## An optimality criterion for sizing members of heated structures with temperature constraints |

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20-element solution aluminum bars area distribution boundary conditions closed-form solution combined plate-bar problem conduction matrix constraint point constraint temperature constraints at points convective heat loss Converged value convergence parameter COOLING FIN PROBLEM design of mechanically design variables displacement constraints equation A1 existing optimality criterion figure 17 finite-element heat conduction heat sink heat-transfer matrix heated structures Iteration histories iterations required ith bar element jth plate element Langley Research Center load vector mathematical programing technique mechanical and thermal mechanically loaded structures MEMBERS OF HEATED minimum fin volume MINIMUM VOLUME minimum-mass design minimum-volume solution Multiple constraints multiple temperature constraints Narayanaswami NASA nodal temperatures nondimensional number of elements number of iterations percent plate volume points c1 PpSj references 9 resizing formulas resizing process shown in figure Single constraint single temperature constraint SIZING MEMBERS square plate step-size parameter temperature distribution thermal conductivities thermal finite elements thermal optimality criterion titanium plate U.S. Customary Units