Sensitivity analysis for Navier-Stokes equations on unstructured meshes using complex variables

Anderson, W. Kyle; Newman, James C.; Whitfield, David L.; Nielsen, Eric J.

doi:10.2514/3.14697

Cited by 53 publications

(70 citation statements)

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“…Until now the CS method has been implemented in multidisciplinary sensitivity analysis (Newman et al, 1998) and for computational fluid dynamics problems (Anderson et al, 1999). It has also been used in validating the adjoint of the aerosol thermodynamic model ISSOROPIA (ANISORROPIA, Capps et al 2012) and other adjoint implementations because of its accuracy in results (Martins et al, 2002;Giles et al, 2000).…”

Section: Chapter 2 -The Complex Step Methodsmentioning

confidence: 99%

Application of the complex step method to chemistry-transport modeling

Constantin¹,

Barrett²

2014

Atmospheric Environment

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Sensitivity analysis in atmospheric chemistry-transport modeling is used to develop understanding of the mechanisms by which emissions affect atmospheric chemistry and composition, to quantify the marginal impact of emissions on air quality, and for other applications including improving estimates of emissions, developing fast first order air quality models, and validating adjoint models. In forward modeling sensitivities have predominantly been calculated using the finite difference approach, i.e. where the results of two separate simulations are subtracted. The finite difference approach incurs truncation and cancellation errors, which mean that exact sensitivities cannot be calculated and even approximate sensitivities cannot always be calculated for a sufficiently small perturbation (e.g. for emissions at a single location or time). Other sensitivity methods can provide exact sensitivities, but require the reformulation of non-linear steps (e.g. the decoupled direct method) or the adjointing of entire codes (partly automatically and partly manually). While the adjoint approach is widely applied and has significant utility in providing source-oriented information, in some applications the receptor-oriented information of forward approaches is needed. Here we apply an alternative method of calculating sensitivities that results in receptor-oriented information as with the finite difference approach, requires minimal reformulation of models, but enables near-exact computation of sensitivities. This approach -the complex step method -is applied for the first time to a complete atmospheric chemistry-transport model (GEOS-Chem). We also introduce the idea of combining complex-step and adjoint sensitivity analysis (for the first time in any context to our knowledge) to enable the direct calculation of near-exact second order sensitivities.Thesis Supervisor:

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Section: Chapter 2 -The Complex Step Methodsmentioning

confidence: 99%

Application of the complex step method to chemistry-transport modeling

Constantin¹,

Barrett²

2014

Atmospheric Environment

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“…(15) is determined using Eq. (14) which requires the solution of the adjoint equations (11)(12)(13). During the solution of the adjoint equations that are integrated backward in time, the sensitivity derivative at each time step is computed and added to its value at the previous time step.…”

Section: Time-dependent Adjoint Formulationmentioning

confidence: 99%

“…Once Eqs. (11)(12)(13) have been solved, the vector of Lagrange multipliers Λ n can be used to evaluate the last two terms in Eq. (10).…”

Section: Time-dependent Adjoint Formulationmentioning

confidence: 99%

“…(11)(12)(13), the derivatives of R and f with respect to Q and D are required to form the adjoint equations and the sensitivity derivative. It is very difficult and time-consuming to obtain these derivatives by manually differentiating a CFD solver, especially if complicated turbulence and physical models are involved.…”

Section: Forming Discrete Adjoint Operators By Using Complex Variamentioning

confidence: 99%

“…Furthermore, any changes in the discretization of the governing equations, boundary conditions, objective functional, or physical models require additional coding and debugging, thus making the software development cycle extremely lengthy. To overcome these difficulties, we use an approach based on complex variables, which has successfully been applied to solving design optimization problems in [ 12,13 ]. The key idea of this technique is to approximate the required real-valued derivatives by using the following formula proposed by Lyness 14 :…”

Section: Forming Discrete Adjoint Operators By Using Complex Variamentioning

confidence: 99%

See 2 more Smart Citations

Adjoint-Based Methodology for Time-Dependent Optimization

Yamaleev

Diskin

Nielsen

2008

12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference

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This paper presents a discrete adjoint method for a broad class of time-dependent optimization problems. The time-dependent adjoint equations are derived in terms of the discrete residual of an arbitrary finite volume scheme which approximates unsteady conservation law equations. Although only the 2-D unsteady Euler equations are considered in the present analysis, this time-dependent adjoint method is applicable to the 3-D unsteady Reynolds-averaged Navier-Stokes equations with minor modifications. The discrete adjoint operators involving the derivatives of the discrete residual and the cost functional with respect to the flow variables are computed using a complex-variable approach, which provides discrete consistency and drastically reduces the implementation and debugging cycle. The implementation of the time-dependent adjoint method is validated by comparing the sensitivity derivative with that obtained by forward mode differentiation. Our numerical results show that O(10) optimization iterations of the steepest descent method are needed to reduce the objective functional by 3-6 orders of magnitude for test problems considered.

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Direct sensitivity analysis for smooth unsteady compressible flows using complex differentiation

Sagaut

2006

Numerical Methods in Fluids

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SUMMARYA method for the direct computation of the instantaneous sensitivities of unsteady compressible flows is proposed. It is based on the complex differentiation of the full compressible Navier-Stokes equations and does not require the storage of the unsteady flow solution to be differentiated. The method does not rely on any assumption on the basic Navier-Stokes solver, and can therefore be implemented in a straightforward way. The method is assessed on several cases, including a two-dimensional subsonic mixing layer. It is observed that the sensitivity patterns can be interpreted thanks to Kovasznay's decomposition for perturbations in a compressible flow.

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Sensitivity analysis for Navier-Stokes equations on unstructured meshes using complex variables

Cited by 53 publications

References 0 publications

Application of the complex step method to chemistry-transport modeling

Application of the complex step method to chemistry-transport modeling

Adjoint-Based Methodology for Time-Dependent Optimization

Direct sensitivity analysis for smooth unsteady compressible flows using complex differentiation

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