Mixed variable optimization of the number and composition of heat intercepts in a thermal insulation system Michael Kokkolaras , Charles Audet y, and John E. Dennis, Jr.
z
fkokkos,charlesa,
[email protected] Rice University, 6100 Main, Houston, Texas 77005-1892 June 22, 2000
Key Words: Optimization, thermal insulation, heat intercepts, categorical variables,
mixed variable programming (MVP), pattern search algorithm.
Abstract: In the literature, thermal insulation systems with a xed number of shields used
as heat intercepts have been optimized with respect to the temperatures and locations of the shields. The number of shields and the type of insulators that surround them were chosen by parametric studies. This was because such categorical variables, which refer to a particular choice from a list of \categories", could only be treated as xed parameters by the optimization methods employed. Here, we show that advances in optimization algorithms make it possible to optimize directly with respect to mixtures of continuous and categorical decision variables. We report mixed variable results for two models from the engineering literature, and in one case, we obtain a 65% reduction in the objective function over the previously published result. The main purpose of this paper is to show that the mixed variable optimization algorithms can be applied pro tably to a broad class of optimization problems in engineering that could not be solved with earlier methods.
Research Scholar, Department of Mechanical Engineering and Materials Science Research Scientist, Department of Computational and Applied Mathematics z Noah Harding Professor, Department of Computational and Applied Mathematics |||||||||||||||||||||||||||||||||||||||||||||| M. Kokkolaras, C. Audet and J.E. Dennis Jr. (2000), Mixed variable optimization of the number and composition of heat intercepts in a thermal insulation system, TR00-21, Department of Computational & Applied Mathematics, Rice University, Houston.
y
1 Introduction Thermal insulation systems are based on shields used as heat intercepts in order to minimize the heat ow from a hot to a cold surface. The design con guration of a multi-shielded thermal insulation system is de ned by the number of shields, their locations and temperatures, and the choices of the insulators that are placed between each pair of shields. Hilal and Boom [8] considered cryogenic engineering applications in which either loadbearing insulators are required in the construction of dewars, or mechanical struts are necessary in the design of solenoids for superconducting magnetic energy storage systems. In such applications, cooled shields, kept at a given temperature by refrigeration, are introduced at certain locations in order to intercept the heat ow. Hilal and Boom formulated an objective function based on a power minimization principle to optimize the con guration of a thermal insulation with respect to the locations and the temperatures of the shields. Their gradient-based optimization algorithm could not handle the categorical variables specifying the number of shields and the insulators placed between each pair. They chose these categorical variables by taking the best values they found as they xed the number of shields and the choice of insulators to sensible selected values and then solved for the resulting optimal temperatures and shield locations. Speci cally, they solved the nonlinear programs obtained by xing the number of shields n to 1; 2 and 3. Moreover, only one insulator type was considered between every pair of shields for each run. After obtaining the minimum refrigeration power for a given type of insulator and a xed number of shields, they performed a system cost optimization study with respect to the length L of the solenoid. Hilal and Eyssa [9] considered a variable cross section for the mechanical supports and reported lower optimal power values than those obtained with a constant cross section. Chato and Khodadadi [7] considered applications in which mechanical supports between the cold and hot ends of the thermal insulation system are optional. They based their optimization approach on a minimum energy production principle formulated by Bejan [6] and they determined optimal locations and temperatures for xed numbers of shields. Instead of considering speci c types of insulators explicitly, they assumed a general function for the e ective thermal conductivity and they varied its parameters. Cryogenic systems of space borne magnets have been considered in more recent publications. In such applications, it is quite important to maximize the insulation eciency of the system so that the available liquid helium used for cooling the shields evaporates with a minimum rate during the period of the mission. Musicki, Hilal, and McIntosh [13] optimized the inlet temperatures and ow rates of the liquid helium for a speci ed number of shields and insulator thicknesses. Yamaguchi, Ohmori and Yamamoto [16] studied the e ect of the number of shields and the e ect of the multi-layer density on the temperature distribution over the shields and the resulting heat losses. Li et al. [12] considered the use 1
