Robust optimization method based on D optimum and considering intake and exhaust influences, medium and program product

Abstract

The invention discloses a robust optimization method based on D optimum and considering intake and exhaust influences, a medium and a program product, and belongs to the technical field of aerodynamic optimization design of aircrafts. According to the method, a representative small sample set is constructed by introducing a D optimal criterion, optimization coverage of a design space is realized by maximizing an information matrix determinant, and an optimization objective function is formed by a weighted sum of a mean value and a standard deviation of aerodynamic performance so as to balance performance improvement and robustness optimization. An FFD method is adopted to carry out geometric parametric modeling on the aerodynamic configuration and the intake and exhaust system of the aircraft, an uncertainty quantitative agent model is established through a GPCE method, statistical characteristics and gradient information of a target function are analyzed and calculated, iterative updating is carried out through a gradient optimization algorithm, and it is ensured that design constraints and convergence criteria are met. According to the method, the uncertainty influence of geometric parameters and intake and exhaust conditions is fully considered, the aerodynamic performance of the aircraft is improved, and the adaptability and robustness under actual flight conditions are enhanced.

Description

Technical Field

[0001] The present invention belongs to the technical field of aircraft aerodynamic optimization design, and involves uncertainty quantification, robust optimization, and coupled optimization of intake and exhaust systems and shape layout. Specifically, it is a robust optimization method, medium, and program product based on D optimality and considering the influence of intake and exhaust, which is used to solve the uncertainty optimization problem of an aircraft including an intake and exhaust system under real flight conditions. Background Art

[0002] Modern high-performance aircraft usually adopt a layout in which the fuselage and the intake and exhaust system are highly integrated. As an important component of the aircraft's aerodynamic layout, the intake and exhaust system not only has a significant impact on the aircraft's external flow field, but is also closely coupled with the engine's internal flow field. The mutual influence between the aircraft's external flow field and the flow inside the intake and exhaust system has an important impact on the overall performance of the aircraft. In addition, under real flight conditions, the aircraft's aerodynamic performance may change significantly due to uncertain factors such as fluctuations in incoming flow conditions, structural deformation, and external environmental disturbances. Therefore, in the process of aircraft design optimization, it is necessary to consider not only the performance under nominal conditions, but also the impact of various uncertain factors on performance to ensure that the aircraft can maintain good aerodynamic performance under various flight conditions.

[0003] At present, the integrated optimization design research considering the influence of intake and exhaust mainly focuses on deterministic single-objective or multi-objective optimization, for example: using the inverse design idea to optimize the conformal nozzle considering the internal and external flow coupling of the aircraft to obtain a nozzle profile with excellent robustness and aerodynamic characteristics, multi-objective optimization of the single-sided expansion surface of the rear body of the flying wing to reduce the adverse effects of the engine jet on the flight stability characteristics of the flying wing, multi-objective, multi-constrained three-dimensional optimization design of the aircraft considering the intake and exhaust system to obtain better comprehensive performance, or the inlet profile of the hybrid wing is optimized based on discrete adjoints to obtain optimization results with significant drag reduction and significantly improved inlet distortion, etc. Although these studies have improved the aerodynamic performance of the aircraft to a certain extent, due to ignoring the influence of uncertain factors, their optimization results may show poor robustness in actual flight conditions.

[0004] In recent years, with the development of mathematical modeling and computing technology, uncertainty optimization design methods have gradually been applied to the aviation field to improve the robustness of optimization results. For example, by constructing a robust optimization design framework for full turbulence, the random perturbations of a small number of geometric design variables are considered, and the non-gradient optimization algorithm is used to optimize the uncertainty of power unit components. There are also studies that combine the gradient optimization method based on discrete adjoints with the polynomial chaos expansion method to achieve aerodynamic performance optimization under random perturbations. However, current research mainly focuses on the optimization of the aerodynamic shape of aircraft, and the research on the coupling between the internal flow field and the external aerodynamic flow field of the intake and exhaust system is relatively limited, and the uncertainty problem under the coupling of internal and external flows has not been effectively solved. On the other hand, although there are a large number of mathematical literatures studying methods for improving the accuracy of uncertainty prediction, the existing optimization methods still have room for improvement in sample efficiency and computational accuracy, especially when dealing with high-dimensional design spaces and complex flow field problems. In addition, traditional uncertainty quantification methods often require a large number of computational samples and have high computational costs.

[0005] In summary, how to achieve efficient and robust optimization design of aircraft while considering the influence of intake and exhaust, and how to improve the prediction accuracy and stability of uncertainty optimization methods in the aerodynamic field are technical problems that need to be solved urgently in the field of aircraft design. Summary of the invention

[0006] 1. Purpose of the invention In view of the above defects and deficiencies of the prior art, the purpose of the present invention is to solve the problem of robust optimization of aircraft uncertainty under real flight conditions taking into account the effects of intake and exhaust, and to construct a robust optimization method, medium and program product based on D-optimality and taking into account the effects of intake and exhaust. By combining an efficient adjoint method, an uncertainty analysis method based on gradient-enhanced polynomial chaos expansion (GPCE) and a D-optimal sampling strategy, efficient and accurate aerodynamic shape optimization design of aircraft including intake and exhaust systems is achieved. The method can effectively quantify and propagate the uncertainty of flight state parameters, and select the optimal sample point combination according to the D-optimality criterion, so as to obtain high-precision prediction results at a relatively low computational cost, and finally achieve aerodynamic optimization including the effects of internal flow in the intake and exhaust system, while taking into account the uncertainty of flight conditions, and meeting the overall performance and robustness requirements of the aircraft in the optimization process.

