阅读说明：本技术 基于dffd网格变形技术的椭圆对称翼型多目标优化方法 (Elliptic symmetric wing type multi-objective optimization method based on DFFD grid deformation technology ) 是由 程诗信 曹伟 李宪开 米百刚 于 2021-09-15 设计创作，主要内容包括：本发明提供了基于DFFD网格变形技术的椭圆对称翼型多目标优化方法,本发明属于气动外形优化领域。该方法步骤如下：1)选定网格变形区域,并布置FFD控制点,四个相邻的控制点与连接它们的粗实线组成一个控制体,所有控制体组合起来即为网格变形区域；2)在外形上选定若干直接操作点,设置B样条函数的阶次,初始化直接操作点和网格点的局部坐标,对于不规则布置的FFD控制框架,需要将它们转换成一个优化问题来求解近似的局部坐标s,t和u值；3)使用直接操作点局部坐标计算B样条系数矩阵；4)用系数矩阵和直接操作点位移反计算FFD控制点的位移；5)利用移动后的控制点计算变形区域空间网格变形后的位置。(The invention provides a DFFD grid deformation technology-based elliptic symmetric wing type multi-target optimization method, and belongs to the field of aerodynamic shape optimization. The method comprises the following steps: 1) selecting a grid deformation area, arranging FFD control points, forming a control body by four adjacent control points and a thick solid line connecting the four adjacent control points, and combining all the control bodies to form the grid deformation area; 2) selecting a plurality of direct operation points on the appearance, setting the order of a B spline function, initializing local coordinates of the direct operation points and grid points, and for the FFD control frame which is irregularly arranged, converting the direct operation points and the grid points into an optimization problem to solve approximate local coordinates s, t and u; 3) calculating a B spline coefficient matrix by using the local coordinates of the direct operation points; 4) calculating the displacement of the FFD control point by using the coefficient matrix and the direct operation point displacement; 5) and calculating the deformed position of the spatial grid of the deformed region by using the moved control point.)

1. The multi-objective optimization method of the elliptic symmetric wing profile based on the DFFD grid deformation technology is characterized by comprising the following steps of:

1) selecting a grid deformation area, arranging FFD control points, forming a control body by four adjacent control points and a thick solid line connecting the four adjacent control points, and combining all the control bodies to form the grid deformation area; the mesh deformation region may be a local region of the mesh, or may include the entire mesh;

2) selecting a plurality of direct operation points on the appearance, setting the order of a B spline function, initializing local coordinates of the direct operation points and grid points, and for the FFD control frame which is irregularly arranged, converting the direct operation points and the grid points into an optimization problem to solve approximate local coordinates s, t and u;

3) calculating a B spline coefficient matrix by using the local coordinates of the direct operation points;

4) calculating the displacement of the FFD control point by using the coefficient matrix and the direct operation point displacement;

5) and calculating the deformed position of the spatial grid of the deformed region by using the moved control point.

2. The elliptic symmetry wing multi-objective optimization method based on the DFFD grid deformation technology is characterized by comprising the following specific processes:

when a designer applies a certain displacement delta X to a direct control point on a target object, corresponding position change can be generated on a control frame; q direct control points are arranged, and the initial position is S_fF is 1,2, …, q, and the local coordinate corresponding to it is(s)_f,t_f,u_f) It needs to be moved to the target position T_fAnd f is 1,2, …, q, then the following constraint holds:

wherein, P_ijkFor the initial coordinates of the control vertices in the control frame, R_ijk(s_f,t_f,u_f)＝B_il(s_f)B_jm(t_f)B_kn(u_f)，δ_ijkIs the displacement amount of each control vertex in the control frame; b is_il(s_f) Is a Bernstein polynomial basis function of degree I, B_jm(t_f) Is a polynomial basis function of Bernstein degree m, B_kn(u_f) Is Bernstein polynomial basis function for n times; l, m and n are respectively the number of control points in the three coordinate directions minus 1;

the goal of this problem is to solve δ_ijkδ satisfying the formula (1)_ijkThere may be many, and it is necessary to minimize the sum of the squared displacements of the individual control vertices under least squares conditions, thus defining the lagrangian function:

wherein λ_f＝(λ_f1，λ_f2，λ_f3)^TF is 1,2, …, q is the lagrange multiplier; thus, there are

