On the design of three-ply nonlinearly elastic composite plates with optimal resistance to buckling

The optimal design of symmetric three-ply sandwich plates is studied in the context of finite deformation incompressible nonlinear elasticity. The overall shape of the plate is dictated, fixed amounts of two materials are at our disposal, and performance is defined solely in terms of resistance to buckling. The problem is characterized by three parameters: the volume ratio of the two materials, the stiffness ratio of the two (neo-Hookean) materials, and one of the aspect ratios of the plate. Two competing constructions are considered: one in which the stiffer material is used in the outer plies and the other in which the stiffer material is used for the central ply. It is found that if the material volume ratio and the material stiffness ratio are fixed, then there is a single aspect ratio dependent transition in the optimal design. The configuration with the stiffer material used for the central ply is the optimal design for plates that are sufficiently short in the direction of thrust, while the configuration with the stiffer material used for the outer plies is the optimal design for plates that are sufficiently long in the direction of thrust.     

On the optimal design of elastic beams

In this paper we investigate the optimal dynamics of simply supported nonlinearly elastic beams with rectangular cross-sections. We consider the elastic beam under the assumption of time-dependent intensive transverse loading. The state of the beam is described by a system of partial differential equations of the fourth order. We deal with the problem of choosing the optimal shape for the beam. The optimal shape is determined in such a way that the deflection of the nonlinearly elastic beam for any given time is minimal. The problem of choosing the optimal shape is formulated as an optimal control problem. To solve the obtained problem effectively, we use the optimality principle of Bellman (Bellman and Dreyfus 1962; Bryson and Ho 1975) and the penalty function method (Polyak 1987). We present a constructive algorithm for the optimal design of nonlinearly elastic beams. Some simple examples of the implementation of the proposed numerical algorithm are given.     

Finite element iterative method for optimal elastic design of circular plates

An iterative procedure is presented for the optimal elastic design of circular or annular plates under axisymmetric loads. The deflection or slope at a given radius is prescribed, and technological constraints are taken into account. Some examples are discussed.     

Mixed elements in the optimal design of plates

The theory of design sensitivity analysis of structures, based on mixed finite element models, is developed for static, dynamic and stability constraints. The theory is applied to the optimal design of plates with minimum weight, subject to displacement, stress, natural frequencies and buckling stresses constraints. The finite element model is based on an eight node mixed isoparametric quadratic plate element, whose degrees of freedom are the transversal displacement and three moments per node. The corresponding nonlinear programming problem is solved using the commercially available ADS (Automated Design Synthesis) program. The sensitivities are calculated by analytical, semi-analytical and finite difference techniques. The advantages and disadvantages of mixed elements in design optimization of plates are discussed with reference to applications.     

Modern Wiener--Hopf design of optimal controllers Part I: The single-input-output case

An analytical feedback design technique is presented here for single-input-output processes which are characterized by their rational transfer functions. The design procedure accounts for the topological structure of the feedback system ensuring asymptotic stability for the closed-loop configuration. The plant or process being controlled can be unstable and/or nonminimum phase. The treatment of feedback sensor noise, disturbance inputs, and process saturation is another major contribution of this work. The cornerstone in the development is the selection of a performance index based on sound engineering considerations. It is these considerations, in fact, which ensure the existence of an optimal compensator for the system and make the performance index a natural one for the problem at hand.     

On the optimal design of the shape of a buckled elastic beam

Analysis of stability, post-buckling bending and vibrations is performed for a beam (a spring element) having an optimal shape. A buckled pin-jointed spring element of a constant thickness and variable width is considered. The optimal shape of this beam is suggested to provide a uniform distribution of maximum bending stresses in its buckled equilibrium configuration for a given value of a supercritical axial force. Sensitivities of a critical force and a buckling mode to variations of the shape of a beam are calculated. A dependence of the static lateral deflection upon an axial force is analysed. Nonlinear equations of large-amplitude oscillations are derived by a use of the Hamilton principle. The natural frequencies of a spring element, compressed by a supercritical force are calculated. Received April 29, 1999     

Multidisciplinary aerospace design optimization: survey of recent developments

The increasing complexity of engineering systems has sparked rising interest in multidisciplinary optimization (MDO). This paper surveys recent publications in the field of aerospace, in which the interest in MDO has been particularly intense. The primary c hallenges in MDO are computational expense and organizational complexity. Accordingly, this survey focuses on various methods used by different researchers to address these challenges. The survey is organized by a breakdown of MDO into its conceptual components, reflected in sections on mathematical modelling, approximation concepts, optimization procedures, system sensitivity, and human interface. Because the authors' primary area of expertise is in the structures discipline, the majority of the references focus on the interaction of this discipline with others. In particular, two sections at the end of this review focus on two interactions that have recently been pursued with vigour: the simultaneous optimization of structures and aerodynamics and the simultaneous optimization of structures with active control.     

A new approach to optimal design of elastic structures

A new approach to the optimization of elastic trusses under stress constraints is discussed. The stress constraints are transformed into compliance constraints, which makes possible the derivation of a simple optimality condition. The condition provides a simple test to check whether a fully stressed design is optimal, for single as well as multiple loading conditions. It is readily extended to optimization problems, in which both stress and displacement constraints are imposed. An iterative routine, that is both simple and efficient, is derived from the optimality condition. Numerical examples of the application of the routine are presented.     