[0007] (II) Technical solution In order to achieve the purpose of the invention and solve the technical problems, the present invention adopts the following technical solutions: The first invention objective of the present invention is to provide a robust optimization method based on D-optimality and taking into account the influence of intake and exhaust, which is used for optimizing the intake and exhaust system and the overall aerodynamic layout of an aircraft under real flight uncertainty conditions, thereby improving aerodynamic performance and design robustness, and is characterized by including: SS1. Definition of optimization objectives and geometric parameterization design Define the optimization objectives, including the improvement of the aerodynamic performance of the aircraft and the optimization of the robustness of the design results. The objective function is composed of the weighted sum of the mean and standard deviation of the target aerodynamic performance, and the weight coefficient is dynamically adjusted according to the design requirements; use the Free-Form Deformation (FFD) method to perform geometric parametric modeling on the aerodynamic shape and intake and exhaust system of the aircraft, and impose constraints based on the geometric parametric model; define the optimization design variables, including at least geometric design variables and aerodynamic design variables, and divide them into deterministic design variables and uncertain design variables according to whether they are affected by uncertain factors; SS2. Construct a small sample set that meets statistical characteristics based on the D optimal criterion According to the probability distribution of uncertain design variables, the initial large-scale candidate sample set is generated by random sampling method. Based on the D optimal criterion, the greedy algorithm of QR decomposition is used to maximize the product of the eigenvalues ​​of the basis function matrix and the information matrix to screen out a representative small sample set from the initial large-scale candidate sample set. The representativeness of the sample points is optimized by adjusting the balance between the number of samples and the maximization of the eigenvalues, forming an optimal sampling set that meets the statistical characteristics and has controllable computational costs. SS3. Internal and external flow field solution and gradient calculation based on discrete adjoint method For each sample point in the small sample set, the internal and external flow fields of the aircraft are numerically simulated by computational fluid dynamics (CFD). The simulation process takes into account the interaction between the intake and exhaust system and the external flow field to ensure the accuracy of the calculation of the aerodynamic parameters. The target aerodynamic performance parameters corresponding to each sample point are obtained through numerical solution, and the gradient information of the target aerodynamic performance parameters for each design variable is calculated using the discrete adjoint method. SS4. Constructing a GPCE uncertainty quantification proxy model Based on the CFD calculation results of a small sample set, the Gradient-Enhanced Polynomial Chaos Expansion (GPCE) method is used to model the uncertainty quantification problem. By combining the objective function value and its gradient information, a GPCE proxy model containing orthogonal polynomial basis functions and their derivatives is constructed. The obtained GPCE proxy model is used to analytically calculate the statistical moments of the objective function and its gradient information relative to the design variables. SS5. Robust optimization design and iterative update of aircraft Based on the statistical moment of the objective function calculated by the GPCE proxy model and its gradient information relative to the design variables, all design variables are iteratively updated using the gradient optimization algorithm. During the optimization iteration, the update of the geometric design variables is achieved by adjusting the position of the FFD control points. The surface mesh of the aircraft aerodynamic shape and the intake and exhaust system is regenerated based on the updated FFD control points, and the volume mesh is updated using the Inverse Distance Weighting (IDW) method. The update of the uncertain design variables is adjusted according to the search direction and step size of the gradient optimization algorithm. During the optimization iteration, the update of all design variables must strictly meet the constraints until the convergence criteria are met. SS6. Optimization result verification and robustness evaluation After the optimization is completed, the final optimization results are output, the optimized aircraft configuration is verified and robustness evaluated, the coupling characteristics of the intake and exhaust system and the external flow field are analyzed, and the comprehensive performance of the optimized configuration is evaluated to ensure that the design meets actual flight requirements.

[0008] The second inventive object of the present invention is to provide a computer program product, including computer instructions, for executing the above-mentioned robust optimization method based on D optimality and considering the influence of intake and exhaust of the present invention.

[0009] The third inventive object of the present invention is to provide a computer-readable storage medium, on which a computer program is stored. When the computer program is executed by a processor, the above-mentioned robust optimization method of the present invention based on D optimality and considering the influence of intake and exhaust is implemented.

[0010] (III) Technical Effect Compared with the prior art, the robust optimization method, medium and program product of the present invention based on D optimality and considering the influence of intake and exhaust have the following beneficial and significant technical effects: (1) The present invention adopts the D optimal criterion and constructs a small sample set from a large number of candidate samples through the greedy method of QR decomposition. It combines the GPCE uncertainty quantification proxy model of gradient enhanced polynomial chaos expansion, introduces the gradient information of the objective function into the proxy model, and combines the change trend of the objective function in the optimization iteration. It can accurately capture the change direction of the design variables, improve the prediction accuracy of the model for the objective function, and significantly improve the prediction accuracy and stability of the robust optimization method in the aerodynamic field. It can perform aerodynamic optimization including the influence of the internal flow of the intake and exhaust system, while considering the uncertainty of the flight conditions, and meet the overall performance and robustness requirements of the flying wing layout in the optimization process.

[0011] (2) In the optimization of the intake and exhaust system, the present invention takes into account the coupling effect between the intake and exhaust system and the overall aerodynamic layout, thereby improving the overall performance of the aircraft. By incorporating the mutual influence of the intake and exhaust system and the aerodynamic layout into the optimization model, the present invention can more accurately capture the coupling effect between the various components of the aircraft during the optimization process, and ultimately achieve more efficient and stable aerodynamic performance optimization. In addition, by introducing uncertainty quantification and robust optimization strategies, the present invention can consider various uncertain factors that the aircraft may encounter under different flight conditions during the design process, ensuring that the optimization results can still maintain high performance and stability under uncertain conditions, which not only improves the aerodynamic performance of the aircraft, but also enhances the adaptability and robustness of the design under uncertain environments.

[0012] (3) The present invention adopts the D optimal experimental design method to generate the initial sampling point set. By maximizing the information matrix determinant to optimize the sampling point distribution, it can effectively screen out the sample points that contribute most to the uncertainty quantification of the objective function, thereby constructing a representative small sample set, ensuring that the sampling points have good spatial filling and uniformity in the design space, and improving the prediction accuracy and generalization ability of the proxy model. At the same time, the present invention uses the GPCE method to construct a high-precision proxy model using the function values ​​and gradient information of a small number of sample points. Under the premise of ensuring the optimization accuracy, the required CFD calculation amount is significantly reduced, and higher-precision optimization results can be obtained under the same computing resources, or the calculation time can be significantly reduced under the same accuracy requirements. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] Figure 1 is a flow chart of a robust optimization method based on D-optimality and considering the effects of intake and exhaust gas of the present invention; Figure 2 Schematic diagram of a robust optimization framework based on D-optimality and considering the effects of intake and exhaust gas according to the present invention; Figure 3 FIG. 2 is a schematic diagram of an FFD block diagram of an initial configuration optimization arrangement in Example 2 of the present invention; Figure 4 The figure shows the distribution of the total pressure recovery coefficient of the inlet before and after optimization and the schematic diagram of the change of the central axis and area of ​​the inlet. In the figure: (a) is the distribution cloud diagram of the end face total pressure recovery coefficient σ of the initial configuration (left half) and the configuration after uncertainty optimization (right half); (b) is the distribution curve of the inlet area along the axial direction x of the initial configuration, the deterministic optimization configuration and the uncertainty optimization configuration; (c) is the height z distribution curve of the central axis of the three configurations along the axial direction; Figure 5The figure shows a schematic diagram of the pressure coefficient distribution and its statistical response at different span positions of the wing in Example 2 of the present invention, where (a) to (d) correspond to the pressure coefficient distribution at four different span positions of Y / b=0.3, Y / b=0.5, Y / b=0.7, and Y / b=0.9, respectively. DETAILED DESCRIPTION