Writing the above equation in matrix form:

wherein D ═ T₁-S₁,…,T_q-S_q]^T，δ＝[δ₀₀₀,δ₀₀₁,…,δ_lmn]^T，t_f＝[s_f,t_f,u_f]，λ_f＝[λ_f1,λ_f2,λ_f3]；

By substituting the following formula in formula (4) into the above formula, D ═ R can be obtained^TR Λ, and therefore Λ ═ R (R)^TR)^-1D; the displacement of the control vertex can be finally obtained as:

δ＝R(R^TR)^-1D＝R(R^TR)^-1(T-S) (5)

after the displacement of each control point of the control frame is obtained, the displacement is substituted into the following formula, and then new coordinates of all points on the target object and the space grid can be calculated;

X′＝X+ΔX＝B(P+δ) (6)

wherein X is an initial coordinate on the target object, and Delta X is the displacement of the target object; vector matrix B is B_{q×(l+1)(m+1)(n+1)}The matrix vector of the B spline basis function is shown, and P is the initial coordinate of the control vertex.

Technical Field

The invention belongs to the field of aerodynamic shape optimization, and particularly relates to a DFFD grid deformation technology-based elliptic symmetry wing type multi-objective optimization method.

Background

An important issue in the aerodynamic shape optimization process is how to generate high quality computational meshes that meet the CFD computational needs quickly and efficiently for new aerodynamic shapes. The mode of recording scripts based on grid generation software such as ICEMCFD and calling the software to generate grids is the most intuitive mode, but a large amount of grid generation time is consumed, the grid generation efficiency is low, and the direct result is that the optimization cost is high. Therefore, it is necessary to develop a robust and fast method for mesh motion deformation to overcome this difficulty in the optimization process. Common grid motion methods are: the method comprises a direct interpolation method based on a Delaunay background grid, a radial basis function interpolation method, an infinite interpolation method and a combination mode of RBF/body sample interpolation and TFI. The methods have good robustness for small-deformation design in pneumatic optimization, but when the object is subjected to large-amplitude bending and torsional deformation, the boundary line of the object plane grid is easy to intersect with the object, so that the negative volume of the grid is caused. In order to improve the large-amplitude movement deformation capability of the grid in pneumatic optimization design, the invention provides a grid movement deformation technology based on a direct control free form deformation (DFFD) method.

Free Form Deformation (FFD) techniques have originated in the field of computer graphics, which allow deformation operations to be performed on objects in two-or three-dimensional space, mainly for generating or deforming complex shapes. Free-form deformation surrounds an object by establishing a space control frame to define a deformation area, parameterizes the surface of the object by using a classical Bezier function, and directly moves the vertex of the control frame to indirectly deform the surface of the object. Since object deformation is achieved by controlling the vertices, it can be applied to deformation operations on arbitrarily complex objects in theory. In this context, the free form deformation technology is widely applied to the field of industrial optimization design. Duvigneau designs a control body to deform the pneumatic computing grid by using free form deformation, and the method uses the vertex of the control body as a design variable to carry out pneumatic optimization design on the wing; menzel uses a similar approach to design a turbine blade; morris proposes a domain element method by taking a free modeling deformation method as a reference, a deformation region is embedded into an FFD control frame, a control frame is moved to fix a point, and RBF interpolation is carried out on an object plane grid and a space grid. In the pneumatic optimization design method, the pneumatic analysis grid is driven to deform by moving the control points, and the grid points are changed randomly in the optimization process, so that the method reduces design variables, has high efficiency and can adapt to any type of computational grid. However, any change of the control point may cause the aerodynamic profile to be unsmooth, thereby degrading aerodynamic performance; in addition, the arbitrarily moved control points do not have a clear design meaning, so that it is difficult to inversely design a real shape after optimization. To avoid creating a matte aerodynamic profile, Li combines control points, establishes kinematic links between these combined points and geometry, and applies this method to aerodynamic optimization of high speed trains. Gagnon et al developed a double-layer free-form deformation method to optimally design the wings, and used an axial curve to control the control points of the FFD control body, so that the sweep, plane shape, torsion, tip ratio and computational grid of the wings are changed. The pneumatic shape optimization design based on the FFD method considers the relationship between the geometric dimension and the control point motion, and provides enough support for the parameterized grid motion, but the pneumatic shape optimization design based on the FFD method does not form the link between the pneumatic design parameter and the grid motion. In order to establish the link between the pneumatic design parameters and the grid motion, Li proposes a novel FFD-based parameterized grid motion mapping method, which reduces the number of design variables and enhances the design effect.