of liquid nitrogen and neon instead of liquid helium. All of the above optimization approaches share an important limitation: Categorical variables, such as the number of shields or the type of insulators surrounding them, are not treated as optimization variables but as parameters due to the lack of appropriate optimization algorithms. Categorical optimization variables are represented here by discrete real values, but they di er from ordinary discrete optimization variables in a fundamental way. We call a discrete optimization variable categorical if the objective function or the constraints can not be evaluated unless the variable has one of a prescribed nite set of values, and thus continuous relaxations are not available. The key issue is that categorical variables can not be treated as continuous variables with a side constraint that they be discrete at the solution. For example, the models used for the insulation systems cannot return an output value for an input value of, say, 1:5 shields or for an insulator that is a mixture of steel and aluminum. This preclusion of continuous relaxation techniques excludes popular branch and bound or branch and cut techniques for mixed integer programming. Hence, we use the term mixed variable programming (MVP) for the methodology that enables the optimization of thermal insulation systems with respect to both continuous and categorical variables. The associated MVP optimization algorithm (a pattern search method that allows bound constraints) is described in [2]. The presence of these categorical variables in the optimization process allows a decrease in the objective function value by as much as 65%. The paper is structured as follows. In the next section, we describe the classical model of thermal insulation systems, and then in Section 3 we present the multi-shield model. Section 4 outlines the principal steps of the mixed variable programming algorithm, and in Section 5 we discuss the results of applying MVP to the mixed variable multi-shield design problem. Finally, we conclude with some remarks proposing the use of the mixed variable programming algorithm for a broad class of previously intractable engineering optimization problems.
2 Classical Model of Thermal Insulation Systems The con guration of a thermal insulation system is de ned by the number of shields used as heat intercepts, their location and temperature, and the types of insulators that surround them. The optimal con guration has been studied traditionally through the following optimization problem min (1) x;T f (x; T); where x = [x1; x2; : : : ; xn]T with xi, i = 1; 2; : : : ; n, being the location of the i-th shield, 2
T = [T1; T2; : : : ; Tn]T with Ti, i = 1; 2; : : : ; n, being the temperature of the i-th shield, f : n) rational matrix whose nonnegative linear combination of the columns spans the whole continuous space N if i i1 < I = fI = [I1; I2; : : :; In+1]T : Ii = > T if i1 < i i2 or i3 < i :E if i2 < i i3; 1 i n + 1; 0 i1 i2 i3 n + 1g: 9
Therefore, this extra information about the nature of the problem leads to the assumption that any solution is composed of such sequences. (b)
(a)
0.6 Tefon Nylon Epoxy−fiberglass (in plane cloth) Epoxy−fiberglass (normal cloth) 304 Stainless steel 6063−T5 Aluminum Low−carbon steel
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Figure 4: Thermal conductivity versus temperature for (a) all insulators and (b) the \better" four insulators. The set of neighboring solutions considered by the poll step include those where
