Discontinuous Fields Research Articles

Cascaded image analysis for dynamic crack detection in material testing

Concrete probes in civil engineering material testing often show fissures or hairline-cracks. These cracks develop dynamically. Starting at a width of a few microns, they usually cannot be detected visually or in an image of a camera imaging the whole probe. Conventional image analysis techniques will detect fissures only if they show a width in the order of one pixel. To be able to detect and measure fissures with a width of a fraction of a pixel at an early stage of their development, a cascaded image analysis approach has been developed, implemented and tested. The basic idea of the approach is to detect discontinuities in dense surface deformation vector fields. These deformation vector fields between consecutive stereo image pairs, which are generated by cross correlation or least squares matching, show a precision in the order of 1/50 pixel. Hairline-cracks can be detected and measured by applying edge detection techniques such as a Sobel operator to the results of the image matching process. Cracks will show up as linear discontinuities in the deformation vector field and can be vectorized by edge chaining. In practical tests of the method, cracks with a width of 1/20 pixel could be detected, and their width could be determined at a precision of 1/50 pixel.

Elliptical cracks arbitrarily oriented in 3D-anisotropic elastic media

An elliptical crack in an infinite anisotropic elastic medium is considered. For a polynomial external stress field, an efficient numerical algorithm of the solution of the problem is developed. The discontinuity of the displacement field on the crack surface (the crack opening vector) is presented in the form of 2D-regular integrals that can be evaluated numerically for any crack orientation with respect to the principal axes of the anisotropy of the medium. The integrand functions in these integrals are expressed via the Fourier transform of the Green function of the medium. The cases of constant and linear external stress fields are considered in detail. Simplifications for particular crack orientations in the cases of orthotropic and transversely isotropic media are indicated. The stress intensity factors are obtained in the form of regular integrals that depend only on the Fourier transform of the Green function of the medium. An equation for the tensor of effective elastic constants of an anisotropic media containing a random set of elliptical cracks is presented.

A Novel “Subset Splitting” Procedure for Digital Image Correlation on Discontinuous Displacement Fields

Digital Image Correlation (DIC) is an easy to use yet powerful approach to measure displacement and strain fields. While the method is robust and accurate for a variety of applications, standard DIC returns large error and poor correlation quality near displacement discontinuities such as cracks or shear bands. This occurs because the subsets used for correlation can only capture continuous deformations from the reference to the deformed image. As a result the regions around discontinuities are typically removed from the area of interest, before or after analysis. Here, a novel approach is proposed which enables the subset to split in two sections when a discontinuity is detected. This method enables the measurement of “displacement jumps”, and also of displacements and strains right by the discontinuity (for example a crack profile or residual strains in the wake). The method is validated on digitally created images based on mode I and mode II asymptotic displacement fields, for both sub-pixel and super-pixel crack opening displacements. Finally, an actual fracture experiment on a high density polyethylene (HDPE) specimen demonstrates the robustness of the method on actual images. Compared to other methods capable of handling discontinuities, this novel “subset-splitting” procedure offers the advantage of being a direct extension of the now popular standard DIC, and can therefore be implemented as an “upgrade” to that method.

Surface waves on MHD tangential discontinuities: case for moderate to low plasma-β values

Variation of phase speed of the magnetoacoustic surface waves on a magnetohydrodynamic (MHD), tangential discontinuity (TD) is examined against the variation of Alfvén speeds on opposite sides of the discontinuity with the acoustic speeds remaining constant. The discontinuous thermal profile of the plasma is employed to classify various surface wavemodes. The existence conditions for the surface waves manifest themselves through the appearance of critical values for various parameters restricting the propagation of such waves on the TD. The limiting situation of extremely low-β plasmas is studied to improve upon earlier results based on simplified calculations. Theoretical understanding of the basic structure of surface waves is then applied to specific observations of discontinuities in the interplanetary magnetic field (IMF) to arrive at simple estimates of phase speed of the surface waves on such discontinuities that may be tested in further observations.

