ma 1 energy 9060 0353-3358 1 1 05-6665 -0. 50749 Eqn. (22) t 25 0.425 0.825 1.225 1.625 1.975 Least squares -0. 02473 0. 00033-0. 00633 0. 01867 0. 00025 0. 40556 Mini ma 1 energy 3926 0089 0354-0581 0065-0. 55306 Eqn. (22) Hence, it is concluded that if [0,t ] is the time interval of f interest, then by solving the problem on an extended time interval results which are in good agreement with the analytical solution over
Boundary Element Technology 83 the earlier time interval are obtained. At this stage, because IHCP is an improperly posed problem it is necessary to investigate the stability of the numerical schemes under discussion. Therefore some noise is introduced into both the internal and boundary data through a small perturbation function f(x,t). Here we take T (x,t) = T(x,t) + f (x,t) (23) i where the perturbation function T(x,t) is given by i i f(x,t) = %(?) /2 cos( 2?^ t - y x - ^ ) exp( -? x ) (24) and? = k^ T^, %(y) = 0 if k = 0, %(y) = 1 if k * 0 and k_ is a constant which has to be prescribed. Clearly, the values T(x,0), T(l,t), q(l,t), T(d,t) can be made as small as one wishes by increasing the value of y. However, at the surface x = 0, T(0,t) and q(0,t) are of order of unity and thus the inequality (15) contains solutions which are discontinuously dependent on the initial data and consequently the problem is ill-posed. Tables 2a and 2b show the numerical results for the surface heat flux obtained by solving the problem with t = 2 and for k = 0 and 3, respectively, and the internal condition (15) is imposed at d = 0.25. It is observed that the results for the surface heat flux differ slightly, but are substantially different from the large values of the heat flux given by the perturbed function T. Consequently, the least squares and the minimal energy methods are able to recognize the correct solution when a small amount of noise is included into the input data. Table 2a. Surface heat flux for Table 2b. Surface heat flux for k = 0. k = 3. t Least squares Mini ma i energy Eqn. (22) t Least squares Mini mal energy Eqn. (23) 25-2571 -3124 25 0. 07802 7436 41. 7000 0.225 0.425-0. 00095 0. 00005 1 002 0. 00 1 48 0.225 0.425 0. 001 16 0. 00025 1099 150.358 0162-137. 647 0.625 0. 00044 0250 0.625 0052 0258-62.5226 0.825 0222 03 1 1 0.825 0. 00227 0316 177. 544 0.975-0. 05749-9427 0.975-0. 05745-9423 -32.5121 Figure 1 shows the numerical results for the surface heat flux obtained by using the numerical methods when a large amount of noise is included into the data in comparison with the unperturbed solution (22). This large noise is simulated by taking 7 = 2n, i.e.
84 Boundary Element Technology k = V 2 in expression (24) which defines the perturbation function T. Some of the numerical results for the surface heat flux exhibit oscillatory behaviour when using the least squares method but these large oscillations are substantially reduced when using the minimal energy technique. 0.3-1 q<o,t) 0.2-0.1- - -0.1- -0.2- -0.3 I I I 1 I 0.2 0.4 0.6 0.8 1.0 Figure 1: The surface heat flux when there is a large amount of noise, y = 2n in the input data, where analytical solution, a a a the least squares method, «the minimal energy technique. REFERENCES [11 Beck, J.V., Blackwell, B. & Clair JR., C.R.St. Inverse Heat Conduction, J.Wiley-Intersc.Publ., New York, 1985. [21 Brebbia, C.A., Telles, J.C.F. & Wrobel, L.C. Boundary Element Techniques, Springer-Verlag, New York, 1984. [31 Pasquetti, R. & Le Niliot, C. Boundary element approach for inverse heat conduction problems: application to a bidimensional transient numerical experiment, Num.Heat Transfer, 1991, 20, 169-189. [41 Han, H., Ingham, D.B. & Yuan, Y. The boundary-element method for the solution of the backward heat conduction equation, submitted to IMA J.Appl.Math., 1993.