Airfoil design for compressible inviscid flow based on shape calculus

Schmidt, Stephan; Ilić, Časlav; Schulz, Volker; Gauger, Nicolas R.

doi:10.1007/s11081-011-9145-3

Cited by 20 publications

(15 citation statements)

References 16 publications

(22 reference statements)

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“…For this kind of test case, the success of the optimization can easily be judged according to whether the shock is removed or not and by the value of the remaining drag which ideally would vanish. Similar lift-constrained shape optimization problems have already been considered in [22,43,44]. In this section, the RAE2882 airfoil is parametrized using 15 Hicks-Henne functions.…”

Section: Shape Optimization Of the Rae2822 Airfoilmentioning

confidence: 99%

Adjoint‐based airfoil optimization with discretization error control

Hartmann

2014

Numerical Methods in Fluids

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SUMMARYIn this article, we develop a new airfoil shape optimization algorithm based on higher‐order adaptive DG methods with control of the discretization error. Each flow solution in the optimization loop is computed on a sequence of goal‐oriented h‐refined or hp‐refined meshes until the error estimation of the discretization error in a flow‐related target quantity (including the drag and lift coefficients) is below a prescribed tolerance. Discrete adjoint solutions are computed and employed for the multi‐target error estimation and adaptive mesh refinement. Furthermore, discrete adjoint solutions are employed for evaluating the gradients of the objective function used in the CGs optimization algorithm. Furthermore, an extension of the adjoint‐based gradient evaluation to the case of target lift flow computations is employed. The proposed algorithm is demonstrated on an inviscid transonic flow around the RAE2822, where the shape is optimized to minimize the drag at a given constant lift and airfoil thickness. The effect of the accuracy of the underlying flow solutions on the quality of the optimized airfoil shapes is investigated. Copyright © 2014 John Wiley & Sons, Ltd.

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Section: Shape Optimization Of the Rae2822 Airfoilmentioning

confidence: 99%

Adjoint‐based airfoil optimization with discretization error control

Hartmann

2014

Numerical Methods in Fluids

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“…UFL is currently unable to symbolically differentiate equation operators with respect to the spatial coordinates; furthermore, differentiating through the mesh generation procedure is also necessary [13]. However, it is possible to directly use the automatically computed adjoint solutions for shape optimisation via the shape calculus approach [48,47], which circumvents the need for discretely differentiating through the mesh generation process.…”

Section: Limitations the First Major Limitation Of Dolfin-adjoint Ismentioning

confidence: 99%

Automated Derivation of the Adjoint of High-Level Transient Finite Element Programs

Farrell¹,

Ham²,

Funke³

et al. 2013

SIAM J. Sci. Comput.

250

233

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Abstract. In this paper we demonstrate a new technique for deriving discrete adjoint and tangent linear models of finite element models. The technique is significantly more efficient and automatic than standard algorithmic differentiation techniques. The approach relies on a high-level symbolic representation of the forward problem. In contrast to developing a model directly in Fortran or C++, high-level systems allow the developer to express the variational problems to be solved in near-mathematical notation. As such, these systems have a key advantage: since the mathematical structure of the problem is preserved, they are more amenable to automated analysis and manipulation. The framework introduced here is implemented in a freely available software package named dolfin-adjoint, based on the FEniCS Project. Our approach to automated adjoint derivation relies on run-time annotation of the temporal structure of the model, and employs the FEniCS finite element form compiler to automatically generate the low-level code for the derived models. The approach requires only trivial changes to a large class of forward models, including complicated time-dependent nonlinear models. The adjoint model automatically employs optimal checkpointing schemes to mitigate storage requirements for nonlinear models, without any user management or intervention. Furthermore, both the tangent linear and adjoint models naturally work in parallel, without any need to differentiate through calls to MPI or to parse OpenMP directives. The generality, applicability and efficiency of the approach are demonstrated with examples from a wide range of scientific applications. . While deriving the adjoint model associated with a linear stationary forward model is straightforward, the development and implementation of adjoint models for nonlinear or time-dependent forward models is notoriously difficult, for several reasons. First, each nonlinear operator of the forward model must be differentiated, which can be difficult for complex models. Second, the control flow of the adjoint model runs backwards, from the final time to the initial time, and requires access to the solution variables computed during the forward run if the forward problem is nonlinear. Since it is generally impractical for physically relevant simulations to store all variables during the forward run, the adjoint model developer must implement some checkpointing scheme that balances recomputation and storage [17]. The control flow of such a checkpointing scheme must alternate between the solution of forward variables and adjoint variables, and is thus highly

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“…These techniques are based on the so‐called shape derivative, which measures the sensitivity of a shape due to infinitesimal deformations, and the topological derivative, which measures the sensitivity of a geometry with respect to the insertion of an infinitesimally small hole, see, e.g., [14, 61] for shape calculus and [45] for topological sensitivity analysis. In recent years these techniques have been applied to many industrial problems, e.g., the shape design of polymer spin packs [27, 37–39], electric motors [20, 21], acoustic horns [5, 57], automobiles [18, 46, 48], aircrafts [41, 55, 56] or pipe systems [25, 28, 58]. To the best of our knowledge, the optimization of a microchannel cooling system by means of shape calculus has only been investigated in our earlier work [6], where we rigorously analyzed the theoretical aspects of this problem.…”

Section: Introductionmentioning

confidence: 99%

Model hierarchy for the shape optimization of a microchannel cooling system

2020

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We model a microchannel cooling system and consider the optimization of its shape by means of shape calculus. A three‐dimensional model covering all relevant physical effects and three reduced models are introduced. The latter are derived via a homogenization of the geometry in 3D and a transformation of the three‐dimensional models to two dimensions. A shape optimization problem based on the tracking of heat absorption by the cooler and the uniform distribution of the flow through the microchannels is formulated and adapted to all models. We present the corresponding shape derivatives and adjoint systems, which we derived with a material derivative free adjoint approach. To demonstrate the feasibility of the reduced models, the optimization problems are solved numerically with a gradient descent method. A comparison of the results shows that the reduced models perform similarly to the original one while using significantly less computational resources.

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Airfoil design for compressible inviscid flow based on shape calculus

Cited by 20 publications

References 16 publications

Adjoint‐based airfoil optimization with discretization error control

Adjoint‐based airfoil optimization with discretization error control

Automated Derivation of the Adjoint of High-Level Transient Finite Element Programs

Model hierarchy for the shape optimization of a microchannel cooling system

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