Multi-Objective Constructal Design for Quadrilateral Heat Generation Body with Vein-Shaped High Thermal Conductivity Channel

Based on the quadrilateral heat generation body (HGB) proposed by previous literature, the multi-objective constructal design is performed. Firstly, the constructal design is performed by minimizing the complex function composed of the maximum temperature difference (MTD) and entropy generation rate (EGR), and the influence of the weighting coefficient (a0) on the optimal constructal is studied. Secondly, the multi-objective optimization (MOO) with the MTD and EGR as optimization objectives is performed, and the Pareto frontier with an optimal set is obtained by using NSGA-II. The optimization results are selected from the Pareto frontier through LINMAP, TOPSIS, and Shannon Entropy decision methods, and the deviation indexes of different objectives and decision methods are compared. The research of the quadrilateral HGB shows that the optimal constructal can be gained by minimizing the complex function with the objectives of the MTD and the EGR, the complex function after the constructal design is reduced by up to 2% compared with its initial value, and the complex function of the two reflects the compromise between the maximum thermal resistance and the irreversible loss of heat transfer. The Pareto frontier includes the optimization results of different objectives, and when the weighting coefficient of a complex function changes, the optimization results obtained by minimizing the complex function will also be distributed in the Pareto frontier. The deviation index of the TOPSIS decision method is 0.127, which is the lowest one among the discussed decision methods.


