Multi-Objective Constructal Design for Square Heat-Generation Body with “Arrow-Shaped” High-Thermal-Conductivity Channel

: Based on the square heat-generation body (HGB) with “arrow-shaped” high-thermal-conductivity channel (HTCC) model established in the previous literature, we performed multi-objective optimization (MOO) with maximum temperature difference (MTD) minimization and entropy-generation rate (EGR) minimization as optimization objectives for its performance. Pareto frontiers with optimal set were obtained based on NSGA-II. TOPSIS, LINMAP, and Shannon entropy decision methods were used to select the optimal results in Pareto frontiers, and the deviation index was used to compare and analyze advantages and disadvantages of the optimal results for each decision method. At the same time, multi-objective constructal designs of the “arrow-shaped” HTCC were carried out through optimization of single degree of freedom (DOF), two DOF, and three DOF, respectively, and the thermal performance of the square heat-generation body under optimizations of different DOF were compared. The results show that constructal design with the MOO method can achieve the best compromise between the maximum thermal resistance and the irreversible loss of heat transfer of the square heat-generation body, thereby improving the comprehensive thermal performance of the square heat-generation body. The MOO results vary with different DOF, and optimization with increasing DOF can further improve the comprehensive thermal performance of square HGBs.


Introduction
A crucial link in the design and manufacture of chips is how to effectively avoid damage to components due to excessive local heat generation. The physical size of electronic components is becoming smaller and smaller while their power is increasing. The thermal stability and reliability of electronic devices may be greatly improved using the basic theories of heat transfer optimization to optimize their heat dissipation capacity. Among these is the entropy generation minimization theory [1][2][3][4][5][6][7], which provides optimization criteria from the perspective of evaluating and decreasing the irreversibility of heat transfer in processes and systems. Constructal theory [8][9][10][11][12][13][14][15][16][17][18] is a newly developed theory based on thermodynamic optimization research. It addresses this problem by following the idea that structure develops in the direction of better internal flow performance, provides a new solution for traditional thermodynamic optimization problems, and makes many bottlenecks that are difficult to explain and solve in theory and engineering possible to solve through new angles. Introducing these cutting-edge theories into the research on the deviation index. Feng et al. [52] performed MOO of a marine boiler by taking the EGR and the pumping power consumption as objectives, and compared the optimization results of three decision methods based on the deviation index. Feng et al. [53] performed MOO for a compound heat-dissipation channel, taking the MTD and the pumping power consumption as objectives. Sun et al. [54] performed MOO for a cooling channel with semicircular sidewall ribs, taking the MTD and the pumping power consumption as objectives. Kalkan et al. [55] performed MOO for a microchannel cooling plate.
In this paper, based on the square HGB with "arrow-shaped" HTCC model established in the previous literature [24], MOO with MTD minimization and EGR minimization as optimization objectives was performed, and Pareto frontiers with optimal set were obtained based on NSGA-II. TOPSIS, LINMAP, and Shannon entropy decision methods were used to obtain the optimal results. The deviation index [56] was used to compare and analyze the advantages and disadvantages of the optimal results under each decision method. At the same time, the multi-objective constructal designs of the "arrow-shaped" HTCC were carried out through the optimization of single DOF, two DOF, and three DOF, and the thermal performances of the square heat-generation body under the optimizations of different DOF were compared. Our results indicate that introducing MOO into constructal theory for design can improve the comprehensive thermal conductivity of HTCCs and meet the needs of engineering design. Figure 1 shows the square HGB with "arrow-shaped" HTCC model [24]. The length of the square HGB (the heat-generation rate is q , the thermal conductivity is k 0 ) is L, and an "arrow-shaped" HTCC (thermal conductivity is k c ) is arranged inside it. The heat flow generated in the area of the square HGB converges into the "arrow-shaped" HTCC, and flows out of the heat-generation area through the left side of the "arrow-shaped" HTCC (temperature is T min ). The remaining boundaries of the square HGB are adiabatic. The thermal conductivity ratio of the high-and low-thermal-conductivity materials is defined as k = k c /k 0 , and the characteristic sizes of the "arrow-shaped" HTCC are L 1 , H 1 , L 2 and H 2 respectively.  [51] performed MOO for marine condensers and compared the optimization results of single-objective optimization and three decision methods based on the deviation index. Feng et al. [52] performed MOO of a marine boiler by taking the EGR and the pumping power consumption as objectives, and compared the optimization results of three decision methods based on the deviation index. Feng et al. [53] performed MOO for a compound heat-dissipation channel, taking the MTD and the pumping power consumption as objectives. Sun et al. [54] performed MOO for a cooling channel with semi-circular sidewall ribs, taking the MTD and the pumping power consumption as objectives. Kalkan et al. [55] performed MOO for a microchannel cooling plate.