Macroscopic Singularity In Morphogen Gradient And Bioelectric Field Of Growth Control

In embryogenesis, every physiological system is developed through a growth control system mediated by organizers and growth control boundaries. The growth control signal transduction is embedded in various physiological functions. Many physiological processes are regulated through growth control mechanisms such as hypertrophy, hyperplasia, atrophy, apoptosis and signal transduction pathways involving growth control genes such as proto-oncogenes. A model of growth control system suggests that a growth control system originates from a network of organizers which distribute at extreme points of structural surface (or interface) curvature. Organizers and growth control boundaries are macroscopic singularities (i.e. discontinuity and abrupt transition) of morphogen gradient field and bioelectric field. Small, nonspecific perturbations around singular points - organizers can have long lasting systemic effect. This offers an efficient way of manipulating the system. The growth control model further suggests that singular points - organizers and separatrices - boundaries in growth control form an undifferentiated, interconnected cellular network that regulates growth and physiology both during and after embryogenesis. Stem cells are important components of this undifferentiated network. Acupuncture points and meridians originate from organizers and growth control boundaries respectively. The model of growth control system has met the gold standard of science - the following predictions of the model have been independently confirmed: 1. Organizers have high electric conductance, high electric current density and high density of gap junctions. 2. Growth control boundaries have high electric conductance and high density of gap junctions. 3. Singularity has important role in morphogenesis. 4. Morphogens and organizers partially retain their regulatory function after embryogenesis. 5. Nonspecific stimulation at acupoints - potential organizers in adult causes extensive growth control effects.Singularity provides a potentially efficient way of manipulating the growth control, stem cells and physiological systems.

Study of Singularities in Nonsmooth Dynamical Systems via Singular Perturbation

In this article we describe some qualitative and geometric aspects of nonsmooth dynamical systems theory around typical singularities. We also establish an interaction between nonsmooth systems and geometric singular perturbation theory. Such systems are represented by discontinuous vector fields on R � , � ≥ 2, where their discontinuity set is a codimension one algebraic variety. By means of a regularization process proceeded by a blow-up technique we are able to bring about some results that bridge the space between discontinuous systems and singularly perturbed smooth systems. We also present an analysis of a subclass of discontinuous vector fields that present transient behavior in the 2-dimensional case, and we dedicate a section to providing sufficient conditions in order for our systems to have local asymptotic stability.

A Multivariate Regression Approach to Adjust AATSR Sea Surface Temperature to In Situ Measurements

The Advanced Along-Track Scanning Radiometer (AATSR) onboard Envisat is designed to provide very accurate measurements of sea surface temperature (SST). Using colocated in situ drifting buoys, a dynamical matchup database (MDB) is used to assess the AATSR-derived SST products more precisely. SST biases are then computed. Currently, Medspiration AATSR SST biases are discrete values and can introduce artificial discontinuities in AATSR level-2 SST fields. The new AATSR SST biases presented in this letter are continuous. They are computed, for nighttime and best proximity confidence data, by linear regression with different MDB covariables (wind speed, latitude, aerosol optical depth, etc.). As found, the difference between dual-view and nadir-only SST products explains most of the variability (26%).

FINITE VOLUME MODEL FOR BOD-DO COUPLED EQUATIONS

ABSTRACT BOD-DO interaction in a river is described by a set of coupled one-dimensional transient partial differential equations, which consider the effects of advection, dispersion, and sources and sinks. A finite volume (FV) model for the solution of the coupled equations using the well-tested QUICK scheme is developed in this paper. The proposed FV model is validated with available analytical solutions for various initial and boundary conditions. The model is for both steady state and transient conditions and inherently conservative, and also does not fail at discontinuities in the concentration field.

Internal representations of the olfactory world

Event Abstract Back to Event Internal representations of the olfactory world 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. In mice, as well as flies, projections from the olfactory bulb (antennal lobe) extend to multiple, higher order processing centers in the brain. We have identified a spatially invariant, dimorphic circuit responsive to pheromones that dictates components of sexually dimorphic courtship behavior in Drosophila. We have also asked how the representation in the bulb is transformed at the next processing center for olfactory information in the mouse, the piriform cortex. Optical imaging of odor responses in the cortex of the mouse reveals that the piriform discards spatial segregation as well as chemotopy and returns to a highly distributed organization in which different odors 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. Conference: Computational and systems neuroscience 2009, Salt Lake City, UT, United States, 26 Feb - 3 Mar, 2009. Presentation Type: Oral Presentation Topic: Oral Presentations Citation: (2009). Internal representations of the olfactory world. Front. Syst. Neurosci. Conference Abstract: Computational and systems neuroscience 2009. doi: 10.3389/conf.neuro.06.2009.03.221 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 03 Feb 2009; Published Online: 03 Feb 2009. Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Google Google Scholar PubMed Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

