Abstract

Translate

Forward-swept wings are more suitable for natural laminar flow than backward-swept wings. However, in order to reduce the difficulty of optimization, most aero-structural optimization studies of forward-swept wings do not consider the automatic laminar–turbulent transition, discrete variables, or large-scale constraints, which may result in undesirable optimization results. In this article, an efficient aero-structural optimization method for the composite forward-swept natural laminar flow (FSNLF) wing is proposed, which can solve MDO problems with those issues. Reynolds-averaged Navier–Stokes (RANS) equations coupled with the dual e^N transition method are used to simulate subsonic viscous flows. A surrogate-based optimization (SBO) algorithm combining a discrete variable handling method is developed to solve the multidisciplinary design optimization (MDO) problem involving many discrete ply thickness variables of predefined angles (0°/±45°/90°). The Kreisselmeier–Steinhauser (KS) method is employed to handle large-scale geometric constraints, ply fraction constraints and material failure constraints. To verify the effectiveness of the proposed method, we perform the aero-structural optimization of an A320-class composite FSNLF wing. Results show that the proposed method offers great potential in the aero-structural optimization of the composite FSNLF wing. It can handle 32 discrete variables and 11,089 constraints, the drag coefficient and mass of the wing are reduced significantly, and the area of the laminar flow region on the wing upper surface is increased by 24.3% compared with the baseline.

Full text

Turn on search term navigation

Translate

1. Introduction

Forward-swept wings have the advantages of high critical Mach number and low wave drag, and they can effectively overcome the wingtip stalling compared with backward-swept wings [1]. Moreover, relevant theories and experimental data have proven that the 3D effects lead to a decrease in the effective sweep angle at the leading edge of a forward-swept wing, which is conducive to maintaining a large area of laminar flow, thereby effectively suppressing the instability induced by the crossflow and the attachment line at the leading edge [2,3,4]. Therefore, forward-swept wings are more suitable for natural laminar flow than backward-swept wings [5].

Although forward-swept wings have so many excellent attributes, they have not been widely used. One reason is that the commonly used isotropic metal materials tend to induce aeroelastic divergence and excessive weight for forward-swept wings under combined bending and torsional loads [1,6]. Nowadays, this problem can be solved by using advanced composite materials in the design of wing structures [7]. Composite materials such as carbon-fiber-reinforced polymer (CRFP) have the advantages of low density, good fatigue resistance, and high specific stiffness and strength, which can reduce the weight and restrain the unfavorable deformation of the wing effectively [8,9]. Another reason is that the design of forward-swept wings is challenging. Although traditional aerodynamic design methods can achieve extensive laminar flow, they fail to consider the coupling relationship between aerodynamics and structure, leading to inaccurate optimization results [10].

Due to its comprehensive consideration of the mutual influence between aerodynamics and structure, multidisciplinary design optimization (MDO) is more suitable for the optimization of composite forward-swept wings, thus garnering widespread attention. First, the high-fidelity coupled aero-structural numerical simulation method using the Reynolds-averaged Navier–Stokes (RANS) equations and the structural finite element method (FEM) was developed, and its accuracy was verified by comparing the wind tunnel test results and simulation results of typical aircrafts, such as the High Reynolds Number Aero-Structural Dynamics (HIRENASD) [11,12,13,14,15] and the Common Research Model (CRM) [16,17,18]. Numerous studies have verified that the numerical results considering the aeroelastic effect are closer to experimental data. Then, due to the reliable analysis results, high-fidelity coupled aero-structural numerical simulation was applied to MDO of composite wings including transport aircraft wings [19,20,21], hydrofoils [22,23,24], and forward-swept wings [25,26,27]. Among the design works for the forward-swept wings, Xue et al. [25] adopted the genetic algorithm to perform aeroelastic tailoring on the composite material forward-swept wing, which improved the wing stiffness. Nevertheless, their work did not take into account the laminar–turbulent transition, which would reduce the fidelity of optimization results. Wunderlich et al. [26,27] proposed an efficient MDO method to design the composite NLF forward-swept wing of a high subsonic civil aircraft and improved the wing performance. However, in their studies the laminar–turbulent transition, positions on the wing are fixed to simplify the optimization problems, which may lead to less reliable optimization results.

The purpose of this article is to propose an efficient aero-structural optimization method for the design of the composite forward-swept NLF wing, in which the dual e^N transition method is employed to perform the laminar–turbulent transition prediction automatically [28]. The auxiliary laminate method is used to simplify the complex ply stacking structure in the laminate. And the shared-layer blending (SLB) method [29,30] is used to determine the ply stacking scheme by selecting shared-layer groups. To solve MDO problems with both aerodynamic shape variables and composite structural variables, an efficient discrete variable handling method is developed and coupled in the surrogate-based optimized algorithm. In addition, the Kreisselmeier–Steinhauser (KS) method [31] is employed to deal with large-scale geometric constraints, ply fraction constraints and material failure constraints; it can aggregate all constraints into one. With all the methods mentioned above, the proposed composite wing aero-structural optimization method becomes efficient and has great potential for engineering design, thus providing some useful guidance for future work.

The article is organized as follows. Section 2 presents the high-fidelity coupled aero-structural simulation method used in this work. Section 3 describes the proposed aero-structural optimization method for the composite forward-swept NLF wing. Section 4 verifies the proposed method by performing the aero-structural optimization of the A320-class composite forward-swept NLF wing. Section 5 presents some conclusions.

2. High-Fidelity Coupled Aero-Structural Simulation

2.1. CFD Solver

An in-house RANS-based computational fluid dynamics (CFD) solver [32,33,34,35,36] is used to simulate the viscous flows over the wing. The RANS equations are spatially discretized by the Jameson–Schmidt–Turkel scheme, and the implicit LU-SGS scheme is used for time integration. The turbulence model is the Spalart–Allmaras (SA) model, which uses artificial viscosity for correction, and introduces accelerated convergence technologies such as local time step, multi-grid, and implicit residual smoothing.