Physical Model
In this paper, based on the square HGB with "arrow-shaped" HTCC model established in the previous literature [24], MOO with MTD minimization and EGR minimization as optimization objectives was performed, and Pareto frontiers with optimal set were obtained based on NSGA-II. TOPSIS, LINMAP, and Shannon entropy decision methods were used to obtain the optimal results. The deviation index [56] was used to compare and analyze the advantages and disadvantages of the optimal results under each decision method. At the same time, the multi-objective constructal designs of the "arrow-shaped" HTCC were carried out through the optimization of single DOF, two DOF, and three DOF, and the thermal performances of the square heat-generation body under the optimizations of different DOF were compared. Our results indicate that introducing MOO into constructal theory for design can improve the comprehensive thermal conductivity of HTCCs and meet the needs of engineering design. Figure 1 shows the square HGB with "arrow-shaped" HTCC model [24]. The length of the square HGB (the heat-generation rate is ''' q , the thermal conductivity is 0 k ) is L , and an "arrow-shaped" HTCC (thermal conductivity is c k ) is arranged inside it. The heat flow generated in the area of the square HGB converges into the "arrow-shaped" HTCC, and flows out of the heat-generation area through the left side of the "arrow-shaped" HTCC (temperature is min T ). The remaining boundaries of the square HGB are adiabatic.

Physical Model
The thermal conductivity ratio of the high-and low-thermal-conductivity materials is de-  Square HGB with "Arrow-shaped" HTCC model [24]. Figure 1. Square HGB with "Arrow-shaped" HTCC model [24].
From Figure 1, the area (A 0 ) of the "arrow-shaped" HTCC can be expressed as A 0 = H 1 L 1 + H 2 L 2 , the area of the entire square is L 2 , and the area fraction (φ) of HTCC Pareto frontiers with optimal set of the entire square HGB is defined as: The L 1 , L 2 , H 1 , and H 2 can be dimensionless as: L 1 = L 1 /L, L 2 = L 2 /L, H 1 = H 1 /L, and H 2 = H 2 /L.

Mathematical Model
For the square HGB with "arrow-shaped" HTCC, the two-dimensional thermal conductivity differential equations in high-and low-thermal-conductivity materials are [24]: Because the square HGB with "arrow-shaped" HTCC model is symmetrical, it is sufficient to analyze the area y ≥ 0 within the square HGB. For the region of y ≥ 0, the boundary conditions are: The MTD (∆T) in the square HGB can be dimensionless as: where T max is the hot-spot temperature in the square HGB. With steady-state heat conduction, the EGR in the entire HGB is: The expressions of σ k 0 and σ k c are: where A k 0 and A k c are the areas of the HGB and the HTCC, respectively. k 0 is thermal conductivity of HGB, and k c is thermal conductivity of HTCC. The EGR in the square HGB can be dimensionless as: where q is heat-generation rate per unit volume, L is length of square HGB, and T min is the lowest temperature of square HGB.

