Variational framework for flow optimization using seminorm constraints.
Variational framework for flow optimization using seminorm constraints.
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使用半范数约束进行流优化的变分框架。
DOI：
10.1103/physreve.86.026306
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
Foures DP
中科院分区：
文献类型：
--
作者：
Foures DP
我们提出了一个通用变分框架，旨在考虑定义为偏微分方程解的空间和时间演变流的约束优化和灵敏度分析。我们特别关注自然产生的非线性约束，例如当我们希望优化的量可以通过特定物理系统中的不同物理机制从PDE中的几个项中做出贡献时。我们表明，这种情况下隐含的要求，约束被放置在互补（相对于第一个约束条件）的初始扰动的范数的大小，这些互补范数的总和定义了一个总的“真”的状态向量的范数。一个简单的（真的）范数约束自然满足这个性质。因此，使用这个框架需要引入新的参数，这些参数描述了使用各种约束互补范数计算的初始扰动状态向量的相对大小，即使对于线性问题，也是使用真范数计算的大小。我们证明，任何所需的优化必须进行规定这些新的参数作为初始条件的容许扰动的影响和意义的每个可接受的组件，分区的初始总范数的扰动，然后可以被认为是定量的。为了证明这个框架的实用性，我们考虑一个理想化的问题，（线性）非模态稳定性分析的平均流的“雷诺平均”的一维随机强迫Burgers方程。我们通过引入湍流粘性来模拟湍流混合，从而关闭平均流方程，并允许其根据新的输运方程进行演化。由于我们感兴趣的是优化扰动动能的相对放大（即，微扰的“增益”）这个问题自然需要使用我们的新框架，因为动能是具有新的可调参数的全状态速度-粘度矢量的λ，该参数描述适当的粘度λ与该粘度λ和动能λ之和的比率。使用这个框架，我们证明了完整的系统的动态，允许湍流粘度演变受其传输方程，是定性不同的行为时，湍流粘度是“冻结”在一个固定的，平均值，因为出现了一个新的机制的扰动能量生产，通过耦合的演变湍流粘度扰动和平均速度场。
We present a general variational framework designed to consider constrained optimization and sensitivity analysis of spatially and temporally evolving flows defined as solutions of partial differential equations. We particularly focus on seminorm constraints which naturally arise for instance when the quantity which we wish to optimize can have contributions from several terms in the PDE through different physical mechanisms in a specific physical system. We show that this case implicitly requires that constraints be placed on the magnitude of complementary (with respect to the first constraining seminorm) seminorms of initial perturbations such that the sum of these complementary seminorms defines a total “true” norm of the state vector. A simple (true) norm constraint naturally satisfies this property. Therefore, the use of this framework requires the introduction of new parameters which describe the relative magnitude of the initial perturbation state vector calculated using the various constrained complementary seminorms to the magnitude calculated using the true norm, even for linear problems. We demonstrate that any required optimization has to be carried out by prescribing these new parameters as initial conditions on the admissible perturbations; the influence and significance of each seminorm component, partitioning the initial total norm of the perturbation, can then be considered quantitatively. To demonstrate the utility of this framework, we consider an idealized problem, the (linear) nonmodal stability analysis of a mean flow given by a “Reynolds averaging” of the one-dimensional stochastically forced Burgers equation. We close the mean flow equation by introducing a turbulent viscosity to model the turbulent mixing, which we allow to evolve subject to a new transport equation. Since we are interested in optimizing the relative amplification of the perturbation kinetic energy (i.e., the perturbation's “gain”) this problem naturally requires the use of our new framework, as the kinetic energy is a seminorm of the full state velocity-viscosity vector, with a new adjustable parameter, describing the ratio of an appropriate viscosity seminorm to the sum of this viscosity seminorm and the kinetic energy seminorm. Using this framework, we demonstrate that the dynamics of the full system, allowing the turbulent viscosity to evolve subject to its transport equation, is qualitatively different from the behavior when the turbulent viscosity is “frozen” at a fixed, mean value, since a new mechanism of perturbation energy production appears, through the coupling of the evolving turbulent viscosity perturbation and the mean velocity field.
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DOI：
10.1016/0167-2789(89)90266-2
发表时间：
1989
期刊：
Physica D: Nonlinear Phenomena
影响因子：
--
作者：
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通讯作者：
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影响因子：
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作者：
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影响因子：
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作者：
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DOI：
--
发表时间：
2006
期刊：
影响因子：
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作者：
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通讯作者：
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表征燃烧不稳定期间的能量增长：奇异值还是特征值？
DOI：
10.1016/j.proci.2008.05.035
发表时间：
2009
期刊：
Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences
影响因子：
--
作者：
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