Inspired by the above research results, and in order to improve the disadvantages in the above method, the patent proposes a mesh motion method based on direct control of free form deformation (DFFD).

Disclosure of Invention

The prior art is based on the method that the grid generating software such as ICEMCFD records scripts and then calls the software to generate the grid, and the method is the most intuitive method for realizing grid motion, but consumes a large amount of grid generating time, has low grid generating efficiency, and has the direct consequences of high optimization cost and long time. The conventional grid motion method can be well suitable for small-deformation pneumatic optimization design, but the boundary line of the object plane grid is easy to intersect with an object when the object plane grid is bent and twisted greatly, so that the negative volume of the grid is caused. The method based on direct control of free form deformation (DFFD) can realize large-scale deformation of geometric shapes, and has incomparable advantages when applied to grid motion deformation.

The standard FFD algorithm has many good characteristics, but there are some limitations: the FFD method is only a deformation tool and cannot directly control the object, namely, a direct mapping relation does not exist between a design variable and the appearance of the object, the physical meaning of the design variable is not clear enough, and the geometric shape required by a designer cannot be accurately generated; in order to deform a complex shape, a large number of control points are needed, so that too many design variables are caused, and the design efficiency is low; the Bernstein polynomial has weak local deformability, insufficient capture of the local deformability, and the like.

The grid motion algorithm based on the DFFD method can directly establish the relation between design variables and geometric shapes and the brought grid motion, can realize shape parameterization and space grid motion deformation by using few design variables, and can realize large-amplitude geometric shape deformation and motion deformation of space CFD calculation grids by adjusting the times of B spline functions in the DFFD algorithm. The method is mainly used for generating the high-quality CFD computational grid with the minimum computational cost in pneumatic optimization.

The technical scheme of the invention is as follows:

the multi-objective optimization method of the elliptic symmetric wing profile based on the DFFD grid deformation technology comprises the following steps:

1) selecting a grid deformation area, and arranging FFD control points, as shown in FIG. 2, wherein a solid point is an FFD control point, four adjacent control points and a thick solid line connecting the four adjacent control points form a control body, and all the control bodies are combined to form the grid deformation area. The mesh deformation region may be a local region of the mesh, or may include the entire mesh;

2) selecting a plurality of direct operation points on the appearance, setting the order of a B spline function, initializing local coordinates of the direct operation points and grid points, and for the FFD control frame which is irregularly arranged, converting the direct operation points and the grid points into an optimization problem to solve approximate local coordinates s, t and u;

3) calculating a B spline coefficient matrix by using the local coordinates of the direct operation points;

4) calculating the displacement of the FFD control point by using the coefficient matrix and the direct operation point displacement;

5) and calculating the deformed position of the spatial grid of the deformed region by using the moved control point.

A comparison of the flow chart of the DFFD method with the basic FFD is shown in figure 1.

When the designer applies a certain displacement ax to the direct control point on the target object, a corresponding position change is produced on the control frame. Q direct control points are arranged, and the initial position is S_fF is 1,2, …, q, and the local coordinate corresponding to it is(s)_f,t_f,u_f) It needs to be moved to the target position T_fAnd f is 1,2, …, q, then the following constraint holds:

wherein, P_ijkFor the initial coordinates of the control vertices in the control frame, R_ijk(s_f,t_f,u_f)＝B_il(s_f)B_jm(t_f)B_kn(u_f)，δ_ijkIs the displacement amount of each control vertex in the control frame. B is_il(s_f) Is a Bernstein polynomial basis function of degree I, B_jm(t_f) Is a polynomial basis function of Bernstein degree m, B_kn(u_f) Is a Bernstein polynomial basis function of degree n. l, m and n are in three coordinate directionsThe number of control points is reduced by 1.

