Assumption Of Periodicity Research Articles

A Binary Model of textile composites: III high failure strain and work of fracture in 3D weaves

Prior experiments have revealed exceptionally high values of the work of fracture (0.4– 1.1 MJm −2 ) in carbon/epoxy 3D interlock woven composites. Detailed destructive examination of specimens suggested that much of the work of fracture arose when the specimens were strained well beyond the failure of individual tows yet still carried loads ∼1 GPa . A mechanism of lockup amongst broken tows sliding across the final tensile fracture surface was suggested as the means by which high loads could still be transferred after tow failure. In this paper, the roles of weave architecture and the distribution of flaws in the mechanics of tow lockup are investigated by Monte Carlo simulations using the so-called Binary Model. The Binary Model was introduced previously as a finite element formulation specialised to the problem of simulating relatively large, three-dimensional segments of textile composites, without any assumption of periodicity or other symmetry, while preserving the architecture and topology of the tow arrangement. The simulations succeed in reproducing all qualitative aspects of measured stress–strain curves. They reveal that lockup can indeed account for high loads being sustained beyond tow failure, provided flaws in tows have certain spatial distributions. The importance of the interlock architecture in enhancing friction by holding asperities on sliding fibre tows into firm contact is highlighted.

Extension of second-order Stokes theory to variable bathymetry

In the present work second-order Stokes theory has been extended to the case of a generally shaped bottom profile connecting two half-strips of constant (but possibly different) depths, initiating a method for generalizing the Stokes hierarchy of second- and higher-order wave theory, without the assumption of spatial periodicity. In modelling the wave–bottom interaction three partial problems arise: the first order, the unsteady second order and the steady second order. The three problems are solved by using appropriate extensions of the consistent coupled-mode theory developed by the present authors for the linearized problem. Apart from the Stokes small-amplitude expansibility assumption, no additional asymptotic assumptions have been introduced. Thus, bottom slope and curvature may be arbitrary, provided that the resulting wave dynamics is Stokes-compatible. Accordingly, the present theory can be used for the study of various wave phenomena (propagation, reflection, diffraction) arising from the interaction of weakly nonlinear waves with a general bottom topography, in intermediate water depth. An interesting phenomenon, that is also very naturally resolved, is the net mass flux induced by the depth variation, which is consistently calculated by means of the steady second-order potential. The present method has been validated against experimental results and fully nonlinear numerical solutions. It has been found that it correctly predicts the second-order harmonic generation, the amplitude nonlinearity, and the amplitude variation due to non-resonant first-and-second harmonic interaction, up to the point where the energy transfer to the third and higher harmonics can no longer be neglected. Under the restriction of weak nonlinearity, the present model can be extended to treat obliquely incident waves and the resulting second-order refraction patterns, and to study bichromatic and/or bidirectional wave–wave interactions, with application to the transformation of second-order random seas in variable bathymetry regions.

Numerical Homogenization of the Rigidity Tensor in Hooke's Law Using the Node-Based Finite Element Method

Combining a geological model with a geomechanical model, it generally turns out that the geomechanical model is built from units that are at least a 100 times larger in volume than the units of the geological model. To counter this mismatch in scales, the geological data model's heterogeneous fine-scale Young's moduli and Poisson's ratios have to be “upscaled” to one “equivalent homogeneous” coarse-scale rigidity. This coarse-scale rigidity relates the volume-averaged displacement, strain, stress, and energy to each other, in such a way that the equilibrium equation, Hooke's law, and the energy equation preserve their fine-scale form on the coarse scale. Under the simplifying assumption of spatial periodicity of the heterogeneous fine-scale rigidity, homogenization theory can be applied. However, even then the spatial variability is generally so complex that exact solutions cannot be found. Therefore, numerical approximation methods have to be applied. Here the node-based finite element method for the displacement as primary variable has been used. Three numerical examples showing the upper bound character of this finite element method are presented.