For the design of NLF wing, a transition module is coupled in the RANS solver to perform automatic laminar–turbulent transition prediction during optimization, which can predict local transition positions on 3D wings by the dual e^N transition method [37,38]. The dual e^N transition prediction method is widely used for predicting the location of boundary layer transition from laminar to turbulent flow. Rooted in linear stability theory, its core premise is that transition occurs when small-amplitude disturbances within the boundary layer are amplified by a factor of e^N (where N is the total amplification factor). The “dual” aspect specifically addresses three-dimensional flows by simultaneously tracking the growth of Tollmien–Schlichting (T-S) instability and crossflow (CF) instability. The method calculates separate N-factors for each instability type (N_TS and N_CF) along the flow path. Transition is predicted to occur at the location where either N_TS or N_CF first exceeds the given threshold value.

The process of automatic transition prediction (shown in Figure 1) can be described as follows:

(1). Firstly, the CFD solver is used to numerically simulate the viscous flow around the wing; flow parameters such as the pressure coefficients at the outer boundary of the wing’s boundary layer are extracted from the computational results and used as boundary conditions for solving the 3D laminar boundary layer equations.

(2). Secondly, the 3D laminar boundary layer equation solver is employed to solve the wing’s boundary layer equations, obtaining velocity profiles, temperature profiles, and their first- and second-order derivatives.

(3). Thirdly, the dual e^N transition method is used to perform a linear stability analysis and predict the transition locations along the wing spanwise direction, and then the transition locations are fed back to the CFD solver.

(4). The process is repeated until the transition location converges.

To validate the current approach, the flow simulation considering transition prediction of the DLR-F4 wing-body configuration is performed. The design condition is that the freestream Mach number Ma = 0.785, the Reynolds number Re = 6.0 × 10⁶, and the angle of attack AoA = −0.87°. The computational grid is a multi-block structured grid with 4.2 million cells, as shown in Figure 2. The height of the first layer of the grid is 1.2 × 10⁻⁶, and y+ is 0.8. The amplification N factors are set as [N_TS, N_CF] = [10.5, 7.5]. Predicted transition locations are in good agreement with experimental results [39], as depicted in Figure 3. Please note that the transition positions along the wing span vary significantly; as a result, the CFD results based on fixed transition positions are unreasonable.

2.2. FEM Solver

ANSYS APDL 16.0 is used to perform FE analysis of composite wings. It is a powerful simulation software widely used for engineering analysis and design. In our studies, it is used to (1) build the wing structural model and define the material properties, (2) generate wing structural grid and perform FE analysis, (3) visualize and extract the results such as the maximum structural deformation and material failure factor.

Shell 181 element (see Figure 4) is used for FE analysis; it is a four-node element with six degrees of freedom at each node, and has good nonlinear property suitable for structural analysis of composite laminates, supporting up to 255 material layers. The material used for building the wing FE model is CRFP. Table 1 lists the material performances including the elastic modulus, shear modulus, density, tensile strength, compressive strength, shear strength and Poisson’s ratio of the composite material. The single layer thickness of the composite material is 0.125 mm.

In addition, the first-ply failure theory is used to define the failure of composite wings under the premise that a specific ply would fail if the failure index at any node within that ply reaches a value of 1. We use Tsai-Wu failure criteria to achieve the failure index, and it can be defined as

(1) $F_{1} σ_{1} + F_{2} σ_{2} + F_{11} σ_{1}^{2} + F_{22} σ_{2}^{2} + F_{66} τ_{12}^{2} + 2 F_{12} σ_{1} σ_{2} < 1 .$

The coefficients F₁, F₂, F₁₁, F₂₂, F₆₆ are determined by experimental results obtained from mono-directional fiber-reinforced specimens under simple loading conditions. τ₁₂ is the laminate shear stress. σ₁, σ₂ are the laminate stress along and transverse to the fiber direction.

2.3. Coupled Aero-Structural Simulation Framework

The coupled aero-structural simulation framework, namely CASSF, is constructed by selecting the CFD solver and the CSM solver as main components, as shown in Figure 5. Furthermore, an in-house code based on radial basis functions (RBFs) is developed to transfer the aerodynamic loads and structural deflections between CFD and CSM. Another in-house code is applied to generate a new volume grid for the deformed aerodynamic shape. One can run CASSF as follows:

(1). Firstly, the flow simulation is performed by our CFD solver. Then, the aerodynamic loads are interpolated to the FE model from the CFD surface grid.

(2). Secondly, structural deformations of the wing are obtained by the CSM solver and interpolated to the CFD surface grid.

(3). Thirdly, a new volume grid is generated according to the new surface grid. Then, it is used to obtain aerodynamic loads again.

(4). All above steps are repeated until one of the termination conditions is satisfied.

To validate the current approach, the coupled aero-structural simulation of the HIRENASD wing is performed. It is a typical transport aircraft wing shape designed by the RWTH Aachen University. The nominal flight condition is Ma = 0.8, Re = 7 × 10⁶, AoA = 1.5°. The CFD and structural computational grids are shown in Figure 6. The material is 18 nickel maraging steel whose Poisson’s ratio is 0.3 and Young’s Modulus is 181.3 GPa. The comparison of sectional pressure distributions among the coupled aero-structural simulation results, experimental data and CFD results of the fine grid at different spanwise locations is shown in Figure 7. It can be seen that the numerical results considering the aeroelastic effect are more consistent with the experimental data compared with the ones obtained by CFD only. Table 2 lists the results including aerodynamic performance, torsions δ_tip and deformations of wing tip u_tip of HIRENASD wing obtained with CFD and coupled aero-structural simulations. The lift coefficient and drag coefficient obtained through the aero-structural simulation are 0.03 and 10 counts less than CFD results due to the deformation of the wing, which shows a significant difference for a typical transport aircraft wing. The maximum displacement of wing tip u_tip obtained by the aero-structural simulations is 12.73 mm, which is very close to the experimental results.

