Efficient Inverse Aerodynamic Design Method for Subsonic Flows

Milholen, William E.

doi:10.2514/2.2852

Cited by 5 publications

(3 citation statements)

References 8 publications

(11 reference statements)

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“…We use only three elements to discretize each side of the airfoil and now perform two computations on this grid using P5 elements on classical and curved triangles (N B = 5). The reference solution was taken from the literature (Milholen, 2000). Looking at the pressure coefficient depicted in Fig.…”

Section: Subsonic Flow Around An Naca0012 Airfoilmentioning

confidence: 99%

On Source Terms and Boundary Conditions Using Arbitrary High Order Discontinuous Galerkin Schemes

Dumbser¹,

Munz²

2007

International Journal of Applied Mathematics and Computer Science

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This article is devoted to the discretization of source terms and boundary conditions using discontinuous Galerkin schemes with an arbitrary high order of accuracy in space and time for the solution of hyperbolic conservation laws on unstructured triangular meshes. The building block of the method is a particular numerical flux function at the element interfaces based on the solution of Generalized Riemann Problems (GRPs) with piecewise polynomial initial data. The solution of the generalized Riemann problem, originally introduced by Toro and Titarev in a finite volume context, provides simultaneously a numerical flux function as well as a time integration method. The resulting scheme is extremely local since it integrates the PDE from one time step to the successive one in a single step using only information from the direct side neighbors. Since source terms are directly incorporated into the numerical flux via the solution of the GRP, our very high order accurate method is also able to maintain very well smooth steady-state solutions of PDEs with source terms, similar to the so-called well-balanced schemes which are usually specially designed for this purpose. Boundary conditions are imposed solving inverse generalized Riemann problems. Furthermore, we show numerical evidence proving that by using very high order schemes together with high order polynomial representations of curved boundaries, high quality solutions can be obtained on very coarse meshes.Keywords: discontinuous Galerkin schemes, ADER approach, source terms, boundary conditions, unstructured meshes Discretization of Source Terms with Arbitrary High Order DG SchemesWe consider the non-homogeneous system of conservation lawswith the fluxS is a source term that depends on the state vector u and that may also explicitly depend on space x = (x, y) and time t. For (1), proper initial conditions at t = 0 have to be provided. On the boundary ∂Ω of the domain, appropriate boundary conditions must be imposed on u or on the fluxes.The computational domain Ω is divided into conforming triangles T (m) addressed by a unique index (m). The numerical solution u h is sought in the space V of piecewise polynomials up to degree N. It is hence approximated inside each triangle T (m) by a linear combination of some time independent polynomial basis functions Φ l ( ξ) of degree N defined on T (m) and with time dependent degrees of freedomû (m) (t):We note that the coordinates ξ refer to a reference element T E , see Fig. 2. The number of degrees of freedom per ele-. For the discretization of the inhomogeneous equation (1), we multiply it by test functions Φ k from the same space V of piecewise polynomials of order N . After integration by parts, the semi-discrete DG scheme for the numerical solution u h is

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Section: Subsonic Flow Around An Naca0012 Airfoilmentioning

confidence: 99%

On Source Terms and Boundary Conditions Using Arbitrary High Order Discontinuous Galerkin Schemes

Dumbser¹,

Munz²

2007

International Journal of Applied Mathematics and Computer Science

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“…1−5 In fact, the precision of iterative inverse methods is generally judged from the graphical differences between the target and design pressure or velocity distributions, and its exact amplitude like the maximum pressure and geometric difference is rarely discussed. Exceptions can also exist, such as in the case of Milholen's paper, 2 in which a precision of ±0.005 was specified for most cases and was numerically determined. In the present study, the solution is accepted at a specified iteration number if the graphical differences between the target and the designed pressure coefficients are small enough as the convergence criterion for the specified maximum pressure coefficient difference cannot always be satisfied.…”

Section: Perturbation Calculation and The Design Processmentioning

confidence: 99%

“…Geometry smoothing is very important and even essential to some inverse methods for which the airfoil smoothing is used during each design iteration. 2 Because airfoil smoothing effects need to be meticulously controlled, it is impossible to directly apply the general method, which tends to smooth an airfoil too much or too less. Therefore, special methods suitable for airfoil smoothing must be developed.…”

