Improved hybrid elements for structural analysis

Hybrid elements, which are based on a two-field variational formulation with the displacements and stresses interpolated separately, are known to deliver very high accuracy, and to alleviate to a large extent problems of locking that plague standard displacement-based formulations. The choice of the stress interpolation functions is of course critical in ensuring the high accuracy and robustness of the method. Generally, an attempt is made to keep the stress interpolation to the minimum number of terms that will ensure that the stiffness matrix has no spurious zero-energy modes, since it is known that the stiffness increases with the increase in the number of terms. Although using such a strategy of keeping the number of interpolation terms to a minimum works very well in static problems, it results either in instabilities or fails to converge in transient problems. This is because choosing the stress interpolation functions merely on the basis of removing spurious energy modes can violate some basic principles that interpolation functions should obey. In this work, we address the issue of choosing the interpolation functions based on such basic principles of interpolation theory and mechanics. Although this procedure results in the use of more number of terms than the minimum (and hence in slightly increased stiffness) in many elements, we show that the performance continues to be far superior to displacement-based formulations, and, more importantly, that it also results in considerably increased robustness.

Spin-1 Kitaev model in one dimension

We study a one-dimensional version of the Kitaev model on a ring of size N, in which there is a spin S > 1/2 on each site and the Hamiltonian is J Sigma(nSnSn+1y)-S-x. The cases where S is integer and half-odd integer are qualitatively different. We show that there is a Z(2)-valued conserved quantity W-n for each bond (n, n + 1) of the system. For integer S, the Hilbert space can be decomposed into 2N sectors, of unequal sizes. The number of states in most of the sectors grows as d(N), where d depends on the sector. The largest sector contains the ground state, and for this sector, for S=1, d=(root 5+1)/2. We carry out exact diagonalization for small systems. The extrapolation of our results to large N indicates that the energy gap remains finite in this limit. In the ground-state sector, the system can be mapped to a spin-1/2 model. We develop variational wave functions to study the lowest energy states in the ground state and other sectors. The first excited state of the system is the lowest energy state of a different sector and we estimate its excitation energy. We consider a more general Hamiltonian, adding a term lambda Sigma W-n(n), and show that this has gapless excitations in the range lambda(c)(1)<=lambda <=lambda(c)(2). We use the variational wave functions to study how the ground-state energy and the defect density vary near the two critical points lambda(c)(1) and lambda(c)(2).

A field-consistent and variationally correct representation of transverse shear strains in the nine-noded plate element

We present a biquadratic Lagrangian plate bending element with consistent fields for the constrained transverse shear strain functions. A technique involving expansion of the strain interpolations in terms of Legendre polynomials is used to redistribute the kinematically derived shear strain fields so that the field-consistent forms (i.e. avoiding locking) are also variationally correct (i.e. do not violate the variational norms). Also, a rational method of isoparametric Jacobian transformation is incorporated so that the constrained covariant shear strain fields are always consistent in whatever general quadrilateral form the element may take. Finally the element is compared with another formulation which was recently published. The element is subjected to several robust bench mark tests and is found to pass all the tests efficiently.

A homotopy approach for stabilizing single-input systems with control structure constraints

A linear state feedback gain vector used in the control of a single input dynamical system may be constrained because of the way feedback is realized. Some examples of feedback realizations which impose constraints on the gain vector are: static output feedback, constant gain feedback for several operating points of a system, and two-controller feedback. We consider a general class of problems of stabilization of single input dynamical systems with such structural constraints and give a numerical method to solve them. Each of these problems is cast into a problem of solving a system of equalities and inequalities. In this formulation, the coefficients of the quadratic and linear factors of the closed-loop characteristic polynomial are the variables. To solve the system of equalities and inequalities, a continuous realization of the gradient projection method and a barrier method are used under the homotopy framework. Our method is illustrated with an example for each class of control structure constraint.

Social and material deprivation among youth in Finland : causes, consequences, and coping.

