Analysis of Mindlin micro plates with a modified couple stress theory and a meshless method

A modified couple stress theory and a meshless method is used to study the bending of simply supported micro isotropic plates according to the first-order shear deformation plate theory, also known as the Mindlin plate theory. The modified couples tress theory involves only one length scale parameter and thus simplifies the theory, since experimentally it is easier to determine the single scale parameter. The equations governing bending of the first-order shear deformation theory are implemented using a meshless method based on collocation with radial basis functions. The numerical method is easy to implement, and it provides accurate results that are in excellent agreement with the analytical solutions.     

Topology optimization of multi-purpose structures

Whilst most of the literature on topology optimization of structures deals with so-called selfadjoint problems involving highly idealized, single-purpose structures, this paper discusses topology optimization of multi-purpose structures which concerns nonselfadjoint problems. General methods based on the so-called layout theory, application to trusses and perforated plates and computational difficulties are discussed.     

Natural frequencies of rectangular Mindlin plates coupled with stationary fluid

The present study is concerned with the free vibration analysis of a horizontal rectangular plate, either immersed in fluid or floating on its free surface. The governing equations for a moderately thick rectangular plate are analytically derived based on the Mindlin plate theory (MPT), whereas the velocity potential function and Bernoulli’s equation are employed to obtain the fluid pressure applied on the free surface of the plate. The simplifying hypothesis that the wet and dry mode shapes are the same, is not assumed in this paper. In this work, an exact-closed form characteristics equation is used for the plate subjected to a combination of six different boundary conditions. Two opposite sides are simply supported and any of the other two edges can be free, simply supported or clamped. To demonstrate the accuracy of the present analytical solution, a comparison is made with the published experimental and numerical results in the literature, showing an excellent agreement. Then, natural frequencies of the plate are presented in tabular and graphical forms for different fluid levels, fluid densities, aspect ratios, thickness to length ratios and boundary conditions. Finally, some 3-D mode shapes of the rectangular Mindlin plates in contact with fluid are illustrated.     

Approximation of functionally graded plates with non-conforming finite elements

In this paper rectangular plates made of functionally graded materials (FGMs) are studied. A two-constituent material distribution through the thickness is considered, varying with a simple power rule of mixture. The equations governing the FGM plates are determined using a variational formulation arising from the Reissner–Mindlin theory. To approximate the problem a simple locking-free Discontinuous Galerkin finite element of non-conforming type is used, choosing a piecewise linear non-conforming approximation for both rotations and transversal displacement. Several numerical simulations are carried out in order to show the capability of the proposed element to capture the properties of plates of various gradings, subjected to thermo-mechanical loads.     

Optimal rectangular MITC finite elements for Reissner–Mindlin plates

In recent years a family of finite elements named mixed interpolated tensorial components (MITC) has been introduced for the numerical approximation of Reissner–Mindlin plates. The elements have been proved to be locking free. In this article, we consider the MITC rectangular finite elements and show that it is possible to reduce the number of internal degrees of freedom in the approximation of the rotation field without losing order of convergence. Our mathematical analysis is carried out combining some results for the Stokes problem with the special features of the MITC finite elements. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 575–585, 1997     

Schrödinger operators on periodic discrete graphs

We consider Schrödinger operators with periodic potentials on periodic discrete graphs. The spectrum of the Schrödinger operator consists of an absolutely continuous part (a union of a finite number of non-degenerated bands) plus a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. We obtain estimates of the Lebesgue measure of the spectrum in terms of geometric parameters of the graph and show that they become identities for some class of graphs. Moreover, we obtain stability estimates and show the existence and positions of large number of flat bands for specific graphs. The proof is based on the Floquet theory and the precise representation of fiber Schrödinger operators, constructed in the paper.     

Construction of a CPA contraction metric for periodic orbits using semidefinite optimization

A Riemannian metric with a local contraction property can be used to prove existence and uniqueness of a periodic orbit and determine a subset of its basin of attraction. While the existence of such a contraction metric is equivalent to the existence of an exponentially stable periodic orbit, the explicit construction of the metric is a difficult problem.In this paper, the construction of such a contraction metric is achieved by formulating it as an equivalent problem, namely a feasibility problem in semidefinite optimization. The contraction metric, a matrix-valued function, is constructed as a continuous piecewise affine (CPA) function, which is affine on each simplex of a triangulation of the phase space. The contraction conditions are formulated as conditions on the values at the vertices.The paper states a semidefinite optimization problem. We prove on the one hand that a feasible solution of the optimization problem determines a CPA contraction metric and on the other hand that the optimization problem is always feasible if the system has an exponentially stable periodic orbit and the triangulation is fine enough. An objective function can be used to obtain a bound on the largest Floquet exponent of the periodic orbit.     