Section: Airfoil Smoothingmentioning

confidence: 99%

Iterative Inverse Design Method Based on Streamline Equations

Paraschivoiu

Saeed

2004

Journal of Aircraft

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Aerodynamic characteristics are very sensitive to airfoil leading-edge geometry, and its accurate treatment is a limitation of many existing design methods. The objective of the present research is to develop an interactive inverse design method that is not only efficient but can also treat the leading-edge region accurately. In the formulation of the method, a small-perturbation geometric equation is deduced from the streamline momentum equations, the continuity equation, and the isentropic relations with the geometry similarity assumption of near streamlines to the airfoil surface. Moreover, the transonic correction is considered in the aforementioned equation with the assumption for the effects of waves reflected from the free surface (sonic line) because the method based on the surface flow values cannot take into account the transonic characteristics such as wave interference. The geometric perturbation normal to the airfoil surface is then calculated by solving this second-order initial value ordinary differential equation iteratively to obtain an airfoil design. Techniques such as airfoil smoothing, nonuniform relaxation, and the strained coordinate transfer are employed to accelerate convergence. The airfoil design cases demonstrate the remarkable efficiency and accuracy of the method not only for compressible flows but also for low-speed flows. Moreover, the method allows the leading-edge shape to be determined accurately and, thus, to overcome the deficiency of many of the related methods.

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Novel Inverse Airfoil Design Utilizing Parametric Equations

Lane¹

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NOVEL INVERSE AIRFOIL DESIGN UTILIZING PARAMETRIC EQUATIONS Kevin A. LaneThe engineering problem of airfoil design has been of great theoretical interest for almost a century and has led to hundreds of papers written and dozens of methods developed over the years. This interest stems from the practical implications of airfoil design. Airfoil selection significantly influences the application's aerodynamic performance. Tailoring an airfoil profile to its specific application can have great performance advantages. This includes considerations of the lift and drag characteristics, pitching moment, volume for fuel and structure, maximum lift coefficient, stall characteristics, as well as off-design performance.A common way to think about airfoil design is optimization, the process of taking an airfoil and modifying it to improve its performance. The classic design goal is to minimize drag subject to required lift and thickness values to meet aerodynamic and structural constraints. This is typically an expensive operation depending on the selected optimization technique because several flow solutions are often required in order to obtain an updated airfoil profile. The optimizer requires gradients of the design space for a gradient-based optimizer, fitness values of the members of the population for a genetic algorithm, etc.An alternative approach is to specify some desired performance and find the airfoil profile that achieves this performance. This is known as inverse airfoil design. Inverse design is more computationally efficient than direct optimization because changes in the geometry can be related to the required change in performance, thus requiring fewer flow solutions to obtain an updated profile. The desired performance for an inverse design method is specified as a pressure or velocity distribution over the airfoil at given flight conditions. The improved efficiency of inverse design comes at a cost. Designing a target pressure distribution is no trivial matter and has severe implications on the end performance. There is also no guarantee a specified pressure or velocity distribution can be achieved. However, if an obtainable pressure or velocity distribution can be created that reflects design goals and meets design constraints, inverse design becomes an attractive option over direct optimization.Many of the available inverse design methods are only valid for incompressible flow. Of those that are valid for compressible flow, many require modifications to the method if shocks are present in the flow. The convergence of the methods are also greatly slowed by the presence of shocks. This paper discusses a series of novel inverse design methods that do not depend on the freestream Mach number. They can be iv applied to design cases with and without shocks while not requiring modifications to the methods. Shocks also do not have a significant impact on the convergence of the methods. Airfoils are represented with parametric equations from the CST method to control shape changes and relate them to th...

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Efficient Inverse Aerodynamic Design Method for Subsonic Flows

Cited by 5 publications

References 8 publications

On Source Terms and Boundary Conditions Using Arbitrary High Order Discontinuous Galerkin Schemes

On Source Terms and Boundary Conditions Using Arbitrary High Order Discontinuous Galerkin Schemes

Iterative Inverse Design Method Based on Streamline Equations

Novel Inverse Airfoil Design Utilizing Parametric Equations

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