1. Introduction
The increase of turbine inlet temperature can improve the efficiency of turbomachinery and cause additional damage to turbine blades. To reach higher inlet temperature and protect turbine blades, many cooling structures have been developed, such as lattice structure (LS). For decades, great attention has been paid to LS because of its unique characteristics of high porosity [
1,
2], ultralight weight [
3,
4], high specific stiffness and strength [
5], and high heat transfer [
3,
5]. Although traditional cooling structures of turbine blades can greatly improve the cooling efficiency, Han et al. [
6] found that the flow resistance coefficient increases rapidly with the improvement of heat transfer capacity in different degrees, and the heat transfer coefficient distribution is inhomogeneous, causing excessive structural thermal stress. In order to solve these issues, Kim et al. [
7] studied the heat transfer performance of pyramid-type lattice structure (PLS) by numerical simulation and experiments, and the results showed that PLS had good homogeneity of heat transfer distribution. Yun et al. [
8] carried out an experiment to study face-centered cubic LS and found that this LS showed a good thermal performance compared to non-LS channel. Due to the complex topology of LS, metal additive manufacturing technology can easily solve this limit [
9,
10].
Numerous studies have been conducted on the mechanical performances of LS, including relative density and specific strength [
11,
12], effective modulus and Poisson’s ratio [
13,
14], yield strength, buckling strength, and plateau strength [
15]. These studies greatly enriched the mechanical theory for LS, supporting the mechanical optimization of LS. Hoang et al. [
16] investigated LS with direct multiscale topology optimization method which provided an inexpensive and efficient method to obtain optimal LS. Zhang et al. [
17] proposed topology optimization of LS to obtain optimal density distribution of LS. The results showed that the compressive and bending stiffness and energy absorption of the optimal LS improved. Wu et al. [
18] provided a novel strengthening method of LS and presented a pyramidal LS with some mechanical performance advantages. Other optimization approaches for LS included semi-automated optimization approach [
19] and inverse identification approach [
20], which greatly improved the mechanical performances of LS. These studies almost focus on mechanical optimization with single objective for LS. The optimization studies on heat transfer or specific mechanical performance (lightweight, vibration, and bearing load capacity), or both, performances of LS are as follows.
Liu et al. [
21] proposed a mathematical optimization method with two objectives of LS, which improved the structural performance of each panel under multiple loading cases and minimized structural mass of LS simultaneously. Valdevit et al. [
22,
23] optimized the design of various weight, mechanical, and heat transfer performance of LS, and they found that hexagon structure had the optimal comprehensive performance. Chen et al. [
24] found that sandwich structure filled with sponge had the strongest absorbing capacity and reached the targets of minimum dimensionless mass and maximum absorbing capacity of LS. Roper et al. [
25] investigated and analyzed the PLS and found that compression modulus, compressive strength, and maximum heat flux increased with the increase of density. Su et al. [
26] presented a multiparameter optimization approach for lightweight fiber-reinforced polymer (FRP) composite triangular LS under nonlinear structural response constraints and achieved a significant improvement in terms of weight saving. Moon et al. [
27] made a comparison among Kagome type LS, PLS, and diamond type LS for high bearing load and lightweight characteristics, and came to the conclusion that Kagome LS had the best performance, which was applied to the wings of unmanned aerial vehicles (UAV). Meng et al. [
28] optimized the strength, vibration, and bird impact load of LS and realized the lightweight design of LS, which was applied to blades. Fazilati et al. [
29] optimized the hexagonal multilayer honeycomb lattice and reached the targets of maximizing energy absorption, and minimizing impact vibration and overall size. The results showed that the performance of the optimized structure was significantly improved compared to that of the single layer of nonoptimized structure. Xu et al. [
30] optimized PLS using a genetic algorithm to reduce weight and improve sound insulation performance, and also studied the effects of materials and upper and lower panel thicknesses on performance. Bailong et al. [
31] carried out an optimal design of body-centered cubic (BCC) LS, considering the relative density, initial stiffness, and plastic failure strength. The research showed that, compared to BCC structure, the elastic modulus and yield limit of the optimized structure increased by 121.5% and 77.3%, respectively. Smardzewski et al. [
32] provided the design of optimized or near-optimized LS by experimentally estimating the mechanical strengths and failure mechanisms and found that with the increase of inclination angle of struts and relative density of LS, the mechanical properties of beams increased. Zhao et al. [
33] proposed a concurrent optimization method to optimize natural frequencies of additive manufacturing-fabricated LS, which increased the first natural frequency (
freq1) of the optimized structure, 38.5% higher than that of the original structure. Akihiro et al. [
34] used a simple basic lattice shape composed of pillars, and only optimized its density distribution by setting the pillar diameter as design variable, and treating steady-state pressure and temperature reductions as multiobjective functions. Adil et al. [
35] investigated the optimal designs of novel LS filled with thin tubes based on multiobjective crashworthiness optimization procedure. Lattice member diameter and tube thickness were set as design variables, and minimizing peak crash force and maximizing specific energy absorption were chosen as design objectives. The research revealed that the optimized BCC LS hybrid designs generally had a better crashworthiness performance than BCC LS with vertical strut counterparts with the same design objectives.