Status of periodic optimization of dynamical systems

The problem of the periodic optimization of dynamical systems, recently considered to some extent in the literature, is reviewed here. The most interesting results among those available are illustrated. Mention is made of the most significant and open aspects of the problem.This research was supported by CNR (Consiglio Nazionale delle Ricerche), Rome, Italy. A preliminary version of this paper has been presented at the 6th International Summer School on Electronics and Automation, Herzeg Novi, Yugoslavia, 1971.     

An accurate mathematical study on the free vibration of stepped thickness circular/annular Mindlin functionally graded plates

An analytical solution based on a new exact closed form procedure is presented for free vibration analysis of stepped circular and annular FG plates via first order shear deformation plate theory of Mindlin. The material properties change continuously through the thickness of the plate, which can vary according to a power-law distribution of the volume fraction of the constituents, whereas Poisson’s ratio is set to be constant. Based on the domain decomposition technique, five highly coupled governing partial differential equations of motion for freely vibrating FG plates were exactly solved by introducing the new potential functions as well as using the method of separation of variables. Several comparison studies were presented by those reported in the literature and the FEM analysis, for various thickness values and combinations of stepped thickness variations of circular/annular FG plates to demonstrate highly stability and accuracy of present exact procedure. The effect of the geometrical and material plate parameters such as step thickness ratios, step locations and the power law index on the natural frequencies of FG plates is investigated.     

The Limits of Porous Materials in the Topology Optimization of Stokes Flows

We consider a problem concerning the distribution of a solid material in a given bounded control volume with the goal to minimize the potential power of the Stokes flow with given velocities at the boundary through the material-free part of the domain.We also study the relaxed problem of the optimal distribution of the porous material with a spatially varying Darcy permeability tensor, where the governing equations are known as the Darcy–Stokes, or Brinkman, equations. We show that the introduction of the requirement of zero power dissipation due to the flow through the porous material into the relaxed problem results in it becoming a well-posed mathematical problem, which admits optimal solutions that have extreme permeability properties (i.e., assume only zero or infinite permeability); thus, they are also optimal in the original (non-relaxed) problem. Two numerical techniques are presented for the solution of the constrained problem. One is based on a sequence of optimal Brinkman flows with increasing viscosities, from the mathematical point of view nothing but the exterior penalty approach applied to the problem. Another technique is more special, and is based on the “sizing” approximation of the problem using a mix of two different porous materials with high and low permeabilities, respectively. This paper thus complements the study of Borrvall and Petersson (Internat. J. Numer. Methods Fluids, vol. 41, no. 1, pp. 77–107, 2003), where only sizing optimization problems are treated.     

Free material optimization via mathematical programming

This paper deals with a central question of structural optimization which is formulated as the problem of finding the stiffest structure which can be made when both the distribution of material as well as the material itself can be freely varied. We consider a general multi-load formulation and include the possibility of unilateral contact. The emphasis of the presentation is on numerical procedures for this type of problem, and we show that the problems after discretization can be rewritten as mathematical programming problems of special form. We propose iterative optimization algorithms based on penalty-barrier methods and interior-point methods and show a broad range of numerical examples that demonstrates the efficiency of our approach. Supported by the project 03ZO7BAY of BMBF (Germany) and the GIF-contract 10455-214.06/95.     

Improvements in the treatment of stress constraints in structural topology optimization problems

Topology optimization of continuum structures is a relatively new branch of the structural optimization field. Since the basic principles were first proposed by Bendsøe and Kikuchi in 1988, most of the work has been dedicated to the so-called maximum stiffness (or minimum compliance) formulations. However, since a few years different approaches have been proposed in terms of minimum weight with stress (and/or displacement) constraints.These formulations give rise to more complex mathematical programming problems, since a large number of highly non-linear (local) constraints must be taken into account. In an attempt to reduce the computational requirements, in this paper, we propose different alternatives to consider stress constraints and some ideas about the numerical implementation of these algorithms. Finally, we present some application examples.     

