Optimal control of a variational inequality with applications to structural analysis. I. Optimal design of a beam with unilateral supports

A class of optimal design problems is considered, where the state problem is governed by a variational inequality. The latter includes an elliptic operator, the coefficients of which are chosen as the design (control) variables.Existence of an optimal design is proven on the abstract level. Some applications are presented to the problems of elastic or elasto-plastic beams with unilateral supports. Finite element approximations are proposed and a theoretical convergence result is proven in case of elastic beams.     

Optimal and approximate boundary controls of an elastic body with quasistatic evolution of damage

In this paper, we study an optimal control problem for the mixed boundary value problem for an elastic body with quasistatic evolution of an internal damage variable. We suppose that the evolution of microscopic cracks and cavities responsible for the damage is described by a nonlinear parabolic equation. A density of surface traction p acting on a part of boundary of an elastic body Ω is taken as a boundary control. Because the initial boundary value problem of this type can exhibit the Lavrentieff phenomenon and non‐uniqueness of weak solutions, we deal with the solvability of this problem in the class of weak variational solutions. Using the convergence concept in variable spaces and following the direct method in calculus of variations, we prove the existence of optimal and approximate solutions to the optimal control problem under rather general assumptions on the quasistatic evolution of damage. Copyright © 2014 John Wiley & Sons, Ltd.     

Flow and heat transfer of double fractional Maxwell fluids over a stretching sheet with variable thickness

This paper presents research on the fractional boundary layer flow and heat transfer over a stretching sheet with variable thickness. Based on the Caputo operators, the double fractional Maxwell model and generalized Fourier's law are introduced to the constitutive relationships. The governing equations are solved numerically by utilizing the finite difference method. The effects of fractional parameters on the velocity and temperature field are analyzed. The results indicate that the larger is the fractional stress parameter, the stronger is the elastic characteristic. However, fluids show viscous fluid-like behavior for a larger value of fractional strain parameter. Moreover, the numerical solutions are in good agreement with the exact solution and the convergence order can achieve the expected first order. The numerical method in this study is reliable and can be extended to other fractional boundary layer problems over a variable thickness sheet.     

Optimal Control of a Variational Inequality with Application to Equilibrium Problem of an Elastic Nonhomogeneous and Anisotropic Plate Resting on Unilateral Elastic Foundation

Several optimal control problems with the same state problem—a variational inequality with a monotone operator—are considered. The inequality represents bending of an elastic, nonhomogeneous, anisotropic Kirchhoff plate resting on some unilateral elasto-rigid foundation and point supports. Both the thickness of the plate and the coefficient of the unilateral elastic foundation play the role of design variables. Cost functionals include the work of external forces (compliance), total reaction forces of the foundation or the weight of the plate. The solvability of all the problems is proved. Moreover, approximate methods for the optimal control and weight minimization problems are proposed, making use of finite elements. The design variables are approximated by piecewise affine functions. The solvability of the approximate problems is proved and some convergence analysis is presented.     

Optimal Control of an Axisymmetric Conical Shell with a Unilateral Elastic Foundation and Rigid Supports

We study an optimal control problem for an elastic conical shell. The displacement vector u =(u,v) is then a function of one variable s∈(a,b), 0<a<b<∞. Here u is the meridional and v the normal displacement. The shell is assumed to be simply supported. Moreover, we consider several unilateral obstacles and an unilateral elastic foundation of Winkler type i.e. the reaction force is proportional of the positive part of the normal displacement. The state problem is formulated in a form of variational inequality for u . The design parameter e =(t,z,F) includes the variable thickness of the shell, the stiffness characteristics of the foundation and the friction coefficient. The existence of the optimal control will be explained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the contact interaction between a strip-shaped punch and a linearly-deformable base through a covering of variable thickness

The contact problem of the frictionless penetration of a punch with strip-shaped section into the surface of a linearly-deformable base protected by a thin elastic layer (covering) of variable thickness, the stiffness of which is comparable to or smaller than that of the supporting elastic body, is investigated. A Fredholm integral equation of the second kind is obtained for the unknown contact pressure with a coefficient in front of the leading term that is a fairly arbitrary function of the longitudinal coordinate. To solve it the Bubnov-Galerkin projection method is used in which the coordinate elements are chosen to be a system of orthogonal polynomials and delta-shaped functions [1, 2] (variational-difference method), together with an algorithm for the required asymptotic expansions [3] when the above-mentioned coefficient is small. In the special case of an elastic half-space protected by a covering of constant thickness, the results obtained are compared with the corresponding characteristics given in [4].     

Dynamic interaction of an elastic medium with a thin-walled curvilinear piezoelectric inclusion under longitudinal vibrations of a composite

Using the method of matched asymptotic expansions, we obtain models of dynamic interaction of a thin-walled curvilinear piezoelectric inclusion of variable thickness with an elastic isotropic matrix under stationary vibrations of the composite. The elastic system is under conditions of longitudinal shear. Different cases of electric boundary conditions on the surface of the heterogeneity are considered. We propose an algorithm for the construction of boundary layer corrections for refining the behavior of displacements and stresses in the vicinity of the edge of the inclusion for its different shapes.     