any of the existing shields and the insulator to its left are removed, a new shield together with an insulator to its right are added, the type of insulator between two shields is changed. Instead of explicitly writing out the tedious rules of de ning the set of neighbors for any con guration, the exibility of the method is illustrated on a single example. Table 1 displays a solution with four shields and the 23 poll points that must be explored before declaring an iteration unsuccessful. Only the entries that di er from the current solution are displayed. The rst spanning set, with a mesh size parameter k = 5 generates 9 polling points that di er only in the continuous variables. The other poll points have to be de ned by the user who has an understanding of the model. The set of neighbors contains four solutions where a shield and an insulator are removed and the thickness of the insulator is distributed among the others. It also contains the 5 trial points associated to adding a shield and an insulator, and 5 more for replacing the type of an insulator by another. Of course, the variable I for all these poll points belongs to the set I n+1. The speed of the algorithm and the quality of the solution produced by it depends on the user-de ned set of neighbors. For example, if one does not realize that the type of insulators of interesting solutions belong to I , then the trial points associated with modifying an insulator could contain all the combinations of insulators that di er from the incumbent solution in any one component. In the example above, there would be ten poll 10
Table 1: Example of a poll set. x T [K] L [%] Current incumbent solution 4 [20 20 20 20] [10 20 40 80] [ Mesh neighbors [25 20 20 20] [20 25 20 20] [20 20 25 20] [20 20 20 25] [15 20 40 80] [10 25 40 80] [10 20 45 80] [10 20 40 85] [15 15 15 15] [5 15 35 75]
n
I
T T EET
]
3 3 3 3 5 5 5 5 5
Remove a shield [20 40 80] [10 40 80] [10 20 80] [10 20 40] Add a shield [10 10 20 20 20] [5 10 20 40 80] [20 10 10 20 20] [10 15 20 40 80] [20 20 10 10 20] [10 20 30 40 80] [20 20 20 10 10] [10 20 40 60 80] [20 20 20 20 10] [10 20 40 80 190] Change an insulator type [25 25 25] [25 25 25] [25 25 25] [25 25 25]
[ [ [ [
T EET T EET T T ET T T ET
] ] ] ]
[ [ [ [ [
T T T EET T T T EET
T T EEET T T EEET T T EET T
[ [ [ [ [
N T EET T EEET T T T ET T T ET T
] ] ] ] ]
] ] ] ] ]
T T EEE
points associated to a change of insulator type instead of ve. For larger n, the algorithm would require signi cantly more function evaluations, and not produce a better solution.
5 Results and Discussion In this section, we present numerical results for our Matlab 5.3.0 implementation of the MVP algorithm described above to both mathematical models (objective functions (14) and (15)). First, we reproduce and compare results that were reported previously in the literature. Then, we report new results in Sections 5.2 - 5.5.
5.1 Optimization with Fixed Insulators and Number of Shields In Table 2, we compare the results presented by Hilal and Boom when using 304 stainless steel for the entire mechanical support and n = 1; 2 and 3 shields to the results obtained by the MVP algorithm when forced to use the same insulator and respective maximum number of shields{. The initial guess for the location of the shield Lx1 is 50%, i.e., it is initially positioned at half distance from the hot (TH = 300 K) and cold (TC = 4:2 K) surfaces. The initial guess for the temperature is 50 K. The thermodynamic cycle eciency We emphasize the fact that was xed to 1, 2, and 3 in Hilal and Boom's method, while the MVP algorithm was initiated with = 1 and converged to 1, 2, and 3 for the three test cases, respectively. {
n
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coecient is a function of the temperature as follows 8 > < 2:5 if T 71 K C=> 4 (16) if 71 K > T > 4:2 K : :5 if T 4:2 K All objective function values are nondimensionalized with respect to unit area and length. Table 2: Optimum temperatures, locations, and refrigeration power when using 304 stainless steel for the entire strut; TC = 4.2 K. W Algorithm n T1, [K] T2, [K], T3, [K] Lx1 [%] Lx2 [%] Lx3 [%] Lx4 [%] PL A , [ cm ] H & B 1 39.7 33.8 66.2 1927 MVP 1 36.2 32.9 67.1 1910 H & B 2 21.5 81.9 18.8 33.5 47.6 1134 MVP 2 18.3 71 18.5 36.2 45.3 1077 H & B 3 11.7 28.7 72.4 9.3 14.7 28.1 47.9 966 MVP 3 10.9 28.3 72.8 9.5 15.2 28.4 46.9 963.6 Hilal and Boom performed their computations in the late seventies. In this regard, and in order to have a common comparison basis, we recalculated the objective function values based on their reported variable values using our function evaluation routine. The objective function values they report (i.e., computed by their function evaluation routine) are 1781, 1265, and 948.7 for n = 1, 2, and 3, respectively, in Table 2 and 142, 94.5, and 71.9 for n = 1, 2, and 3, respectively, in Table 3. Table 3 tabulates the results of Hilal and Boom when using epoxy- berglass (in plane cloth) for the entire strut and n = 1; 2 and 3 shields, and the results obtained by the MVP algorithm when forced to use the same insulator and number of shields. The cold surface temperature was changed to