The Two-Fold Singularity of Discontinuous Vector Fields

When a vector field in $\mathbb{R}^3$ is discontinuous on a smooth codimension one surface, it may be simultaneously tangent to both sides of the surface at generic isolated points (singularities). For a piecewise-smooth dynamical system governed by the vector field, we show that the local dynamics depends on a single quantity—the jump in direction of the vector field through the singularity. This quantity controls a bifurcation, in which the initially repelling singularity becomes the apex of a pair of parabolic invariant surfaces. The surfaces are smooth except where they intersect the discontinuity surface, and they divide local space into regions of attraction to, and repulsion from, the singularity.

Numerical Fracture analysis of prestressed concrete beams

Fracture of prestressed concrete beams is studied with a novel and robust three-dimensional meshfree method. The meshfree method describes the crack as a set of cohesive crack segments and avoids the representation of the crack surface. It is ideally suited for a large number of cracks. The crack is modeled by splitting particles into two particles on opposite sides of the crack segment and the shape functions of neighboring particles are modified in a way the discontinuous displacement field is captured appropriately. A simple, robust and efficient way to determine, on which side adjacent particles of the corresponding crack segment lies, is proposed. We will show that the method does not show any "mesh" orientation bias and captures complicated failure patterns of experimental data well.

Finite elements with embedded branching

This paper is concerned with the numerical modeling of crack branching in brittle materials using finite elements with embedded strong discontinuities, that is, discontinuities in the displacement field defining the solution of the underlying boundary-value problem. In particular, new finite elements are developed in this framework accommodating the different branches of the bifurcating discontinuity in the element interior. The key aspect of these developments is the correct representation of the kinematics of these configurations. This is accomplished through the identification of the proper separation modes characterizing these solutions and their incorporation in the discrete strain field of the finite element. The resulting enhanced modes are activated based on a branching criterion depending on the velocity of the crack tip. The performance of the new elements is illustrated with several numerical simulations involving other approaches for the treatment of branching and comparisons with available experimental results.

Explicit piecewise smooth solutions of Landau-Lifshitz equation with discontinuous external field

In this paper, we shall construct some explicit piecewise smooth (global continuous) solutions as well as blow up solutions to the multidimensional Landau-Lifshitz equation, subject to the external magnetic fields being both discontinuous and unbounded. When the external magnetic field is continuous, some explicit exact smooth solutions and blow up solution are also constructed. We also establish some necessary and sufficient conditions to ensure that the solution of multidimensional Landau-Lifshitz equation with external magnetic field converges to the solution of equation without external magnetic field when the external magnetic field tends to zero.

A new continuum FE approach for fracture mechanics discontinuous problems

Abstract A continuum finite elements (FE) formulation involving discontinuity of the displacement field to simulate fracture mechanics problems for brittle or quasi-brittle solids is proposed. A homogeneous discontinuity is assumed in a cracked finite element, and a new simple stress-based implementation of the displacement discontinuity is introduced by an appropriate stress field correction to simulate, as usually done in an elastic–plastic classical FE formulation, the mechanical effects of the crack, i.e. the proposed formulation does not introduce any discontinuous displacement field by mean of special or modified shape functions. Both linear elastic and elastic–plastic behaviour of the non-cracked material can be considered. 2D fracture problems are solved by the proposed procedure, to predict the load–displacement behaviour as well as the crack patterns in brittle structures. The new proposed simple FE formulation for discontinuous problems is computationally economic, and maintains the internal continuity of the numerical model with well-known numerical benefits.

Simulations of Dam-Break Flows Using Free Surface Capturing Method

AbstractA model adopting the surface capturing method is developed for the simulation of dam-break flows by solving the Navier-Stokes equations of weakly compressible and variable density flows in open channels. Due to the characteristics of weakly compressible flow equations, a compressibility parameter describing the compressibility of fluid is determined to obtain the time-accurate flow fields in both liquid and gas regions simultaneously. Accordingly, the location of free surface can be captured as a discontinuity of the density field for dam-break flow simulations. The numerical algorithm in the proposed method is based on the framework of the finite volume method for discretization in space. To deal with the discontinuity property of fluid density near the free surface, the TVD-MUSCL scheme is adopted to overcome numerical oscillations and dissipation. For discretization in time, the explicit 4-stage Runge-Kutta scheme is employed in the model. Finally, several typical dam-break flow problems in open channel are simulated to demonstrate the validation and applicability of the proposed model.