[0014] The present invention aims to provide a robust optimization method, medium and program product based on D-optimality and considering the influence of intake and exhaust. By combining an efficient adjoint method, a GPCE uncertainty analysis method based on gradient-enhanced polynomial chaos expansion and a D-optimal sampling strategy, aerodynamic optimization including the influence of internal flow in the intake and exhaust system is achieved, while considering the uncertainty of flight conditions, and meeting the overall performance and robustness requirements of the aircraft during the optimization process. In order to make the purpose, technical scheme and advantages of the implementation of the present invention clearer, the technical scheme in the embodiment of the present invention will be described in more detail below in conjunction with the drawings in the embodiment of the present invention. The described embodiments are part of the embodiments of the present invention, not all of the embodiments, and the described embodiments are exemplary and are intended to be used to explain the present invention, and cannot be understood as limiting the present invention. Based on the embodiments in the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0015] Example 1: Robust optimization method implementation process As a specific example, Figure 1 The implementation process of the robust optimization method based on D optimality and considering the influence of intake and exhaust provided by the present invention is shown. Figure 2 The implementation framework of the robust optimization method is further shown. Figure 1 , 2 As shown, the robust optimization method mainly includes the following steps when implemented: SS1. Definition of optimization objectives and geometric parameterization design Define the optimization objectives, including the improvement of the aerodynamic performance of the aircraft and the optimization of the robustness of the design results. The objective function is composed of the weighted sum of the mean and standard deviation of the target aerodynamic performance, and the weight coefficient is dynamically adjusted according to the design requirements. Use the FFD method to perform geometric parametric modeling on the aerodynamic shape and intake and exhaust system of the aircraft, and impose constraints based on the geometric parametric model. Define the optimization design variables, including geometric design variables and aerodynamic design variables, and divide them into deterministic design variables and uncertain design variables according to whether they are affected by uncertain factors.

[0016] In the embodiment of the present invention, the objective function I ( m , s ) is defined as the weighted sum of the statistical moments of the target aerodynamic performance and satisfiesI ( m , s )= K 1. m + K 2. s ,in m is the mean, s is the standard deviation, K 1. K 2 are weight coefficients and satisfy K 1+ K 2=1, when the optimization goal focuses more on performance improvement, it is preferred to set K 1> K 2. When the optimization goal focuses more on robustness, it is preferred to set K 1< K 2; The mean of the objective function m and standard deviation s The statistical moments of aerodynamic performance based on the analytical calculation of the proxy model are calculated and obtained by weighted averaging and standard deviation calculation of the sample point responses under various flight conditions. In addition, the optimization state conditions are set to the actual flight state under cruise flight conditions and the influence of uncertainty is considered. The flight state parameters include at least Mach number, incoming flow angle of attack, sideslip angle and / or inlet flow parameters. For Mach number and incoming flow angle of attack with uncertain quantities, their probability distribution type and statistical characteristics are given to ensure that the variable distribution in the optimization process can accurately reflect the actual flight conditions.

[0017] In an embodiment of the present invention, the FFD method used in geometric parametric modeling is implemented by generating a control framework around the geometric shape of the aircraft, wherein the control framework is composed of B-spline control points, and the FFD control points perform parametric modeling on the aerodynamic shape, inlet lip shape, and internal flow path shape of the inlet of the aircraft by adjusting their positions; the number and distribution of FFD control points are allocated according to the complexity of the aircraft design area, with a focus on arranging higher density control points in the wing and inlet areas to meet the refined requirements of geometric deformation during the optimization process.

[0018] In an embodiment of the present invention, the constraints imposed on the geometric parameterized model include aerodynamic performance constraints, geometric shape constraints and / or intake and exhaust system-related constraints, etc. The aerodynamic performance constraints include moment balancing constraints and / or lift coefficient constraints, etc. The geometric shape constraints include thickness constraints and / or volume constraints, etc. The constraints related to the intake and exhaust system include inlet duct total pressure recovery coefficient constraints and / or distortion coefficient constraints, etc.

[0019] In an embodiment of the present invention, the geometric design variables are defined by the FFD method, covering the aerodynamic shape of the aircraft and the geometry of the intake and exhaust system, including at least the FFD control points that control the aerodynamic shape of the aircraft, the FFD control points that control the geometry of the inlet duct, and the FFD control points that control the deformation of the inlet duct lip; the aerodynamic design variables include parameters related to the flight state, including at least the Mach number, the incoming flow angle of attack, the sideslip angle and / or the inlet flow parameters; the deterministic design variables are used to define fixed geometry and aerodynamic characteristics, and the uncertain design variables are used to characterize the random effects of flight conditions and environmental disturbances on aerodynamic performance, and the probability density distribution of the uncertain design variables is determined by statistical analysis, and it is assumed that they obey a normal distribution or a uniform distribution to meet the random response modeling requirements.

[0020] SS2. Construct a small sample set that meets statistical characteristics based on the D optimal criterion According to the probability distribution of uncertain design variables, the random sampling method is used to generate the initial large-scale candidate sample set. Based on the D optimal criterion, the greedy algorithm of QR decomposition is adopted to maximize the product of the eigenvalues ​​of the basis function matrix and the information matrix to screen out a representative small sample set from the initial large-scale candidate sample set. The representativeness of the sample points is optimized by adjusting the balance between the number of samples and the maximization of the eigenvalue, forming an optimal sampling set that meets the statistical characteristics and has controllable computational costs.