Introduction
Arranging high thermal conductivity channels (HTCCs) is a common method to perform the heat dissipation design of electronic devices so that the internal heat can be concentrated and dissipated to the outside through the HTCCs. Therefore, optimizing the arrangement of HTTCs is an important study. Bejan established entropy generation minimization theory [1,2], which measures the performance of the heat transfer process by evaluating the degree of irreversible loss of energy and provides a new evaluation method for thermodynamic optimization. Since the entropy generation minimization theory was proposed, it has been widely used in heat conduction [3][4][5][6][7][8], fins [9][10][11][12][13][14], heat exchangers [15][16][17][18][19][20], and heat sinks [21][22][23][24][25][26]. The constructal theory [27][28][29] optimizes problems by following the idea that the structure of things develops in the direction of better internal flow performance, which provides a new method for traditional thermodynamic optimization problems. Constructal theory proved to be fully interdisciplinary and versatile, it can be used in heat transfer problems [30][31][32][33][34][35][36][37][38][39][40], fluid flow problems [36][37][38][39][40][41][42][43], solar cell [44], and stiffened plates [45] designs. The maximum temperature difference (MTD) is one of the optimization objectives for the constructal design of a heat generation body (HGB). Bejan [30] first performed the constructal design of the two-dimensional rectangular HGB embedded with HTCCs by minimizing the MTD and assembled a new first-level structure according to its optimal constructal. Ghodoossi and Eǧrican [46] obtained the optimal constructal of the triangular HGB by minimizing the MTD with the analytical method. da Silva et al. [47] studied the "disc-point" thermal conductivity problem, arranged strip-shaped HTCCs on the circumferential side of the disc HGB, and the optimal constructal by taking MTD as the objective was obtained. Zhang et al. [48] obtained the optimal constructal of arrow-shaped HTCCs in square HGB through a three-degree-of-freedom constructal design, which further reduced the MTD of the square HGB. Hajmohammadi et al. [49] established an annular fin model embedded with HTCCs and obtained the optimal constructal of the HTCCs by minimizing the MTD. According to the common plant leaf veins in nature, Li and Feng [50] proposed a quadrilateral HGB model embedded with vein-shaped HTCCs and obtained its optimal constructal by minimizing the MTD.
The MTD reflects the maximum thermal resistance in the HGB, while the entropy generation rate (EGR) can reflect the irreversible loss of heat transfer in the HGB. Some scholars further studied the EGR performance of different HGBs based on EGR. Ghodoossi [51] studied the optimal constructal of the rectangular HGB by minimizing the MTD and further analyzed the EGR performance of the HGB. Tescari et al. [52] studied the rectangular HGB with the objective of minimizing the EGR and compared it with the optimal constructal obtained by minimizing the MTD. You et al. [53] performed the constructal design of the non-uniform triangular HGB with the objective of the EGR. Feng et al. [54] obtained the optimal constructal of the disc HGB with the objective of the EGR. Ribeiro and Queiros-Condé [55] performed the constructal optimization of the square HGB with I-shaped HTCC, and further analyzed the local EGR performance. Zhu et al. [56] further gained the optimal constructal of the quadrilateral HGB established in Ref. [50] with the minimum EGR. The research showed that the optimal constructal with the minimum EGR is different from that obtained by minimizing the MTD.
The above constructal designs are all single-objective optimizations, which can only meet a single design requirement, but the actual engineering design often needs to meet multiple design requirements. Therefore, the multi-objective optimization (MOO) not only adapts to the engineering design requirements but also promotes the update and replacement of the heat dissipation design strategy of electronic devices. The Non-dominated Sorting Genetic Algorithm II (NSGA-II) [57] with an elite strategy has been successfully applied to many engineering designs [58][59][60][61][62][63][64][65][66][67][68][69]. In particular, some scholars apply the NSGA-II algorithm to the study of constructal design with different optimization objectives. Chen et al. [70] proposed a non-uniform disc HGB model considering the thermal and flow performance and performed the constructal design by minimizing the complex function with the objectives of the MTD and pumping power consumption (PPC) in the HGB. Zhang et al. [71] obtained the optimal constructal of the trapezoidal HGB with heat conduction and flow by minimizing the complex function with the objectives of the EGR and PPC. Furthermore, the NSGA-II is used to perform MOO on this problem to obtain the Pareto frontier. The research shows that the optimization result with the minimum complex function is distributed in the Pareto frontier. Feng et al. [72] obtained the optimal constructal of marine condensers with single-objective optimization and MOO and compared the optimization results of single-objective optimization and three decision methods based on deviation index. Feng et al. [73] used the NSGA-II to perform the constructal design with the minimum EGR and the constructal design with the minimum PPC and compared the optimization results of three decision methods based on the deviation index.
In this paper, a multi-objective constructal design of the quadrilateral HGB established in Ref. [50] will be performed. Firstly, the constructal design will be performed by minimizing a complex function with the objectives of the MTD and EGR, and the influence of the weighting coefficient (a 0 ) on the optimal constructal will be studied. Secondly, the MOO with MTD and EGR as optimization objectives will be performed, and the Pareto frontier with optimal settings will be obtained by using NSGA-II. The optimization results will be selected from the Pareto frontier through LINMAP, TOPSIS, and Shannon Entropy decision methods, and the deviation indexes of different objectives and decision methods will be compared. Figure 1 shows the quadrilateral HGB model [50]. The quadrilateral HGB (heat generation rate is q , thermal conductivity is k 0 ) is symmetrical about OA, the length of OA is L 1 , the height from point B to OA is H 1 , and the total area is A 1 = H 1 × L 1 . The shape of the quadrilateral HGB changes with the aspect ratio H 1 /L 1 and the angle θ between BA and OA. A series of branch HTCCs M i D i (i = 1, 2, 3, . . . , n) (width is D 0 ) are equidistantly distributed on the central HTCC (width is D 1 ), and the central HTCC is divided into n intervals (n 1). The point M i of the branch HTCC is located in the middle of the ith intervals, and M i D i (i = 1, 2, 3, . . . , n) is parallel to BA. The thermal conductivity of the HTCC is k c k 0 . The periphery of the quadrilateral HGB is adiabatic, and its internal heat is concentrated through HTCCs and dissipated from the point A (temperature is T 0 ) to the outside. The elemental unit based on any branch HTCC M i D i is shown in Figure 2 [50]. The height of the trapezoidal elemental unit is δ i (δ i = (L 1 sin θ)/n). When n 1, δ i w i . The trapezoidal "abcd" is similar to rectangular "1234". When δ i w i , the heat flow is perpendicular to M i D i , and the differential equation can be expressed as:

Quadrilateral Heat Generation Body Model
The boundary conditions are: When y > 0, solving Equation (1) yields: The heat conduction differential equation of the M i D i can be expressed as: The boundary conditions are: Substituting T i (x) into Equation (4) yields: The porosity of HTCC in the elemental unit is: The porosity of the HTCCs of the quadrilateral HGB is: From Equations (9) and (10), one has: Figure 3 shows the central HTCC [50]. The heat flows to point A from point M i . The temperature difference distribution between points M i and M i+1 can be obtained as: The boundary conditions are:

