On the proper treatment of grid sensitivities in continuous adjoint methods for shape optimization

Kavvadias, Ioannis S.; Papoutsis‐Kiachagias, Evangelos M.; Giannakoglou, Kyriakos C.

doi:10.1016/j.jcp.2015.08.012

Cited by 42 publications

(41 citation statements)

References 21 publications

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“…In particular, we are interested in the UQ of the sensitivities of these quantities with respect to the random system parameters (randomness affects not only the time-averages, but also their sensitivities). The latter are useful to know when performing for example gradient-based optimization under uncertainty [17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Uncertainty quantification of sensitivities of time-average quantities in chaotic systems

2020

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We consider time-average quantities of chaotic systems and their sensitivity to system parameters. When the parameters are random variables with a prescribed probability density function, the sensitivities are also random. The central aim of the paper is to study and quantify the uncertainty of the sensitivities; this is useful to know in robust design applications. To this end, we couple the nonintrusive polynomial chaos expansion (PCE) with the multiple shooting shadowing (MSS) method, and apply the coupled method to two standard chaotic systems, the Lorenz system and the Kuramoto-Sivashinsky equation. The method leads to accurate results that match well with Monte Carlo simulations (even for low chaos orders, at least for the two systems examined), but it is costly. However, if we apply the concept of shadowing to the system trajectories evaluated at the quadrature integration points of PCE, then the resulting regularization can lead to significant computational savings. We call the new method shadowed PCE (sPCE).

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Section: Introductionmentioning

confidence: 99%

Uncertainty quantification of sensitivities of time-average quantities in chaotic systems

2020

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“…Since

\frac{δ}{δ bold-italicb}

includes the effect of changes in the computational domain due to a change in b ,

\frac{δ}{δ bold-italicb}

and

\frac{\partial}{\partial x_{j}}

do not permute. Instead, they are linked through

\frac{δ}{δ bold-italicb} ((), \frac{\partial false(\cdot false)}{\partial x_{j}}) = \frac{\partial}{\partial x_{j}} ((), \frac{δ false(\cdot false)}{δ bold-italicb}) - \frac{\partial false(\cdot false)}{\partial x_{k}} \frac{\partial}{\partial x_{j}} ((), \frac{δ x_{k}}{δ bold-italicb}) .

…”

Section: The Unsteady Continuous Adjoint Methodsmentioning

confidence: 99%

“…Their expression reads

\begin{matrix} alignleft \\ align-1 \frac{δ L}{δ b} & align-2 = \int_{T} \int_{S_{W}} j_{S_{W}, i} \frac{δ (n_{i} d S)}{δ b} + \int_{T} \int_{Ω} A_{j k} \frac{\partial}{\partial x_{j}} (\frac{δ x_{k}}{δ b}) d Ω d t - \int_{T} \int_{S_{W}} (- u_{k} n_{k} + \frac{\partial j_{S_{W}, k}}{\partial τ_{l z}} n_{k} n_{l} n_{z}) τ_{i j} \frac{δ (n_{i} n_{j})}{δ b} d S d t \\ align-1 & align-2 - \int_{T} \int_{S_{W}} \frac{\partial j_{S_{W}, k}}{\partial τ_{l z}} n_{k} t_{l}^{I} t_{z}^{I} τ_{i j} \frac{δ (t_{i}^{I} t_{j}^{I})}{δ b} \end{matrix}

…”

Section: The Unsteady Continuous Adjoint Methodsunclassified

“…Since b includes the effect of changes in the computational domain due to a change in b, b and x do not permute. Instead, they are linked through 17,18 b…”

Section: The Unsteady Continuous Adjoint Methodsmentioning

confidence: 99%

“…S W ,i is the part of j S,i defined along the wall boundary S W ; t I and t II are the two tangential unit vectors along S W forming a Frenet system with n; and S et W is the part of S W , where the jets are defined. In shape optimization problems, where the domain and thus the computational fluid dynamics (CFD) grid change whenever the shape undergoes modifications, the second term on the RHS of Equation (11) containing grid sensitivities 18 This term is computed analytically since a volumetric B-splines parameterization is used for both the geometry and the computational grid. 19 For flow control problems, the wall geometry and the computational grid are not altered, nullifying all but the last integral on the RHS of Equation (11).…”

Section: The Unsteady Continuous Adjoint Methodsmentioning

confidence: 99%

See 2 more Smart Citations

On the incremental singular value decomposition method to support unsteady adjoint‐based optimization

Vezyris

Papoutsis‐Kiachagias

Giannakoglou

2019

Numerical Methods in Fluids

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Summary This paper proposes and evaluates an approximation model based on an incremental Singular Value Decomposition (iSVD) algorithm, for unsteady flow field reconstructions, needed for integrating the unsteady adjoint equations backward in time, within a gradient‐based optimization loop. Due to the iSVD algorithm, the computational cost of solving the unsteady adjoint equations is reduced considerably, without practically affecting the accuracy of the computed gradient. Approximations to the unsteady flow fields are constructed while solving the time‐varying flow equations (moving forward in time) and used to reconstruct these fields during the backward‐in‐time integration of the continuous adjoint equations. Optimization results obtained using the proposed method are compared to those computed using the binomial checkpointing technique, which acts as the reference method. Test cases for both flow control and shape optimization problems are presented.

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Adjoint variable‐based shape optimization with bounding surface constraints

Damigos

Papoutsis‐Kiachagias

Giannakoglou

2020

Numerical Methods in Fluids

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Summary This article presents an algorithm for constraining shape deformations in adjoint‐based aerodynamic shape optimization. The algorithm considers known bounding surfaces and constrains the shape undergoing optimization not to intersect with them. For each and every node on the shape, its signed distance to the bounding surfaces is computed. The signed distance function returns a positive value, in case a node lies outside the bounds, or a negative one otherwise. It can, therefore, serve as an inequality constraint, the satisfaction of which ensures that this node does not violate the no‐intersection criteria. To increase the efficiency and make this imposition of constraints usable with any parameterization scheme, the resulting inequality constraints are transformed to a single‐equality constraint by means of a penalty function which is, then, summed over the shape to be optimized. The shape parameterization used in this article is spline‐based but this is not restrictive and any other parameterization type (node‐based, RBF, etc) can be used instead. The gradient of the objective function of the aerodynamic problem with respect to the design variables defined by the parameterization is computed using the continuous adjoint method. The method is demonstrated in the optimization of a 2D manifold case and two industrial‐like cases including the shape optimization of a U‐bend cooling duct, constrained to stay within a box, and the side mirror of a passenger car constrained to retain an almost undeformed reflector glass.

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On the proper treatment of grid sensitivities in continuous adjoint methods for shape optimization

Cited by 42 publications

References 21 publications

Uncertainty quantification of sensitivities of time-average quantities in chaotic systems

Uncertainty quantification of sensitivities of time-average quantities in chaotic systems

On the incremental singular value decomposition method to support unsteady adjoint‐based optimization

Adjoint variable‐based shape optimization with bounding surface constraints

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