[0021] In the embodiment of the present invention, the process of constructing a small sample set based on the D optimal criterion includes the following sub-steps: SS21. Generation of initial large-scale candidate sample set: According to the probability distribution type and statistical characteristics of the uncertain design variables, the initial large-scale candidate sample set is generated by Monte Carlo sampling or Latin hypercube sampling method. The sample points cover the full range of the uncertain design variables and the distribution characteristics are consistent with the probability distribution of the uncertain design variables. The number of samples is reasonably set according to the number of uncertain design variables and the order of the subsequent polynomial chaos expansion PCE to ensure that the candidate sample set can fully characterize the statistical characteristics of the uncertain variables.

[0022] SS22. Construction of basic information matrix of small sample set: Based on the generated initial large-scale candidate sample set, a polynomial basis function matrix related to the uncertain design variables is constructed. The basis function adopts the orthogonal polynomial basis function of the polynomial chaos expansion PCE. Each element of the basis function matrix is ​​composed of the value of the basis function at the candidate sample point, reflecting the correlation between each candidate sample point and the objective function. Based on the basis function matrix, a basic information matrix of the small sample set for D optimal criterion screening is formed. The information matrix is ​​defined as the product of the transposed matrix of the polynomial basis function matrix and itself. The eigenvalues ​​and determinant values ​​of the information matrix directly determine the representativeness and prediction accuracy of the candidate sample set for the uncertainty quantification of the objective function.

[0023] SS23. Greedy algorithm screening based on QR decomposition: Using the constructed information matrix, the greedy algorithm of QR decomposition is used to screen the candidate sample points. By decomposing the basis function matrix QR into an orthogonal matrix Q and an upper triangular matrix R, the sample points with the largest absolute value of the diagonal elements of the upper triangular matrix R are preferentially selected. According to the greedy algorithm and combined with the distribution uniformity among the sample points, the sample points with the largest determinant value of the information matrix are iteratively selected to add to the small sample set to update the basis function matrix information matrix, so as to maximize the product of the eigenvalues ​​of the basis function matrix information matrix, until the small sample set reaches the preset size or meets the stopping criterion.

[0024] SS24. Optimize the balance between sample size and eigenvalues: During the screening process, the number of small sample sets is dynamically adjusted according to the complexity of the objective function and the computational cost requirements to ensure that the sample sets can effectively improve the accuracy of the proxy model without increasing the computational cost due to an excessive number of samples. After the optimization is completed, a small sample set that meets the statistical characteristics is formed.

[0025] SS25. Sample set quality verification: Perform quality verification on the small sample set that has been screened. By calculating the eigenvalue distribution of the information matrix and the minimum distance between samples, evaluate the coverage and distribution uniformity of the sample points, ensure that the sample set can fully reflect the impact range of the uncertain design variables, and support the construction and optimization iteration of the subsequent GPCE proxy model.

[0026] Preferably, the method for generating the initial large-scale candidate sample set is Monte Carlo sampling or Latin hypercube sampling, wherein: Monte Carlo sampling is applicable to the case where the uncertain design variables obey a complex or non-uniform distribution, and the probability distribution characteristics of the variables are simulated by randomly generating sample points, and the sampling density is dynamically adjusted according to the distribution range of the variables and the importance of their impact on the objective function to ensure that the candidate sample set fully covers the true distribution; Latin hypercube sampling is applicable to the case where the uncertain design variables are uniformly distributed or normally distributed, and the variable range is divided into several equal intervals by stratified sampling, and a sample point is randomly generated in each interval to ensure the uniformity and representativeness of the candidate sample set, while reducing the possibility of sample point overlap and improving sample utilization efficiency.

[0027] SS3. Internal and external flow field solution and gradient calculation based on discrete adjoint method For each sample point in the small sample set, computational fluid dynamics (CFD) numerical simulation of the internal and external flow fields of the aircraft is performed based on the Reynolds-Averaged Navier-Stokes (RANS) equation and combined with the turbulence model. The simulation process considers the interaction between the intake and exhaust system and the external flow field to ensure the calculation accuracy of the aerodynamic parameters. The target aerodynamic performance parameters corresponding to each sample point are obtained through numerical solution, and the discrete adjoint method is used to calculate the gradient information of the target aerodynamic performance parameters for each design variable.

[0028] In an embodiment of the present invention, CFD numerical simulation is based on the finite volume method to solve the RANS equations and combines the Spalart-Allmaras turbulence model to simulate the influence of turbulent viscosity. The viscous flux in the flow field calculation is discretized using the standard central difference, and the inviscid flux is discretized using the artificial dissipative central difference. When solving the residual equation, the Runge-Kutta method and the Newton-Krilov method are combined to dynamically switch and optimize the solution efficiency, and ensure that the residual norm is reduced by at least five orders of magnitude to ensure the stability and accuracy of the flow field solution. The interaction between the intake and exhaust system and the external flow field is simulated using the overlapping grid technology, and the exchange of internal and external flow field information is realized through the interpolation method, thereby coupling the solution of the internal and external flow fields. The discrete adjoint method is used to calculate the gradient of the target aerodynamic performance parameters to the design variables, including geometric design variables and aerodynamic design variables.

[0029] In an embodiment of the present invention, the target aerodynamic performance parameters include aircraft aerodynamic shape performance parameters and intake and exhaust system performance parameters, wherein the aircraft aerodynamic shape performance parameters are drag coefficient, lift coefficient and / or moment coefficient, and the intake and exhaust system performance parameters include inlet duct total pressure recovery coefficient and / or distortion coefficient, wherein the inlet duct total pressure recovery coefficient is defined as the ratio of the average total pressure at the inlet duct outlet to the free flow total pressure, and the distortion coefficient is composed of the sum of the circumferential non-uniformity of the total pressure and the surface average pulsation intensity, the circumferential non-uniformity of the total pressure is calculated based on the total pressure distribution of the inlet duct outlet cross section, reflecting the circumferential non-uniformity of the inlet duct outlet flow field, and the surface average pulsation intensity is a time-varying random variable associated with the total pressure fluctuation on the inlet duct outlet plane.

[0030] SS4. Constructing a GPCE uncertainty quantification proxy model Based on the CFD calculation results of a small sample set, the Gradient-Enhanced Polynomial Chaos Expansion (GPCE) method is used to model the uncertainty quantification problem. A surrogate model containing orthogonal polynomial basis functions and their derivatives is constructed by combining the objective function value and its gradient information. The GPCE surrogate model is then used to analytically calculate the statistical moments of the objective function and its gradient information relative to the design variables.