TC = 1:8 K. Due to the presence of local optima, it is possible to converge to a di erent solution if the algorithm is initiated at a di erent starting point. For example, it has been observed that the algorithm produces di erent results when the initial guess for the temperature is 150 K. In order to investigate the existence of local optima, we plot the objective function for n = 1 shield over the possible temperature and location ranges. When looking at the left of Figure 5, the objective function looks quite smooth. However, zooming in reveals (at the right of Figure 5) the presence of a local optimum. It is clear that the local optimum is associated with the discontinuity at T =71 K caused by the discontinuous change of the thermodynamic cycle eciency C at this point. This points to an interesting compromise issue between the two mathematical models. The entropy production rate 12
Table 3: Optimum temperatures, locations, and refrigeration power when using epoxy berglass (in plane cloth) for the entire strut; TC = 1.8 K. Algorithm H&B MVP H&B MVP H&B MVP
n 1 1 2 2 3 3
T1, [K] 29.6 21.7 11.1 10.5 5.28 5.8
T2, [K], T3, [K] 70.3 64.1 20 71.7 19.9 77.4
x1 [%] x2 [%] x3 [%] x4 [%] PL , [ W ] L
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cm 145 140 91.9 89.7 68.6 66
objective function does not take into account the thermodynamic cycle eciencies and does not exhibit any discontinuities; mathematically the entropy model is smoother since it will not yield any local optima. However, the power model is physically more accurate since it takes into account the thermodynamic eciency as a function of temperature. With the initial guess of the temperature being T1 = 150 K, the MVP algorithm converges to the local optimum caused by the discontinuity. When we change the initial guess for T1 to 50 K, the MVP algorithm yields the results displayed in Table 2. Note that Hilal and Boom did not comment on either the initial guesses or the convergence properties of the gradient-based optimization algorithm they used. 3500
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Figure 5: The refrigeration power objective function for n = 1 shield when using 304 stainless steel for the entire strut. An alternative to switching starting points for increasing the likelihood of converging to the global optimum is to incorporate a more sophisticated search strategy and/or increase the local mesh size parameter k at successful iterations. 13
5.2 Categorical Variable Optimization Before proceeding with further numerical results, we rst need to address some more computational implementation issues: As mentioned in Section 4.2, the search step is minimal; all results are obtained mostly by polling. In practice, the algorithm terminates when the following criterion is satis ed k < ; (17) where k is the mesh size parameter at the k-th iteration and is some nonnegative small tolerance. When condition (17) is met for a small , the current solution satis es optimality conditions on a ne mesh; a local optima is probably found or nearby. For the speci c applications of this paper, we start with the initial mesh size parameter 0 = 10, and then re ne the mesh according to the rule (18) k+1 = 2`k when the `-th local mesh optimizer is found. This means that when the rst local mesh optimizer is found, the mesh size parameter decreases to 5, then when another is found it drops to 1:25 and nally to 0:15625. Since a nal mesh size parameter of 0:15625 is sucient for practical engineering purposes, we stop at ` = 4, and accept the associated local mesh optimizer as our nal solution. The parameter that triggers the extended poll step is set to 1% and 0:1% of the incumbent objective function value for the refrigeration power and entropy production rate models, respectively. The following initial guess and parameter values are used for all calculations in the remaining sections: n = 1, Lx1 = 50%, I1 = N , I2 = T , T1 = 150 K, TC = 4:2 K, and TH = 300 K. We now apply the MVP algorithm to the optimization problem with the maximum number of shields being quite large (for practical purposes we set nmax = 100). In addition, any appropriate combination of the following three insulators can be chosen for the spaces between the heat intercepts: (N)ylon, (T)e on, or (E)poxy- berglass (normal cloth). For later comparison, the available information on the nature of the problem is not exploited when de ning the neighbors for the poll step, i.e., it is not assumed that the solutions are composed insulators in the set I . We depict the evolution of the objective function versus the number of function evaluations in Figure 6. It can be seen on the left of Figure 6 that the minimum refrigeration power is approached rapidly in the early stages of the algorithm; this is typical behavior for derivative-free algorithms. The numbers next to the objective function values curve indicate the number of shields at local mesh minimizers, i.e., the local optimum number of shields for k = 10; 5; 1:25, and 0:15625 was n = 7; 11; 23, and 29, respectively. The drop in the objective function can be examined better on the right of Figure 6, where the 14
