Optimal Elastic Design for Prescribed Maximum Deflection

ABSTRACT

For the problems of the optimal elastic design with prescribed maximum deflection, a variational formulation is proposed, with reference to the one-or two-dimensional bending structures.

Necessary optimum conditions are found, and the physical features of the optimal solutions are discussed for the “absolute” minimum cost problems, and, when dealing with beams, for the solutions with piece-wise constant design function.

Some examples are solved by using numerical methods that are directly derived from the variational formulation.     

Optimal Force Transmission by Flexure-Clamped Boundaries

ABSTRACT

Grillages of maximum strength and maximum stiffness and fiber-reinforced plates of maximum strength are considered. Using a technique developed earlier, optimal solutions are given for a large number of boundary shapes. In all problems discussed, the flexural systems are clamped along all boundaries and the loading is given by an arbitrary nonnegative function.     

Optimal Shape Design of Multicomposite Structures

Abstract

The problem of the optimal shape design of an elastic composite structure with unspecified interfaces is discussed for the case of mean compliance constraint. The optimality conditions are derived, and the optimization of a circular plate with segmentwise constant thickness is considered in order to illustrate these conditions. The case of optimal plastic design is obtained as a limiting case of mean compliance design.     

Optimal Control of Turbulent Flow as Measure Solutions

Abstract

We use the concept of relaxed or measure-valued solutions for control problems of turbulent flow related to Navicr-Stokes equation. Sufficient conditions guaranteeing the existence of measure solutions arc presented. Results on existence of optimal controls for Blow up time and Bolza problems for such systems are also presented. New results on relaxed necessary conditions of optimality are proved. Further it is shown that the relaxed necessary conditions reduce to classical Pontryagin type necessary conditions if measure solutions degenerate into Dirac structure. The paper is concluded with an algorithm based on the new necessary conditions for computing optimal controls.     

On Optimality Conditions for Trusses with Nonuniform Stress Constraints*

ABSTRACT

The optimal design of a truss subjected to a single loading system and stress constraints, which are not necessarily the same in each bar, is considered. Sufficient conditions for global optimality are derived by variational methods. While these optimality criteria lead to a linear programming formulation of the problem, they show in a clear physical way how the optimal design is found, and that advantages accrue from incorporating the optimality criteria in a numerical scheme.     

Some Comments on the Coupling Effects of Angle-Ply Laminates

ABSTRACT

It is shown that by making a judicious choice of thicknesses of the laminae of angle-ply laminates, the degrading effect of coupling between bending and extension can be eliminated altogether for alternating laminates, and partially reduced for other types. This results in optimal use of materials for given conditions.     

Duality among Optimal Design Problems for Torsional Vibration

ABSTRACT

Four types of mass and frequency optimization problems are stated for free torsional vibration of thin-walled cylinders subject to constraints on wall thickness and frequencies of vibration. It is shown, using Pontryagin's method, that the mathematical structure of all four problems is similar and leads to identical classes of optimal thickness distributions. These duality relations are used in an example to construct an optimal frequency solution from the solutions for both maximum and minimum mass problems. General relations among the governing parameters for the four problems are stated. The results of Grinev and Filippov and of Thermann for the abnormal optimization problems are verfied as a specific limiting example of the general results.     

A NUMERICAL STUDY OF OPTIMAL DRAG REDUCTION FOR TURBULENT TRANSONIC PROJECTILES USING A PASSIVE CONTROL

Abstract

The purpose of this research is to numerically study a drag reduction method—passive control of shock/boundary layer interaction, which is applied to the boattail portion of a secant-ogive-cylinder-boattail projectile in turbulent transonic flows. The flow pattern and the components of aerodynamic drag computed from numerical data are analyzed. The effectiveness of this method is studied by varying the values of parameters such as porosity distribution, maximum porosity factor and size of porous region. The conditions for optimal drag reduction are investigated and reported. The present results show that the use of this passive control method can not only reduce the boattail drag but also the base drag, and results in an additional 8% total drag reduction compared to that without the passive control technique. This passive control method can be an effective approach for the design of high-performance projectiles in the transonic regime.     