[0031] In the embodiment of the present invention, the process of constructing the GPCE uncertainty quantification agent model at least includes: SS41. Determining orthogonal polynomial basis functions and model framework First, according to the probability distribution type of the uncertain design variables, select the multivariate orthogonal polynomial basis function form that is suitable for it. If the variables obey the normal distribution, choose the Hermite polynomial; if the variables obey the uniform distribution, choose the Legendre polynomial. Secondly, the effect sparsity principle and hyperbolic truncation method are applied to limit the interaction order of multivariate orthogonal polynomial basis functions, retain the main effects and important low-order interaction effects, and eliminate high-order nonlinear interaction terms; After that, the uncertainty quantification problem of the objective function is modeled and the random response of the objective function is Y ( D , x ) is expanded into a linear combination of orthogonal polynomial basis functions: in: D is the deterministic design variable vector, x for n dimensional uncertain design variable vector and x =( x 1, x 2,..., x n ); P j ( x ) is the j The order basis functions, c j ( D ) is the j The expansion coefficients of the order basis functions, P +1 is the total number of basis functions and , n for x The dimension of p is the highest order of expansion; SS42. Introducing objective function gradient information to construct GPCE proxy model The gradient information of the objective function relative to the design variables is introduced into the model framework to construct a gradient-enhanced GPCE agent model. The gradient expansion form of the objective function is: in, Designing variables for uncertainty x i The standard random variable form of m i , s i Respectively i Uncertain design variables x i The mean and standard deviation of i =1,2,… n , n for x The dimension of SS43. Solving the expansion coefficients of orthogonal polynomials Constructing the augmented linear system ,in[ P ] is the augmented basis function matrix composed of basis functions and their derivatives, c is the expansion coefficient vector to be solved, Y is a vector consisting of the function value and gradient value of the sample point; the system of equations is based on N s The sampling points were constructed N s (1+ n ) equations, n for x The dimension of is the number of sample points in the small sample set, the basis function P j Design variables for uncertainty x i The partial derivative of The standard random variables are calculated by the analytical expression of orthogonal polynomials. X i right x i The partial derivative of s i The probability density function of the uncertain design variables is determined analytically and solved using the least squares method. Get the coefficients of orthogonal polynomial expansion c j ( D ); SS44. Analytically calculate the statistical moments of the objective function and its gradient Based on the constructed GPCE proxy model and the solved expansion coefficient, the statistical moment of the objective function is calculated analytically, where the mean of the objective function is m Y By the constant coefficient c 0( D ) is given and , the variance of the objective function satisfy , is the normalization factor of the basis function, and on this basis, the statistical moment is calculated for the deterministic design variables D Gradient and ; SS45. Verification and optimization of GPCE proxy model accuracy Calculate the normal root mean square deviation , verify the prediction accuracy of the GPCE proxy model, where Y i is the true value of the objective function, is the predicted value of the GPCE proxy model. The closer the NRMSD value is to 0, the higher the prediction accuracy is. According to the verification results, the highest order of the orthogonal polynomial basis function in the GPCE proxy model is adjusted. p , truncation strategy and / or number of sampling points in small sample sets N s , in order to improve model accuracy and reduce computational costs.

[0032] SS5. Robust optimization design and iterative update of aircraft Based on the statistical moments of the objective function calculated by the GPCE proxy model and its gradient information relative to the design variables, all design variables are iteratively updated using the gradient optimization algorithm. During the optimization iteration, the update of the geometric design variables is achieved by adjusting the positions of the FFD control points. The surface meshes of the aerodynamic shape of the aircraft and the intake and exhaust systems are regenerated based on the updated FFD control points, and the volume meshes are updated using the Inverse Distance Weighting (IDW) method. The update of the uncertain design variables is adjusted according to the search direction and step size of the gradient optimization algorithm. During the optimization iteration, the update of all design variables must strictly meet the constraints until the convergence criteria are met.

[0033] Preferably, during the iterative update process of the gradient optimization algorithm, a line search or trust region method is used to determine the appropriate step size to ensure the convergence and efficiency of the algorithm, and the constraint conditions are processed using a penalty function method or a barrier function method to ensure that the optimization results meet the design requirements; if the change in the objective function value is less than a set threshold, the gradient norm of the objective function equivalent to the design variable is less than a set threshold, the change in the design variable is less than a set threshold, the residuals of all constraints are less than a preset allowable error and / or the number of iterations reaches a preset maximum value, then the uncertainty optimization design process is considered to meet the convergence criterion.

[0034] SS6. Optimization result verification and robustness evaluation After the optimization is completed, the final optimization results are output, the optimized aircraft configuration is verified and robustness evaluated, the coupling characteristics of the intake and exhaust system and the external flow field are analyzed, and the comprehensive performance of the optimized configuration is evaluated to ensure that the design meets actual flight requirements.

[0035] Preferably, the optimization result verification adopts the CFD numerical simulation method, and the optimized aircraft configuration is subjected to CFD calculation under deterministic conditions, and the calculation results are compared with the initial design to evaluate the optimization effect; the robustness evaluation adopts the Monte Carlo method to perform uncertainty analysis on the optimized aircraft configuration, and evaluate the changes in its aerodynamic performance under different values ​​of uncertainty variables to verify the robustness of the design; and in the process of robustness evaluation, the coupling characteristics of the intake and exhaust system and the external flow field are analyzed, including the changes in the total pressure recovery coefficient and / or distortion coefficient of the inlet duct, and the influence of the intake and exhaust system on the overall aerodynamic performance of the aircraft.

[0036] The above embodiment 1 provides a robust optimization method based on D-optimality and considering the influence of intake and exhaust of the present invention, which significantly improves the performance and robustness of aircraft design by combining accurate aircraft aerodynamic performance modeling and uncertainty analysis. Through efficient optimization algorithms and proxy models, this method can ensure that the optimization results have excellent aerodynamic performance and reliability under actual flight conditions, meet the design requirements of high precision and high stability, and have important engineering application value.

[0037] Example 2: Uncertainty Optimization of a Flying Wing Configuration Including an Inlet Based on the above-mentioned Example 1, as a further application, Example 2 performs uncertainty optimization on the X1 flying wing configuration including the air inlet. By comparing the results of deterministic optimization and uncertainty optimization, the aerodynamic performance and robustness improvement after optimization are analyzed.