Predicting the Duration of Sequential Survival Studies

Interim analyses are a common feature of clinical trial design, especially for large trials in high mortality conditions such as cancer or cardiovascular disease in which the primary endpoint is often the survival time from randomization to death. A plan for a series of interim analyses in which the criteria for stopping are specified in advance is known as a sequential design, and can be constructed to prevent patients from being randomized to an evidently inferior treatment and avoid continuation of a trial that is obviously futile.In this paper, methods for predicting the final sample size and total duration of a sequential survival study are described, and the play-off between speed of recruitment and length of follow-up is examined. The use of interim analyses to review the event rate, recruitment period, and model assumptions is discussed and software for the implementation of the methods is described. The approach is illustrated in the context of a trial seeking to establish noninferiority.

Numerical Study of Steady Forced Convection in a Grooved Annulus Using a Design of Experiments

Numerical simulations have been performed to study the flow and heat transfer characteristics in the annular space between an inner smooth-cylinder rotating at constant angular velocity and an outer stationary grooved one. A first series of calculations were conducted to determine the ranges of Taylor number and longitudinal aspect ratio for which a two-dimensional modeling can be considered as valid. To this end, a map showing the domain of validity of two dimensional computations is presented. The assumption of circumferential periodicity of the flow is then justified in the range of Taylor numbers investigated. A number of simulations were carried out to investigate the effects of the geometrical parameters describing the cross section of the grooved annulus for various Taylor numbers. To end, correlations describing the influences on the main parameters both on the friction factor and Nusselt numbers are derived by using a design of experiments.

Numerical homogenization of the absolute permeability using the conformal-nodal and mixed-hybrid finite element method

The modeling of hydrocarbon reservoirs and of aquifer-aquitard systems can be separated into two activities: geological modeling and fluid flow modeling. The geological model focuses on the geometry and the dimensions of the subsurface layers and faults, and on its rock types. The fluid flow model focuses on quantities like pressure, flux and dissipation, which are related to each other by rock parameters like permeability, storage coefficient, porosity and capillary pressure. The absolute permeability, which is the relevant parameter for steady single-phase flow of a fluid with constant viscosity and density, is studied here. When trying to match the geological model with the fluid flow model, it generally turns out that the spatial scale of the fluid flow model is built from units that are at least a hundred times larger in volume than the units of the geological model. To counter this mismatch in scales, the fine-scale permeabilities of the geological data model have to be 'upscaled' to coarse-scale permeabilities that relate the spatially averaged pressure, flux and dissipation to each other. The upscaled permeabilities may be considered as 'complicated averages,' which are derived from the spatially averaged flow quantities in such a way that the continuity equation, Darcy's law and the dissipation equation remain valid on the coarse scale. In this paper the theory of upscaling will be presented from a physical point of view aiming at understanding, rather than mathematical rigorousness. Under the simplifying assumption of spatial periodicity of the fine-scale permeability distributions, homogenization theory can be applied. However, even then the spatial distribution of the permeability is generally so intricate that exact solutions of the homogenized permeability cannot be found. Therefore, numerical approximation methods have to be applied. To be able to estimate the approximation error, two numerical methods have been developed: one based on the conventional nodal finite element method (CN-FEM) and the other based on the mixed-hybrid finite element method (MH-FEM). CN-FEM gives an upper bound for the sum of the diagonal components of the homogenized mobility matrix, while MH-FEM gives a lower bound. Three numerical examples are presented.

Towards a micromechanics based inelastic and damage modeling of composites

Micromechanics based models are considered for application to viscoplasticity and damage in metal matrix composites. The method proposes a continuation and development of Dvorak's transformation field analysis, considering piecewise uniform eigenstrains in each material phase. Standard applications of the method for a two-phase model (only one sub-volume per phase) deliver too stiff responses. Two modifications are proposed that improve considerably the overall stress–strain response, at least for materials sustaining a linear hardening under large strains. Introduction of damage effects in the TFA method is considered, within the CDM framework, using two modeling approaches: the “Direct Simplified Model”, using only two damaged phases, and the “Generalized Eigenstrain Model”, with more degrees of freedom, that allows the description of interfacial damage. The different aspects of the proposed methods are systematically checked by comparing with finite element unit cell analyses, made through periodic homogenization assumptions, for a SiC/Ti unidirectional MMC.

On Rayleigh-Taylor stability

We study the nonlinear stability of the rest state for the Rayleigh-Taylor problem related to a plane layer of viscous, heavy fluid under periodicity assumptions in the horizotal directions, with nonzero surface tension. Such a problem is studied for both incompressible and isothermal fluids where, in the class of regular flows, it is proved the exponential decay to the rest state for perturbations, in absence of hypotheses on the characteristic coefficients of the flow, and on the initial conditions as well.