Optimal Design of Supports of Elastic Structures Subjected to Loads and Initial Distortions

ABSTRACT

A problem of optimal location, stiffness, and prestress of supports in an elastic frame structure that is subjected to external loads, displacements, and initial distortions is formulated. Sensitivity analysis and optimality conditions are discussed for the assumed objective functional, accounting for conflicting design requirements that correspond to stiffness and stress constraints. A simple example is presented that illustrates the general theory.     

Optimization of Internal Line Support Positions for Plates Against Vibration∗

ABSTRACT

This paper presents a simple numerical method for determining optimal positions of internal line supports for an arbitrarily shaped plate, so as to maximize the fundamental frequency of its transverse vibration. The vibration analysis is performed using the pb-2 Rayleigh-Ritz method. Because this method does not require discretization, since it treats the entire plate with its boundary conditions as a kind of superelement, the optimization problem becomes relatively easy to solve. To illustrate the method, trapezoidal, elliptical, and semi-circular plates with at most two internal line supports are considered. The optimization exercise, for optimal locations of internal line supports, demonstrates significant improvement in the value of fundamental frequency when compared to that of plates with specified positions of internal supports.     

Shape optimization of two equal holes in an infinite elastic plate

Abstract

The optimal design of the stress state in elastic plate structures with openings is a problem of great significance in engineering practice. Achieving proper shape of hole can reduce stress concentration around the boundaries remarkably. The optimal shape of a single hole in an infinite plate under uniform stresses has been obtained by complex variable method based on different optimal criteria. The complex variable method is particularly suitable for the hole shape optimization in infinite plate, in which the continuous hole boundary can be represented by the mapping function. It can also be used to solve the shape optimization problems of two or more holes. However, because of the difficulty of finding the mapping function for multi connected domain, the holes are mapped onto slits or separately mapped onto a circle. In this article, the two symmetrical and identical holes are mapped onto an annulus simultaneously by the newly found mapping function, which has a general form. The maximum tangential stress around the boundaries is minimized to achieve the optimal hole shape. And the coefficients of mapping function which describe the boundary are calculated by differential-evolution algorithm.     

Analytical Treatment of Some Extended Problems in Structural Optimization,Part II

ABSTRACT

The Prager-Shield theory of optimal plastic design is applied to systems of preassigned topography (configuration) subject to either a single or several alternate load conditions. For the particular case of pin-jointed frames and a single load system, the criteria derived are shown to reduce to a condition obtained recently by Prager. The method is extended to cover joints of nonzero cost and it is illustrated with examples of trusses and grillages.

Finally, the optimization of discrete grillages having movable beams in preassigned directions is considered.     

An Optimal Control Approach to Minimum-Weight Vibrating Beam Design

ABSTRACT

A gradient projection algorithm is applied to a class of vibrating cantilever beam optimization problems which are formulated as optimal control problems. The cross-section area is distributed along the beam for minimum total weight subject to fixed natural frequency constraints and a minimum allowed area limit. Three topics receive major emphasis: the effects of shear deformation and rotary inertia, higher-mode frequency constraints, and multiple frequency constraints. Computational results are presented for the cases of fixed fundamental frequency, fixed second-mode frequency, and fixed fundamental and second-mode frequencies under a variety of conditions.     

Structural Responses of Underground Pipelines to Dynamic Loadings

ABSTRACT

This study develops a simple analytical model to assess the dynamic response of a pipeline under ground shaking. Since many observed failures of buried pipes are attributable to pullout or crushing at joints and breaks along the pipeline itself, the present model is proposed to estimate the maximum relative joint displacement and the maximum structural strain, under various soil conditions and material properties. Outcomes from both the quasi-static approach and dynamic analysis are compared, to examine the effect of inertia and damping, which are generally neglected in dealing with such a structure. The effect of excitation frequency, joint stiffness, pipe size, and pipe length on structural response is studied. Numerical results may serve as a guide for design, as well as for reliability analysis.     