rst few hundred function evaluations are excluded. The numbers in parentheses next to the objective function values curve on the right of Figure 6 indicate that the number of shields was increased and then decreased by the algorithm, which shows that shields are not only added during the optimization process, but also removed. The improvement in the objective function value after the rst 20000 evaluations is marginal. 27
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Figure 6: Evolution of the objective function for minimum refrigeration power when the set of neighbors does not take into account the extra information on the problem. The con guration of the thermal insulation system appears in Table 4; denotes the number of function evaluations. It can be seen that combining di erent insulators decreases the objective function value by 65% (compared to using only epoxy- berglass for the entire strut and only three shields). Observe that the selection of insulators is optimal with respect to their thermal conductivity over the chosen temperature intervals, i.e., the solution produced by the algorithm is composed of the speci ed sequence of insulators, even if this was not imposed. In fact, all solutions produced after the tenth function evaluation are composed of the speci ed sequence of insulators. The speed of convergence can be improved by exploiting the available information on the nature of the problem when de ning the neighbors for the poll step, i.e., assuming that the variable I belongs to the set I . These results are tabulated in Table 5. The total number of function evaluations is signi cantly smaller (by approximately 90%) than when the extra information on the problem is not taken into account. However, the gain in function evaluations compensates for a small loss (approximately 5.7%) in the objective function value. The solution is di erent, a fact that points to the presence of local optima (both are local mesh optimizers for the same mesh size parameter). Di erent starting points and/or di erent de nitions of the set of neighbors may lead the algorithm to di erent local solutions. It is also clear that obtaining an improved solution is correlated to higher 15
Table 4: Con guration of the thermal insulation system for minimum refrigeration power when the set of neighbors does not take into account the extra information on the problem. W = 29, PL A = 23.511151 [ cm ], = 221399 x L ,[%] = [ 0.3125 0.8099 0.8264 1.6139 2.4008 1.9348 1.7915 1.8013 1.6609 3.2293 3.0639 2.9001 1.3368 1.3363 1.3307 1.3258 2.0807 2.8253 4.5312 4.8438 7.3438 10.4688 0.7812 5.6218 4.6874 2.5059 2.6619 2.3492 2.3490 19.2751] T,[K] = [4.2188 4.6875 5.1562 6.0938 7.5 8.75 10. 11.25 12.5 15. 17.5 20. 21.25 22.5 23.75 25. 27.0312 30. 35.1562 41.0938 51.5625 70.9375 71.0938 88.5938 106.5564 117.6415 130.4517 142.6425 155.3019] I = [N N N N N N NN NN NN NN NNN NN NT EEEEEEEET ] n
computational expenses. However, it has to be emphasized that the current, preliminary implementation of the MVP algorithm is intended to be as simple as possible and does not include major cost saving features, like a non trivial search strategy, the ability of increasing the mesh size parameter k at successful iterations, or avoiding the function evaluation of same points in di erent iterations. In this regard, the reader should look at the number of necessary function evaluations in a qualitative and not quantitative manner. Table 5: Optimum con guration of the thermal insulation system for minimumrefrigeration power when de ning the neighbors by taking into account available information on the problem. W = 20, PL A = 24.850771 [ cm ], = 18310 x L ,[%] = [0.3125 1.4062 1.4062 2.204 2.0734 1.7786 3.6564 3.1802 2.8699 4.1221 3.8111 3.9688 5.6886 5.8195 19.2536 9.7443 5.0536 3.0194 1.4543 1.452] T,[K] = [4.2188 5. 5.7812 7.1875 8.5938 9.8438 12.5 15. 17.5 21.25 25. 29.2126 35.7697 43.3896 71.0131 105.2367 128.8392 144.9409 153.0728 161.3605] I = [N N N N N N NNN NN NN T EEEEEET ] n
All runs in the following sections use the information in the set I for the poll step.
