Entropy-Constrained Topology Optimization for Thermoelastic Wear Resistance

Frictional mechanical structures present a demanding design problem because their performance depends on stiffness and load transfer as well as on surface degradation, heat generation, and irreversible energy dissipation at moving interfaces. Components exposed to sliding contact must often satisfy competing requirements: they must remain mechanically efficient, use material economically, and resist wear in localized surface regions where contact pressure, frictional work, and temperature rise are concentrated. Conventional topology optimization has become a mature tool for distributing material within a design domain, yet the problems most commonly addressed are compliance, frequency response, heat transfer, and stress.  A structural layout that is optimal for stiffness may not be optimal for a frictional surface whose degradation is driven by local energy dissipation. At the same time, direct treatment of wear in topology optimization is complicated by the need for a physically meaningful scalar measure that can be differentiated with respect to design variables and used to guide the optimization. In a recently published research paper in Thin-Walled Structures Dr. Guikai Guo, Jie Lei, Jichang Yang, and Professor Huanhuan Gao from Jilin University developed a SIMP-based thermoelastic topology optimization method that minimizes structural compliance while limiting material use and wear-related entropy generation at the sliding surface. Its technically distinct element is the use of the degradation entropy generation theorem to represent mechanical wear rate through frictional entropy production. They derived finite element formulations for steady-state heat conduction and static thermoelastic deformation, together with adjoint sensitivities for compliance and entropy generation rate. The new method was tested on cantilever, half MBB, and L-shaped structures under frictional heating and wear-surface loading conditions. The research team used steady-state heat conduction to compute the temperature field produced by frictional heat input, while static thermoelastic analysis accounts for the deformation and stiffness response under mechanical loading and thermally induced loads. Within the finite element framework, the global thermal conductivity matrix, thermoelastic coupling matrix, and structural stiffness matrix are assembled from element contributions. This construction matters because the design variables alter material density, and through density interpolation they also alter thermal conductivity, Young’s modulus, and the thermal stress coefficient.

The authors introduced wear through entropy generation during sliding. For the frictional contact process considered in their study, the dominant entropy contribution is taken to arise from plastic deformation work converted into heat. Under the stated assumptions of steady conditions, localized entropy generation, negligible energy transport by material loss, and complete conversion of frictional work into thermal energy inside the control volume, the entropy generation rate at the wear surface becomes a function of friction coefficient, normal force, sliding velocity, and contact temperature. With friction coefficient, normal pressure, and sliding velocity fixed in the instantaneous optimization problem, the design sensitivity enters mainly through the temperature field. A change in material distribution changes heat conduction, contact temperature, and ultimately the entropy generation rate. The authors used the SIMP material interpolation scheme and solve the resulting gradient-based problem using the method of moving asymptotes. Adjoint sensitivity analysis is derived for both the compliance objective and the entropy generation constraint, avoiding direct computation of displacement and temperature derivatives for every design variable and by formulating the entropy constraint in adjoint form, the thermodynamic wear measure becomes computationally compatible with density-based topology optimization.

The team used three benchmark structures to examine the behavior of the method: a cantilever beam, a half MBB beam, and an L-shaped beam. In the cantilever case, the wear region is placed near the upper right surface, with frictional force and heat generated from the imposed normal force and sliding velocity. When the entropy generation constraint is tightened, the optimized layout changes its material distribution near the wear region and alters the temperature field. The study reports that enforcing the constraint reduces the entropy generation rate associated with wear, while compliance increases only modestly. The design response is not a simple addition of material at the hot surface; rather, the optimizer redistributes material in a way that manages thermal transfer and local stiffness simultaneously. Additionally, they noticed as the entropy generation rate limit decreases, the topology becomes more elaborate, with added supporting members and modified internal branching. Material distribution is refined in regions where the thermal and mechanical fields interact, and the lower right region uses less material to suppress excessive heat transfer and entropy production.  There, material tends to concentrate more in vertical members, while stricter entropy constraints increase low-temperature regions and reduce high-temperature regions near the wear-related thermal field. The optimization introduces additional voids near the wear surface, adjusts supporting members near the fixed boundary, and creates load-bearing paths that reduce deformation in the wear region. They also found that increasing the filtering radius produces simpler and more uniform structures, lowers entropy production in the examined cantilever model, and raises compliance. Density penalization factors influence the clarity and continuity of the optimized layout, especially because thermal conductivity, elastic modulus, and thermal stress coefficient are interpolated separately. The friction coefficient has a direct role in both frictional force and heat source intensity; as it rises, entropy production and compliance increase, and the topology shifts toward the friction surface. Wear region size also changes the design response: when the wear surface expands in the half MBB beam, the heat-generation and load-application regions expand as well, internal trusses decrease, and the branch structure shifts toward the enlarged wear area.

The findings of Professor Huanhuan Gao  and colleagues are directly relevant to the design of mechanical components whose service performance is limited by sliding contact, frictional heating, and progressive wear. In such components, structural stiffness alone is not a sufficient design target, because the same material layout that provides efficient load transfer may also intensify thermal accumulation or entropy generation near the contact surface. The entropy-constrained topology optimization approach developed in this work offers engineers a way to treat wear resistance as an active design requirement rather than as a later material-selection or surface-treatment issue. One immediate application lies in frictional machine elements such as gears, brake pads, cams, pistons, and sliding bearings, where contact surfaces experience repeated mechanical loading and heat generation. The proposed method can guide material distribution around wear-prone regions so that stiffness is retained while entropy generation at the sliding interface is reduced. This is particularly valuable for components in which local surface degradation can shorten service life, alter contact geometry, or reduce mechanical reliability over time.

The approach also has practical value for lightweight structural design. Conventional topology optimization often removes material to satisfy a volume constraint while maintaining stiffness, but frictional components require a more careful balance. By incorporating entropy generation rate as a wear-related constraint, designers can identify layouts that do not simply minimize compliance, but also regulate temperature fields and thermal pathways associated with frictional work. The numerical examples show that this can be achieved with only a modest increase in compliance, suggesting that wear-aware stiffness optimization can be integrated into early-stage engineering design without abandoning structural efficiency. Another important application is in simulation-driven design of structures exposed to both frictional heating and mechanical loading. The method provides a finite-element framework in which heat conduction, thermoelastic deformation, frictional loading, and degradation-related entropy production are evaluated together. This allows engineers to compare design alternatives under controlled assumptions before manufacturing or testing physical prototypes. Parameter studies on filtering radius, interpolation factors, friction coefficient, and wear-region size further show how operating conditions and modeling choices influence optimized layouts. Finally, for engineering practice, the main value is that wear resistance can be brought into the topology optimization stage itself. Components exposed to sliding contact can therefore be shaped for stiffness, material economy, and reduced entropy generation at critical wear surfaces.