The goal of this problem is to solve δ_ijkδ satisfying the formula (1)_ijkThere may be many, and it is necessary to minimize the sum of the squared displacements of the individual control vertices under least squares conditions, thus defining the lagrangian function:

wherein λ_f＝(λ_f1，λ_f2，λ_f3)^TAnd f is 1,2, …, and q is a lagrange multiplier. Thus, there are

Writing the above equation in matrix form:

wherein D ═ T₁-S₁,…,T_q-S_q]^T，δ＝[δ₀₀₀,δ₀₀₁,…,δ_lmn]^T，t_f＝[s_f,t_f,u_f]，λ_f＝[λ_f1,λ_f2,λ_f3]。

By substituting the following formula in formula (4) into the above formula, D ═ R can be obtained^TR Λ, and therefore Λ ═ R (R)^TR)^-1D. The displacement of the control vertex can be finally obtained as:

δ＝R(R^TR)^-1D＝R(R^TR)^-1(T-S) (5)

after the displacement of each control point of the control frame is obtained, the displacement is substituted into the following formula, so that new coordinates of all points on the target object and the space grid can be calculated.

X′＝X+ΔX＝B(P+δ) (6)

Wherein X is an initial coordinate on the target object, and Delta X is the displacement of the target object; vector matrix B is B_{q×(l+1)(m+1)(n+1)}The matrix vector of the B spline basis function is shown, and P is the initial coordinate of the control vertex.

The key point of the invention is to apply the direct control free form deformation (DFFD) method to the grid motion deformation in the pneumatic optimization design. The core idea of the novel grid motion technology based on the DFFD method is as follows: selecting a plurality of points on the surface of the object as direct control points, moving to the required position, calculating the position change of the vertex of the control frame of the computational grid in a reverse way, and calculating new coordinates of other spatial grid points in the control frame. The invention realizes the integrated implementation of the shape parameterization and the grid motion, and realizes the control of the grid deformation capacity and the deformation area by adjusting the times of the B splines in the DFFD.

The invention has the beneficial effects that: the technology solves the problem that the traditional grid motion method cannot adapt to severe shape deformation in the wing pneumatic optimization process, and greatly improves the motion deformation capability of the grid. The free modeling deformation method based on direct control has the greatest advantages that a user can realize the deformation of a complex shape only by defining a control frame around a deformation target, and can realize accurate deformation in a specific area on the surface of an object, so that the user can pay attention to a key area; in addition, the deformation process of the grid is not related to the spatial distribution characteristic of the grid, and the characteristic is very beneficial to realizing motion deformation for the grids with various structure types, so that the grid motion can be realized for the grids with any spatial structure; then, for the area where the grids are overlapped, the quality of the grids can be improved by adding a direct control point; finally, the grid deformation technology based on the direct control points can be added with geometric constraints conveniently, and has stronger adaptability and practicability. The grid deformation shows that the grid motion technology based on the DFFD method has strong deformation capability, strong robustness and higher deformation efficiency for complex configuration or grid motion.

Drawings

FIG. 1 compares the flow chart of direct control FFD with the basic FFD;

FIG. 2 grid deformation regions and FFD control frames;

FIG. 3 is a schematic diagram of two point movements;

FIG. 4 is a two-point motion profile and mesh deformation wherein (a) is cubic B-spline, (B) is quintic B-spline, and (c) is heptatic B-spline;

FIG. 5 illustrates the deformation of the seven-point motion profile and the mesh for three B-spline;

FIG. 6 illustrates a seven-point motion profile and mesh deformation for a quintic B-spline;

FIG. 7 illustrates seven-point motion profile and mesh deformation for a seven-degree B-spline;

FIG. 8 optimizes design space and direct control points;

FIG. 9 is a grid deformation region and FFD control frame of an elliptically symmetric airfoil;

FIG. 10 parametric airfoil and spatial grid motion for two extreme deformation cases, where (a) is case one and (b) is case two;