On the effects of coupling between in-plane and out-of-plane vibrating modes of smart functionally graded circular/annular plates

In recent years many articles concerned with the mechanics of functionally graded plates have been published. The variation in material properties through the thickness of the plate introduces a coupling between in-plane and transverse displacements, the coupling is important in the vibration of functionally graded plates (FGPs), but none have produced an exact closed-form solution for the in-plane as well as transverse vibrations of smart circular/annular FGPs. Therefore, this paper develops an exact closed-form solution for the free vibration of piezoelectric coupled thick circular/annular FGPs subjected to different boundary conditions on the basis of the Mindlin’s first-order shear deformation theory. Through the comparison of present results with those available, the accuracy of the present method was verified. The effects of coupling between in-plane and transverse displacements on the frequency parameters are proved to be significant. It is concluded that the developed model can describe vibrational behavior of smart FGM plates more realistic. Due to the inherent features of the present solution, all findings will be a useful benchmark for evaluating other analytical and numerical methods developed by researchers in the future.     

关于模糊数空间的Skorokhod拓扑的注记

首先考察模糊数空间中Skorokhod度量与紧承下方图度量之间的关系,然后说明了文献[4]中的关于Skorokhod拓扑紧致性的例子是错误的并给出了正确的例子.     

Acoustic band gaps in arrays of neutral inclusions

We analyse numerically the acoustic stop band properties of an array of orthotropic coated cylinders whose elastic parameters are deduced from a geometric transform [H. Chen, C.T. Chan, Acoustic cloaking in three dimensions using acoustic metamaterials, Appl. Phys. Lett. 91 (2007) 183518]. We find that whereas a single coated inclusion is acoustically neutral at any frequency, an array of them might display some stop bands. More precisely, an array of freely vibrating coated voids is always neutral, whereas an array of clamped coated inclusions might display a zero frequency stop band. Interestingly, an array of radially symmetric coated inclusions behaves as local Helmholtz resonators, for which the eigenfield within each cloak is obtained in closed form, leading to a frequency estimate associated with the lower edge of the low frequency stop band. A finite phononic crystal of such coated cylinders behaves either as an invisible material or a reflector depending on the frequency of an acoustic source.     

Adaptive mesh method for topology optimization of fluid flow

In order to solve the topology optimization problems of fluid flow and obtain higher resolution of the interface with a minimum of additional expense, an automatic local adaptive mesh refinement method is proposed. The optimization problem is solved by a simple but robust optimality criteria (OC) algorithm. A material distribution information based adaptive mesh method is adopted during the optimization process. The optimization procedure is provided and verified with several benchmark examples.     

Bimodal optimal design of vibrating plates using theory and methods of nondifferentiable optimization

This paper is concerned with an optimal design problem of vibrating plates. The optimization problem consists in maximizing the smallest eigenvalue of the elliptic eigenvalue problem describing the free plate vibration. The thickness of the plate is the variable subject to optimization. The volume of the plate is constant and the thickness of the plate is bounded.In this paper, we consider the case where the smallest eigenvalue is multiple. This implies that the optimization problem is nondifferentiable. A necessary optimality condition is formulated. The finite-element method is employed as an approximation method. A nonsmooth optimization method is used to solve this optimization problem. Numerical examples are provided.This work was supported by the Polish Academy of Sciences and the Education Ministry of Japan. Lemarechal's implementation of his method was used for numerical computations.on leave from Systems Research Institute, Warsaw, Poland.     

Homotopical theory of periodic points of periodic homeomorphisms on closed surfaces

In this work we show that the Wecken theorem for periodic points holds for periodic homeomorphisms on closed surfaces, which therefore completes the periodic point theory in such a special case. Using it we derive the set of homotopy minimal periods for such homeomorphisms. Moreover we show that the results hold for homotopically periodic self-maps of closed surfaces. This let us to re-formulate our results as a statement on properties of elements of finite order in the group of outer automorphisms of the fundamental group of a surface with non-positive Euler characteristic.     

Inductive inference of ultimately periodic sequences

An algorithm is presented for inductive inference of ultimately periodic sequences. It can be implemented in linear time and space.Dedicated to Peter Naur on the occasion of his 60th birthday