As outlined above, optimization methods of LS with two or more objectives of mechanical or heat transfer, or both, greatly expanded the application of LS in heat pipes and tubes. However, few studies on optimization method of LS concerned both heat transfer and mechanical performances used at region “1” of turbine blades, as shown in
Figure 1. Although, in [
36], Gao et al. developed an effective optimization method of LS concerning both heat transfer and mechanical performances, the application background was for composite structure instead of turbine blades. Therefore, the aim of this study was to construct an optimization model of PLS and X-type LS (XLS) to obtain the optimal LS used at the trailing edge of turbine blades based on functions establishment between geometric parameters (diameter and inclination angle of LS) and representative parameters (Nusselt number, relative density, the first-order natural frequency, and elastic modulus). The boundary conditions were simplified from turbine blades, such as thin-wall structure; applying a constant heat flux on the target surface; mass flow inlet; and pressure outlet. The results show that the heat transfer and mechanical (lightweight, vibration, and bearing load) performances of the optimal LS are better than that of the initial LS obtained from [
36] i.e.,
Nu increases by 24.1% and
decreases by 31% in the optimal LS of the first selected problem; and
Nu increases by 28.8% while
freq1 and
are almost unchanged in the optimal LS of the second selected problem, compared to initial LS.
3. The Mathematical Optimization Model and Two Selected Optimization Problems
Previous studies of LS were mainly on the application of LS’s mechanical performances, and few studies used LS in turbine blades to improve both thermal and mechanical performances of blades. We simplified the application of LS on turbine blades as an LS channel with variable aspect ratio and developed a corresponding optimization method. As shown in
Figure 1, to set more LS on thin walls of regions “1” and “2”, bearing load capacity, vibration, and heat transfer performances of turbine blade would be better. However, it would enlarge weight of turbine blades and pressure drop of the channel. Therefore, the optimization method aimed to seek a reasonable topology of LS which can improve both heat transfer and mechanical performances. Herein, based on the given functions which represented relevance between heat transfer and mechanical performances and geometric parameters of the LS channel, a mathematical optimization model was established; two selected optimization problems were proposed; NSGA-II algorithm was used to solve these problems, and optimal structural parameters were obtained.
3.1. The Mathematical Optimization Model
An LS channel may improve both heat transfer and mechanical performances of turbine blades. Relevance between heat transfer and mechanical performances and geometric parameters of a LS channel was established, and their functions were obtained. Based on these functions, functional integration design and the multidisciplinary performance requirements were taken into account, and a mathematical optimization design model was established at the start of structural design.
Diameter (D) and inclination angle (ω) of the LS channel were taken as geometric variables; Nu, freq1, E, and were selected as objectives to characterize the performance of heat transfer, vibration, load bearing, and light weight, respectively, and the corresponding mathematical optimization model was developed as follows.