3. Optimization Strategy

3.1. Surrogate-Based Composite Wing Aero-Structural Optimization

The surrogate-based optimization (SBO) algorithm is used to find global optimum in our studies. To solve MDO problems with many discrete structural variables and large-scale constraints such as geometric constraints, ply fraction constraints and material failure constraints, an efficient discrete variable handling method is developed and coupled in the SBO algorithm. And the KS method is introduced to the SBO algorithm to deal with large-scale constraints. The SBO algorithm and the aero-structural simulation method are combined as the framework of the composite wing aero-structural optimization. Figure 8 sketches the optimization framework, and the optimization progress can be described as follows:

(1). Latin hypercube sampling (LHS) is selected to generate initial samples. The aerodynamic variables are used to drive the free-form deformation (FFD) method [40] to parametrize the wing and generate the new aerodynamic shape. All the structural variables are discretized by the developed discrete variable handling method, and they are used to generate the composite structural model.

(2). Then, responses of the initial samples are achieved by CASSF. Initial kriging models are built based on the samples and their responses.

(3). Five new samples are selected by the parallel infill-sampling criteria; the FFD method is used to generate new aerodynamic models, and all the structural variables are discretized by the developed discrete variable handling method, and they are used to generate new composite structural models.

(4). The corresponding responses of these new aerodynamic shapes and composite structural models are solved by CASSF.

(5). Kriging models are updated based on these samples and their responses.

(6). Steps 3 and 4 are repeated until one of the termination conditions is satisfied.

Finally, the shared-layer blending (SLB) method is adopted to analyze the optimization results obtained by the auxiliary laminate method, and provide a set of reasonable ply stacking sequences satisfying the following composite design rules:

(1). The laminate contains plies in 0°/±45°/90° orientations;

(2). At least one set of ±45° plies should be contained in the outer skin due to the damage tolerance requirements;

(3). The number of plies in the same orientation placed sequentially in the stack is limited to four;

(4). The ply number of each ply orientation is no less than 10% of total numbers in a laminate.

3.2. Discrete Variable Handling Method

In the optimization of composite structures, the design variables are generally integers, such as the number of plies in each ply orientation. If such variables cannot be effectively handled, the accuracy of the optimization results is difficult to guarantee, and the optimization results may even fail to satisfy the constraints. Therefore, an efficient integer variable handling method applicable to composite structural optimizations is developed. It transforms some real variables generated by DoE and sub-optimization into the nearest integer values, which means that SBO can be used to directly solve optimization problems with integer variables, avoiding tedious manual rounding work after the optimization. As shown in Figure 9, 12 integer samples are randomly generated using the developed method, which proves the effectiveness of this method.

To validate the developed discrete variable handling method, we perform the optimization of the pressure vessel design test case with four design variables and four constraints. This case can be defined as

(2) $\begin{array}{l} M i n . & f (x) = 0.6224 x_{1} x_{3} x_{4} + 1.7781 x_{2} x_{3}^{2} + 3.1661 x_{1}^{2} x_{4} + 19.84 x_{1}^{2} x_{3} \\ s . t . & g_{1} (x) = - x_{1} + 0.0193 x_{3} \leq 0 \\ g_{2} (x) = - x_{2} + 0.00954 x_{3} \leq 0 \\ g_{3} (x) = - π x_{3}^{2} x_{4}^{2} - 4 / 3 π x_{3}^{3} + 1296000 \leq 0 \\ g_{4} (x) = x_{4} - 240 \leq 0 \\ 10 \times 0.0625 \leq x_{1}, x_{2} \leq 99 \times 0.0625, 10.0 \leq x_{3}, x_{4} \leq 200.0 \end{array} .$

The first two design variables x₁, x₂ must be multiples of 0.0625. The best solution of the objective function has been observed to be f (x*) = 6059.714335 [41].

To eliminate the randomness, we repeated the optimization five times. Figure 10 shows the convergence history of the objective function, and the optimization result is listed in Table 3. The best objective functional of 6059.758484 is very close to the reference optimum, which indicates that the developed method is effective.

3.3. Large-Scale Constraint Handling Method

Composite wings have complex stacking structures, and their design needs to consider material properties, failure mechanisms, and processing requirements, resulting in a large number of geometric constraints, strength constraints, as well as machining process constraints in the composite wing aero-structural optimization, which brings a great challenge to the optimization method due to the large computational cost.

To efficiently handle large-scale constraints involved in composite wing aero-structural optimization, the KS method is introduced in our studies; it can lump a large number of constraints into one or a few, and handle multiple kinds of constraints simultaneously, thus greatly saving computational costs of an SBO [42]. The KS method can be defined as

(3) $K S [g (x)] = g_{\max} (x) + \frac{1}{ρ} \ln [\sum_{j = 1}^{n} e^{ρ (g_{j} (x) - g_{\max} (x))}],$

where g_max(x) is the maximum constraint value, and ρ is the aggregation parameter.

