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Figure 2. Three-dimensional feasible design space of the thermoelastic rod as a function of diameter D, length L, and allowable rod temperature change ΔT_rod. Feasible designs satisfy both the yielding and Euler-buckling constraints, whereas designs outside the admissible region violate at least one structural criterion.

Figure 3. Governing structural failure modes in the thermoelastic rod design space. The map distinguishes safe designs from regions governed by Euler buckling, material yielding, or simultaneous violation of both constraints.

Figure 4. Thermoelastic structural failure maps in the D - L design space at (a) ΔT_rod = 50 K, (b) ΔT_rod = 120 K, (c) ΔT_rod = 300 K, and (d) ΔT_rod = 500 K. Each map identifies safe, buckling-controlled, yield-controlled, and combined-failure regions under the same structural assumptions and safety factors.

Figure 5. Mass-power design space and Pareto front for the feasible thermoelastic rod configurations. Feasible solutions are shown together with the non-dominated Pareto designs, illustrating the trade-off between structural mass and average mechanical power.

Figure 6. Log-log relationship between average mechanical power and rod mass along the Pareto front. The fitted power law closely follows the analytical scaling P_avg ∝ m^2/³ , confirming the buckling-controlled scaling behavior of the optimized thermoelastic rods.

Figure 7. Selection of the compromise design from the mass-power Pareto front. (a) Location of the selected compromise design in the physical mass-power space. (b) Corresponding normalized Pareto representation and distance-to-ideal selection criterion.

Figure 8. Effect of the load-transfer parameter λ on the maximum achievable average mechanical power and the corresponding optimal rod geometry. (a) Maximum average power as a function of λ, showing the numerical optimum at λ_opt = 0.473 with P_max = 0.170454 W and the reference case λ = 0.500, for which P_max = 0.166245 W. (b) Corresponding optimal rod diameter D_opt and length L_opt over the investigated λ range. The optimal diameter remains at the upper design bound of 150 mm, whereas the optimal length decreases as the buckling constraint becomes increasingly restrictive.