Variables of the model are two geometric variables of the LS channel, which are:
Three functions of heat transfer and mechanical performance of mathematical optimization model are given as:
Constraints of the variables of the model are:
Additionally, its equivalent elastic modulus represented bearing load capacity of LS channel, and the value of equivalent elastic modulus was taken greater than the value of its crushing strength, which was also a constraint for LS. The function is expressed as:
3.2. Two Selected Optimization Problems
According to the different emphases of specific engineering applications on turbine blades, two kinds of combinatorial optimization problems were proposed. The optimization problem I (Op-I) was a problem based on Nu and , focusing on the optimization of structural performance of heat transfer and light weight of LS channel. The corresponding mathematical model is as follows:
Op-I:
where “Max” and “Min” mean that the structural performances of heat transfer and light weight of the LS channel should reach relatively maximum and minimum, respectively, and “s.t.” refers to the constraint of the optimization problem.
The optimization problem II (Op-II) was a problem based on Nu, , and freq1, focusing on the optimization of the structural performances of heat transfer, light weight, and vibration. The corresponding mathematical model is as follows:
Op-II:
where “Max” and “Min” mean that the structural performances of heat transfer, light weight, and vibration of LS channel should reach relatively maximum and minimum, respectively.
3.3. Establishment and Procedure of Solution for Mathematical Optimization Model
The ISIGHT2018 optimization software was used to build the mathematical optimization model for PLS and XLS channel. As shown in
Figure 6, models of PLS and XLS core were established by SOLIDWORKS; value of
freq1 was calculated by ABAQUS; and
Nu, relative density (
), and
E were directly calculated by the given functions. Herein, we mainly investigated the heat transfer and mechanical performances of the LS channel. Moreover, in future study, pressure loss, porosity, and friction coefficient of the LS channel can be added at this mathematical optimization model.
3.3.1. Establishment of Approximate Model
In ISIGHT, model objectives (
freq1,
Nu and
) were the input values. Before the optimization work, the relevance between model variables (
D and
ω) and these model objectives was established by the third-order response surface model (RSM), as shown in
Figure 6a, and RSM was used to establish the approximate model to improve the calculation efficiency. At the same time, in order to make all the test points evenly distributed in the design space, the DOE optimal Latin hypercube (opt. LHD) sampling mode was selected, with the sample number of 50, and the sample number for error analysis was 25.
The RSM cannot guarantee to pass all the sample points; therefore, there was a certain error compared to the actual model. In order to ensure the calculation accuracy of the approximate model, the fitting accuracy
R2 was used to verify the accuracy of the approximate model (
Table 2).
The calculation formula of fitting accuracy
R2 is:
where
SSR represents the sum of regression squares,
;
SST represents the sum of total squares,
;
is the average value of response;
is the predicted value at the design point;
is the real value of response, and
k is the number of sample points.
It can be observed from
Table 2 that the fitting accuracy values for the four optimization objectives were all above 0.97. Therefore, the model can well approximate the actual model, and this model can be used for the following optimization work of the LS channel.
3.3.2. Optimization Procedure
As shown in
Figure 6a, the NSGA-II algorithm, with good exploration performance, was selected for optimization. The specific procedure is shown in
Figure 6b and described as follows:
At the beginning of the solution, an accurate approximate model was established with RSM.
The population was initialized. Value of the objective function was solved.
Pareto sorting was carried out to obtain its elite individuals.
Step 3 was iterated repeatedly until the optimal solution set was found.
At the end of the solution, the optimum results were obtained.
3.3.3. NSGA-II Algorithm Setting and Stability Verification
As mentioned above, the NSGA-II algorithm was used to solve the mathematical optimization model of the LS channel, and the PLS channel was used to describe this algorithm as follows:
The specific parameters of this algorithm were set as sizes of population, 80; algebras of heredity, 100; crossover of probability, 0.9; and index for distribution of mutation, 20.
Figure 7 reveals that when the sizes of population increased from 12 to 24, the Pareto front tended to be consistent. When the sizes of population were 20, the Pareto fronts of PLS and XLS were basically stable, as shown in
Figure 7a. With the increase of algebras of heredity from 20 to 40, the Pareto frontier tended to be consistent. When the algebras of heredity were 30, the Pareto fronts of PLS and XLS were basically stable, as shown in
Figure 7b. Therefore, to avoid only obtaining the local optimal solution, and to improve the accuracy of the optimization results, the sizes of population and the algebras of heredity were selected as 20 and 30, respectively.