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Research Papers: Fundamental Issues and Canonical Flows

Optimization Using Arbitrary Lagrangian–Eulerian Formulation of the Navier–Stokes Equations

[+] Author and Article Information
Eysteinn Helgason

Division of Fluid Dynamics,
Department of Applied Mechanics,
Chalmers University of Technology,
Gothenburg S-412 96, Sweden
e-mail: eysteinn@chalmers.se

Siniša Krajnović

Division of Fluid Dynamics,
Department of Applied Mechanics,
Chalmers University of Technology,
Gothenburg S-412 96, Sweden

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received January 23, 2014; final manuscript received February 2, 2015; published online March 9, 2015. Assoc. Editor: Zhongquan Charlie Zheng.

J. Fluids Eng 137(6), 061202 (Jun 01, 2015) (9 pages) Paper No: FE-14-1039; doi: 10.1115/1.4029724 History: Received January 23, 2014; Revised February 02, 2015; Online March 09, 2015

In this paper, we present a new shape optimization method by using sensitivities obtained from the Arbitrary Lagrangian–Eulerian (ALE) form of the Navier–Stokes equations. In the ALE description, the nodes of the computational domain may be moved with the fluid as in the Lagrangian description, held fixed in space as in the Eulerian description, or moved in some arbitrary way in between. Applying the adjoint method with respect to mesh motion allows the whole sensitivity field for the shape changes to be calculated using only two solver calls, a primal solver call and an adjoint solver call. We show that the sensitivities with respect to the mesh motion can be calculated in a postprocessing step to the primal and adjoint flow simulations. The resulting ALE sensitivities are compared to sensitivities obtained using a finite difference approach. Finally, the sensitivities are coupled to a mesh motion smoothing algorithm, and a duct is optimized with respect to the total pressure drop using the proposed method.

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Figures

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Fig. 1

The geometry used for the validation. The value of α is varied in both the x- and y-directions for each of the six cells marked on the figure, and the results are compared to the gradient obtained using the adjoint method.

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Fig. 2

Gradient obtained using finite difference compared to the gradient obtained using the adjoint method. The step size varies from 1 × 10–8 to 1 × 10–4 depending on the location of the cell; (a) x-direction and (b) y-direction.

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Fig. 3

(a) Artificial gradients are added to the domain and the resulting mesh motion is shown in (b)–(f). (b) After one loop through the mesh coupling algorithm. (c)–(f) show the mesh motion for different criteria on I. (c) I = 0.95, (d) I = 0.9, (e) I = 0.5, and (f) I = 0.1. Note that the vectors are not in the same scale.

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Fig. 4

Flow chart of the optimization process. The dotted box contains the optimization loop used for smoothing the gradients. The cell sensitivities are used as an input to the mesh motion. When the convergence criterion is reached the mesh motion, s, is directly added to the location of the points in the mesh, denoted with x, and the loop starts again.

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Fig. 5

The S-bend seen from the side and below. The flow direction is from left to right, as indicated by the arrows.

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Fig. 6

The total pressure drop compared to the original design for two different criteria on I; (a) I = 0.5 and (b) I = 0.95

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Fig. 7

(a) The original geometry seen from the side and top. The inlet is at the left and the outlet at the right. The original geometry can be compared to the final geometry obtained using two different conditions on I; (b) I = 0.5, and (c) I = 0.95.

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Fig. 8

A cut along the center of the domain showing the velocity vectors (left) and contours of total pressure (right) for the original geometry, (a), and the final geometry obtained using two different conditions on I; (b) I = 0.5, and (c) I = 0.95

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Fig. 9

A cut along the duct showing the impact on the structure of the mesh for the upper part of the first bend for the original geometry (a) and two different conditions on I, (b), and (c).

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