On the design of P-I-D controllers using optimal linear regulator theory

The purpose of this note is, using a very simple example, to indicate the proper formulation of a class of tracking problems, so that the solution by means of optimal linear regulator theory leads to a proportional-integral-derivative (P-I-D) type of controller. This controller has the property of eliminating steady-state errors and leads to an insensitive design.     

Numerical optimum design of elastic annular plates with respect to buckling

Annular plates of fixed volume, under uniform radial edge compression are considered. The plate is divided into N segments of linearly variable or constant thickness. Geometrical constrains for the segment thickness and for the effective stress are assumed. Two types of objective functions are explored: (I) maximum value of critical load at the fixed number of circumferential buckling half-waves j, (II) unimodal optimum design for such a number j that buckling load equals max

. The shooting method is applied to compute the distribution of axial forces in the prebuckling state and the basic buckling load for the plate of variable thickness. The optimal distribution of segment thicknesses Xk for k = 1, …, N are computed by means of Rosenbrock's method with internal penalty function. Results of numerical analysis are reported for the both optimum design problems I and II.     

An application of stochastic optimal control theory to the optimal rescheduling of airplanes

A model for the air traffic flow between two airports subject to random constraints on the takeoff and landing capacities is set up. For a simple case a dynamic programming algorithm is used to compute the optimal solutions explicitly.     

Application of multiplier update method to the minimum weight design of elastic plates

This study deals with the minimum weight optimization of arbitrary shaped plates under size, stress, and displacement constraints. The ‘8-nodes isoparametric’ plate element is incorporated with the optimization procedure by using Lagrange multipliers and the Kuhn-Tucker optimality condition. An optimization computer code ARS 2 (Automatic Resizing System 2) is developed and applied to some practical examples. Results reveal that the procedure presented in this paper is quite efficient.     

On thea priori model reduction: Overview and recent developments

Summary Karhunen-Loève expansion and snapshot POD are based on principal component analysis of series of data. They provide basis vectors of the subspace spanned by the data. All the data must be taken into account to find the basis vectors. These methods are not convenient for any improvement of the basis vectors when new data are added into the data base. We consider the data as a state evolution and we propose an incremental algorithm to build basis functions for the decomposition of this state evolution. The proposed algorithm is based on the APHR method (A Priori Hyper-Reduction method). This is an adaptive strategy to build reduced order model when the state evolution is implicitely defined by non-linear governing equations. In case of known state evolutions the APHR method is an incremental Karhunen-Loève decomposition. This approach is very convenient to expand the subspace spanned by the basis functions. In the first part of the present paper the main concepts related to the “a priori” model reduction technique are revisited, as a previous task to its application in the cases considered in the next sections. Some engineering problems are defined in domains that evolve in time. When this evolution is large the present and the reference configurations differ significantly. Thus, when the problem is formulated in the total Lagrangian framework frequent remeshing is required to avoid too large distortions of the finite element mesh. Other possibility for describing these models lies in the use of an updated formulation in which the mesh is conformed to each intermediate configuration. When the finite element method is used, then frequent remeshing must be carried out to perform an optimal meshing at each intermediate configuration. However, when the natural element method, a novel meshless technique, is considered, whose accuracy does not depend significantly on the relative position of the nodes, then large simulations can be performed without any remeshing stage, being the nodal position at each intermediate configuration defined by the transport of the nodes by the material velocity or the advection terms. Thus, we analyze the extension of the “a priori” model reduc tion, based on the use in tandem of the Karhunen-Loève decomposition (that extracts significant information) and an approximation basis enrichment based on the use of the Krylov's subspaces, previously proposed in the framework of fixed mesh simulation, to problems defined in domains evolving in time. Finally, for illustrating the technique capabilities, the “a priori” model reduction will be applied for solving the kinetic theory model which governs the orientation of the fibers immersed in a Newtonian flow.     

Martingales and their application to optimal state estimation

The main properties of martingales »are discussed from an engineering point of view. This is followed by applications of the theory to optimal linear as well as non-linear estimation, problems?     

A novel application of grey system theory to information security (Part I)

This study applies the grey data generating techniques in grey system theory on a novel cryptosystem, which is guiding a new research in the field of information security. In this paper, we present the concepts of sum-lock, difference-lock, sum-ladder, and difference-ladder. Using these concepts, we can obtain a cryptosystem with lock generation and sum-difference mixed ladder. The cryptographic algorithms for our cryptosystem are also presented and an illustrative example is used to verify it.     

Application of game theory to the sensitivity design of optimal systems

The theory of games is applied to the design of systems with unknown plant parameters. It is assumed that a controller structure is known and furthermore that this controller is optimal when the controller parameters are equal to the plant parameters. The performance index then becomes a function of plant and controller parameters. This function is treated as a pay-off function with the antagonists represented by the controller and plant parameters. The theory is illustrated with some simple samples.     

On the Wiener-Hopf approach to optimal feedback design

A simple solution is given to the problem of designing a proper feedback which minimizes the ²-norm of the sensitivity function multiplied by a suitable weighting function.