Generally, a wing aero-structural optimization problem can be defined as

(4) $\begin{array}{l} \min & f (x) \\ s . t . & g_{i} (x) \leq 0, i = 1, \dots, N_{1} \\ p_{j} (x) \leq 0, j = 1, \dots, N_{2} \\ η_{k} (x) \leq 0, k = 1, \dots, N_{3} \\ c (x) \leq 0 \end{array},$

where

f (x)

denotes the objective function, and

g_{i} (x), p_{j} (x), η_{k} (x)

represent a large number of geometric constraints, ply fraction constraints and material failure constraints. c(x) denotes the remaining constraints, such as lift constraint and structure deformation constraint. With the KS method, Equation (4) can be transformed into

(5) $\begin{array}{l} \min & \hat{f} (x) \\ s . t . & {\hat{g}}_{K S} (x) \leq 0 \\ {\hat{p}}_{K S} (x) \leq 0 \\ {\hat{η}}_{K S} (x) \leq 0 \\ \hat{c} (x) \leq 0 \end{array},$

The computational cost of training surrogate models before and after using the KS method is compared to evaluate how much benefit this method brings to the optimization efficiency. Taking a wing aero-structural optimization problem with 80 design variables as an example, there are 525 geometric constraints, 40 ply fraction constraints and 10,522 material failure constraints, and another 2 constraints for limiting the lift and wing-tip deformation. A totala of 100 samples are generated by DoE to build the initial surrogate models, and 5 samples are selected at each updating cycle and evaluated in parallel to update the surrogate models. This process is repeated until maximum number of sample points of 380 is reached. As list in Table 4, the CPU time for training one surrogate model based on 380 samples is about 2.5 min on a personal computer; in contrast, the CPU time of training 11,089 surrogate models is as large as 2.77 × 10⁴ min (462 h). The computational cost of training the surrogate models, for the optimizations in Equations (4) and (5), respectively, is compared in Table 2, which confirms the large benefit of using the KS method.

3.4. Auxiliary Laminate Method

For a multi-zone composite laminate, as shown in Figure 11, the panel thickness and ply orientations in each zone are different, which leads to a large number of discrete structural design variables. It is computationally very expensive to optimize the composite structure using the genetic algorithm that can handle the discrete variables. Moreover, as the complexity of the structural model increases, the size of the variables will increase dramatically, which results in the prohibitive computational cost.

To solve this issue, the auxiliary laminate method is developed. By using this method, an auxiliary laminate that equates a symmetric laminate with eight plies in the orientations of 0°/±45°/90° can be constructed. As shown in Figure 12, this method can significantly reduce the number of design variables, such as ply thickness or number of plies in the laminate. There are only four plies on one side of the symmetry plane of the laminate, corresponding to the four orientations ±45°/0°/90°. Assuming that the thickness of each ply is t1, t2, t3 and t4, respectively, the total thickness of the laminate is t = 2(t1 + t2 + t3 + t4), which greatly reduces the difficulty of the optimization.

3.5. Shared-Layer Blending Method

Once the optimization of the composite wing structure is completed, the number of plies in each design zone of the wing are determined according to the variable values. In order to determine the ply stacking sequences, and avoid stress concentration between two adjacent laminates, the SLB method is used in our studies. One can perform the SLB method as follows:

(1). Ply numbers for each orientation of 0°/±45°/90° of all design regions are ranked. For each ply orientation, the minimum ply numbers from all zones are defined as the first shared layers.

(2). Then, finding the minimum ply numbers for each ply orientation from the remaining plies after excluding the first shared plies, they are defined as the second shared plies.

(3). The second step is repeated until all shared layers are identified.

(4). Lastly, the shared plies are placed in the stacks for all panels according to the design criteria.

4. Examples and Results

In this section, the proposed aero-structural optimization method is used to design an A320-class composite forward-swept NLF wing.

4.1. CFD Model

The baseline geometry used for optimizations is an A320-class forward-swept NLF wing (see Figure 13) configured using the NPU-LSC-72613 NLF airfoil. It has a half wingspan of 17.5 m, the wing taper ratio is 0.3, and the root chord length is 5.25 m. The sweep angle of the leading edge is −19°. The location of the reference point is (x, y, z) = (−0.411, 0, 0) m and the reference length is 3.749 m. The nominal flight condition is C_L = 0.5, Ma = 0.76, H = 10 km in viscous transonic flow.

Figure 14 shows two levels of multi-block CFD structured grids. The L1 grid is used for optimizations, and the L0 grid with 2.273 million cells is used for assessing the optimization results. The L1 grid has 0.778 million cells; the height of the first layer of the grid is 1.6 × 10⁻⁶, and y+ is 1.0. We increase cells of the surface and normal direction of the wing, so the size of the L0 grid reaches 2.273 million, the height of the first layer of the grid is 1.1 × 10⁻⁶, y+ is 0.9.

4.2. FE Model

We use ANSYS APDL to generate the structure of the forward-swept NLF wing. As shown in Figure 15, spars are arranged at 20 and 60% chordwise locations, and 21 ribs are evenly distributed spanwise. The FE model with 11,832 elements and 10,522 nodes is shown as Figure 16; the element type is Shell181 and the material is CFRP, as listed in Table 1. The 0° fiber orientation is defined at an angle with reference to 50% of the chord of the wing.

4.3. Optimization Problem Formulation

The aero-structural optimization of the composite forward-swept NLF wing is performed by using the surrogate-based multi-round optimization strategy. The first-round optimization is executed to reduce the mass and drag of the wing. Then, we use the SLB method to analyze shared plies and give determined ply stacking sequences. The second-round optimization aims to achieve a further reduction in drag. Finally, the optimization results are evaluated using the fine CFD mesh.

For the first round, the angle of attack and 48 wing shape variables are selected as the aerodynamic design variables. As shown in Figure 17, the displacements of the 48 FFD control points in the y-axis direction are selected as design variables to change the wing shape; the remaining 8 (in black) are fixed to keep the trailing edge unchanged. To obtain a practical design, 525 (21 spanwise × 25 chordwise) thickness constraints (see Figure 18) and a volume constraint are added to the optimization. The thickness constraints are enforced to be no less than 97% of those of the baseline wing, and the volume constraint is enforced to be no less than that of the baseline wing. The structural design variables are the numbers of the plies in 0°/±45°/90° orientations of the 10 design regions of the wing skins (see Figure 19). The structural constraints include 40 ply fraction constraints, a wing tip deflection constraint and 10,522 failure constraints. The ply fraction constraints require the ply number of each ply orientation to be no less than 10% of total numbers. The wing deformation constraint requires that the deformation of the wing tip be no less than 10% of the wing half-span (e.g., 1.75 m).