A new Orthonormal Polynomial Series Expansion Method in vibration analysis of thin beams with non-uniform thickness

In this article, OPSEM (Orthonormal Polynomial Series Expansion Method) is developed as a new computational approach for the evaluation of thin beams of variable thickness transverse vibration. Capability of the OPSEM in assessing the free vibration frequencies and mode shapes of an Euler–Bernoulli beam with varying thickness is discussed. Multispan continuous beams with various classical boundary conditions are included. Contribution of BOPs (Basic Orthonormal Polynomials) in capturing the beam vibrations is also illustrated in numerical examples to give a quantitative measure of convergence rate. Furthermore, OPSEM is adopted for the forced vibration of a thin beam caused by a moving mass. Dynamics of beams supported by flexible elastic base like free to free beam on elastic foundation are also regarded. Verifications are made via eigenfunction expansion method and GMLSM (Generalized Moving Least Square Method). The very close observed agreement between the results of the two recently mentioned methods and that of OPSEM can be regarded as a guarantee of validity for the newly introduced technique. In comparison with eigenfunction expansion method, the simplicity and handiness of OPSEM in coping with different boundary conditions of the beam can be considered as its benefit for engineering practitioners.     

Optimal design of an elastic beam with a unilateral elastic foundation: Semicoercive state problem

A design optimization problem for an elastic beam with a unilateral elastic foundation is analyzed. Euler-Bernoulli’s model for the beam and Winkler’s model for the foundation are considered. The state problem is represented by a nonlinear semicoercive problem of 4th order with mixed boundary conditions. The thickness of the beam and the stiffness of the foundation are optimized with respect to a cost functional. We establish solvability conditions for the state problem and study the existence of a solution to the optimization problem.     

Calculation of microstrains and microstresses in a thick non-symmetric heterogeneous plate by homogenization

We consider the thermoelastic behaviour of a thick heterogeneous plate containing in its thickness a large number of periodically distributed transverse holes or inclusions. We use the Reissner-Mindlin thermoelastic linear model of thick plates with a known temperature and we distinguish displacements in the upper and lower part of the plate with respect to the middle plane. Due to the structure of the plate, thermal and elastic coefficients are non-uniformly and rapidly oscillating functions of the space variable. Two-scale convergence, which is the state of the art in mathematical homogenization technics, is used and gives convergence results and formulae allowing to calculate the distribution of microstrains and microstresses inside the plate when a macroscopic behaviour is given.     

非均匀变截面弹性圆环在任意载荷下的弯曲问题

本文在等刚度弹性圆环的初参数公式的基础上,利用[2]提出的阶梯折算法,进一步研究非均匀变截面弹性圆环的弯曲,得到了这类问题的通解,应当指出,这组通解对非均匀变截面圆柱拱的相应问题也是适用的.为验证所得的公式并说明这种方法的应用,文末给出了示例并进行了求解,圆环、圆拱是工程上经常采用的结构,它们的弯曲,Timoshenko,S.[5],Barber,J.R.[3],Roark,R J[4],津村利光[6]等曾作过很多研究.然而,迄今只求得了均匀材料、等截面圆环的通解。对变截面问题,仅仅求得了抗弯刚度是坐标的线性函数这一特殊情况的解.由于非均匀变截面问题常常导出变系数微分方程,它们的求解遇到很大的数学困难.本文通过阶梯折算法把非均匀变截面弹性圆环弯曲问题的变系数微分方程转化成一等效的等刚度圆环弯曲的常系数微分方程.为保证内力连续,引入虚拟内力,并以[1]导出的初参数公式为影响函数,通过积分构造出了非齐次解,从而求得了非均匀变截面弹性圆环弯曲问题的通解.     

一类机器人系统的最优控制

本文通过把结构阻尼系数当作控制变量来讨论一类弹性机器人系统的最优控制问题 ,并利用Banach空间几何性质证明了最优控制元的存在唯一性     

Dynamics of elastic bodies connected by a thin soft viscoelastic layer

A dynamic study was performed on a structure consisting of two three-dimensional linearly elastic bodies connected by a thin soft nonlinear Kelvin–Voigt viscoelastic adhesive layer. The adhesive is assumed to be viscoelastic of Kelvin–Voigt generalized type, which makes it possible to deal with a relatively wide range of physical behavior by choosing suitable dissipation potentials. In the static and purely elastic case, convergence results when geometrical and mechanical parameters tend to zero have already been obtained using variational convergence methods. To obtain convergence results in the dynamic case, the main tool, as in the quasistatic case, is a nonlinear version of Trotter?s theory of approximation of semigroups acting on variable Hilbert spaces. The limit problem involves a mechanical constraint imposed along the surface to which the layer shrinks. The meaning of this limit with respect to the relative behavior of the parameters is discussed. The problem applies in particular to wave phenomena in bonded domains.     