Multicomponent AM-FM demodulation via periodicity-based algebraic separation and energy-based demodulation

Previously investigated multicomponent AM-FM demodulation techniques either assume that the individual component signals are spectrally isolated from each other or that the components can be isolated by linear time-invariant filtering techniques and, consequently, break down in the case where the components overlap spectrally or when one of the components is stronger than the other. In this paper, we present a nonlinear algorithm for the separation and demodulation of discrete-time multicomponent AM-FM signals. Our approach divides the demodulation problem into two independent tasks: algebraic separation of the components based on periodicity assumptions and then monocomponent demodulation of each component by instantaneously tracking and separating its source energy into its amplitude and frequency parts. The proposed new algorithm avoids the shortcomings of previous approaches and works well for extremely small spectral separations of the components and for a wide range of relative amplitude/power ratios. We present its theoretical analysis and experimental results and outline its application to demodulation of cochannel FM voice signals.

Experimental verification of response of embedded optical fiber Bragg grating sensors in non-homogeneous strain fields

One of the advantages of optical fiber sensors is their ease of embedment within a structure for non-destructive strain monitoring. In particular, Bragg grating sensors are written directly into an optical fiber hence remaining unobtrusive. In addition, several gratings can be written in series along a single fiber, permitting sensing at discrete points throughout the strain field. However, in regions of strong strain gradients, measuring the strain at discrete points may not be sufficient. One solution is to write a Bragg grating longer than the strain region of interest and use the change in its spectral response to determine the applied strain field as a function of position along the fiber. This paper presents an experimental verification of the response of an embedded optical fiber Bragg grating (OFBG) to applied non-homogeneous strain fields. Optical fiber Bragg grating sensors were embedded in four epoxy specimens of different forms so as to apply known strain functions along the gauge length when the specimen is under uniaxial tension. The complete spectral response of the Bragg gratings was then measured as a function of increasing load. The results are compared with analytical calculations, based on the piecewise-uniform period assumption for chirped gratings. Finally, the use of these spectra is discussed as possible basis functions for the resolution of an arbitrary applied strain distribution.

The geometry and symmetries of magnetohydrodynamic turbulence: Anomalies of spatial periodicity

It has become common to formulate theories and computations of magnetohydrodynamic turbulent effects in rectangular periodic boundary conditions, proceeding by analogy with what is seen as a useful framework for Navier–Stokes fluid turbulence. It is shown here that because of certain features of Maxwell’s equations for electrodynamics, it is inconsistent to invoke three-dimensional, rectangular, periodic boundary conditions and symmetry at the same time that the displacement current is neglected. The difficulty does not arise in the two-dimensional case. In three dimensions, the difficulty can be remedied by a reformulation in cylindrical geometry, imposing symmetry in the azimuthal and axial directions, but not in the radial one; a geometry that is closer to laboratory possibilities than the wholly three-dimensional periodic assumption. The reformulation seems particularly necessary in cases with a net flux of magnetic field and/or electric currents through the system. These cases no longer seem discontinuous from those without net magnetic fluxes or currents. The price paid is a loss of some possibilities for dimensional analysis and identification of similarity variables.

Finite deformation plasticity for composite structures: Computational models and adaptive strategies

We develop computational models and adaptive modeling strategies for obtaining an approximate solution to a boundary value problem describing the finite deformation plasticity of heterogeneous structures. A nearly optimal mathematical model consists of an averaging scheme based on approximating eigenstrains and elastic concentration factors in each micro phase by a constant in the portion of the macro-domain where modeling errors are small, whereas elsewhere, a more detailed mathematical model based on a piecewise constant approximation of eigenstrains and elastic concentration factors is utilized. The methodology is developed within the framework of ‘statistically homogeneous’ composite material and local periodicity assumptions.

Optimal treatment for duodenal ulcer disease: a cost-decision analysis in Malaysian patients.