Maximum Temperature Difference
From Equations (12)-(14), one has: Substituting (i − 0.5)L 1 /n for x in Equation (15), the temperature difference between M i+1 and M i is: The temperature difference distribution between points M n and A is: The boundary conditions are: From Equations (17)- (19), one has: Substituting (n − 0.5)L 1 /n for x in Equation (20), the temperature difference between M n and A is: From Equations (8), (16) and (21), the temperature distribution of the elemental unit in the quadrilateral HGB can be obtained as: According to the Ref. [50], maximum temperature point T max is on the boundary of the elemental unit. Therefore, the MTD can expressed as: where T max is obtained by bringing the interval number i of the elemental unit where the maximum temperature point is located into Equation (22).

Entropy Generation Rate
According to the Ref. [56], the EGRs (σ k 0 and σ k c ) of heat generating area and HTCCs area in quadrilateral HGB can be obtained as: where A k 0 and A k c are the areas of heat generating area and HTCCs area in quadrilateral HGB.
The total EGR of quadrilateral HGB can be obtained as:
The MTD reflects the maximum thermal resistance of the quadrilateral HGB and the EGR reflects the irreversible loss of heat transfer of the quadrilateral HGB. Optimizing the MTD or EGR of the quadrilateral HGB alone cannot fully reflect the comprehensive heat transfer performance of the quadrilateral HGB. Therefore, a complex function composed of the MTD and the EGR based on the linear weighting method [70,71] is established: where a 0 is the weighting coefficient, and∆T int (= 1.77 K) and σ int (= 1.59 × 10 −3 W · K −1 ) are the MTD and EGR of the HGB under the initial structure, respectively.   Figure 6 shows the effects of a 0 on F ST,min and (H 1 /L 1 ) opt . From Figure 5, when a 0 = 0.5, (H 1 /L 1 ) opt and F ST,min are 0.905 and 0.980, respectively. Compared with the initial structure, F ST , H 1 /L 1 and ∆T are reduced by 2.0%, 9.5% and 6.07%, respectively, while σ increased by 2.06%. When H 1 /L 1 reaches (H 1 /L 1 ) opt , ∆T and σ achieve the best compromise. From Figure 6, when a 0 = 0, (H 1 /L 1 ) opt is equal to (H 1 /L 1 ) S,opt . When a 0 = 1,(H 1 /L 1 ) opt is equal to (H 1 /L 1 ) T,opt . when a 0 = 0.16, F ST,min = 1. The optimal constructal can be gained by minimizing the complex function with the objectives of the MTD and the EGR, which is better than the initial design point. The selection of the weighting coefficient has a great influence on the optimal construct, and the optimal complex function gets smaller as the weighting coefficient of EGR decreases. Therefore, design with a complex function relies on the selection of an appropriate weighting coefficient.

