Optimization design method of aircraft boundary characteristics based on upwind scheme adjoint equation
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摘要:
飞行器的边界特性决定了其安全性和飞行性能，是飞行器设计的难点和重点。引入迎风格式伴随方程和不同的通量限制器处理形式，提高复杂流动问题的伴随方程求解精度、效率和鲁棒性，拓展伴随优化方法在飞行器边界特性设计的应用范围。介绍了离散伴随求解梯度的基本原理，在此基础上推导了伴随方程的无黏项及其边界条件的变分形式，根据通量限制器的处理方式，形成一阶精度、二阶精度和混合精度的伴随方程。对伴随方程的边界处理措施进行了研究。通过ONERA M6机翼梯度精度和鲁棒性验证算例，对比迎风格式和中心格式的伴随方程求解性能，分析限制器和边界处理措施对伴随方程的收敛性和梯度精度的影响。通过CRM翼身组合体巡航气动优化设计和边界特性优化设计算例，验证了求解器对飞行器巡航性能设计和边界特性设计的有效性。计算和设计结果表明，建立的迎风格式伴随方程求解方法鲁棒和梯度精度高，能够适用于飞行器边界特性设计难题的求解。
Abstract:A challenging and crucial aspect of aircraft design is defining the boundary characteristics of the aircraft, which dictate their safety and flying performance. In order to increase the precision, effectiveness, and resilience of complex flow problems, we suggested in this study that adjoint optimization methods be extended to the design of aircraft boundary features. We did this by developing an upwind scheme adjoint equation and various flux limiter treatments. Firstly, the basic principle of discrete adjoint gradient solution was introduced. This served as the foundation for deriving the adjoint equation's inviscid term and its variational form of boundary conditions. According to the processing method of the flux limiter, adjoint equations with first-order accuracy, second-order accuracy, and mixed accuracy were formed. Subsequently, the boundary treatments for the adjoint equation were studied. By using the ONERA M6 wing for gradient accuracy and robustness validation examples, the performance of solving adjoint equations using upwind and central schemes was compared, and the effects of limiter and boundary treatments on the convergence and gradient accuracy of the adjoint equations were analyzed. The effectiveness of the solver in designing aircraft cruise performance and boundary characteristics was verified through examples of CRM wing body at cruise and boundary characteristics optimization design. The computation and design outcomes demonstrated the robustness and high gradient accuracy of the upwind approach developed in the article, suggesting its potential for resolving design issues pertaining to aircraft boundary characteristics.
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图 2 基于离散伴随方法的梯度求解流程与优化设计框架
Figure 2. Gradient solving process based on discrete concomitant methods and a framework for optimization design
图 4 巡航状态下迎风格式与中心格式伴随方程收敛性和梯度精度对比
Figure 4. Convergence and gradient accuracy comparison of upwind scheme and central scheme adjoint equation in cruise state
图 5 边界状态下迎风格式与中心格式伴随方程收敛性对比
Figure 5. Convergence comparison of upwind scheme and central scheme adjoint equation in boundary state
图 6 边界状态下迎风格式与有限差分梯度精度对比
Figure 6. Gradient accuracy comparison of upwind scheme and central scheme adjoint equation in boundary state
图 7 限制器处理方式对收敛性和梯度精度的影响
Figure 7. Influence of limiter treatment on convergence and gradient accuracy
图 8 限制器全变分时壁面通量修正变分对伴随方程收敛性和梯度精度的影响
Figure 8. Influence of wall modify on convergence and gradient accuracy in limiter variation
图 9 限制器部分变分时壁面通量修正变分对伴随方程收敛性和梯度精度的影响
Figure 9. Influence of wall modify on convergence and gradient accuracy in fix part of limiter
图 10 限制器固定时壁面通量修正变分对伴随方程收敛性和梯度精度影响
Figure 10. Influence of wall modify on convergence and gradient accuracy in fix limiter
图 11 混合精度伴随方程对收敛性和梯度精度的影响
Figure 11. Influence of mixed adjoint equation on convergence and gradient accuracy
图 15 基础外形（左）与优化外形（右）表面压力分布
Figure 15. Pressure distribution of baseline model (left) and opt model (right)
图 16 基础外形与优化外形各站位压力分布
Figure 16. Distributions at different sections of baseline model and opt model
图 17 基础外形(左)与优化外形(右)表面压力分布
Figure 17. Pressure distribution of baseline model (left) and opt model (right)
图 18 基础外形与优化外形各站位压力分布
Figure 18. Distributions at different sections of baseline model and opt model
图 19 抖振边界状态下混合精度伴随方程与中心格式伴随方程收敛性对比
Figure 19. Convergence comparison of mixed adjoint equation and central scheme adjoint equation in buffeting boundary state
图 22 基础外形(左)与优化外形(右)表面压力分布
Figure 22. Pressure distribution of baseline model (left) and opt model (right)
图 23 基础外形（左）与优化外形（右）各站位压力分布
Figure 23. Distributions at different sections of baseline model (left) and opt model (right)
图 24 基础外形（左）与优化外形（右）表面极限流线
Figure 24. Limiting streamline of surface of baseline model (left) and opt model (right)
表 1 求解器性能验证计算状态
Table 1. Solver performance validate status
计算状态 Ma Re CL 巡航状态 0.84 6.8×106 0.245 0 边界状态 0.84 6.8×106 0.400 0 
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