The aim of the present study was to determine the cost-efficiency of different duodenal ulcer disease treatment practices in Malaysia. Six Malaysian gastroenterologists met to discuss the direct costs related to Helicobacter pylori (HP) eradication treatment. Five treatment strategies were compared: (i) histamine H2 receptor antagonists (H2RA), acid suppression therapy for 6 weeks followed by maintenance therapy as needed; (ii) bismuth triple + proton pump inhibitor (PPI), bismuth (120 mg, q.i.d.), metronidazole (400 mg; t.i.d.), tetracycline (500 mg, q.i.d.) for 7 days and PPI, b.i.d., for 7 days; (iii) OAC, omeprazole (20 mg, b.i.d.), amoxycillin (1000 mg, b.i.d.) and clarithromycin (500 mg, b.i.d.) for 7 days; (iv) OMC, omeprazole (20mg, b.i.d.), metronidazole (400mg, b.i.d.) and clarithromycin (500 mg, b.i.d.) for 7 days; and (v) OAM, omeprazole (20 mg, b.i.d.), amoxycillin (1000 mg, b.i.d.) and metronidazole (400 mg, b.i.d.) for 7 days. A decision tree model was created to determine which therapy would be the most cost-effective. The model considered eradication rates, resistance to anti-microbial agents, compliance and cost implications of treatment regimens, physician visits and ulcer recurrences during a 1 year time period assumption. The H2RA maintenance therapy was the most expensive treatment at Malaysian Ringgit (MR) 2335, followed by bismuth triple therapy (MR 1839), OMC (MR 1786), OAM (MR 1775) and OAC, being the most cost-effective therapy, at MR 1679. In conclusion, HP eradication therapy is superior to H2RA maintenance therapy in the treatment of duodenal ulcer disease. Of the HP eradication regimens, OAC is the most cost-effective.

A review of homogenization and topology opimization II—analytical and numerical solution of homogenization equations

This is the second part of a three-paper review of homogenization and topology optimization. In the first paper, we focused on the theory and derivation of the homogenization equations. In this paper, motives for using the homogenization theory for topological structural optimization are briefly explained. Different material models are described and the analytical solution of the homogenization equations for the so called “rank laminate composites” is presented. The finite element formulation is explained for the material model, based on a miscrostructure consisting of an isotropic material with rectangular voids. Using the periodicity assumption, the boundary conditions are derived and the homogenization equations are solved, and the results to be used in topology optimization are presented. The third paper deals with the use of homogenization for structural topology optimization by using optimality criteria methods.

Exact Solutions for Transient Wave Propagation in a Two-Dimensional Array of Particles

We derive exact analytical solutions for transient waves propagating in an array of particles occupying the half-plane. x ≥0, forced by general external excitations applied at the top horizontal row x = 0. We consider only nearest neighbour interactions between the particles, resulting in four-particle interactions throughout the array. First, we assume spatially periodic forces and internal displacements, and apply Fourier and Z-transforms to obtain closed form transformed solutions. Next, we relax the spatial periodicity assumption by resorting to the following limiting analysis: We let the wavelength of the force distribution tend to infinity, and employ continuity arguments to obtain the exact transformed wave responses. These transformed solutions are Fourier-inverted in closed form by employing previous results of WANG and LEE [10]. We perform numerical simulations using the exact analytical formulas, and compute transient waves in the array due to a single transient force of trapezoidal form applied at the top row. Assuming spatially harmonic force we compute propagation and attenuation zones of the array. Finally, we consider a second limiting process by allowing the horizontal distances between particles to tend to zero. Performing rescalings and limiting calculations we derive an additional closed form solution for transient waves propagating in a half-plane continuum of borizontal strings transversely coupled by distributed elasticities, due to forces applied to the top string.

Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling

An accurate homogenization method that accounts for large deformations and viscoelastic material behavior on microscopic and macroscopic levels is presented. This method is based on the classical homogenization theory, assuming local spatial periodicity of the microstructure. Consequently, the microstructure is identified by a representative volume element (RVE) with conformity of opposite boundaries at any stage of the deformation process. The local macroscopic stress is obtained by applying the local macroscopic deformation (represented by the deformation tensor) on a unique RVE through imposing appropriate boundary conditions and averaging the resulting RVE stress field. If the assumption of local periodicity of the morphology is valid, this homogenization procedure supplies a consistent objective relationship between the local macroscopic deformation and the microstructural deformation. The homogenization method was implemented in a multi-level finite element program with meshes on macroscopic level (mesh of entire structure) and microscopic level (meshes of RVEs). The performance was successfully verified by the comparison of the deformation of a perforated macroscopic sheet to the response of a homogenized sheet.