Design with NSGA-II
In order to adapt the engineering design requirements, The MOO of the quadrilateral HGB is performed by using the "gamultiobj" algorithm that comes from the MATLAB software based on the NSGA-II. Figure 7 shows the complete process of NSGA-II [74]. In the NSGA-II, the decision variable is H 1 /L 1 , and the optimization objectives are ∆T and σ. LINMAP, TOPSIS, and Shannon Entropy decision methods [75] are used to select three results from the Pareto frontier that are suitable for the actual needs of the project.  Figure 8 shows the Pareto frontier of the dimensionless MTD (∆ T = ∆T/∆T int ) and the dimensionless EGR ( σ = σ/σ int ) gained by MOO. From Figure 8, points A and B of the Pareto frontier represent the results of the optimal constructal of the quadrilateral HGB with the minimum σ and minimum ∆ T, respectively. Although point A and point B correspond to the minimum σ and the minimum ∆ T, respectively, they also correspond to the maximum∆ T and the maximum σ, respectively. Decreasing σ (or ∆ T) on the Pareto frontier will inevitably lead to ∆ T(or σ) increase, so it is necessary to find the best com-promise between σ and ∆ T to optimize the comprehensive heat transfer performance of quadrilateral HGB. Point C is an ideal point, which is the minimum point that σ and ∆ T can reach. Since the minimum σ cannot be obtained when ∆ Treach the minimum point, the ideal point cannot be reached. Point D is a non-ideal point, which is the maximum point that σ and ∆ T can reach. Since the maximum σ cannot be obtained when ∆ T reach the maximum, the non-ideal point D cannot be reached. Point E is the result of the optimization constructal of the quadrilateral HGB based on the complex function with σ and ∆ T as the optimization objective, and point E is an optimal result in the Pareto frontier. When the a 0 of F ST changes, the optimal results of the optimization constructal of the quadrilateral HGB obtained by minimizing the complex function composed of σ and ∆ T is also distributed in the Pareto frontier. The remaining optimal results of the Pareto frontier are selected by using other decision methods, and the choice of decision methods needs to be decided by the decision maker according to the actual needs of the project. Therefore, the Pareto frontier can provide a better choice for the performance optimization and constructal design of the quadrilateral HGB. Figure 9 shows the distribution of H 1 /L 1 in the Pareto frontier within its value range. From Figure 9, the two endpoints on the left and right of the abscissa are the lower limit and upper limit of H 1 /L 1 in the Pareto frontier, respectively, and the corresponding optimal results are points B and A in Figure 7, respectively. Therefore, individual optimization may not be the substantive optimal result, because the optimal variable is on the boundary of the Pareto frontier.  Table 1 lists the optimization results of different objectives. From Table 1, the optimal result of the Shannon Entropy decision method is the same as that of single-objective optimization with σ as optimization objective. The optimal result by minimizing the F ST , and the optimal results obtained through the LINMAP and TOPSIS decision methods are a compromise of the optimal results obtained with the minimum σ and minimum ∆ T. The constructal design goes through the LINMAP decision methods is similar to that goes through the TOPSIS decision methods, and the corresponding ∆ T decreased by 0.49% and 0.69% compared with the optimal result of F ST , respectively, while the corresponding σ increases by 0.75% and 0.96%, respectively. The deviation index [76] of the optimization constructal obtained by TOPSIS decision methods is 0.127, which is better than other decision methods and objectives.

Conclusions
Based on the quadrilateral HGB proposed in the previous literature, the multi-objective constructal design is performed. Firstly, the constructal design is performed by minimizing the complex function with the objectives of the MTD and EGR, and the influence on the optimal constructal is studied. Secondly, the MOO with the MTD and EGR as optimization objectives is performed. The optimization results are selected from the Pareto frontier through LINMAP, TOPSIS, and Shannon Entropy decision methods, and the deviation indexes of different objectives and decision methods are compared. The results show:

1.
The optimal constructal can be gained by minimizing the complex function with the objectives of the MTD and the EGR. Compared to the initial structure, F ST , H 1 /L 1 and ∆T are reduced by 2.0%, 9.5% and 6.07%, respectively, while σ increased by 2.06%. The complex function of the two reflects the compromise between the maximum thermal resistance and the irreversible loss of heat transfer. The selection of the weighting coefficient has a great influence on the optimal constructal, and the optimal complex function gets smaller as the weighting coefficient of EGR decreases. Therefore, design with a complex function relies on the selection of an appropriate weighting coefficient.

2.
The Pareto frontier includes the optimization results of different objectives, and when the weighting coefficient of complex function changes, the optimization results obtained will also be distributed in the Pareto frontier. The constructal design goes through the LINMAP decision methods is similar to that goes through the TOPSIS decision methods, and the corresponding ∆ T decreased by 0.49% and 0.69% compared with the optimal result of F ST , respectively, while the corresponding σ increases by 0.75% and 0.96%, respectively. 3.
The deviation index of the optimization constructal obtained by TOPSIS decision methods is 0.127, which is better than other decision methods and objectives. Compared to the optimal construct with minimum MTD and minimum EGR, the optimal construct obtained by using NSGA-II and decision methods has a smaller deviation index and smaller conflict between the two objectives.

4.
Constructal theory and NSGA-II are powerful tools for comprehensive thermal performance improvements of the high thermal conductivity channels. By increasing the optimization objectives, high thermal conductivity channels can be better used in engineering applications considering multiple design requirements. Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.

Acknowledgments:
The authors wish to thank the editor and reviewers for their careful, unbiased, and constructive suggestions, which led to this revised manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.