As a result, the first-round optimization contains 80 design variables and 11,089 constraints. The second-round optimization contains 48 variables and 11,049 constraints, since the structural design variables are not considered. Table 5 gives the summary of the optimization models. The thickness constraints, ply fraction constraints and failure constraints are lumped into three during optimization using the KS method. As a result, the cost of optimization is significantly reduced.

4.4. Results of Optimization

The convergence history of the optimization is shown in Figure 20, and the results are listed in Table 6. The wing mass is reduced from 2057.970 kg to 1549.755 kg, resulting in the mass reduction of 508.215 kg. The drag coefficient drops from 151.39 counts to 138.84 counts; about 12.55-count drag-reducing benefits are achieved.

Then, the optimization results are evaluated by using the L0 mesh. The evaluated results are listed in Table 7. It still maintains about 10-count drag-reducing benefits. Meanwhile, other performance attributes, such as the moment coefficient and the displacement, change very little. Figure 21 shows the comparison of the pressure distribution between the baseline and optimal wing shape. It is obvious that the shock wave on the suck surface at the wing root is significantly weakened, which is due to the increase in the thickness of the leading edge, leading to the acceleration of the airflow and the rapid reduction in the favorable pressure gradient to the adverse pressure gradient. The laminar–turbulent transition is significantly delayed on the upper surface of the wing. The areas of the laminar flow regions on the baseline and optimal wing are calculated and listed in Table 8. The laminar flow region on the suck surface of the optimal wing is 6.579 m² larger than that of the baseline wing, while the laminar flow region on the lower surface of the optimal wing is comparable to that of the baseline wing, with a negligible difference of about 0.2 m².

Figure 22 shows the amplification of TS and CF waves on the upper surface of the baseline and optimal wing. As can be seen from Figure 21 and Figure 22, the strong favorable pressure gradient (cp value decreases from large to small) inhibits the amplification of TS waves on the upper surface of the baseline wing. As a result, the transition at the root of the wing is caused by the CF wave. As the wing twist angle gradually increases along the span, the favorable pressure gradient on the upper surface becomes more moderate, and CF waves are effectively suppressed, which causes the transition position to occur at the location where the shock wave is generated. Compared with the baseline wing, the favorable pressure gradient is weakened on the upper surface at the root of the optimal wing, CF waves are suppressed, and TS waves are slightly enhanced but do not exceed the prescribed threshold N_CF. The favorable pressure gradient at the leading edge is enhanced at other spanwise positions of the wing, leading to the enhancement of CF waves, but they do not exceed the prescribed threshold N_CF. As a result, the transition position is predicted to occur at the location where the shock wave is generated.

Figure 23 shows the amplification of TS and CF waves on the lower surface of the baseline and optimal wing. The strong favorable pressure gradient inhibits the amplification of TS waves on the lower surface of the baseline wing, and the transition is caused by the CF wave. Compared with the baseline wing, the favorable pressure gradient is weakened on the lower surface of the optimal wing; both CF waves and TS waves are suppressed. As a result, the transition occurs at the location where the shock wave is generated. These results show that the proposed composite wing aero-structural optimization method is efficient and can handle a large number of constraints, and it can provide definite ply stacking sequences of the wing skin (see Table 9).

5. Conclusions

In this article, an efficient aero-structural optimization method for the composite forward-swept NLF wing was investigated within the framework of an SBO. To verify the proposed method, the aero-structural optimization of an A320-class composite forward-swept NLF wing in viscous subsonic flow was performed. Some conclusions can be drawn as follows.

(1). It is necessary to consider the aeroelastic effect when optimizing the wing, since the numerical results considering the aeroelastic effect are more consistent with the experimental data compared with those of CFD simulations.

(2). Surrogate-based optimization combining the discrete variable handling method, the SLB method and the KS method can solve MDO problems with many discrete variables and large-scale constraints effectively compared with existing surrogate-based aero-structural optimization methods, and it can provide the definitive ply stacking scheme as well.

(3). The proposed method offers great potential in the aero-structural optimization of the composite forward-swept NLF wing. Both the mass and drag of the wing are reduced significantly. And due to the use of the dual e^N transition method, the proposed method can accurately predict the transition position on the wing compared with existing methods, and the area of the laminar flow region on the upper surface of the wing is increased by 24.3%.

In future studies, the aero-structural optimization of a full aircraft configuration with composite forward-swept NLF wings will be investigated.

Author Contributions

Conceptualization, K.W.; methodology, K.W. and X.W.; software, K.W.; validation, X.W., X.H. and B.X.; formal analysis, K.W., X.W. and B.X.; investigation, K.W., X.H. and Z.S.; data curation, K.W. and Z.S.; writing—original draft preparation, K.W.; writing—review and editing, K.W., X.W., J.D. and T.W.; visualization, K.W. and X.H.; project administration, K.W. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

Authors Kai Wang, Xiaoguang Wang, Xiujie Han, Bo Xiao, Zhiyuan Shan, Jie Ding were employed by the company Norinco Group Air Ammunition Research Institute Co., Ltd. Author Tao Wu was employed by the company Northwest Institute of Mechanical and Electrical Engineering. The author declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1 RANS solver coupled with the automatic transition prediction module.

Figure 2 O-H multiblock structured grid of DLR-F4 wing-body configuration.

Figure 3 Predicted transition line and experimental results [39] on upper surface of DLR-F4 wing.

Figure 4 Shell 181 element.

Figure 5 High-fidelity composite wing aero-structural simulation framework.

Figure 6 Computational grids of the HIRENASD configuration.

Figure 7 Comparison of sectional pressure distributions of the HIRENASD wing.

Figure 8 Framework of surrogate-based composite wing aero-structural optimization.

Figure 9 Twelve integer samples randomly generated using the developed method.

