This paper demonstrates the application of correlated Gaussian process based approximations to optimization where multiple levels of analysis are available, using an extension to the geostatistical method of co-kriging. An exchange algorithm is used to choose which points of the search space to sample within each level of analysis. The derivation of the co-kriging equations is presented in an intuitive manner, along with a new variance estimator to account for varying degrees of computational ‘noise’ in the multiple levels of analysis. A multi-fidelity wing optimization is used to demonstrate the methodology.

/pdf/multi-fidelity-optimization-via-surrogate-modelling-1f0ifoi8dw.pdf

Multi-fidelity optimization via surrogate modelling

Applications of the unsteady vortex-lattice method in aircraft aeroelasticity and flight dynamics

The classical solution to Prandtl's well-known lifting-line theory applies only to a single lifting surface with no sweep and no dihedral. However, Prandtl's original model of a finite lifting surface has much broader applicability. A general numerical lifting-line method based on Prandtl's model is presented. Whereas classical lifting-line theory is based on applying the two-dimensional Kutta-Joukowski law to a three-dimensional flow, the present method is based on a fully three-dimensional vortex lifting law. The method can be used for systems of lifting surfaces with arbitrary camber, sweep, and dihedral. The accuracy realized from this method is shown to be comparable to that obtained from numerical panel methods and inviscid computational fluid dynamics solutions, but at a small fraction of the computational cost

Modern Adaptation of Prandtl's Classic Lifting-Line Theory

in the field of computational fluid dynamics that the more generally the physics of the flow is modeled by a method, the less general is the boundary geometry to which the method is applicable. Part of the difficulty lies in the stringent requirements on the calculational grid in the field of flow. According to the above observation, the simp lest meaningful flow physics- that of invi scid, in compressible potential flow­ should be applicable to the most general geometries. Indeed, as is well known, this problem can be formulated as a linear integral equation over the boundary, thus eliminating the need for a field grid and allowi ng the possibility that a potential-flow method formulated in this way can be capable 'of obtaining flow solutions about completely arbitrary con­ figurations. Happily, this generality substantially can be achieved in prac­ tice, although with more analytical and programming effort than most publications on the subject are prepared to admit. At the same time, the predictions of such methods have been found to agree well with experiment over a surprisingly large range of flow conditions. Even when their results fail to give the proper experi mental values, they are frequently useful in predicting the incremental effect of a proposed design change or in ordering various designs in terms of effectiveness. This perhaps fortuitous agreement with real flow, combined with their geometric generality, has made numeri­ cal potential-flow methods indispensable design tools in many fields. For example, in the author's company a major design calculation (e.g. flow

Panel Methods in Computational Fluid Dynamics

Modern helicopter rotor aerodynamics

A low-order panel method is presented for the calculation of subsonic aerodynamic characteristics of general configurations. The method is based on piecewise constant doublet and source singularities. Two forms of the internal Dirichlet boundary condition are discussed and the source distribution is determined by the external Neumann boundary condition. Calculations are compared with higher-order solutions for a number of cases. It is demonstrated that for comparable density of control points where the boundary conditions are satisfied, the low-order method gives comparable accuracy to the higher-order solutions. It is also shown that problems associated with some earlier low-order panel methods, e.g., leakage in internal flows and junctions and also poor trailing-edge solutions, do not appear for the present method. Further, the application of the Kutta condition is extremely simple; no extra equation or trailing-edge velocity point is required. The method has very low computing costs and this has made it practical for application to nonlinear problems requiring iterative solutions and to three-dimensional unsteady problems using a time-stepping approach. In addition, the method has been extended to model separated flows in three dimensions, using free vortex sheets to enclose the separated zone.

Prediction of Subsonic Aerodynamic Characteristics: A Case for Low-Order Panel Methods

A method for converting steady state, potential panel methods into a time-dependent mode is derived and applied to several test cases. For this development, an improved vortex wake model was constructed which was also suitable for simulating the leading edge separation of slender wings at high angles of attack. Computed flow-field simulations are presented for various unsteady, and high-angle-of-attack conditions, involving geometries such as simple wings, rotors, and complete aircraft configurations.

Unsteady low-speed aerodynamic model for complete aircraft configurations

Prediction of subsonic aerodynamic characteristics - A case for low order panel methods

The close-approach problem associated with flow calculation methods based on vortex-lattice theory was examined numerically using two-dimensional discretized vortex sheets. The analysis first yields a near-field radius of approximately the distance apart of the vortices in the lattice; only within this distance from the sheet are the errors arising from the discretization significant. Various modifications to the discrete vortices are then considered with the objective of reducing the errors. This leads to a near-field model in which a vortex splits into an increasing number of subvortices as it is approached. The subvortices, whose strengths vary linearly from the vortex position, are evenly distributed along an interpolated curve passing through the basic vortices. This subvortex technique can be extended to the three-dimensional case and is efficient because the number of vortices is effectively increased, but only where and when needed.

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A subvortex technique for the close approach to a discretized vortex sheet

The close-approach problem associated with vortex-lattice methods was examined numerically with the objective of calculating velocities at arbitrary points, not just at midpoints, between the vortices. The objective was achieved using a subvortex technique in which a vortex splits into an increasing number of subvortices as it is approached. The technique, incorporated in a two-dimensional potential flow method using "submerged" vortices and sources, was evaluated for a cambered Joukowski airfoil. The method could be extended to three dimensions, and should improve non-linear methods, which calculate interference effects between multiple wings and vortex wakes, and which include procedures for force-free wakes.

/pdf/a-subvortex-technique-for-the-close-approach-to-a-55a5szj7y1.pdf

Brian Maskew

Papers

Prediction of Subsonic Aerodynamic Characteristics: A Case for Low-Order Panel Methods

Unsteady low-speed aerodynamic model for complete aircraft configurations

Prediction of subsonic aerodynamic characteristics - A case for low order panel methods

A subvortex technique for the close approach to a discretized vortex sheet

A subvortex technique for the close approach to a discretized vortex sheet