Background: Social and material deprivation is associated with poor health, decreased subjective well-being, and limited opportunities for personal development. To date, little is known about the lived experiences of Finnish low-income youths and the general purpose of this study is to fill this gap. Despite the extensive research on socioeconomic income disparities, only a few scholars have addressed the question of how low socioeconomic position is experienced by disadvantaged people themselves. Little is known about the everyday social processes that lead to decreased well-being of economically and socially disadvantaged citizens. Data: The study is based on the data of 65 autobiographical essays written by Finnish low-income youths aged 14-29 (M=23.51, SD=3.95). The research data were originally collected in a Finnish nationwide writing contest “Arkipäivän kokemuksia köyhyydestä” [Everyday Experiences of Poverty] between June and September of 2006. The contest was partaken by 850 Finnish writers. Methods and key concepts: Autobiographical narratives (N=65) of low-income youths were analyzed based on grounded theory methodology (GTM). The analysis was not built on specific pre-conceived categorizations; it was guided by the paradigm model and so-called “sensitizing concepts”. The concepts this study utilized were based on the research literature on socioeconomic inequalities, resilience, and coping. Socioeconomic inequalities refer to unequal distribution of resources, such as income, social status, and health, between social groups. The concept of resilience refers to an individual’s capacity to cope despite existing risk factors and conditions that are harmful to health and well-being. Coping strategies can be understood as ways by which a person tries to cope with psychological stress in a situation where internal or externals demands exceed one’s resources. The ways to cope are cognitive or behavioral efforts by which individual tries to relieve the stress and gain new resources. Lack of material and social resources is associated with increased exposure to health-related stressors during the life-course. Aims: The first aim of this study is to illustrate how youths with low socioeconomic status perceive the causes and consequences of their social and material deprivation. The second aim is to describe what kind of coping strategies youths employ to cope in their everyday life. The third aim is to build an integrative conceptual framework based on the relationships between causes, consequences, and individual coping strategies associated with deprivation. The analysis was carried out through systematic coding and orderly treatment of the data based on the grounded theory methodology. Results: Finnish low-income youths attributed the primary causes of deprivation to their family background, current socioeconomic status, sudden life changes, and contextual factors. Material and social deprivation was associated with various kinds of negative psychological, social, and material consequences. Youths used a variety of coping strategies that were identified as psychological, social, material, and functional-behavioral. Finally, a conceptual framework was formulated to link the findings together. In the discussion, the results were compared and contrasted to the existing research literature. The main references of the study were: Coping: Aldwin (2007); Lazarus & Folkman (1984); Hobfoll (1989, 2001, 2002). Deprivation: Larivaara, Isola, & Mikkonen (2007); Lister (2004); Townsend (1987); Raphael (2007). Health inequalities: Dahlgren & Whitehead (2007); Lynch. et al. (2000); Marmot & Wilkinson (2006); WHO (2008). Methods: Charmaz (2006); Flick (2009); Strauss & Corbin (1990). Resilience: Cutuli & Masten (2009); Luthar (2006).

Non-linearity in piezo-fiber reinforced composites: an asymptotic approach

An asymptotically correct analysis is developed for Macro Fiber Composite unit cell using Variational Asymptotic Method (VAM). VAM splits the 3D nonlinear problem into two parts: A 1D nonlinear problem along the length of the fiber and a linear 2D cross-sectional problem. Closed form solutions are obtained for the 2D problem which are in terms of 1D parameters.

Coupled electrostatic-elastic analysis for topology optimization using material interpolation

In this paper, we present a novel analytical formulation for the coupled partial differential equations governing electrostatically actuated constrained elastic structures of inhomogeneous material composition. We also present a computationally efficient numerical framework for solving the coupled equations over a reference domain with a fixed finite-element mesh. This serves two purposes: (i) a series of problems with varying geometries and piece-wise homogeneous and/or inhomogeneous material distribution can be solved with a single pre-processing step, (ii) topology optimization methods can be easily implemented by interpolating the material at each point in the reference domain from a void to a dielectric or a conductor. This is attained by considering the steady-state electrical current conduction equation with a `leaky capacitor' model instead of the usual electrostatic equation. This formulation is amenable for both static and transient problems in the elastic domain coupled with the quasi-electrostatic electric field. The procedure is numerically implemented on the COMSOL Multiphysics (R) platform using the weak variational form of the governing equations. Examples have been presented to show the accuracy and versatility of the scheme. The accuracy of the scheme is validated for the special case of piece-wise homogeneous material in the limit of the leaky-capacitor model approaching the ideal case.

Variable Elimination in Linear Constraints

Gauss and Fourier have together provided us with the essential techniques for symbolic computation with linear arithmetic constraints over the reals and the rationals. These variable elimination techniques for linear constraints have particular significance in the context of constraint logic programming languages that have been developed in recent years. Variable elimination in linear equations (Guassian Elimination) is a fundamental technique in computational linear algebra and is therefore quite familiar to most of us. Elimination in linear inequalities (Fourier Elimination), on the other hand, is intimately related to polyhedral theory and aspects of linear programming that are not quite as familiar. In addition, the high complexity of elimination in inequalities has forces the consideration of intricate specializations of Fourier's original method. The intent of this survey article is to acquaint the reader with these connections and developments. The latter part of the article dwells on the thesis that variable elimination in linear constraints over the reals extends quite naturally to constraints in certain discrete domains.

