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首页> 外文期刊>Journal of Computational Physics >The effect of shocks on second order sensitivities for the quasi-one-dimensional Euler equations
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The effect of shocks on second order sensitivities for the quasi-one-dimensional Euler equations

机译：冲击对拟一维Euler方程的二阶灵敏度的影响

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摘要

The effect of discontinuity in the state variables on optimization problems is investigated on the quasi-one-dimensional Euler equations in the discrete level. A pressure minimization problem and a pressure matching problem are considered. We find that the objective functional can be smooth in the continuous level and yet be non-smooth in the discrete level as a result of the shock crossing grid points. Higher resolution can exacerbate that effect making grid refinement counter productive for the purpose of computing the discrete sensitivities. First and second order sensitivities, as well as the adjoint solution, are computed exactly at the shock and its vicinity and are compared to the continuous solution. It is shown that in the discrete level the first order sensitivities contain a spike at the shock location that converges to a delta function with grid refinement, consistent with the continuous analysis. The numerical Hessian is computed and its consistency with the analytical Hessian is discussed for different flow conditions. It is demonstrated that consistency is not guaranteed for shocked flows. We also study the different terms composing the Hessian and propose some stable approximation to the continuous Hessian.
机译：在离散级别上的准一维欧拉方程中研究了状态变量中的不连续性对优化问题的影响。考虑压力最小化问题和压力匹配问题。我们发现，目标函数在连续水平上可以是平滑的，而在离散水平上由于冲击点的交叉点是不平滑的。更高的分辨率会加剧这种影响，使网格细化无法发挥作用，从而无法计算离散的灵敏度。一阶和二阶灵敏度以及伴随解在冲击波及其附近精确计算，并与连续解进行比较。结果表明，在离散水平上，一阶灵敏度在冲击位置处包含一个尖峰，随着网格的细化，收敛到一个三角函数，这与连续分析一致。计算了数值Hessian，并讨论了其在不同流动条件下与解析Hessian的一致性。已经证明，不能保证冲击流的一致性。我们还研究了组成粗麻布的不同术语，并对连续的粗麻布提出了一些稳定的近似。

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