FIG. 11 is a Pareto solution after elliptical airfoil optimization;

FIG. 12 optimized airfoil versus baseline airfoil profile;

FIG. 13 design point-airfoil pressure distribution comparison;

FIG. 14 design point two wing pressure profile comparison;

fig. 15 optimizes Cl, Cd and Cm comparisons (Ma 0.6, Re 4.8e6) for an airfoil and a reference airfoil, where (a) is Cl, (b) is Cd, and (c) is Cm;

fig. 16 optimizes Cl, Cd and Cm comparisons (Ma 0.5, Re 4.0e6) for an airfoil and a reference airfoil, where (a) is Cl, (b) is Cd, and (c) is Cm;

FIG. 17 compares the reference airfoil and the optimized airfoil pole curves by one;

FIG. 18 comparison of reference airfoil and optimized airfoil pole curves;

FIG. 19 is a Pareto solution after four-target optimization of elliptical airfoils;

FIG. 20A pareto airfoil profile in comparison to a reference airfoil profile;

FIG. 21 is a Pareto solution after five-target optimization of elliptical airfoil profiles;

FIG. 22ParetoA airfoil in comparison to a reference airfoil profile.

Detailed Description

The technical solution of the present invention is further explained below with reference to examples and drawings.

The mesh motion morphing process and capability of the DFFD method is detailed using a two-dimensional airfoil-shaped mesh motion. Wherein, a B spline function is adopted to obtain the coefficient matrix. Fig. 2 shows a quadrilateral mesh generated around an elliptical airfoil with unit chord length, and the mesh region and the airfoil to be deformed are placed in a simple mesh control frame, and control points of the control frame are uniformly distributed along coordinate axes, and are shown as square points in fig. 2. The implementation process of the mesh deformation motion and the influence of the B spline times on the deformation capability of the mesh motion are mainly described.

In the optimization design process, two or more points are generally directly controlled to perform the deformation design, so that enough design variables can be provided for the appearance design. To study the effect of the grid motion morphing technique based on the DFFD method, four direct control points were defined in fig. 3, where the elliptical airfoil leading and trailing edge direct control points were kept fixed and the two control points in the middle of the airfoil were each moved upward by 0.1 times the single chord length. Fig. 4 shows the deformation of the mesh using 3-, 5-and 7-fold B-spline basis functions, respectively. It can be seen that the airfoil surface generated by the 3-time B-spline has an uneven shape; when the number of times of the splines is increased to 5, the concave-convex part on the surface of the airfoil shape is reduced; the number of splines continues to increase to 7, and the upper and lower surfaces of the airfoil are only convex upward, and become smoother. The corresponding mesh deformation area and the control frame change area are enlarged, the number of control vertexes generating displacement is increased, and the displacement amplitude of a single control vertex is reduced. The quality of the motion mesh was checked using the mesh quality check function of the mesh generation software ICEMCFD, and the mesh quality comparison data is listed in table 1. It can be seen that the grid Quality after the grid movement is reduced to a certain extent, but the minimum angles are all larger than 50 degrees, the minimum Quality index is also higher than 0.8, and the grid Quality index is improved along with the increase of the number of times of B splines.

Table 1 grid quality check

In order to better generate the grid large deformation motion, seven movable direct control points and one position constraint point respectively at the front edge and the rear edge of the airfoil are arranged on the upper airfoil surface and the lower airfoil surface of the airfoil, so that the airfoil shape change as shown in FIG. 5 is generated. It can be seen that at the position of 0.6 chord direction, the airfoil shape is severely changed by 0.4 times of chord length, and the space grid is also severely changed, so that when a cubic B-spline basis function is used, the space grid has a large-area overlapping region; as the splines increase 5 times (shown in fig. 6), the mesh overlap area becomes dramatically smaller; when the spline number is 7, the overlap area disappears completely as shown in fig. 7.

Example (b):

many design factors and design indexes need to be considered in the aerodynamic design of the elliptical airfoil profile, the aerodynamic performance of the elliptical airfoil profile is difficult to be improved in multiple aspects by the traditional design method, and the elliptical airfoil profile aerodynamic multi-objective optimization design needs to be developed. An oval wing section with the thickness of 16% is selected as a reference wing section to carry out multi-objective aerodynamic optimization design.