3-D Warping in Four-Bar Laminated Linkages

This paper deals with the evaluation of the component-laminate load-carrying capacity, i.e., to calculate the loads that cause the failure of the individual layers and the component-laminate as a whole in four-bar mechanism. The component-laminate load-carrying capacity is evaluated using the Tsai-Wu-Hahn failure criterion for various layups. The reserve factor of each ply in the component-laminate is calculated by using the maximum resultant force and the maximum resultant moment occurring at different time steps at the joints of the mechanism. Here, all component bars of the mechanism are made of fiber reinforced laminates and have thin rectangular cross-sections. They could, in general, be pre-twisted and/or possess initial curvature, either by design or by defect. They are linked to each other by means of revolute joints. We restrict ourselves to linear materials with small strains within each elastic body (beam). Each component of the mechanism is modeled as a beam based on geometrically nonlinear 3-D elasticity theory. The component problems are thus split into 2-D analyses of reference beam cross-sections and nonlinear 1-D analyses along the three beam reference curves. For the thin rectangular cross-sections considered here, the 2-D cross-sectional nonlinearity is also overwhelming. This can be perceived from the fact that such sections constitute a limiting case between thin-walled open and closed sections, thus inviting the nonlinear phenomena observed in both. The strong elastic couplings of anisotropic composite laminates complicate the model further. However, a powerful mathematical tool called the Variational Asymptotic Method (VAM) not only enables such a dimensional reduction, but also provides asymptotically correct analytical solutions to the nonlinear cross-sectional analysis. Such closed-form solutions are used here in conjunction with numerical techniques for the rest of the problem to predict more quickly and accurately than would otherwise be possible. Local 3-D stress, strain and displacement fields for representative sections in the component-bars are recovered, based on the stress resultants from the 1-D global beam analysis. A numerical example is presented which illustrates the failure of each component-laminate and the mechanism as a whole.

Ionic Polymer Metal Composite Flapping Actuator Mimicking Dragonflies

In this study, variational principle is used for dynamic modeling of an Ionic Polymer Metal Composite (IPMC) flapping wing. The IPMC is an Electro-active Polymer (EAP) which is emerging as a useful smart material for `artificial muscle' applications. Dynamic characteristics of IPMC flapping wings having the same size as the actual wings of three different dragonfly species Aeshna Multicolor, Anax Parthenope Julius and Sympetrum Frequens are analyzed using numerical simulations. An unsteady aerodynamic model is used to obtain the aerodynamic forces. A comparative study of the performances of three IPMC flapping wings is conducted. Among the three species, it is found that thrust force produced by the IPMC flapping wing of the same size as Anax Parthenope Julius wing is maximum. Lift force produced by the IPMC wing of the same size as Sympetrum Frequens wing is maximum and the wing is suitable for low speed flight. The numerical results in this paper show that dragonfly inspired IPMC flapping wings are a viable contender for insect scale flapping wing micro air vehicles.

Solid finite elements through three decades

conventionally, solid finite elements have been looked upon as just generalizations of two-dimensional finite elements. In this article we trace their development starting from the days of their inception. Keeping in tune with our perceptions on developing finite elements, without taking recourse to any extra variational techniques, we discuss a few of the techniques which have been applied to solid finite elements. Finally we critically examine our own work on formulating solid finite elements based on the solutions to the Navier equations.

Symmetry-breaking transition in a two-level system coupled to an Ohmic bath: A squeezed-state approach

Displaced squeezed states are proposed as variational ground states for phonons (Bose fields) coupled to two-level systems (spin systems). We have investigated the zero-temperature phase diagram for the localization-delocalization transition of a tunneling particle interacting with an Ohmic heat bath. Our results are compared with known existing approximate treatments. A modified phase diagram using the displaced squeezed state is presented.

The structure of amorphous platinum disulfide as studied by anomalous x-ray-scattering

Anomalous X-ray scattering (AXS) has been applied to study the structure of amorphous platinum disulfide, Pt1-xS2, prepared by the precipitation process. The local atomic arrangement in amorphous Pt1-xS2 was determined by the least-squares variational method so as to reproduce the experimental differential interference function at the Pt L(III) absorption edge by the AXS method as well as the ordinary interference function by MoK alpha. The structural unit in amorphous Pt1-xS2 is found to be a PtS6 octahedron, similar to that in crystalline PtS2. These octahedra share both their corners and edges, while only edge-sharing linkages occur in crystalline PtS2.

Stability analysis of stochastically parametered nonconservative columns

Nonconservatively loaded columns. which have stochastically distributed material property values and stochastic loadings in space are considered. Young's modulus and mass density are treated to constitute random fields. The support stiffness coefficient and tip follower load are considered to be random variables. The fluctuations of external and distributed loadings are considered to constitute a random field. The variational formulation is adopted to get the differential equation and boundary conditions. The non self-adjoint operators are used at the boundary of the regularity domain. The statistics of vibration frequencies and modes are obtained using the standard perturbation method, by treating the fluctuations to be stochastic perturbations. Linear dependence of vibration and stability parameters over property value fluctuations and loading fluctuations are assumed. Bounds for the statistics of vibration frequencies are obtained. The critical load is first evaluated for the averaged problem and the corresponding eigenvalue statistics are sought. Then, the frequency equation is employed to transform the eigenvalue statistics to critical load statistics. Specialization of the general procedure to Beck, Leipholz and Pfluger columns is carried out. For Pfluger column, nonlinear transformations are avoided by directly expressing the critical load statistics in terms of input variable statistics.

Generation of an eight node brick element using Papcovitch-Neuber functions

Some conventional finite elements suffer from drawbacks, such as shear locking, membrane locking, etc. To overcome them researchers have developed various techniques, termed as tricks by some and variational crimes by others. Many attempts have been made, but satisfactory explanations for why some of these techniques work have not been obtained, especially in the case of solid elements. This paper attempts a simple non-conforming solid element using assumed displacement fields which satisfy the Navier equation exactly. Its behaviour under simple loadings like bending, torsion and tension is examined and comparisons are made with existing elements.