Discontinuous Fields Research Articles

Small-scale instabilities in dynamical systems with sliding

We demonstrate with a minimal example that in Filippov systems (dynamical systems governed by discontinuous but piecewise smooth vector fields) stable periodic motion with sliding is not robust with respect to stable singular perturbations. We consider a simple dynamical system that we assume to be a quasi-static approximation of a higher-dimensional system containing a fast stable subsystem. We tune a system parameter such that a stable periodic orbit of the simple system touches the discontinuity surface: this is the so-called grazing-sliding bifurcation. The periodic orbit remains stable, and its local return map becomes piecewise linear. However, when we take into account the fast dynamics the local return map of the periodic orbit changes qualitatively, giving rise to, for example, period-adding cascades or small-scale chaos.

Representations of Odor in the Piriform Cortex

Olfactory perception is initiated by the recognition of odorants by a large repertoire of receptors in the sensory epithelium. A dispersed pattern of neural activity in the nose is converted into a segregated map in the olfactory bulb. How is this representation transformed at the next processing center for olfactory information, the piriform cortex? Optical imaging of odorant responses in the cortex reveals that the piriform discards spatial segregation as well as chemotopy and returns to a highly distributed organization in which different odorants activate unique but dispersed ensembles of cortical neurons. Neurons in piriform cortex, responsive to a given odorant, are not only distributed without apparent spatial preference but exhibit discontinuous receptive fields. This representation suggests organizational principles that differ from those in neocortical sensory areas where cells responsive to similar stimulus features are clustered and response properties vary smoothly across the cortex.

Modeling cohesive crack growth via a truly-mixed formulation

An alternative approach for cohesive crack growth in elastic media is proposed. Standard methods move from displacement-based formulations that are enriched to handle discontinuities in the inherently continuous displacement field. The herein adopted formulation is conversely based on a truly-mixed discretization that has stresses as main regular variables, while discontinuous displacements play the role of Lagrangian multipliers. The approach directly handles the analysis of propagating cohesive cracks in elastic media through the appropriate inclusion of interface energy terms that enrich the formulation when a crack is growing. Notably, no edge element is introduced but simply the inherent discontinuity of the displacement field is taken advantage of. Furthermore, the continuity of traction vectors is imposed in an exact fashion within the formulation and not as an additional weak constraint, as classically done. The work has the main aim of investigating the features of the approach through numerical simulations that refer to well-known experimental results on concrete specimens. The capability of modeling size effect is firstly tested in case of a pure mode I growth. The accuracy of truly-mixed stress interpolation is also exploited to recover crack path and to handle energy dissipation in mixed mode simulations.

Limit analysis decomposition and finite element mixed method

This paper proposes an original decomposition approach to the upper bound method of limit analysis. It is based on a mixed finite element approach and on a convex interior point solver, using linear or quadratic discontinuous velocity fields. Presented in plane strain, this method appears to be rapidly convergent, as verified in the Tresca compressed bar problem in the linear velocity case. Then, using discontinuous quadratic velocity fields, the method is applied to the celebrated problem of the stability factor of a Tresca vertical slope: the upper bound is lowered to 3.7776–value to be compared to the best published lower bound 3.7752–by succeeding in solving a nonlinear optimization problem with millions of variables and constraints.

Automated Modelling of Evolving Discontinuities

The automated approximation of solutions to differential equations which involve discontinuities across evolving surfaces is addressed. Finite element technology has developed to the point where it is now possible to model evolving discontinuities independently of the underlying mesh, which is particularly useful in simulating failure of solids. However, the approach remains tedious to program, particularly in the case of coupled problems where a variety of finite element bases are employed and where a mixture of continuous and discontinuous fields may be used. We tackle this point by exploring the scope for employing automated code generation techniques for modelling discontinuities. Function spaces and variational forms are defined in a language that resembles mathematical notation, and computer code for modelling discontinuities is automatically generated. Principles underlying the approach are elucidated and a number of two- and three-dimensional examples for different equations are presented.

