Aerodynamic design via control theory.

*(English)*Zbl 0676.76055The purpose of this paper is to propose that there are benefits in regarding the design problem as a control problem in which the control is the shape of the boundary. A variety of alternative formulations of the design problem can then be treated systematically by using the mathematical theory for control of systems governed by partial differential equations. Suppose that the boundary is defined by a function f(x), where x is the position vector. As in the case of optimization theory applied to the design problem, the desired objective is specified by a cost function I, which may, for example, measure the deviation from a desired surface pressure distribution, but could also represent other measures of performance such as lift and drag. The introduction of a cost function has the advantage that if the objective is unattainable, it is still possible to find a minimum of the cost function. Now a variation in the control \(\delta\) f leads to a variation \(\delta\) I in the cost.

It is shown that \(\delta\) I can be expressed to first order as an inner product of a gradient function g with \(\delta\) f: \(\delta I=(g,df)\). Here g is independent of the particular variation \(\delta\) f in the control, and can be determined by solving an adjoint equation. Now choose \(\delta f=-\lambda g\) where \(\lambda\) is a sufficiently small positive number. Then \(\delta I=-\lambda (g,g)<0\) assuring a reduction in I. After making such a modification, the gradient can be recalculated and the process repeated to follow a path of steepest descent until a minimum is reached. In order to avoid violating constraints, such as a minimum acceptable wing thickness, the steps can be taken along the projection of the gradient into the allowable subspace of the control function.

In this way one can devise design procedures that must necessarily converge at least to a local minimum, and which might be accelerated by the use of more sophisticated descent methods. While there is a possibility of more than one local minimum, the cost function can be chosen to reduce the likelihood of difficulties caused by such a contingency, and in any case the method will lead to an improvement over the initial design.

In order to illustrate the application of control theory to design problems in more detail, the article presents design procedures for three examples. Section 2 discusses the design of two-dimensional profiles for compressible potential flow when the profile is generated by conformal mapping. Section 3 discusses the same problem when the flow is governed by the inviscid Euler equations. Finally, Section 4 addresses the three- dimensional design problem for a wing, assuming the flow to be governed by the inviscid Euler equations. The procedures that are presented require the solution of several partial differential equations at each step. The question of the most efficient discretization of these equations is deferred for future investigation.

It is shown that \(\delta\) I can be expressed to first order as an inner product of a gradient function g with \(\delta\) f: \(\delta I=(g,df)\). Here g is independent of the particular variation \(\delta\) f in the control, and can be determined by solving an adjoint equation. Now choose \(\delta f=-\lambda g\) where \(\lambda\) is a sufficiently small positive number. Then \(\delta I=-\lambda (g,g)<0\) assuring a reduction in I. After making such a modification, the gradient can be recalculated and the process repeated to follow a path of steepest descent until a minimum is reached. In order to avoid violating constraints, such as a minimum acceptable wing thickness, the steps can be taken along the projection of the gradient into the allowable subspace of the control function.

In this way one can devise design procedures that must necessarily converge at least to a local minimum, and which might be accelerated by the use of more sophisticated descent methods. While there is a possibility of more than one local minimum, the cost function can be chosen to reduce the likelihood of difficulties caused by such a contingency, and in any case the method will lead to an improvement over the initial design.

In order to illustrate the application of control theory to design problems in more detail, the article presents design procedures for three examples. Section 2 discusses the design of two-dimensional profiles for compressible potential flow when the profile is generated by conformal mapping. Section 3 discusses the same problem when the flow is governed by the inviscid Euler equations. Finally, Section 4 addresses the three- dimensional design problem for a wing, assuming the flow to be governed by the inviscid Euler equations. The procedures that are presented require the solution of several partial differential equations at each step. The question of the most efficient discretization of these equations is deferred for future investigation.

##### MSC:

76G25 | General aerodynamics and subsonic flows |

76H05 | Transonic flows |

93C20 | Control/observation systems governed by partial differential equations |

##### Keywords:

design problem; control problem; optimization theory; cost function; two- dimensional profiles; compressible potential flow; conformal mapping; inviscid Euler equations; three-dimensional design problem
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DOI

##### References:

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