11 variable direct control points are respectively arranged on the upper surface and the lower surface of the elliptical airfoil, a fixed direct control point is respectively arranged on the front edge and the rear edge, and the total number of the direct control points is 24, and the specific positions of the direct control points are shown as solid dots in a figure 8. Considering that the elliptical airfoil needs to keep front-back symmetry, the first six movable direct control points of the upper airfoil surface and the lower airfoil surface are selected as design variables, the total number of the design variables is 12, the variation interval of each design variable is +/-30% of the z-direction coordinate of each design variable, and the specific design space range is determined by the interval represented by the dotted line in fig. 8.

The DFFD control framework is shown in fig. 9, where the solid larger vertices are the control vertices, numbered 45x 31. Because the optimization design of the invention adopts more direct control points, the control vertexes near the wing profile are encrypted, and the dragon lattice phenomenon is prevented; in order to symmetrically deform the airfoil back and forth after the direct control point movement, all control vertex coordinates are bilaterally symmetric about the (0.5,0) position. When the direct control point shown in fig. 8 moves, the vertex of the control frame generates a corresponding displacement, so that the indirect control point in the control frame moves. The process simultaneously realizes the parameterization of the wing profile and the space grid motion, and simplifies the optimization steps.

A6-time B-spline DFFD method is adopted to carry out parameterization and grid motion on the airfoil profile, and the parameterization and space grid motion of the airfoil profile under two extreme deformation conditions which possibly occur in the optimization process are verified. The final wing profile parameterization and grid motion results are shown in fig. 10, and it can be seen from the figure that no dragon grid phenomenon occurs in the wing profile, and the grid distribution near the wing profile and in the space is good, which indicates that the grid motion deformation technology based on the DFFD method is suitable for the optimization calculation example.

Comprehensively considering the main working state of the elliptical symmetrical airfoil, 2 design points and 2 design targets are summarized as listed in Table 2, wherein the minimization target is to reduce the drag coefficient of the airfoil when hovering; the second objective is to reduce the drag coefficient at fixed-wing cruise conditions. The optimized mathematical model is established as follows:

min f₁(x)＝Cd₁

min f₂(x)＝Cd₂

s.t.x∈(x_l,x_u)

g_m(x)≥0，m＝1,2,…,N_m

subscripts 1 and 2 correspond to design point serial numbers in table 2, and "+" in table 2 represents a aerodynamic value of a reference elliptical airfoil, and maximum thickness constraint is performed on the airfoil.

An LMPSO algorithm is adopted as an optimization algorithm, parameters of the LMPSO are kept the same as those of a reference function during testing, the population scale is 60, the cycle number is 50, and the number of times of calling a target function is 3000. The Pareto frontiers obtained after multi-objective optimization are shown in fig. 11, from which it can be seen that all Pareto solutions are greatly improved in both design objectives compared to the original airfoil profile, which indicates that the multi-objective optimization design for the elliptically symmetric airfoil profile is effective. In the graph, the Pareto solution distribution indicates that the optimization improvement amount on the second target is larger than the first optimization target, so that the third Pareto solution from the left is selected as the optimal solution to be compared and analyzed, and the wing profiles are all called as optimized wing profiles.