Analysis of phase-lag effects on wave propagation in a thick plate under axisymmetric temperature distribution

The main objective of the present paper is to analyze the effects of phase lags on wave propagation in a two dimensional thick plate due to an axisymmetric temperature distribution. The problem is formulated by employing three recent theories of thermoelasticity in a unified way, namely thermoelasticity of type III (Green and Naghdi in J Therm Stress 15:253–264, 1992), thermoelasticity with dual phase lags (Tzou in ASME J Heat Transf 117:8–16, 1995) and thermoelasticity with three phase lags (Roychoudhuri in J Therm Stress 30:231–238, 2007). The lower and upper surfaces of the plate are assumed to be traction free and subjected to a given axisymmetric temperature distribution. A potential function approach together with the Laplace and Hankel transform method is employed to derive the solution in the transform domain. The Hankel inversion is performed analytically and the solution in the Laplace transform domain is obtained. Detailed analyses on wave propagation and discontinuities of different fields of the medium are presented by using a method due to Boley (Quart Appl Math 19:273–284, 1962). By employing a numerical method for the Laplace inversion the distributions of different fields like temperature, displacement and stresses in the middle plane of the plate have been computed and depicted graphically. Findings are analyzed along with the comparison with the corresponding results obtained in earlier works.

Monotone Piecewise Affine Systems

Piecewise affine (PWA) systems are autonomous systems with discontinuous vector fields which are affine ordinary differential equations at the points of continuity. These systems have applications to many fields of engineering, including systems biology and traffic engineering. We define what it means for a PWA system to be monotone, and we provide a set of sufficient conditions for monotonicity of PWA systems with hyperrectangular invariants. Such sufficient conditions are useful for understanding the dynamics of such PWA systems and for designing controllers for qualitative, reference tracking. We apply these results towards the drug-discovery problem for the cancer-related p53 pathway.

Turbulent motions and shocks waves in galaxy clusters simulated with adaptive mesh refinement

We have implemented an Adaptive Mesh Refinement criterion explicitly designed to increase spatial resolution around discontinuities in the velocity field in ENZO cosmological simulations. With this technique, shocks and turbulent eddies developed during the hierarchical assembly of galaxy clusters are followed with unprecedented spatial resolution, even at large distances from the clusters center. By measuring the spectral properties of the gas velocity field, its time evolution and the properties of shocks for a reference galaxy cluster, we investigate the connection between accretion processes and the onset of chaotic motions in the simulated Inter Galactic Medium over a wide range of scales

Approximate analysis of two-dimensional photonic crystals with rectangular geometry II H-polarization

An approximate analytical approach for calculating the dispersion relations of two-dimensional photonic crystals, which was earlier developed for E-polarization, is extended for H-polarization (which is usually problematic for analytical treatment, because of field discontinuities). Useful analytical formulas, for calculating the dispersion relations and the magnetic fields, are developed. We show that the presented approach and the derived expressions provide a good approximation for a wide range of photonic crystal parameters. The results are also compared with accurate numerical calculations to check the validity of the approximations. This approach provides not only a fast way for photonic crystal calculations, but it also can be useful for the investigation of various physical effects as well as for the design and analysis of new photonic crystal devices.

Stereo matching using colour consistency and hierarchical stochastic diffusion

A stereo matching algorithm is proposed using colour consistency and hierarchical stochastic diffusion. The colour consistency is introduced to the prior Markov random field (MRF) model to consider both the smoothness and the discontinuity of the disparity field. The energy function derived from the MRF model is minimised in the hierarchical stochastic diffusion, which first propagates the probability distribution of the reliable disparity field into the neighbourhood. Experimental results show good disparity maps compared with other propagation based methods.

Estimation of sheet-rolling parameters on the basis of discontinuous and continuous velocity fields

Estimation of sheet-rolling parameters on the basis of discontinuous and continuous velocity fields

Short-period variability in the galactic cosmic ray intensity: High statistical resolution observations and interpretation around the time of a Forbush decrease in August 2006