5.3 Limiting the Number of Shields The results in the previous section suggest that after some point, additional shields do not improve the objective function value signi cantly. Moreover, if the mesh size parameter k is allowed to get smaller than 0.15625, the algorithm will converge to larger numbers of shields n. In addition, as n increases, so does the computational work since the problem 16
size increases. In this regard, one would like to be able to terminate the optimization process sooner, but with a good solution. One way to accomplish that is by setting an upper bound on the number of shields to be used, i.e., by using smaller values for nmax. Figure 7 displays the associated objective function plot for nmax = 10. 300
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Figure 7: Evolution of the objective function for minimum refrigeration power when the set of neighbors takes into account the extra information on the problem; nmax = 10. The optimal associated con guration is documented in Table 6. It can be seen that, when hitting the upper bound for n, the MVP algorithm converges in 87% less function evaluations with a loss in optimality at the order of 2%, compared to the run with information detailed in Table 5. Table 6: Optimum con guration of the thermal insulation system for minimumrefrigeration power when the set of neighbors takes into account the extra information on the problem; nmax = 10. W ], = 2350 = 25.363362 [ cm x L ,[%] = [0.3125 4.8724 4.4201 6.3014 5.8519 8.0527 6.1672 5.0552 20.5262 18.1773 20.2631] T,[K] = [4.2188 7.0312 10.0000 14.8438 20.0000 27.9688 34.9899 41.5262 71.0443 146.3616] I = [N N N N N N NT EET ] n=10, PL A
5.4 Adding an Extra Cost Term to the Objective Function Keeping the number of shields n reasonably low, without sacri cing optimality substantially, can alternatively be accomplished by including an extra cost term in the objective 17
function such that a shield is not added unless a certain percentage gain in the objective function value can be achieved. Similarly, cost or weight (especially for use in space missions) functions can be de ned for speci c applications in the future in order to keep the number of shields low. Figure 8 summarizes pictorially the results obtained for di erent extra cost term coef cients. On the left, middle, and right of Figure 8, a shield is added only if the objective function is reduced by at least 0.1%, 0.5%, or 1%, respectively; the rst few function evaluations are excluded to facilitate the examination of the objective function behavior. It is clear that a high value for the extra cost term will yield a lower number of shields and a higher objective function value. Note that the extra cost term coecient cannot be larger than the extended poll triggering coecient . γ = 1%
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Figure 8: Evolution of the objective function for minimum refrigeration power when the set of neighbors takes into account the extra information on the problem and including an extra cost term of 0.1% (left), 0.5% (middle), and 1% (right) in the objective function; nmax = 100. Losses in optimality and gains in computational work with respect to not including the extra cost term and using the polling information are tabulated in Table 7; denotes the extra cost term coecient. The optimum con gurations are reported in Table 8. Table 7: Losses in optimality and gains in computational work when including an extra cost term in the objective function for minimum refrigeration power. W
, [%] n f = PL Loss in f , [%] Gain in , [%] A , [ cm ] 0 20 24.850771 18310 0.1 14 25.432415 15912 2.3 13 0.5 9 27.060122 4206 8.9 77 1 7 27.403100 11513 10.3 37
18
Table 8: Optimum con guration of the thermal insulation system for minimumrefrigeration power when the set of neighbors takes into account the extra information on the problem and including an extra cost term in the objective function. W = 0.1%, n = 14, PL A = 25.432415 [ cm ], = 15912 x L ,[%] = [0.3125 2.8125 2.5000 3.9062 3.2949 3.3054 6.9060 5.0458 5.0494 5.2155 5.5634 17.2878 6.7941 6.1267 25.8798] T, [K] = [4.2188 5.7812 7.3438 10.0000 12.5000 15.1562 21.2500 26.2500 31.8670 38.2664 45.6513 71.0062 94.1206 120.9594] I = [N N N N N NN NN NN EEET ] W