Convex Analysis of the Eigenvalues of a 3D Second-Order Symmetric Tensor

Within the framework of nonsmooth convex analysis, the subdifferentials of the maximum eigenvalue and the negative of the minimum eigenvalue of a three-dimensional second-order symmetric tensor A are determined for all possible cases in an explicit and coordinate-free way. In particular, the expressions obtained for the subdifferentials show that: (i) An eigenvalue of A is differentiable if and only if it is simple; (ii) the maximum eigenvalue of A is subdifferentiable when it is double or triple; (iii) the negative of the minimum eigenvalue of A is subdifferentiable when it is double or triple. These results can be applied directly to elasticity and continuum mechanics where three-dimensional second-order symmetric tensor eigenvalues are frequently involved.     

The 3-layered explicit difference scheme for 2-D heat equation

A 3-layered explicit difference scheme for the numerical solution of 2-D heat equation is proposed. Firstly, a general symmetric difference scheme is constructed and its optimal error is obtained. Then two kinds of condition for choosing the parameters for optimal error and stable difference scheme are given. Finally, some numerical results are presented to show the advantage of the schemes Foundation items: the Science Foundation of Chinese Postdoctoral (2002031224); the Science Foundations of Southeast University (9209011148, 3007011043) Biography: Liu Ji-jun (1965-)     

A superposition principle in optimal plastic design for alternative loads

The paper is concerned with the optimal plastic design of sandwich beams, frames and trusses for alternative loading conditions. Upper and lower bounds for the optimal weight of a beam are derived, for single as well as for alternative loading conditions. These bounding theorems are used to establish a superposition principle. If no explicit bounds on the cross-sectional areas are prescribed, the optimal design for alternative loading conditions P₁ and P₂ can be obtained by superposition of the optimal designs for the single loading conditions and . If the cross-sections are to have at least given non-zero values, the principle furnishes upper and lower bounds to the optimal weight.The principle is illustrated by a simple example.     

On the Design of Structure and Controls for Optimal Performance of Actively Controlled Flexible Structures

ABSTRACT

The simultaneous design of scruccure and active controls is considered for structures equipped with force controllers. Both requirements for mission control (prescribed terminal conditions) and for the control of structural response (control damping) are reflected in developments for the design of

distributed parameter and large scale structures. As an example of problems with simple modal control, the optimal design is predicted for a cantilevered beam and for the feedback gains and actuator positions of its discrete controllers. Also the additional criterion to limit control spillover is incorporated into the formulation for modal control with a prescribed number of controlled modes. Computational results indicate that in some cases the use of a fully coupled model for the design of structure and its control devices leads to a considerable improvement in performance.     

Stretching Graphene to 3.3% Strain Using Formvar-Reinforced Flexible Substrate

Background

As a one-atom-thick material, the mechanical loading of graphene in large scale remains a challenge, and the maximum tensile strain that can be realized is through a flexible substrate, but only with a value of 1.8% due to the weak interfacial stress transfer.

Objective

Aims to illustrate the interface reinforcement brought by formvar resins as a buffering layer between graphene and substrates.

Methods

Single crystal graphene transferred to different substrates, applied with uniaxial stretching to compare the interface strength, and finite element analysis was performed to simulate tensile process for studying the influence of Poisson’s ratio of the buffering layer for interface reinforcement.

Results

In this work we use formvar resins as a buffering layer to achieve a maximum uniaxial tensile strain of 3.3% in graphene, close to the theoretical limit (3.7%) that graphene can achieve by flexible substrate stretching. The interface reinforcement by formvar is significantly higher than that by other polymers, which is attributed to the liquid–solid phase transition of formvar for more conformal interfacial contact and its suitable Poisson’s ratio with graphene to avoid its buckling along the transverse direction.

Conclusions

We believe that these results can provide guidance for the design of substrates and interfaces for graphene loading, as well as the support for mechanics analysis of graphene-based flexible electronic devices.