On 20 August 2006 a Forbush decrease observed at Polar in the Earth's magnetosphere was also seen at the INTEGRAL spacecraft outside the magnetosphere during a very active time in the solar wind. High-resolution energetic particle data from ACE SIS, the Polar high-sensitivity telescope, and INTEGRAL's Ge detector saturation rate, which measures the galactic cosmic ray (GCR) background with a threshold of similar to 200 MeV, show similar, short-period GCR variations in and around the Forbush decrease. Focusing upon the GCR intensity within a 3-day interval from 19 August 2006 to 21 August 2006 reveals many intensity variations in the GCR on a variety of time scales and amplitudes. These intensity variations are greater than the 3 sigma error in all the data sets used. The fine structures in the GCR intensities along with the Forbush decrease are propagated outward from ACE to the Earth with very little change. The solar wind speed stays relatively constant during these periods, indicating that parcels of solar wind are transporting the GCR population outward in the heliosphere. This solar wind convection of GCR fine structure is observed for both increases and decreases in GCR intensity, and the fine structure increases and decreases are bracketed by solar wind magnetic field discontinuities associated with interplanetary coronal mass ejection (ICME) magnetosheath regions, clearly seen as discontinuous rotations of the field components at ACE and at Wind. Interestingly, the electron heat flux shows different flux tube connectivity also associated with the different regions of the ICME and magnetosheath. Gosling et al. (2004) first discussed the idea that solar energetic particle intensities commonly undergo dispersionless modulation in direct association with discontinuous changes in the solar wind electron strahl. The observations show that the intensity levels in the GCR flux may undergo a similar partitioning, possibly because of the different magnetic field regions having differing magnetic topologies.

Fracture behaviour of brittle fibre-reinforced solids by a new FE formulation

Fibre-reinforced materials are nowadays used in many different fields of applications due to their good load bearing capacity under general stress state and to their fracture and fatigue resistance. The use of reinforcing fibres, especially in brittle or quasi-brittle materials, generally allows a reduction of the manufacturing costs and to get more elevated performance with respect to traditional materials. In the present paper, a continuum finite element (FE) formulation to analyse fracture mechanics problems for fibre-reinforced and in plain brittle or quasi-brittle materials involving discontinuity of the displacement field is developed. By employing an energy approach, an appropriate stress field correction is introduced to simulate the displacement discontinuity (crack) and the mechanical effects of the fibres on the matrix material, and a new FE formulation is used to simulate the behaviour of such a class of materials. Finally, some simple 2D examples are presented in order to evaluate the capability of the proposed computational approach.

Finite-Difference Modeling of Dielectric Waveguides With Corners and Slanted Facets

With the help of an improved finite-difference (FD) formulation, we investigate the field behaviors near the corners of simple dielectric waveguides and the propagation characteristics of a slant-faceted polarization converter. The formulation is full-vectorial, and it takes into consideration discontinuities of fields and their derivatives across the abrupt interfaces. Hence, the limitations in conventional FD formulation are alleviated. In the first investigation, each corner is replaced with a tiny arc rather than a really sharp wedge, and nonuniform grids are adopted. Singularity-like behavior of the electric fields emerge as the arc becomes smaller without specific treatment such as quasi-static approximation. Convergent results are obtained in the numerical analysis as compared with results from the finite-element method. In the second investigation, field behaviors across the slanted facet are incorporated in the formulation, and hence the staircase approximation in conventional FD formulation is removed to get better modeling of the full-vectorial properties.

Modelling fingerprint ridge orientation using Legendre polynomials

The estimation of fingerprint ridge orientation is an essential step in every automatic fingerprint verification system. The importance of ridge orientation can be deflected from the fact that it is inevitably used for detecting, describing and matching fingerprint features such as minutiae and singular points. In this paper we propose a novel method for fingerprint ridge orientation modelling using Legendre polynomials. One of the main problems it addresses is smoothing orientation data while preserving details in high curvature areas, especially singular points. We show that singular points, which result in a discontinuous orientation field, can be modelled by the zero-poles of Legendre polynomials. The models parameters are obtained in a two staged optimization procedure. Another advantage of the proposed method is a very compact representation of the orientation field, using only 56 coefficients. We have carried out extensive experiments using a state-of-the-art fingerprint matcher and a singular point detector. Moreover, we compared the proposed method with other state-of-the-art fingerprint orientation estimation algorithms. We can report significant improvements in both singular point detection and matching rates.