= 0.5%, n=9, PL A = 27.060122 [ cm ], = 4206 x L ,[%] = [0.3125 5.3125 3.7500 5.9375 5.6250 5.4688 8.2812 9.2188 27.9688 28.1249] T, [K] = [4.2188 7.3438 10.0000 14.6875 19.6875 25.1562 34.5312 47.1875 108.5938] I = [N N N N N N NT ET ] W
= 1%, n=7, PL A = 27.403100 [ cm ], = 11513 x L ,[%] = [0.3125 6.4478 7.7061 7.3496 7.7470 11.9552 30.9685 27.5133] T, [K] = [4.2188 8.0677 13.8063 20.2572 28.4304 43.4259 110.6041] I = [N N N N N NET ]
5.5 Minimizing the Entropy Production Rate In all the previous sections, the refrigeration power was used as the mathematical model. In this section, we apply the MVP algorithm to the objective function used by Chato and Khodadadi [7]. They presented results obtained by varying the coecients and the exponents of equation (12), so we cannot compare them directly to the results obtained by MVP. In addition, rather than numbers or tabulated data, Chato and Khodadadi reported mainly nondimensional design curves. Despite this, we present the MVP results in this section with the hope that they can be used as a reference for future thermal insulation optimization studies of a more general nature. As mentioned before, the media that surround the heat intercepts do not necessarily have to be insulators or more complicated multi-layer insulation components; for example, they can be vacua. As long as the MVP algorithm is provided with appropriate e ective thermal conductivity information, it can indicate the optimum combination of media for a general thermal insulation system. Figure 9 illustrates the behavior of the objective function during the optimization process when de ning the neighbors by taking into account available information on the problem (i.e., assuming that the variable I belongs to I ): On the left and center of Figure 9, nmax is limited to 10, and on the right, nmax is set back to 100 and an extra cost term with
=0.05% is included in the objective function. 19
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0.0235
2000
4000
6000 8000 10000 12000 Number of function evaluations
14000
16000
Figure 9: Evolution of the objective function for minimum entropy production rate when the set of neighbors takes into account the extra information on the problem. The associated optimum con gurations are reported in Table 9. It can be seen that even in the presence of an extra cost term, approximately 90% more function evaluations and 8 additional shields improve the objective function by less than 1%. Table 9: Optimum con guration of the thermal insulation system for minimum entropy production rate when the set of neighbors takes into account the extra information on the problem. _ = 0.023831 [ W ], = 1684 0%, nmax = 10, n = 10, sL A K cm x ,[%] = [3.4375 3.5938 6.2500 5.7812 5.0000 5.9375 9.2188 L 15.0000 11.4062 11.7188 22.6562] T, [K] = [6.2500 8.7500 13.5938 18.7500 23.7500 30.4688 42.5000 69.8438 103.7500 153.9062] I = [N N N N N NN EEET ]
= 0.05%, nmax = 100, n = 18, s_AL = 0.023772 [ KW cm ], = 17526 x L ,[%] = [1.6083 1.7825 1.9201 1.7537 3.3098 3.0045 2.8543 2.5368 5.0329 5.6548 5.4826 7.9381 11.3938 11.2876 6.4416 1.4347 1.4330 2.6816] T,[K] = [5.1562 6.25 7.5 8.75 11.25 13.75 16.25 18.5938 23.5938 29.8438 36.6988 48.1763 70.1617 103.1837 128.6944 135.1135 141.5268 154.1850] I = [N N N N N N NNN NN T EEEEEET ]
=
6 Conclusions A recently developed mixed variable programming algorithm is used to optimize thermal insulation systems with respect to both continuous and categorical variables. The advantages of the pattern search MVP algorithm include the following: 20
Categorical variables are treated as optimization variables and not as parameters. In
this manner, important components of the optimal con guration, such as the number of shields used as heat intercepts and the types of insulator that surround them, are taken into consideration directly during the optimization process. The objective function value is reduced by as much as 65%. It is emphasized that categorical optimization variables di er from ordinary discrete ones in that the former do not qualify for continuous relaxation.