Asymptotic analysis of a thin fluid layer-elastic plate interaction problem

We study the nonstationary flow of an incompressible fluid in a thin rectangle with an elastic plate as the upper part of the boundary. The flow is governed by a time-dependent pressure drop and an external force and it is modeled by Stokes equations. The dynamic of this fluid–structure interaction problem is studied in the limit when the thickness of the fluid domain tends to zero. Using the asymptotic techniques, we obtain for the effective plate displacement a sixth-order parabolic equation with a non standard boundary conditions. Results on existence, uniqueness and regularity of the solution are proved. The approximation is justified through a weak convergence theorem.     

Justification of two dimensional model of shallow shells using gamma convergence

In this paper we derive the two dimensional model of elastic shallow shell using gamma convergence. We consider thin elastic shallow shells of “very small” thickness and we show that the sequence of functions minimizing the energy associated with the three-dimensional elastic shallow shells converges to the function which minimizes the energy associated with the two dimensional elastic shallow shell as the thickness of the shell goes to zero.     

On the uniform stress state inside an inclusion of arbitrary shape in a three-phase composite

The stress field inside a two-dimensional arbitrary-shape elastic inclusion bonded through an interphase layer to an infinite elastic matrix subjected to uniform stresses at infinity is analytically studied using the complex variable method in elasticity. Both in-plane and anti-plane shear loading cases are considered. It is shown that the stress field within the inclusion can be uniform and hydrostatic under remote constant in-plane stresses and can be uniform under remote constant anti-plane shear stresses. Both of these uniform stress states can be achieved when the shape of the inclusion, the elastic properties of each phase, and the thickness of the interphase layer are properly designed. Possible non-elliptical shapes of inclusions with uniform hydrostatic stresses induced by in-plane loading are identified and divided into three groups. For each group, two conditions that ensure a uniform hydrostatic stress state are obtained. One condition relates the thickness of the interphase layer to elastic properties of the composite phases, while the other links the remote stresses to geometrical and material parameters of the three-phase composite. Similar conditions are analytically obtained for enabling a uniform stress state inside an arbitrary-shape inclusion in a three-phase composite loaded by remote uniform anti-plane shear stresses.     

Asymptotic analysis of Poisson's equation in a thin domain and its application to thin-walled elastic beams and tubes

We study the limit behaviour of solution of Poisson's equation in a class of thin two-dimensional domains, both simply connected or single-hollowed, as its thickness becomes very small. The method is based on a transformation of the original problem into another posed on a fixed domain, obtention of a priori estimates and convergence results when thickness parameter tends to zero. As an important application of abstract results we obtain the limit expressions for functions appearing in elastic beam theories as torsion and warping functions. In this way, we provide a mathematical justification and a correct definition of torsion, warping and Timoshenko functions and constants that should be used in the open and closed thin-walled elastic beam theories. © 1998 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

弹性地基上正交各向异性变厚度圆薄板的大挠度问题

本文推出了均布载荷下弹性基地上的正交各向异性变厚度圆薄板大挠度问题的基本方程。利用修正迭代法获得了该问题的二阶近似解。     

Stresses of a rotating circular disk of variable thickness carrying a current and bearing a coaxial viscoelastic coating

In the present work we consider a circular elastic disk (conductor) of variable thickness under the influence of a steady coaxial current and bearing a coaxial viscoelastic coating (insulator). In both conductor and insulator there exist a heat source generation. As a first step, the solution of purely elastic conductor and insulator is obtained. Then the problem of model with viscoelastic coating is solved using the correspondence principle and Ilyushin’s approximation method. A numerical example is given and the results are discussed in order to investigate the influences of time, temperature, and rotation on the stresses and displacements.     

Optimal Scaling of a Gradient Method for Distributed Resource Allocation

We consider a class of weighted gradient methods for distributed resource allocation over a network. Each node of the network is associated with a local variable and a convex cost function; the sum of the variables (resources) across the network is fixed. Starting with a feasible allocation, each node updates its local variable in proportion to the differences between the marginal costs of itself and its neighbors. We focus on how to choose the proportional weights on the edges (scaling factors for the gradient method) to make this distributed algorithm converge and on how to make the convergence as fast as possible.We give sufficient conditions on the edge weights for the algorithm to converge monotonically to the optimal solution; these conditions have the form of a linear matrix inequality. We give some simple, explicit methods to choose the weights that satisfy these conditions. We derive a guaranteed convergence rate for the algorithm and find the weights that minimize this rate by solving a semidefinite program. Finally, we extend the main results to problems with general equality constraints and problems with block separable objective function.The authors are grateful to Professor Paul Tseng and the anonymous referee for their valuable comments that helped us to improve the presentation of this paper.Communicated by P. Tseng