[0038] Based on Example 1, this Example 2 selects an X1 flying wing configuration including an air inlet for uncertainty optimization. The Mach number in the optimized state parameter Yes Angle of attack AoAFor the uncertain design variables, we assume that they follow a normal distribution. Yes ~N(0.75,0.03 2 ), AoA ~N(3,0.4 2 ). The optimization objective is the objective function under the optimization state. I ( m , s ) is the smallest, where the mean value of aerodynamic performance m and standard deviation s The weighted sum of constitutes the objective function, and the aerodynamic constraint is the moment balance under the optimized state. C my =-0.002. The optimization design variables are: ① 266 FFD control points for controlling wing deformation; ② 64 FFD control points for controlling inlet lip deformation; ③ 72 FFD control points for controlling inlet deformation; ④ cruise angle of attack. During the optimization process, thickness constraints were imposed on the wing and fuselage, with the constraint position ranging from 5% to 95%, and the constraint range being such that the thickness was not less than 98% of the initial thickness. A total of 154 thickness constraints were imposed. The FFD frame used for initial configuration optimization is as follows: Figure 3 shown.

[0039] At the same time, in order to compare and analyze the optimization effect, this embodiment 2 also performs a deterministic optimization with the optimization parameters consistent with the uncertainty optimization. The optimization state is set as: Yes =0.75, C l =0.20, the optimization target is the objective function 5 in the optimized state C D - s AV Minimum, among which C D is the drag coefficient, s AV is the standard deviation of the lift coefficient, the aerodynamic constraint is the moment balance under the optimized state, C my =-0.002, ensuring that the aircraft has balanced aerodynamic characteristics in the optimized state. By comparing the results of uncertainty optimization and deterministic optimization, this embodiment 2 analyzes in detail the differences between the two in robustness, performance improvement and number of optimization iterations, thereby providing guidance for robustness optimization of aircraft design.

[0040] In this Example 2, the uncertainty optimization results (marked as Unopt) and the deterministic optimization results (marked as Deopt) are shown in Table 1. Compared with the initial configuration, the statistical moments of the configuration after uncertainty optimization are m ( C D )+ s ( CD ) decreased by 6.8%, indicating that the optimized design not only improved the mean value of aerodynamic performance m ( C D ), while reducing the standard deviation s ( C D ), which effectively enhances the robustness of the design. Compared with the initial configuration, the optimized configuration based on uncertainty has a lower standard deviation.

[0041] Table 1 Comparison of deterministic and uncertain optimization before and after

[0042] Drag coefficient of the uncertainty optimized configuration under benchmark conditions C D The drag coefficient of the deterministic optimization configuration is 139.44, which is 5.3 counts lower than the initial configuration of 144.74, while the drag coefficient of the deterministic optimization configuration is reduced by 5.5 counts. Although the drag of the uncertainty optimization is slightly higher, its advantage in robustness is more prominent. The main reason for this difference is that the deterministic optimization has undergone more than 200 rounds of optimization, but due to time constraints, the robust optimization has only 4 rounds. Nevertheless, the uncertainty optimization can provide a significant improvement in the stability and robustness of the aircraft performance while ensuring lower drag.

[0043] Figure 4 The schematic diagram of the distribution of the total pressure recovery coefficient of the inlet before and after optimization and the change of the central axis and area of ​​the inlet is presented. In the figure: (a) is the distribution cloud map of the end face total pressure recovery coefficient σ of the initial configuration (left half) and the configuration after uncertainty optimization (right half). The color of the cloud map changes from blue to red, corresponding to the change range of σ value from 0.85 to 1.0; (b) is the distribution curve of the inlet area along the axial direction (x) of the initial configuration, deterministic optimization configuration and uncertainty optimization configuration; (c) is the height z distribution curve of the central axis of the three configurations along the axial direction. It can be seen from the figure that for the initial and uncertainty optimization configurations, the low total pressure recovery area ( s <0.91) concentrated at the edge of the end face, high total pressure recovery area ( s >0.97) are concentrated in the center of the end surface. Compared with the initial configuration, the high total pressure recovery area of ​​the uncertainty optimized configuration is larger, indicating that the aerodynamic performance of the inlet is improved after optimization. Figure 4 From the area distribution curve in Figure (b), we can see that the air inlet area of ​​the uncertainty optimization configuration is larger than that of the initial configuration, with two peaks near x=0.8 and x=1.2. Figure 4The central axis distribution in Figure (c) shows that the optimized configuration shifts downward more significantly in the x>0.8 region, but the overall trend remains similar to the initial configuration. This moderate geometric change helps maintain a smooth transition of the flow channel and avoids drastic flow separation.

[0044] Figure 5 The figure shows the distribution of pressure coefficient and its statistical response at different span positions of the wing. (a) to (d) are the pressure coefficient distribution at spans of Y / b=0.3, Y / b=0.5, Y / b=0.7, and Y / b=0.9, respectively. Each sub-figure contains the pressure coefficient curves of the initial configuration (solid line), the deterministic optimized configuration (red dashed line), and the uncertain optimized configuration (blue dotted line), and also shows the statistical response range of the pressure coefficient of each configuration (±2 times the standard deviation, represented by the gray shaded area). In order to explore the robust design more deeply, the statistical response of the pressure coefficient and lift distribution at different span positions of the wing is analyzed. For the pressure coefficient, the results of the initial configuration show the worst robustness. Since the robust optimization results have not yet fully converged, it can be seen that the pressure distribution is not smooth enough, but the variance perturbation range of the uncertainty optimization (blue shaded area) is significantly better than that of the initial configuration. The statistical response of the pressure coefficient distribution further explains the robustness of the uncertainty-based optimization.

[0045] Through the above embodiments, the purpose of the present invention is fully and effectively achieved. Those skilled in the art can understand that the present invention includes but is not limited to the contents described in the drawings and the above specific embodiments. Although the present invention has been described with respect to the most practical and preferred embodiments currently considered, it should be understood that the present invention is not limited to the disclosed embodiments, and any modification that does not deviate from the functional and structural principles of the present invention will be included in the scope of the claims.