The algorithm requires from the user only a function evaluation routine (black box)
and no derivative information. Moreover, there are no continuity restrictions on the objective function; as a matter of fact, the black box is allowed to return in nite or no value at all for the objective function (for example, it returns the value +1 if the discrete variables are relaxed to continuous ones). Finally, general constraints can also be handled by a \ lter" version of the algorithm [3].
The algorithm consists of two main components: a) the search step that can employ any strategy based on available information of the problem (including none) and may accelerate convergence and/or lead to the global optimum and b) the poll step that guarantees local convergence for any initial guess. As illustrated in this paper, the poll step also should be designed based on the knowledge of the problem.
The algorithm can be applied to a broad class of optimization problems in engineering that could not be solved before due to the presence of such categorical variables.
The algorithm can be readily implemented to perform the necessary function evalua-
tions in parallel; such an implementation would a) be highly scalable and b) decrease computational time dramatically.
Although the main purpose of the paper is to promote the MVP algorithm's features and to propose it for solving complex engineering optimization problems, the considered application is not to be neglected. A number of issues have been addressed and conclusions can be drawn.
The size of the optimization problem, i.e. the number of optimization variables, is a
function of an optimization variable. Speci cally, the number of continuous variables is equal to 2n and the number of categorical variables is equal to n + 2, where n is the number of shields. The total number of variables, 3n + 2, can be quite large for a moderate amount of shields, making the nonlinear optimization problem quite challenging even for the purely continuous case. 21
Di erent types of insulators are optimal for di erent temperature ranges. It was shown that the algorithm is choosing the correct insulators.
As expected [14], the algorithm may converge to di erent local optima depending on
the initial guess. However, a good search strategy combined perhaps with a less trivial way of modifying the mesh size parameter would increase the likelihood of converging to the global optimum.
The algorithm requires much less function evaluations when available information on
the problem is exploited in de ning the set of neighbors. It is also clear that obtaining improved extrema is highly correlated with paying a higher computational price, e.g., executing the algorithm with a higher extended poll triggering parameter will require more function evaluations but yield a better solution. It is emphasized that the current, preliminary, and unsophisticated Matlab implementation of the algorithm does not feature cost saving characteristics like avoiding the function evaluation at same points in di erent iterations. In this regard, the number of function evaluations is merely a qualitative, and not quantitative, algorithm performance indicator.
The important decrease of the objective function value in the early stages of the algo-
rithm, followed by a plateau, is in accordance with the typical behavior of derivativefree methods. The termination criterion depends on the mesh size parameter. It has been observed that the addition of shields in the late stages of the algorithm reduces the objective function value only marginally. In this regard, an extra cost has been included, causing the algorithm to add a shield only if a certain percentage gain is achieved in the objective. More sophisticated cost and/or weight functions can be developed for speci c applications.
Two mathematical models used previously by other researchers have been used to
de ne alternate objective functions. The refrigeration power model is physically more accurate since it takes into account the thermodynamic cycle eciency but exhibits discontinuities that cause local optima. The entropy production rate model is smoother but inferior physically. The numerical results presented by other researchers for the power model for xed number of shields and type of insulator were reproduced by a special case of the MVP algorithm. Moreover, when the number of shields is treated as an optimization variable and di erent types of insulators are combined, the objective function value decreases dramatically. Further improvement is possible by treating the cross section area as an optimization variable. A comparison for the entropy model was not possible due to the incompatibility of the mechanism accounting for the e ective thermal conductivity. However, the results presented by the MVP algorithm are included for future reference. 22
The algorithm can be used for optimizing the con guration of any general thermal
insulation system (with or without mechanical supports and for any kind of media) as long as it is provided with e ective thermal conductivity data.
Acknowledgments Work of the second author was supported by NSERC (Natural Sciences and Engineering Research Council) fellowship PDF-207432-1998 and all three authors were supported by DOE DE-FG03-95ER25257, AFOSR F49620-98-1-0267, The Boeing Company, Sandia LG4253, Exxon-Mobil and CRPC CCR-9120008.
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