Multi-Objective Constructal Design for "Arrow-Shaped" HTCC
From Figure 1, the "arrow-shaped" HTCC area was determined by four characteristic sizes: L 1 , L 2 , H 1 , and H 2 . Under the given condition of area fraction (φ) of the HTCC, the heat conduction problem of the square HGB has three DOF. The NSGA-II algorithm program was run through MALAB software, and the connection between MATALB and COMSOL was realized through COMSOL LiveLink for MATLAB. In this way, the multiobjective constructal design of the "arrow-shaped" HTCC could be carried out through the optimization of the single DOF, two DOF and three DOF, and the thermal performances of the square HGB under the optimization of different DOF could be compared. In the NSGA-II algorithm, the population size, mutation probability, and evolutionary generation were set to 100, 0.8, and 10, respectively. The TOPSIS decision method was selected based on the distances from each point in the Pareto frontiers to the ideal and non-ideal points. The LINMAP decision method was selected based on the weighted distance from each point in the Pareto frontiers to the ideal point. The Shannon entropy decision method was selected based on the best feasible result of different objectives.
The deviation index is defined as [56]: where G positive j is the normalized and weighted value of the jth objective of the ideal point.G negative j is the normalized and weighted value of the jth objective of the non-ideal point, and i opt is the best feasible result.

Multi-Objective Constructal Design under Single-DOF Optimization
First, we considered the constructal design of the "arrow-shaped" HTCC when the area of the arrowhead was fixed. Next, taking the width of the rectangular area of the "arrowshaped" HTCC L 1 as the optimization variable, the multi-objective constructal design of the "arrow-shaped" HTCC in the square HGB could be carried out with the objective of minimizing the MTD and the EGR. According to Ref. [24], the relevant parameters are as follows: Figure 2 shows the Pareto fronters of the dimensionless MTD (∆ T) and the dimensionless EGR ( σ) obtained by MOO under single-DOF optimization. Figure 3 shows the distribution of L 1 in the Pareto frontiers under single-DOF optimization. From Figure 2, with the increase of ∆ T, σ decreased continuously. ∆ T and σ cannot reach their minimum values at the same time, so it is necessary to find the best compromise between ∆ T and σ to make the comprehensive thermal performance of the square HGB optimal. Having at least one objective function ( σ or ∆ T) in the Pareto frontiers is better than having the other solutions outside the Pareto frontiers. Decreasing σ (or ∆ T) will inevitably lead to ∆ T (or σ) increases, so the Pareto frontiers have the least conflict of objectives compared to other solutions. From Figure 3, in the Pareto frontiers under single-DOF optimization, L 1 was distributed between 0.0903 and 0.0981, and the optimal solutions with minimum ∆ T and σ of L 1 were on the boundary of the distribution range. Table 1 shows the optimization results of the square HGB with different objectives under single-DOF optimization. The three decision methods in the table, comprising TOPSIS, LINMAP, and Shannon entropy, are commonly used for solving multi-objective decision-making problems. From Table 1, the optimal result with the Shannon entropy decision method was the same as that obtained with the objective of minimizing the ∆ T. The optimal results with the TOPSIS and LINMAP decision methods represent the optimal compromise of ∆ T and σ. In the single-DOF optimization, the deviation index of the optimal result with the TOPSIS decision method was the smallest, so the optimal result with the TOPSIS decision method was selected as the design scheme for the best compromise between the maximum thermal resistance and the loss of heat transfer irreversibility of the square HGB.   Table 1 shows the optimization results of the square HGB with different objectives under single-DOF optimization. The three decision methods in the table, comprising TOP-SIS, LINMAP, and Shannon entropy, are commonly used for solving multi-objective decision-making problems. From Table 1, the optimal result with the Shannon entropy decision method was the same as that obtained with the objective of minimizing the T Δ  . The optimal results with the TOPSIS and LINMAP decision methods represent the optimal compromise of T Δ  and σ . In the single-DOF optimization, the deviation index of the optimal result with the TOPSIS decision method was the smallest, so the optimal result with the TOPSIS decision method was selected as the design scheme for the best compromise between the maximum thermal resistance and the loss of heat transfer irreversibility of the square HGB.   Table 1 shows the optimization results of the square HGB with different objectives under single-DOF optimization. The three decision methods in the table, comprising TOP-SIS, LINMAP, and Shannon entropy, are commonly used for solving multi-objective decision-making problems. From Table 1, the optimal result with the Shannon entropy decision method was the same as that obtained with the objective of minimizing the T Δ  . The optimal results with the TOPSIS and LINMAP decision methods represent the optimal compromise of T Δ  and σ . In the single-DOF optimization, the deviation index of the optimal result with the TOPSIS decision method was the smallest, so the optimal result with the TOPSIS decision method was selected as the design scheme for the best compromise between the maximum thermal resistance and the loss of heat transfer irreversibility of the square HGB.