Two optimization design goals for a table 216% thickness elliptical airfoil

Figure 12 shows a comparison of profiles for a baseline elliptical airfoil profile and a selected optimized airfoil profile. The front edge and the rear edge of the optimized wing profile are thinned, the shapes of the upper wing surface and the lower wing surface are not symmetrical any more, the middle of the upper wing surface is raised, and the maximum thickness is increased; the maximum thickness of the lower airfoil surface is slightly reduced, and two symmetrical loading areas are formed; the central line of the optimized airfoil profile is a horizontal straight line relative to the central line of the standard airfoil profile and is convex upwards. Table 3 shows the aerodynamic comparison before and after optimization for two design points, the maximum thickness of the optimized airfoil is slightly increased and the aerodynamic characteristics are improved compared to the baseline airfoil. The optimized target resistance coefficient of the design point I is reduced by 11.60% (-20.6counts), and the resistance reduction effect is very obvious; the attack angle is reduced by 0.695 degrees under the same lift force, which is also the main reason for reducing the resistance; the absolute value of the moment coefficient is reduced by 84.86%. The drag coefficient of the second optimized airfoil design point is reduced by 14.21% (-21.4 counts); the attack angle is reduced by 0.767 degrees under the same lift force; the absolute value of the moment coefficient is reduced by 66.53%. It can be seen that the reduction in the absolute value of the pitch moment coefficient, while reducing the drag of the optimized airfoil at two design points, is significant, which is very beneficial for flight control.

FIGS. 13 and 14 are a comparison of the airfoil surface pressure coefficient distributions before and after optimization at design point one and design point two, showing the reduction of suction near the leading edge and trailing edge and the increase of suction in the middle of the optimized airfoil; the two aerodynamic force loading areas on the lower airfoil surface have obvious effects, the area of the fishtail-shaped intersection area of the trailing edge is reduced, the lift force of the rear half section of the airfoil is increased, and the main reason for optimizing the reduction of the moment coefficients of the two design points of the airfoil is. Fig. 15 and 16 show the variation curves of the lift coefficient, the drag coefficient and the moment coefficient with the attack angle at two design points, and it can be seen that the lift characteristic of the first-time optimized airfoil profile at the design points is obviously superior to that of the reference airfoil profile, the maximum lift coefficient is increased, but the stall attack angle is slightly reduced; the optimized airfoil moment is changed into a low head moment at a small attack angle, and has a smaller absolute value in a larger attack angle range; the drag coefficient of the optimized airfoil is better than that of the reference airfoil at small angles of attack, but is worse than that of the reference airfoil at large angles of attack. The performance of a lift coefficient curve linear section of the optimized airfoil profile at a design point II is obviously superior to that of a reference airfoil profile, but the maximum lift coefficient and the stall attack angle are inferior to those of the reference airfoil profile; the drag coefficient and the moment coefficient behave similarly to the design point. The optimized airfoil lift and drag coefficient deterioration at a large attack angle is mainly caused by the fact that the radius of the front edge of the airfoil is reduced, which also shows that lift and drag characteristics of the elliptical airfoil at the large attack angle possibly conflict with mechanical characteristics at the small attack angle, and the elliptical airfoil should be optimally designed as an optimization target.

TABLE 3 comparison of aerodynamic performance of optimized airfoil vs. baseline airfoil design point (Cl ═ 0.6)

Fig. 17 and 18 show lift and drag pole curves of optimized front and rear wing profiles in a hovering state and a straight wing forward flying state, respectively. It can be seen from the figure that the optimum airfoil drag coefficient is significantly reduced both at a lift coefficient of 0.6 and at a lift coefficient of 0.4.

1) Elliptical airfoil four-target optimization design result

Table 4 shows the design points and optimization objectives for four-objective optimization, based on which a mathematical model is built:

min f₁(x)＝Cd₁

min f₂(x)＝Cd₂

min f₃(x)＝Cd₃

min f₄(x)＝Cd₄

s.t.x∈(x_l,x_u)

g_m(x)≥0，m＝1,2,…,N_m

four optimization design goals for a table 416% thickness elliptical airfoil

TABLE 5 aerodynamic Performance comparison of four design points for optimized and baseline airfoils

2) Elliptical airfoil five-target optimization design result

Table 6 gives seven design points and optimization objectives for five-objective optimization, based on which a mathematical model was built:

min f₁(x)＝Cd₁

min f₂(x)＝Cd₂

min f₃(x)＝Cd₃

min f₄(x)＝Cd₄

min f₅(x)＝[(Cd₅-Cd₆)²+(Cd₆-Cd₇)²]/(ΔMa)²

s.t.x∈(x_l,x_u)，g_m(x)≥0，m＝1,2,…,N_m

five optimization design goals for table 616% thickness elliptical airfoil

TABLE 7 aerodynamic Performance comparison of seven design points for optimized airfoils versus baseline airfoils