Economic analysis of earthquake retrofit options: an application to welded steel moment frames

Decisions on whether to modify the seismic performance of an existing building may depend on the comparative economic consequences of the approaches under consideration. The measures of economic consequences assessed include benefit–cost ratio, probable maximum loss, and insurance payments with different deductibles. These measures are applied to a three building complex of welded steel moment frames. Fourteen different repair schemes for the damaged buildings were evaluated for a typical Los Angeles site. The repair and retrofit approaches include: repair to former condition; modify the connections; modify the structural system with braces or shear walls; and, add dampers. The evaluation indicates that for the specific buildings some repair and modification approaches are cost–beneficial, depending on the time period and interest rate assumptions. The preferred options were then examined for PML and insurance indicators of economic impact. Many financial institutions require the probable maximum loss to be less than a threshold number and/or that insurance be purchased to be considered for financing. A PML analysis indicates that only some of the options should be considered further. The best of the near cost–beneficial options reduced the PML by a factor of two. If insurance was bought for a 5-year term, then repairing the buildings to their former condition yields a 19·5% probability of an insurance loss (damage less the amount deductible) being paid in the 5-year period, with a median payment of 6%, and a 10% probability of a payment more than 17%. Modifying the structural system in the most cost-effective manner yields a 5·2% probability of an insurance loss being paid in the 5-year period, with a median payment of 5%, and a 10% probability of a payment more than 15%. The analysis indicates that the key question in making such assessments is whether earthquake insurance is purchased or not. If insurance costs are included in the economic analysis, then more of the approaches to repair are cost effective than if not. © 1998 John Wiley & Sons, Ltd.

A Homogenization Method for Nonlinear Materials undergoing Large Deformation. 1st Report, Mathematical Formulations, Which Can Rigorously Satisfy the Assumption of Periodicity.

The homogenization method has been known as a powerful tool to investigate the macroscopic material properties of composite materials, whose microstructures are periodic. This methodology can also be considered to be an advanced version of unit cell approach. However, the assumption of periodic structure becomes somewhat suspicious after the material has undergone a large deformation, because each unit cell also deforms finitely. In the context of homogenization method presented by Guedes and Kikuchi, the periodicity condition has to be satisfied rigorously. In this paper, a formulation which can rigorously satisfy the condition of periodicity even after the microstructure has undergone a finite deformation, is presented. It is accomplished by referring the initial undeformed configuration of microstructure, when we define the periodicity condition. Some simple numerical results of two-dimensional finite strain elasto-plastic problem are also presented and discussed. The outcome of the present research may find some application in analyzing Metal Matrix Composite Materials.

Constructing wall laws with Domain Decomposition or asymptotic expansion techniques

This paper is concerned with the numerical simulation of flows of viscous fluids over rough surfaces. The proposed methodology introduces consistant equivalent wall laws using Domain Decomposition techniques with nonmatching grids and local periodicity assumptions. The paper describes this approach with its theoretical background, studies its limitations and its potential applicability and compares its results to the boundary conditions that are obtained by alternative asymptotic strategies.

High-definition temporal/frequency analysis of sampled data sets

The FFT is a successful steady-state analysis tool, but falls short when temporal analysis is needed. A class of filters designed around a Gaussian windowed DFT is described that, when applied to signal data sets in a way similar to that of a wavelet analysis, yields high-definition, fractional octave bandwidth, temporal analysis. The filter set’s desired properties are temporal resolution that is inversely proportional to bandwidth and directly proportional to frequency, no assumption of signal periodicity, and the trade-off of accuracy against temporal window width. The main undesirable property is the computation needed to perform an analysis. Examples are chosen to show the need of a good display technique. They include analysis of a mathematical impulse, a hearing aid output as driven by a transient tone burst, and a room reflection with a broadband chirp signal. The author wishes to acknowledge Mike Day and Ilker Gungor of Frye Electronics for help in basic concepts and software programming.