Homogenization of delamination growth in an ACA flip-chip joint based on micropolar theory

Abstract In the present paper the delamination mechanism of a typical internal structure of the anisotropic conductive adhesive (ACA) interconnect for electronic packaging is modeled on the basis of micropolar theory and computational homogenization. The interface is treated as a finite Representative Volume Element (RVE), across which the macroscopic deformation is expressed in terms of regularized strong displacement and rotational discontinuities. For the microstructure of the RVE, the micro-macro kinematical coupling is considered as a Taylor series expansion in the regularized macroscopic discontinuities, and, connected to that, a discontinuous fluctuation field representing the microstructural variation is included to describe delamination on the microlevel. As to the microlevel delamination modeling, on the basis of the discontinuous fluctuation field, a damage coupled to slip and dilation formulation is used to model the interface degradation. The constitutive relations are established in a thermodynamic setting, where the interfacial free energy involves internal variables of damage and plastic deformation. The parameters of the interface are calibrated so that a predefined amount of fracture energy is dissipated in mode I. In the numerical example, the response of a planar interface is considered when it is subjected to the basic modes I-II and also the non-conventional rotational discontinuity mode. Case studies on fracture and geometry parameters have also been carried out. Finally, an uncoupled thermomechanical analysis of a microsystem involving a representative ACA microstructure has been made for the understanding of the microscopic delamination during a thermal cycling procedure.

Computer analysis of the configurations of magnetic fields of subsurface discontinuities of finite dimensions and arbitrary shapes in tested objects of finite length by the method of spatial integral equations

Using software developed by the authors and based on the method of spatial integral equations, the spatial configurations of the magnetic fields of tested objects of finite length that contain subsurface discontinuity flaws of finite dimensions and arbitrary shapes are studied. It is possible to take into account the influence of both uniform and nonuniform magnetizing fields, including real sources; an arbitrary geometry of tested objects; and a nonlinear character of the magnetic properties of materials. It is shown that software can be used in studies of the influence of the geometrical parameters of a defect on the topography of the informative magnetic field in the testing zone.

High-frequency Dispersion Characteristics of Smart Delaminated Composite Beams

Many promising techniques for structural health monitoring rely on the usage of in situ piezoelectric actuators and sensors, which provide the ability of self-excitation and monitoring of the damage effect on the structural vibration and wave propagation. This article presents layerwise mechanics and a finite element capable of describing the response of delaminated composite beams with piezoelectric actuators and sensors. The layerwise beam theory approximates the through-thickness, in-plane displacements, and electrical field as a continuous assembly of piecewise linear layerwise fields through the thickness. This theory further describes the field discontinuities across the delamination as additional degrees of freedom. The introduction of a variable transverse displacement field provides the capability to capture important symmetric and antisymmetric high-frequency modes, guided waves, and other complex phenomena. Pseudo Wigner-Ville distributions provide the frequency dispersion characteristics of predicted and measured time responses of beams with single delaminations and an active piezoelectric sensor pair. Time—frequency plots obtained from the experimental and numerical data are correlated and their ability to reveal the presence and size of delamination is investigated.

Two approaches to the optimal design of composite flywheels

In this article, two approaches to the design of reinforced composite flywheels are presented. The main goal of the optimization procedure is to maximize the accumulated kinetic energy of a flywheel. The first approach is based on a discrete model of reinforcement, causing the discontinuity of static fields along reinforcement and preserving the continuity of kinematic fields. In the second approach, the material of the reinforced flywheel is subjected to the homogenization procedure using the Halpin–Tsai assumption and the continuity of both static and kinematic fields is preserved within the flywheel domain. The evolutionary algorithm was used in both cases to determine the optimal shape of reinforcements, while the finite-element method was applied in order to analyse the mechanical response of a flywheel.

A localized mixed‐hybrid method for imposing interfacial constraints in the extended finite element method (XFEM)

AbstractThe paper proposes an approach for the imposition of constraints along moving or fixed immersed interfaces in the context of the extended finite element method. An enriched approximation space enables consistent representation of strong and weak discontinuities in the solution fields along arbitrarily‐shaped material interfaces using an unfitted background mesh. The use of Lagrange multipliers or penalty methods is circumvented by a localized mixed hybrid formulation of the model equations. In a defined region in the vicinity of the interface, the original problem is re‐stated in its auxiliary formulation. The availability of the auxiliary variable enables the consideration of a variety of interface constraints in the weak form. The contribution discusses the weak imposition of Dirichlet‐ and Neumann‐type interface conditions as well as continuity requirements not fulfilled a priori by the enriched approximation. The properties of the proposed approach applied to two‐dimensional linear scalar‐ and vector‐valued elliptic problems are investigated by studying the convergence behavior. Copyright © 2009 John Wiley & Sons, Ltd.