Adjoint sensitivity analysis of chaotic dynamical systems with non-intrusive least squares shadowing

Blonigan, Patrick

doi:10.1016/j.jcp.2017.08.002

Cited by 43 publications

(53 citation statements)

References 36 publications

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“…NILSS has been applied successfully to a chaotic flow over a backwards facing step [17]. A discrete adjoint version was derived in [18] and applied for the first time to a 3D DNS simulation of turbulent channel flow at Re τ = 140.…”

Section: Introductionmentioning

confidence: 99%

A preconditioned Multiple Shooting Shadowing algorithm for the sensitivity analysis of chaotic systems

Shawki

Papadakis

2019

Journal of Computational Physics

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We propose a preconditioner that can accelerate the rate of convergence of the Multiple Shooting Shadowing (MSS) method [1]. This recently proposed method can be used to compute derivatives of time-averaged objectives (also known as sensitivities) to system parameter(s) for chaotic systems. We propose a block diagonal preconditioner, which is based on a partial singular value decomposition of the MSS constraint matrix. The preconditioner can be computed using matrix-vector products only (i.e. it is matrix-free) and is fully parallelised in the time domain. Two chaotic systems are considered, the Lorenz system and the 1D Kuramoto Sivashinsky equation. Combination of the preconditioner with a regularisation method leads to tight bracketing of the eigenvalues to a narrow range. This combination results in a significant reduction in the number of iterations, and renders the convergence rate almost independent of the number of degrees of freedom of the system, and the length of the trajectory that is used to compute the time-averaged objective. This can potentially allow the method to be used for large chaotic systems (such as turbulent flows) and optimal control applications. The singular value decomposition of the constraint matrix can also be used to quantify the effect of the system condition on the accuracy of the sensitivities. In fact, neglecting the singular modes affected by noise, we recover the correct values of sensitivity that match closely with those obtained with finite differences for the Kuramoto Sivashinsky equation in the light turbulent regime.

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Section: Introductionmentioning

confidence: 99%

A preconditioned Multiple Shooting Shadowing algorithm for the sensitivity analysis of chaotic systems

Shawki

Papadakis

2019

Journal of Computational Physics

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“…In particular, the problem is the determination of the derivative with respect to control or design inputs, of a long-time average or the ensemble mean of objective functions of interest in statistically stationary turbulent fluid flows. The current methods based on shadowing [4,5] and those based on ensemble averaging of linearized sensitivities [1] suffer from lack of consistency and prohibitive computational cost, respectively. In this work, we propose the space split sensitivity (S3) algorithm as a potential solution to this problem that circumvents the pitfalls of these previous approaches.…”

Section: Discussionmentioning

confidence: 99%

Sensitivity computation of statistically stationary quantities in turbulent flows

Chandramoorthy

Wang

2019

AIAA Aviation 2019 Forum

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It is well-known that linearized perturbation methods for sensitivity analysis, such as tangent or adjoint equation-based, finite difference and automatic differentiation are not suitable for turbulent flows. The reason is that turbulent flows exhibit chaotic dynamics, leading to the norm of an infinitesimal perturbation to the state growing exponentially in time. As a result, these conventional methods cannot be used to compute the derivatives of long-time averaged quantities to control or design inputs. The ensemble-based approaches [1, 2] and shadowingbased approaches ([3-5]) to circumvent the problems of the conventional methods in chaotic systems, also suffer from computational impracticality and lack of consistency guarantees, respectively. We introduce the space-split sensitivity, or the S3 algorithm, which is a Monte Carlo approach to the chaotic sensitivity computation problem. In this work, we derive the S3 algorithm under simplifying assumptions on the dynamics and present a numerical validation on a low-dimensional example of chaos.

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“…This is due to the inherent nature of the DG approach used and underscores the need for an adjoint-driven unstructured mesh adaptation technique. The development of chaotic adjoints for turbulent flows [46] and high-order unstructured hex-dominant mesh generation tools [47] are the focus of current research.…”

Section: Turbulent Kinetic Energy Budgetmentioning

confidence: 99%

Scale-Resolving Simulations of Low-Pressure Turbine Cascades With Wall Roughness Using a Spectral-Element Method

Garai

Diosady

Murman

et al. 2018

Volume 2C: Turbomachinery

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The accurate prediction of wall-roughness effects in turbomachinery is becoming critical as turbine designers address airfoil surface quality and degradation concerns arising from the shift to advanced ceramic matrix composite (CMC) or additively-manufactured airfoils operating in higher temperature environments. In this paper, a recently developed computational capability for accurate and efficient scale-resolving simulations of turbomachinery is extended to analyze the boundarylayer separation and transition characteristics in a rough-wall low-pressure turbine (LPT) cascade. The computational capability is based on an entropy-stable discontinuous-Galerkin spectral-element approach that extends to arbitrarily high orders of spatial and temporal accuracy, and is implemented in an efficient manner for a modern high performance computer architecture. Results from the scale-resolving simulations of both smooth and rough airfoil cascades are presented and compared to previous experiments and numerical simulations. The results show that the suction surface boundary layer undergoes laminar separation, transition, and turbulent reattachment for the smooth airfoil cascade, while in the presence of roughness the separation and transition behavior of the suction surface boundary layer is substantially modified. The differences between the smooth and rough airfoil cascades are then highlighted by a detailed analysis of their respective turbulent flow fields.

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Adjoint sensitivity analysis of chaotic dynamical systems with non-intrusive least squares shadowing

Cited by 43 publications

References 36 publications

A preconditioned Multiple Shooting Shadowing algorithm for the sensitivity analysis of chaotic systems

A preconditioned Multiple Shooting Shadowing algorithm for the sensitivity analysis of chaotic systems

Sensitivity computation of statistically stationary quantities in turbulent flows

Scale-Resolving Simulations of Low-Pressure Turbine Cascades With Wall Roughness Using a Spectral-Element Method

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