Multi-Objective Constructal Design under Two-DOF Optimization
By further releasing the constraints fixed by the arrowhead shape and taking the width ( L 1 ) of the rectangular area and the width ( L 2 ) of the arrowhead as the optimization variables while keeping the area of the arrowhead ( L 2 H 2 = 0.04) unchanged, the multiobjective constructal design of the "arrow-shaped" HTCC in the square HGB could be carried out with the objective of minimizing the MTD and the EGR.  Figure 4 shows the Pareto frontiers of ∆ T and σ obtained by MOO under two-DOF optimization. Figure 5 shows the distributions of L 1 and L 2 in the Pareto frontiers under two-DOF optimization, respectively. From Figure 4, with the increase of ∆ T, σ decreased continuously. Therefore, it was necessary to find the optimal constructs of L 1 and L 2 . Compared with Figure 2, the minimum values of ∆ T and σ on Pareto frontiers were reduced. From Figure 5, in the Pareto frontiers under two-DOF optimization, L 1 was distributed between 0.0911 and 0.0991, L 2 was distributed between 0.4876 and 0.4980, and the optimal solutions with minimum ∆ T and σ of L 1 and L 2 were on the boundary of the distribution range.
width ( 1 L  ) of the rectangular area and the width ( 2 L  ) of the arrowhead as the optimization variables while keeping the area of the arrowhead ( 2 2 0.04 L H =   ) unchanged, the multiobjective constructal design of the "arrow-shaped" HTCC in the square HGB could be carried out with the objective of minimizing the MTD and the EGR. Figure 4 shows the Pareto frontiers of T Δ  and σ obtained by MOO under two-DOF optimization. Figure 5 shows the distributions of 1 L  and 2 L  in the Pareto frontiers under two-DOF optimization, respectively. From Figure 4, with the increase of T Δ  , σ decreased continuously. Therefore, it was necessary to find the optimal constructs of 1 L  and 2 L  . Compared with Figure Table 2 shows the optimization results of the square HGB with different objectives under two-DOF optimization. From Table 2, the optimal result with the Shannon entropy decision method was the same as that obtained with the objective of minimizing the T Δ  , which has smaller deviation indexes compared to the optimal result with minimum σ .
The optimal results with the TOPSIS and LINMAP decision methods were represent the optimal compromise of T Δ  and σ . In the two-DOF optimization, the deviation index of the optimal result with the TOPSIS decision method was the smallest, so the optimal result with the TOPSIS decision method was selected as the design scheme of the square HGB.  Table 2 shows the optimization results of the square HGB with different objectives under two-DOF optimization. From Table 2, the optimal result with the Shannon entropy decision method was the same as that obtained with the objective of minimizing the ∆ T, which has smaller deviation indexes compared to the optimal result with minimum σ. The optimal results with the TOPSIS and LINMAP decision methods were represent the optimal compromise of ∆ T and σ. In the two-DOF optimization, the deviation index of the optimal result with the TOPSIS decision method was the smallest, so the optimal result with the TOPSIS decision method was selected as the design scheme of the square HGB.

