Abstract: This paper treats shape optimal design of two-dimensional structures. Sensitivity of cost and constraint functions to changes in the shape of the structure are obtained by applying theorems from the calculus of variations and by using consistent first-order approximations of functions arising in an optimal design problem. A generalized steepest descent algorithm is presented for determination of the optimum shape. The method is applied to a load carrying shear plate that models a component of the high pressure seal in a gun launched high velocity projectile.

Abstract: A technique is presented for the calculation of Pareto-optimal solutions to a multiple-objective constrained optimization problem by solving a series of single-objective problems. Threshold-of-acceptability constraints are placed on the objective functions at each stage to both limit the area of search and to mathematically guarantee convergence to a Pareto optimum.

Abstract: This paper deals with the shape optimizing problem for a structural beam where its allowable displacement is restricted under multiple loads. A Growing-Reforming Procedure proposed by the authors is employed. The Procedure is extended to the cases where a beam has to support a number of possible loads ; that is, a method to evaluate the envelope of the stress induced by the possible loads. In order to make allowable displacement within a restricted value, a combined method of sensitivity with the Growing-Reforming Procedure is introduced. A computational algorithm based on this method is established. Some typical applications are shown to demonstrate the effectiveness of these methods.

Abstract: A constant-pressure tokamak model with skin currents on the plasma surface is used to optimize the plasma shape for fixed values of the aspect ratio and poloidal beta so as to maximize the total beta. The constraint is imposed for the plasma to be stable with respect to axisymmetric MHD modes. Although the absolute value of the maximum attainable beta is strongly determined by non-axisymmetric instabilities, the omission of these is demonstrated to be of minor influence on the optimum shape for the plasma model considered. The energy principle is used to determine axisymmetric stability. For shape optimization a modified method of steepest descent is used which adjusts the Fourier coefficients representing the plasma boundary. Since for small aspect ratio there is a preference for doublet-type shapes, a theory of axisymmetric stability for surface current doublets is included in the paper.

Abstract: This paper demonstrates a numerical technique for canard-wing shape optimization at two operating conditions. For purposes of simplicity, a mean surface wing paneling code is employed for the aerodynamic calculations. The optimization procedures are based on the method of feasible directions. The shape functions for describing the thickness, camber, and twist are based on polynomial representations. The primary design requirements imposed restrictions on the canard and wing volumes and on the lift coefficients at the operating conditions. Results indicate that significant improvements in minimum drag and lift-to-drag ratio are possible with reasonable aircraft geometries. Calculations were done for supersonic speeds with Mach numbers ranging from 1 to 6. Planforms were mainly of a delta shape with aspect ratio of 1.

Abstract: Weakening the constraints in a general non-linear optimal control problem we obtain a convex optimization program with the same value.

Abstract: THIS paper demonstrates a numerical technique for canard-wing shape optimization at two operating conditions. For purposes of simplicity, a mean surface wing paneling code1 is employed for the aerodynamic calculations. The optimization procedures2 are based on the method of feasible directions. The shape functions for describing thickness, camber, and twist are based on polynomial representations. The primary design requirements imposed restrictions on the canard and wing volumes and on the lift coefficients at the operating conditions. Results indicate that significant improvements in minimum drag and lift-to-drag ratio are possible with reasonable aircraft geometries. Calculations were done for supersonic speeds with Mach numbers ranging from 1 to 6. Planforms were mainly of a delta shape with aspect ratio of 1, with the canard and wing in the same plane. Contents The shape functions for the thickness,3 and camber, and twist4 were each expressed as a ten-term polynomial function of the Cartesian coordinates defined in the canard-wing plane. The coefficients of these polynomials had the status of optimization variables. Volumes of the wing and canard are constrained to specified values and correspond to the volumes of the base configuration in which both the canard and wing have 5% parabolic sections. The study initially explored minimizing wave drag through wing-canard shaping by calculating the optimum thickness distribution with zero camber and twist. The results are shown in Fig. 1 for two canard sizes as well as for a wing-along case. Wave drag reductions of up to 50%, relative to the base configuration with constant thickness ratio airfoils, are feasible while still meeting canard-wing internal volume limits. The improvements in drag become more pronounced at high Mach numbers. The optimum shapes were found to be similar to those reported by Strand 3 for the wing-alone case, indicating that the presence of the canard does not introduce significant perturbations in the shape functions. The second study explored the reduction in drag due to lift through optimization of the camber and twist of the lifting surface with zero thickness (Fig. 2). Again, results are shown for two canard sizes as well as for a wing-alone case. The configurations with subsonic leading edges show drag reductions of up to 36% by use of optimum camber and twist of the lifting surface. The potential for improvement tends to diminish with increasing Mach number in contrast with the results for the optimization of thickness. Figure 3 shows, in terms of LID, the data of Fig. 2 and includes a simple flat

Abstract: A METHOD for solving non-linear optimal control problems, and in particular, problems with discontinuous phase trajectories, is described; it is based on expressions for the partial derivatives of the functional with respect to the parameters of the discretized control. A model problem of optimization of a production system is solved.