A qualitative computational study of mass transfer in upward bubble train flow through square and rectangular mini-channels

In this paper we present a new method for numerical simulation of conjugate mass transfer of a dilute species with resistance in both phases and an arbitrary equilibrium distribution coefficient. The method is based on the volume-of-fluid technique and accounts for the concentration jump at the interface by transforming the discontinuous physical concentration field into a continuous numerical one. The method is validated by several test problems and is used to investigate the mass transfer in upward bubble train flow within square and rectangular channels. Computations are performed for a single flow unit cell and a channel hydraulic diameter of 2 mm. The simulations consider the transfer of a dilute species from the dispersed gas into the continuous liquid phase. Optionally, the mass transfer is accompanied by a first-order homogeneous chemical reaction in the liquid phase or a first-order heterogeneous reaction at the channel walls. The results of this numerical study are qualitative in nature. First, because periodic boundary conditions in axial direction are not only used for the velocity field but also for the concentration field and second, because the species diffusivity in the liquid phase is arbitrarily increased so that the liquid phase Schmidt number is 0.8 and the thickness of the concentration and momentum boundary layer is similar. Two different equilibrium distribution coefficients are considered, one where the mass transfer is from high to low concentration, and one where it is vice versa. The numerical study focuses on the influence of the unit cell length, liquid slug length and channel aspect ratio on mass transfer. It is found that for the exposure times investigated the liquid film between the bubble and the wall is saturated and the mass transfer occurs by the major part through the bubble front and rear so that short unit cells are more efficient for mass transfer. Similar observations are made for the homogeneous reaction and for the heterogeneous reaction when the reaction is slow. In case of a fast heterogeneous reaction and when the main resistance to mass transfer is in the gas phase, it appears that for square channels long unit cells are more efficient, while large aspect ratio rectangular channels are more efficient than square channels, suggesting that for these conditions they might be more appropriate for use in monolithic catalysts.

Limit analysis and convex programming: A decomposition approach of the kinematic mixed method

AbstractThis 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 with the best published lower bound 3.772—by succeeding in solving non‐linear optimization problems with millions of variables and constraints. Copyright © 2008 John Wiley & Sons, Ltd.

Application of local operators for numerical reconstruction of the singular support of a vector field by its known ray transforms

The paper is devoted to the localization problem of a set of vector field discontinuities, and also discontinuities of its derivatives. We propose new approaches to the reconstruction of vector field singularities by its ray transforms. Alongside modification the Vainberg operator, for solution of this problem we use certain operators of vector analysis in different combinations.

Shape optimization of piezoelectric devices using an enriched meshfree method

AbstractWe present an enriched reproducing kernel particle method for shape sensitivity analysis and shape optimization of two‐dimensional electromechanical domains. This meshfree method incorporates enrichment functions for better representation of discontinuous electromechanical fields across internal boundaries. We use cubic splines for delineating the geometry of internal/external domain boundaries; and the nodal coordinates and slopes of these splines at their control points become the design parameters. This approach enables smooth manipulations of bi‐material interfaces and external boundaries during the optimization process. It also enables the calculation of displacement and electric‐potential field sensitivities with respect to the design parameters through direct differentiation, for which we adopt the classical material derivative approach. We verify this implementation of sensitivity calculations against an exact solution to a variant of Lamé's problem, and also, finite‐difference approximations. We follow a sequential quadratic programming approach to minimize the cost function; and demonstrate the utility of the overall technique through a model problem that involves the shape optimization of a piezoelectric fan. Copyright © 2008 John Wiley & Sons, Ltd.

Low modal volume dipole-like dielectric slab resonator

We propose a novel geometry in a silicon planar resonator with an ultra-small modal volume of 0.01(lambda/2n)(3). The geometry induces strong electric field discontinuities to decrease the modal volume of the cavity below 1(lambda/2n)(3) The proposed structure and other common resonators such as 1D and 2D photonic crystal resonators are compared for tradeoffs in confinement and quality factors.