Figure 10 Convergence histories of the optimizations.

Figure 11 Multi-zone composite laminate.

Figure 12 Auxiliary laminate with 8 plies.

Figure 13 A320-class forward-swept NLF wing model.

Figure 14 C-H multi-block CFD structured grids of the NLF wing.

Figure 15 The structure of the composite NLF wing.

Figure 16 NLF wing structural computational grid.

Figure 17 FFD parameterization of NLF wing.

Figure 18 The 525 thickness constraints enforced on the wing.

Figure 19 The 10 structural design zones on the wing.

Figure 20 Optimization convergence history.

Figure 21 Comparison of pressure distributions of the baseline and optimal wing.

Figure 22 The amplification of the TS and CF waves on the upper surface of the baseline and optimal wings.

Figure 23 The amplification of the TS and CF waves on the lower surface of the baseline and optimal wings.

Table 1

Material properties of the wing.

E₁[GPa]	E₂[GPa]	G₁₂[GPa]	G₁₃[GPa]	G₂₃[GPa]	ρ[kg/m³]	X_t[MPa]	X_c[MPa]	Y_t[MPa]	Y_c[MPa]	S[MPa]	v ₁₂
117.9	9.7	4.8	4.8	4.8	1550	1648	1304	64	228	71	0.34

Table 2

Comparison of CFD and coupled aero-structural simulation results.

Analysis Type	C_L	C_D	u_tipEXP. = 12.5 mm	δ _tip
CFD	0.39508	0.02231	0 mm	0°
Aero-structural simulation	0.36587	0.02137	12.73 mm	−0.52°

u_tip: The displacement of wing tip. The experimental result is 12.5 mm. δ_tip: The twist of wing tip.

Table 3

Optimization results of the pressure vessel design test case.

	Number of Runs	Optimum	Surrogate Prediction Error at Optimum	Optimal Values of Design Variables x
	Number of Runs	Optimum	Surrogate Prediction Error at Optimum	x ₁	x ₂	x ₃	x ₄
Reference optimum	/	6059.714335	/	0.8125	0.4375	42.0984	173.736366
SBO	5	6059.758484	1.94 × 10⁻⁷%	0.8125	0.4375	42.0981	174.641011

Table 4

Computational cost of training surrogate models before and after using the KS method.

Optimization Problem	Number of Surrogate Models	CPU Time for Training Surrogate Models/min
Optimization Problem	Number of Surrogate Models	1 Iteration	All the Iterations
Equation (4)	11,089	2.77 × 10⁴	1.55 × 10⁶
Equation (5)	5	12.5	700

Table 5

Optimization models of the composite forward-swept NLF wing.

Round	Design Variables		Objective	Constraints
Round	Description	Quantity	Objective	Function	Description	Quantity
Round1	Wing shape	48	$\begin{matrix} M i n . 0.5 * O b j 1 \\ + 0.5 * O b j 2 \\ O b j 1 = \frac{M a s s}{M a s s_{0}} \\ O b j 2 = \frac{C_{D}}{C_{D, 0}} \end{matrix}$	C_L = 0.5	Lift constraint	1
				t − 0.97 t₀ ≥ 0	Geometric constraints	525
				g ≥ 10%	ply fraction constraints	40
	Ply numbers	32		u ≤ 1.75 m	Wing deformation constraint	1
	Ply numbers	32		η ≤ 1	failure constraints	10,522
	Total	80				11,089
Round2	Wing shape	48	Min. C_D	C_L = 0.5		1
				t − 0.97 t₀ ≥ 0		525
				u ≤ 1.5 m		1
				η ≤ 1		10,522
	Total	48				11,049

Table 6

Aero-structural optimization results of the composite forward-swept NLF wing.

	AoA/°	C_L	C_D	C_M	Mass/kg	u_tip/m	δ_tip/°
Baseline	0.772	0.5	0.015139	−0.08377	2057.970	1.53881	2.57460
Opt.	0	0.5	0.013884	−0.10626	1549.755	1.58762	0.64540

u_tip, δ_tip: the displacement and twist angle of the wing tip.

Table 7

Aero-structural optimization results assessed by using L0 CFD grid.

	AoA/°	C_L	C_D	C_M	Mass/kg	u_tip/m	δ_tip/°
Baseline	0.72	0.5	0.013913	−0.08030	2057.970	1.54505	2.54507
Opt.	−0.15	0.5	0.012914	−0.10546	1549.755	1.58956	0.58890

u_tip, δ_tip: the displacement and twist angle of the wing tip.

Table 8

Areas of the laminar flow regions of the baseline and optimal wing.

		Areas of the Laminar Flow Regions/m²
baseline	upper surface	27.0437
baseline	lower surface	25.7484
Opt.	upper surface	33.6227 (+6.579)
Opt.	lower surface	25.5332 (−0.2152)

Table 9

Ply stacking sequences of the optimal wing skins of the composite forward-swept NLF wing.

Skin	No. of Design Zones Along Span	Ply Stacking Sequences
	1	[45/−45₂/0₃/45₂/0₄/45/90₂/0/−45/0₄/−45₂/90]_s
Upper surface	2	[±45/0₃/45₂/0₄/45/90₂/0/−45/0₃/−45₂/90]_s
	3	[45/−45₂/0₃/45/0₄/90₂/0₄/−45₂/90]_s
	4	[45/−45₂/0₂/45₂/0₂/45/90₂/0/−45/0₂/−45₂/90]_s
	5	[45/−45/0₃/45/0₄/90/0₃/−45/90]_s
	1	[±45/0/45/0₃/90/0₄/−45/45/−45/0₃/45₂/90₃/0₄/−45/90]_s
Lower surface	2	[±45/0/45/0₃/90/45/0/45/0₃/−45/45/−45/0₃/45₂/90₃/0₄/−45/90]_s
	3	[±45/45/0/90/0₄/−45₂/0₃/90/0₄/90]_s
	4	[±45/45/0/90/45/0/45/0₂/−45/45/−45/0₃/45₂/90₂/0_3/−45/90]_s
	5	[±45/45/0/90/0₂/−45₂/0/45/90₂/0₂/90]_s

References

1. Krone, N.J. Forward-Swept Wing Technology. Proceedings of the Evolution of Aircraft Wing Design Symposium; Dayton, OH, USA, 18–19 March 1980.

