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So far the literature on inverse shape design in aerodynamics is still confined to the single‐point (nominal design point) design and to steady flow. This situation cannot cope with the modern development of internal and external aerodynamics and aerothermoelasticity, especially turbomachinery and aircraft flows. Accordingly, in recent years a new generation of inverse shape design problem has been suggested and investigated theoretically and computationally, consisting mainly of: unsteady inverse and hybrid problems; multipoint inverse and hybrid problems; and inverse problem in aerothermoelasticity. It opens a new area of research in fluid mechanics and aerothermoelasticity. An overview of its status and perspective is given herein, emphasizing the new concepts, theory and methods of solution involved.

This article has been withdrawn as it was published elsewhere and accidentally duplicated. The original article can be seen here: 10.1108/00022660410545483. When citing the article, please cite: Zhe-Min Wu, Chi Chen, Gao-Lian Liu, (2004), “Multipoint inverse shape design of airfoils based on variable-domain variational principle”, Aircraft Engineering and Aerospace Technology, Vol. 76 Iss: 4, pp. 376-38.

Presents a brief overview of some new concepts and research results concerning aerodynamic computation and design of jet‐propulsion engines with emphasis on turbomachinery (TM) developed in China, without any attempt to be exhaustive.

Provides two hybrid methods for the aerodynamic design of cascade profiles, of which the design constraints are the combination of aerodynamic and geometric conditions.

In the first method, the design constrain is composed of the velocity (or pressure) distribution on part of the blade surface and the geometry of the rest part. In the second method, the aerodynamic load distribution, i.e. the pressure difference between the suction and pressure surfaces, and the blade thickness distribution are employed as the design constrain. These constraints, together with all the other boundary conditions, are involved in the stationary conditions of a variational principle. The solution domain, i.e. the blade‐to‐blade passage, is transformed into a square in the image plane, while the blade contour is projected to a straight line; thus, the difficulty caused by the unknown geometry of profile is avoided. Finite element method is employed to produce the calculation code.

Applications show the accuracy and the flexibility of the two methods, which can satisfy the different needs from blade design. Finally, the possibility of combining the hybrid methods with the through‐flow method is discussed, which would develop the present methods to three‐dimensional design of cascades.

The design methods are limited to frictionless flow.

A design software of cascade profiles based on this method has been developed, and will be provided to the engineering users for cascade design.

The hybrid methods developed in this paper can satisfy the demands from different aspects of engineering designs: aerodynamics, strength, manufacture, etc.

This paper seeks to develop an approach for the unsteady inverse problem of two‐dimensional oscillating airfoils based on the finite difference method (FDM) solution of the transient Euler equations.

The solution strategies are determined according to the mathematical model for the inverse‐problem of oscillating airfoils. Then the unsteady nonreflecting far field boundary condition and the permeable wall boundary condition are employed to treat the boundary conditions. The applications are carried out for the modification of an oscillating airfoil according to the design targets of the unsteady pressure distribution in an oscillating period.

The results show that the pressure distributions over the new airfoils coincide with the design objects indicating that the mathematical model and solution strategy developed in this paper is rational and reliable.

This method is limited to frictionless flow.

The paper provides a new FDM solution of unsteady inverse problem for oscillating airfoils, which can be extended to treat the multipoint problem of airfoil design.

As a kind of free (unknown) boundary problem, inverse shape design of airfoils has attracted extensive attention in recent years. By variable‐domain variational theory, free boundary condition can be coupled with the governing equations for flow field, which makes it possible to calculate the flow field with the free boundary simultaneously. In this paper, the variational principle (VP) of 2D airfoil shape design problem is obtained from the basic dimensionless velocity potential equations for 2D compressible flow by using a systematic approach and variable‐domain variational formula. The deformable finite element method based on the VP is used to segmentally design airfoil at two design points (angles of attack) and four design points, respectively. The results show that the present method is highly effective and accurate for solving multipoint inverse problem of airfoils.

A generalized variational principle of 2D unsteady compressible flow around oscillating airfoils is established directly from the governing equations and boundary/initial conditions via the semi‐inverse method proposed by He. In this method, an energy integral with an unknown F is used as a trial‐functional. The identification of the unknown F is similar to the identification of the Lagrange multiplier. Based on the variational theory with variable domain, a variational principle for the inverse problem (given as the time‐averaged pressure over the airfoil contour, while the corresponding airfoil shape is unknown) is constructed, and all the boundary/initial conditions are converted into natural ones, leading to well‐posedness and the unique solution of the inverse problems.

Using the semi‐inverse method proposed by the present author, a family of variational principle for direct problem of S2‐flow in mixed‐flow turbomachinery is obtained; then, applying the functional variation with variable domain, two families of variational principles are established for the hybrid problems of determining the unknown shape of bladings, where pressure or velocity is over‐specified. The present variational models are well posed for redundant data at boundaries. The theory provides both rational ways for best contouring the hub/casing walls to meet various practical design requirements and a theoretical basis for introducing the finite element method into computational aerodynamics of turbomachinery.