Claims

1. A robust optimization method based on D-optimality and considering the influence of intake and exhaust, characterized in that: include: SS1. Define the optimization objective, where the objective function is the weighted sum of the mean and standard deviation of the target aerodynamic performance; use the FFD method to perform geometric parameterized modeling of the aerodynamic shape and intake and exhaust system of the aircraft, and impose constraints accordingly; Define optimization design variables, including geometric design variables and aerodynamic design variables, and divide them into deterministic design variables and uncertain design variables; SS2. Generate an initial large-scale candidate sample set using random sampling method according to the probability distribution of uncertain design variables; Based on the D-optimal criterion, the greedy algorithm of QR decomposition is used to screen out a small sample set from the candidate sample set by maximizing the product of the eigenvalues ​​of the basis function matrix and the information matrix. SS3. For each sample point in the small sample set, perform CFD numerical simulation on the external aerodynamic flow field of the aircraft and the internal flow field of the intake and exhaust system. The internal and external flow fields include the external aerodynamic flow field of the aircraft and the internal flow field of the intake and exhaust system, obtain the target aerodynamic performance parameters, and use the discrete adjoint method to calculate the gradient information of the target aerodynamic performance parameters; SS4. Use the GPCE method to model the uncertainty quantification problem. By combining the objective function value and its gradient information, a GPCE proxy model containing orthogonal polynomial basis functions and their derivatives is constructed. The proxy model is used to calculate the objective function statistical moment and its gradient information. SS5. Based on the objective function statistical moment and its gradient information, the design variables are iteratively updated using the gradient optimization algorithm. The geometric design variables are updated by adjusting the position of the FFD control points, the surface mesh is regenerated based on the updated FFD control points, and the volume mesh is updated using the IDW method. The update of the uncertain design variables is adjusted according to the search direction and step size of the gradient optimization algorithm. The update of all design variables must strictly meet the constraints until the objective function meets the convergence criteria; SS6. Output the optimization results and verify and evaluate the robustness of the optimized aircraft configuration.

2. A robust optimization method based on D-optimality and considering the effects of intake and exhaust according to claim 1, characterized in that: In step SS1, the objective function I ( μ , σ ) is defined as the weighted sum of the statistical moments of the target aerodynamic performance and satisfies I ( μ , σ )= K 1. μ + K 2. σ , μ is the mean, σ is the standard deviation; K 1. K 2 are weight coefficients and satisfy K 1+ K 2=1; mean μ and standard deviation σ The calculation is based on the statistical moment of the objective function calculated analytically by the proxy model; the optimization state condition is set to the actual flight state under cruise flight conditions and the influence of uncertainty is taken into account. The flight state parameters include at least Mach number, incoming flow angle of attack, sideslip angle and / or inlet flow parameters. For the Mach number and incoming flow angle of attack with uncertain quantities, their probability distribution type and statistical characteristics must be given.

3. A robust optimization method based on D-optimality and considering the effects of intake and exhaust according to claim 1, characterized in that: In step SS1, the FFD method is implemented by generating a control frame around the geometric shape of the aircraft, and the FFD control points perform geometric parameterized modeling of the aircraft's aerodynamic shape, inlet lip shape, and inlet internal flow path shape by adjusting their positions; the number and distribution of FFD control points are allocated according to the complexity of the aircraft design area; the constraints include at least aerodynamic performance constraints, geometric shape constraints, and / or intake and exhaust system-related constraints, the aerodynamic performance constraints include moment balancing constraints and / or lift coefficient constraints, the geometric shape constraints include thickness constraints and / or volume constraints, and the intake and exhaust system-related constraints include inlet total pressure recovery coefficient constraints and / or distortion coefficient constraints.

4. A robust optimization method based on D-optimality and considering the effects of intake and exhaust according to claim 3, characterized in that: In step SS1, the geometric design variables are defined by the FFD method, including at least the FFD control points that control the aerodynamic shape of the aircraft, the FFD control points that control the geometric shape of the inlet, and the FFD control points that control the deformation of the inlet lip; the aerodynamic design variables include parameters related to the flight state, including at least the Mach number, the incoming flow angle of attack, the sideslip angle and / or the inlet flow parameters; the deterministic design variables are used to define the fixed geometric shape and aerodynamic characteristics, and the uncertain design variables are used to characterize the random effects of flight conditions and environmental disturbances on the aerodynamic performance. The probability density distribution of the uncertain design variables is determined by statistical analysis, and it is assumed that they obey the normal distribution or uniform distribution.

5. A robust optimization method based on D-optimal and considering the influence of intake and exhaust according to claim 4, characterized in that: In step SS2, the small sample set construction process includes at least the following sub-steps: SS21. Generate an initial large-scale candidate sample set based on the probability distribution type and statistical characteristics of the uncertain design variables by Monte Carlo sampling or Latin hypercube sampling method. The sample points cover the full range of the uncertain design variables and the distribution characteristics are consistent with the probability distribution of the variables. The number of samples is set according to the number of uncertain design variables and the order of the subsequent polynomial chaos expansion. SS22. Based on the initial large-scale candidate sample set, a polynomial basis function matrix related to the uncertain design variables is constructed. The basis function adopts the orthogonal polynomial basis function of the polynomial chaos expansion PCE. Each element of the basis function matrix consists of the value of the orthogonal polynomial basis function at the candidate sample point. Based on the basis function matrix, a basic information matrix of the small sample set for D optimal criterion screening is formed. The information matrix is ​​defined as the product of the transposed matrix of the basis function matrix and itself. SS23. Using the constructed information matrix, the greedy algorithm of QR decomposition is used to screen the candidate sample points. By decomposing the basis function matrix QR into an orthogonal matrix Q and an upper triangular matrix R, the sample points with the largest absolute value of the diagonal elements of the upper triangular matrix R are selected. According to the greedy algorithm and the distribution uniformity among the sample points, the sample points with the largest determinant value of the information matrix are iteratively selected to add to the small sample set to update the basis function matrix information matrix, so as to maximize the product of the eigenvalues ​​of the basis function matrix information matrix, until the small sample set reaches the preset size or meets the stopping criterion; SS24. During the screening process, the number of small sample sets is dynamically adjusted according to the complexity of the objective function and the computational cost requirements, and after the optimization is completed, a small sample set that meets the statistical characteristics is formed; SS25. Perform quality verification on the small sample set that has been screened, and evaluate the coverage and distribution uniformity of the sample points by calculating the eigenvalue distribution of the information matrix and the minimum distance between samples.

6. A robust optimization method based on D-optimal and considering the influence of intake and exhaust according to claim 5, characterized in that: In step SS2, the initial large-scale candidate sample set is generated by Monte Carlo sampling or Latin hypercube sampling, where Monte Carlo sampling simulates the probability distribution characteristics of variables by randomly generating sample points, and the sampling density is dynamically adjusted according to the distribution range of the variables and the importance of their impact on the objective function; Latin hypercube sampling divides the variable range into several equal intervals by stratified sampling, and randomly generates a sample point in each interval.