2. Redeker, G.; Wichmann, G. Forward sweep—A favorable concept for a laminar flow wing. J. Aircr.; 1991; 28, pp. 97-103. [DOI: https://dx.doi.org/10.2514/3.45997]

3. Rossow, C.C. Aerodynamics—A discipline swept away?. Aeronaut. J.; 2010; 114, pp. 599-609. [DOI: https://dx.doi.org/10.1017/S0001924000004085]

4. Jaffal, H.M.; Darwish, A.S.; Hassan, I.A. Theoretical and Experimental Study of a Forward Swept Wing. Anbar J. Eng. Sci.; 2010; 3, pp. 15-30. [DOI: https://dx.doi.org/10.37649/aengs.2010.14202]

5. Frota, J. NACRE novel aircraft concepts. Aeronaut. J.; 2010; 114, pp. 399-404. [DOI: https://dx.doi.org/10.1017/S0001924000003869]

6. Seitz, A.; Kruse, M.; Wunderlich, T.; Bold, J.; Heinrich, L. The DLR Project Lam AiR: Design of a NLF Forward Swept Wing for Short and Medium Range Transport Application. Proceedings of the 29th AIAA Applied Aerodynamics Conference; Honolulu, HI, USA, 27–30 June 2011.

7. Chen, G.S.; Dugundji, J. Experimental aeroelastic behavior of forward-swept graphite/epoxy wings with rigid-body freedom. J. Aircr.; 1987; 24, pp. 454-462. [DOI: https://dx.doi.org/10.2514/3.45501]

8. Tian, W.; Yang, Z.C.; Gu, Y.S.; Ouyang, Y. Aeroelastic tailoring of a composite forward-swept wing using a novel hybrid pattern search method. J. Aerosp. Eng.; 2016; 29, 04016056. [DOI: https://dx.doi.org/10.1061/(ASCE)AS.1943-5525.0000652]

9. Bach, C.; Jebari, R.; Viti, A.; Hewson, R. Composite stacking sequence optimization for aeroelastically tailored forward-swept wings. Struct. Multidiscip. Optim.; 2017; 55, pp. 105-119. [DOI: https://dx.doi.org/10.1007/s00158-016-1477-3]

10. Krüger, W.R.; Klimmek, T.; Liepelt, R.; Schmidt, H.; Waitz, S.; Cumnuantip, S. Design and aeroelastic assessment of a forward-swept wing Aircraft. CEAS Aeronaut. J.; 2014; 5, pp. 419-433.

11. Chwalowski, P.; Heeg, J. FUN3D Analyses in Support of the Second Aeroelastic Prediction Workshop. Proceedings of the 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition; Grapevine, TX, USA, 7–10 January 2013.

12. Heeg, J. Overview of the Aeroelastic Prediction Workshop. Proceedings of the 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition; Grapevine, TX, USA, 7–10 January 2013.

13. Chwalowski, P.; Florance, J.P.; Heeg, J.; Wieseman, C.D.; Perry, B.P. Preliminary computational analysis of the HIRENASD configuration in preparation for the aeroelastic prediction workshop. Proceedings of the International Forum on Aeroelasticity and Structural Dynamics (IFASD) 2011; Paris, France, 26–30 June 2011.

14. Ritter, M. Static and forced motion aeroelastic simulations of the HIRENASD wind tunnel model. J. Mater. Sci.; 2006; 42, pp. 5452-5457.

15. Wieseman, C.; Chwalowski, P.; Heeg, J.; Boucke, A.; Castro, J. Structural Dynamics Modeling of HIRENASD in support of the Aeroelastic Prediction Workshop. Proceedings of the 54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference; Boston, MA, USA, 8–11 April 2013.

16. Keye, S.; Brodersen, O.; Rivers, M.B. Investigation of Aeroelastic Effects on the NASA Common Research Mode. J. Aircr.; 2014; 51, pp. 1323-1329. [DOI: https://dx.doi.org/10.2514/1.C032598]

17. Tinoco, E.N.; Brodersen, O.P.; Keye, S.; Laflin, K.R.; Feltrop, E.; Vassberg, J.C.; Mani, M.; Rider, B.; Wahls, R.A.; Morrison, J.H. . Summary data from the sixth AIAA CFD drag prediction workshop: CRM cases. J. Aircr.; 2018; 55, pp. 1352-1379. [DOI: https://dx.doi.org/10.2514/1.C034409]

18. Keye, S.; Mavriplis, D. Summary of case 5 from sixth drag prediction workshop: Coupled aerostructural simulation. J. Aircr.; 2018; 55, pp. 1380-1387. [DOI: https://dx.doi.org/10.2514/1.C034427]

19. Kennedy, G.J.; Kenway, G.K.W.; Martins, J.R.R.A. A Comparison of Metallic, Composite and Nanocomposite Optimal Transonic Transport Wings; NASA/CR-2014-218185 Langley Research Center: Hampton, VA, USA, 2014.

20. Brooks, T.R.; Martins, J.R.R.A.; Kennedy, G.J. High-fidelity aerostructural optimization of a high aspect ratio tow-steered composite wing. Proceedings of the 58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference; Grapevine, TX, USA, 9–13 January 2017.

