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Variational Methods in Design Optimization and Sensitivity Analysis for Two-Dimensional Euler Equations

机译:二维Euler方程设计优化和灵敏度分析的变分方法

摘要

Variational methods (VM) sensitivity analysis employed to derive the costate (adjoint) equations, the transversality conditions, and the functional sensitivity derivatives. In the derivation of the sensitivity equations, the variational methods use the generalized calculus of variations, in which the variable boundary is considered as the design function. The converged solution of the state equations together with the converged solution of the costate equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The application of the variational methods to aerodynamic shape optimization problems is demonstrated for internal flow problems at supersonic Mach number range. The study shows, that while maintaining the accuracy of the functional sensitivity derivatives within the reasonable range for engineering prediction purposes, the variational methods show a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference sensitivity analysis.
机译:变分方法(VM)敏感性分析用于推导肋骨(伴随)方程,横向条件和功能敏感性导数。在推导灵敏度方程式时,变分方法使用广义的变分演算,其中将可变边界视为设计函数。状态方程的收敛解与肋式方程的收敛解一起沿着区域边界积分,以唯一地确定相对于设计函数的功能灵敏度导数。针对超音速马赫数范围内的内部流动问题,论证了变分方法在空气动力学形状优化问题中的应用。研究表明,与有限差分灵敏度分析相比,变分方法在将功能灵敏度导数的精度保持在合理范围内以进行工程预测的同时,在计算效率(即计算机时间和内存)方面显示出可观的收益。

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