Multi-Objective Constructal Design under Three-DOF Optimization
Considering that the "arrow-shaped" HTCC shape can be changed, taking the width ( L 1 ) of the rectangular area, the width ( L 2 ) of the arrowhead, and the length ( H 1 ) of the arrowhead as the optimization variables, the multi-objective constructal design of the "arrow-shaped" HTCC in the square HGB can be carried out with the objective of minimizing the MTD and the EGR. Figure 6 shows the Pareto frontiers of ∆ T and σ obtained by MOO under three-DOF optimization. Figure 7 shows the distributions of L 1 , L 2 , and H 1 in the Pareto optimal frontiers under three-DOF optimization, respectively. From Figure 6, with the increase of ∆ T, σ decreased continuously. Therefore, it was necessary to find the optimal constructs of L 1 , L 2 , and H 1 . Compared with Figure 4, the minimum values of ∆ T and σ on the Pareto frontiers were further reduced. From Figure 7, in the Pareto frontiers with three-DOF optimization, L 1 was distributed between 0.1170 and 0.1198, L 2 was distributed between 0.4794 and 0.4911, and H 1 was distributed between 0.6258 and 0.7019. L 1 , L 2 and H 1 of the "arrow-shaped" HTCC in the square HGB were based on multi-objective    Table 3 shows the optimization results of the square HGB with different objectives under three-DOF optimization. From Table 3, the optimal result with the Shannon entropy decision method was the same as that obtained by taking the objective of minimizing the T Δ  . The optimal results with the TOPSIS and LINMAP decision methods represent the optimal compromise of T Δ  and σ . In three-DOF optimization, the deviation index of the optimal result with the TOPSIS decision method was the smallest, so the optimal result with the TOPSIS decision method was selected as the design scheme of the square HGB.  Figure 8 shows the optimal constructs of the "arrow-shaped" HTCC with the TOPSIS decision method based on single-DOF, two-DOF, and three-DOF optimization. From Figure 8, the corresponding T Δ  to the optimal construct under single-DOF, two-DOF, and three-DOF optimizations was 0.0932, 0.0903, and 0.0886, respectively, and the corresponding σ was 0.0431, 0.0416, and 0.0398, respectively. Compared with the two-DOF and single-DOF optimizations, the T Δ  of square HGB under three-DOF optimization was reduced by 1.9% and 4.9%, respectively, and the σ was reduced by 4.3% and 7.7%, respectively. The results show that T Δ  and σ of the square HGB can be further reduced by increasing the DOF optimization of the "arrow-shaped" HTCC, which can improve the comprehensive thermal performance of the square HGB. Compared to what was reported in [24], by taking into account the thermal conductivity of the heat-generation body, the optimization process may not be as obvious as optimizing for only one objective.  Table 3 shows the optimization results of the square HGB with different objectives under three-DOF optimization. From Table 3, the optimal result with the Shannon entropy decision method was the same as that obtained by taking the objective of minimizing the ∆ T. The optimal results with the TOPSIS and LINMAP decision methods represent the optimal compromise of ∆ T and σ. In three-DOF optimization, the deviation index of the optimal result with the TOPSIS decision method was the smallest, so the optimal result with the TOPSIS decision method was selected as the design scheme of the square HGB.  Figure 8 shows the optimal constructs of the "arrow-shaped" HTCC with the TOPSIS decision method based on single-DOF, two-DOF, and three-DOF optimization. From Figure 8, the corresponding ∆ T to the optimal construct under single-DOF, two-DOF, and three-DOF optimizations was 0.0932, 0.0903, and 0.0886, respectively, and the corresponding σ was 0.0431, 0.0416, and 0.0398, respectively. Compared with the two-DOF and single-DOF optimizations, the ∆ T of square HGB under three-DOF optimization was reduced by 1.9% and 4.9%, respectively, and the σ was reduced by 4.3% and 7.7%, respectively. The results show that ∆ T and σ of the square HGB can be further reduced by increasing the DOF optimization of the "arrow-shaped" HTCC, which can improve the comprehensive thermal performance of the square HGB. Compared to what was reported in [24], by taking into account the thermal conductivity of the heat-generation body, the optimization process may not be as obvious as optimizing for only one objective. single-DOF optimizations, the T Δ  of square HGB under three-DOF optimization was reduced by 1.9% and 4.9%, respectively, and the σ was reduced by 4.3% and 7.7%, respectively. The results show that T Δ  and σ of the square HGB can be further reduced by increasing the DOF optimization of the "arrow-shaped" HTCC, which can improve the comprehensive thermal performance of the square HGB. Compared to what was reported in [24], by taking into account the thermal conductivity of the heat-generation body, the optimization process may not be as obvious as optimizing for only one objective.

Conclusions
Based on the square HGB with "arrow-shaped" HTCC model established in the previous literature, in this study, MOO was performed with MTD minimization and EGR minimization as optimization objectives, and the Pareto frontiers with optimal set were