21. Brooks, T.R.; Martins, J.R.R.A.; Kennedy, G.J. High-fidelity aerostructural optimization of tow-steered composite wings. J. Fluids Struct.; 2019; 88, pp. 122-147. [DOI: https://dx.doi.org/10.1016/j.jfluidstructs.2019.04.005]

22. Garg, N.; Kenway, G.K.W.; Martins, J.R.R.A.; Young, Y.L. High-fidelity multipoint hydrostructural optimization of a 3-D hydrofoil. J. Fluids Struct.; 2017; 71, pp. 15-39. [DOI: https://dx.doi.org/10.1016/j.jfluidstructs.2017.02.001]

23. Liao, Y.; Martins, J.R.R.A.; Young, Y.L. Sweep and anisotropy effects on the viscous hydroelastic response of composite hydrofoils. Compos. Struct.; 2019; 230, 111471. [DOI: https://dx.doi.org/10.1016/j.compstruct.2019.111471]

24. Liao, Y.; Martins, J.R.R.A.; Young, Y.L. 3-D high-fidelity hydrostructural optimization of cavitation-free composite lifting surfaces. Compos. Struct.; 2021; 268, 113937. [DOI: https://dx.doi.org/10.1016/j.compstruct.2021.113937]

25. Xue, R.R.; Ye, Z.Y.; Ye, K.; Wang, G. Composite material structure optimization design and aeroelastic analysis on forward swept wing. J. Aerosp. Eng.; 2019; 233, pp. 4679-4695.

26. Wunderlich, T.; Dähne, S. Aeroelastic tailoring of an NLF forward swept wing. CEAS Aeronaut. J.; 2017; 8, pp. 461-479. [DOI: https://dx.doi.org/10.1007/s13272-017-0251-6]

27. Wunderlich, T.; Dähne, S.; Heinrich, L.; Reimer, L. Multidisciplinary optimization of an NLF forward swept wing in combination with aeroelastic tailoring using CFRP. CEAS Aeronaut. J.; 2017; 8, pp. 673-690. [DOI: https://dx.doi.org/10.1007/s13272-017-0266-z]

28. Arnal, D. Boundary Layer Transition: Predictions Based on Linear Theory; Special Course on Progress in Transition Modelling AGARD: Neuilly sur Seine, France, 1994.

29. Liu, D.Z.; Toropov, V.V.; Querin, O.M.; Barton, D.C. Bi-level optimization of blended composite wing panels. J. Aircr.; 2011; 48, pp. 107-118. [DOI: https://dx.doi.org/10.2514/1.C000261]

30. Liu, D.Z.; Toropov, V.V.; Querin, O.M.; Barton, D.C. Bi-level optimization of blended composite panels. Proceedings of the 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference; Palm Springs, CA, USA, 4–7 May 2009.

31. Kreisselmeier, G.; Steinhauser, R. Systematic control design by optimizing a vector performance index. Proceedings of the IFAC Symposium on Computer Aided Design of Control Systems; Zurich, Switzerland, 29–31 August 1979.

32. Han, S.Q. High-Accurate Numerical Approaches for Coaxial Rotor Blade-Tip-Vortex Simulation. Ph.D. Thesis; Northwestern Polytechnical University: Xi’an, China, 2021.

33. Han, S.Q.; Song, W.P.; Han, Z.H.; Li, S.B.; Lin, Y.F. Hybrid inverse/optimization design method for rigid coaxial rotor airfoils considering reverse flow. Aerosp. Sci. Technol.; 2019; 95, 105488. [DOI: https://dx.doi.org/10.1016/j.ast.2019.105488]

34. Zhang, Y.; Han, Z.H.; Shi, L.X.; Song, W.P. Multi-round Surrogate-based Optimization for Benchmark Aerodynamic Design Problems. Proceedings of the 54th AIAA Aerospace Sciences Meeting; San Diego, CA, USA, 4–8 January 2016.

35. Liu, J.; Song, W.P.; Han, Z.H.; Zhang, Y. Efficient aerodynamic shape optimization of transonic wings using a parallel infilling strategy and surrogate models. Struct. Multidiscip. Optim.; 2017; 55, pp. 925-943. [DOI: https://dx.doi.org/10.1007/s00158-016-1546-7]

36. Han, Z.H.; Chen, J.; Zhang, K.S. Aerodynamic shape optimization of natural-laminar-flow wing using surrogate-based approach. AIAA J.; 2018; 56, pp. 2579-2593. [DOI: https://dx.doi.org/10.2514/1.J056661]

37. Smith, A.M.O.; Gamberonin, G. Transition, Pressure Gradient, and Stability Theory; Douglas Aircraft Company: Long Beach, CA, USA, 1956.

38. Ingen, J.L.V. A Suggested Semi-Empirical Method for the Calculation of the Boundary Layer Transition Region: VTH-74; Delft University of Technology: Delft, The Netherlands, 1956.

39. Fey, U.; Egami, Y.; Engler, R. High Reynolds number transition detection by means of temperature sensitive paint. Proceedings of the 44th AIAA Aerospace Sciences Meeting and Exhibit; Reno, NV, USA, 9–12 January 2006.

40. Sederberg, T.W.; Parry, S.R. Free-form deformation of solid geometric models. ACM Siggraph Comput. Graph.; 1986; 20, pp. 151-160. [DOI: https://dx.doi.org/10.1145/15886.15903]

41. Yang, X.S.; Gandomi, A.H. Bat algorithm: A novel apporach for global engineering optimization. Eng. Comput.; 2012; 29, pp. 464-483. [DOI: https://dx.doi.org/10.1108/02644401211235834]

42. Wang, K.; Han, Z.H.; Zhang, K.S.; Song, W.P. An Efficient Geometric Constraint Handling Method for Surrogate-Based Aerodynamic Shape Optimization. Eng. Appl. Comput. Fluid Mech.; 2023; 17, e2153173. [DOI: https://dx.doi.org/10.1080/19942060.2022.2153173]

© 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.

Global Aero-Structural Optimization of Composite Forward-Swept Wings Considering Natural Laminar Flow

Content area