7. A robust optimization method based on D-optimality and considering the effects of intake and exhaust according to claim 1, characterized in that: In step SS3, the CFD numerical simulation solves the RANS equations based on the finite volume method and combines the turbulence model to simulate the influence of turbulent viscosity. The viscous flux in the flow field calculation is discretized using the standard central difference, and the inviscid flux is discretized using the artificial dissipative central difference. When solving the residual equation, the Runge-Kutta method and the Newton-Krilov method are dynamically switched to optimize the solution efficiency to ensure that the residual norm is reduced by at least five orders of magnitude to ensure the stability and accuracy of the flow field solution. The interaction between the intake and exhaust system and the external flow field is simulated using the overlapping grid technology, and the exchange of internal and external flow field information is achieved through the interpolation method. The discrete adjoint method is used to calculate the gradient of the target aerodynamic performance to the design variables.

8. The robust optimization method based on D-optimal and considering the influence of intake and exhaust according to claim 1, characterized in that: In step SS3, the target aerodynamic performance parameters include aircraft aerodynamic shape performance parameters and intake and exhaust system performance parameters, wherein the aircraft aerodynamic shape performance parameters are drag coefficient, lift coefficient and / or moment coefficient, and the intake and exhaust system performance parameters include inlet duct total pressure recovery coefficient and / or distortion coefficient, the inlet duct total pressure recovery coefficient is defined as the ratio of the average total pressure at the inlet duct outlet to the free flow total pressure, the distortion coefficient is composed of the sum of the circumferential non-uniformity of the total pressure and the surface average pulsation intensity, the circumferential non-uniformity of the total pressure is calculated based on the total pressure distribution of the inlet duct outlet cross section, and the surface average pulsation intensity is a time-varying random variable associated with the total pressure fluctuation on the inlet duct outlet plane.

9. A robust optimization method based on D-optimal and considering the influence of intake and exhaust according to claim 1, characterized in that: In step SS4, the construction process of the GPCE proxy model includes: SS41. Select the appropriate multivariate orthogonal polynomial basis function form according to the probability distribution type of the uncertain design variables; apply the effect sparsity principle and hyperbolic truncation method to limit the interaction order of the multivariate orthogonal polynomial basis function, retain the main effect and important low-order interaction effects; model the uncertainty quantification problem of the objective function and convert the random response of the objective function into Y ( D , ξ ) is expanded into a linear combination of orthogonal polynomial basis functions: in: D is the deterministic design variable vector, ξ =( ξ 1, ξ 2,..., ξ n )for n dimensional uncertain design variable vector; Ψ j ( ξ ) is the j The order basis function, c j ( D ) is the j The expansion coefficients of the order basis functions, the total number of basis functions is , n for ξ The dimension of p is the highest order of expansion; SS42. The gradient information of the objective function relative to the design variables is introduced to construct a gradient-enhanced GPCE agent model. The gradient expansion form of the objective function is: in, Designing variables for uncertainty ξ i The standard random variable form of μ i , σ i Respectively i Uncertain design variables ξ i The mean and standard deviation of i =1,2,… n , n for ξ The dimension of SS43. Constructing Augmented Linear Systems , [ Ψ ] is the augmented basis function matrix composed of basis functions and their derivatives, c is the expansion coefficient vector to be solved, Y is a vector consisting of the function value and gradient value of the sample point; the system of equations is based on N s The sampling points were constructed N s (1+ n ) equations, n for ξ The dimension of is the number of sample points in the small sample set, the basis function Ψ j Design variables for uncertainty ξ i The partial derivative of The standard random variables are calculated by the analytical expression of orthogonal polynomials. X i right ξ i The partial derivative of σ i The probability density function of the uncertain design variables is determined analytically and solved using the least squares method. Get the expansion coefficient c j ( D ); SS44. Based on the GPCE proxy model and the solved expansion coefficients, the statistical moments and mean of the objective function are calculated analytically. μ Y By the constant coefficient c 0( D ) is given and , the variance of the objective function satisfy , is the normalization factor of the basis function, and on this basis, the statistical moment is calculated for the deterministic design variables D Gradient and ; SS45. Calculate the normal root mean square deviation , verify the prediction accuracy of the GPCE proxy model, where Y i is the true value of the objective function, is the predicted value of the GPCE proxy model; according to the verification results, adjust the highest order of the orthogonal polynomial basis function in the GPCE proxy model p , truncation strategy and / or number of sampling points in small sample sets N s .

10. A robust optimization method based on D-optimality and considering the effects of intake and exhaust according to claim 1, characterized in that: In step SS5, during the iterative update process, a line search or trust region method is used to determine the appropriate step size to ensure the convergence and efficiency of the algorithm, and the constraint conditions are processed using a penalty function method or a barrier function method to ensure that the optimization results meet the design requirements; if the change in the objective function value is less than a set threshold, the gradient norm of the objective function relative to the design variable is less than a set threshold, the change in the design variable is less than a set threshold, the residuals of all constraints are less than the preset allowable error and / or the number of iterations reaches a preset maximum value, then the convergence criterion is considered to be met.

11. A robust optimization method based on D-optimality and considering the effects of intake and exhaust according to claim 1, characterized in that: In step SS6, the optimization result is verified by using the CFD numerical simulation method, and the optimized aircraft configuration is subjected to CFD calculation under deterministic conditions, and the calculation results are compared with the initial design to evaluate the optimization effect; the robustness evaluation adopts the Monte Carlo method to perform uncertainty analysis on the optimized aircraft configuration, evaluate the changes in its aerodynamic performance under different values ​​of uncertainty variables, and verify the robustness of the design; and in the process of robustness evaluation, analyze the coupling characteristics of the intake and exhaust system and the external flow field, including the changes in the total pressure recovery coefficient and / or distortion coefficient of the inlet duct, and the influence of the intake and exhaust system on the overall aerodynamic performance of the aircraft.

12. A computer program product, characterized in that: It includes computer instructions for executing the robust optimization method based on D optimality and considering the influence of intake and exhaust as described in any one of claims 1 to 11.

13. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the robust optimization method based on D optimality and taking into account the influence of intake and exhaust as described in any one of claims 1 to 11 is implemented.

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