Discontinuous Fields Research Articles

A change in boundary conditions induces a discontinuity of tissue flow in chicken embryos and the formation of the cephalic fold

The morphogenesis of vertebrate body parts remains an open question. It is not clear whether the existence of different structures, such as a head, can be addressed by fundamental laws of tissue movement and deformation, or whether they are only a sequence of stop-and-go genetic instructions. I have filmed by time-lapse microscopy the formation of the presumptive head territory in chicken embryos. I show that the early lateral evagination of the eye cups and of the mesencephalic plate is a consequence of a sudden change in boundary conditions of the initial cell flow occurring in these embryos. Due to tissue flow, and collision of the two halves of the embryo, the tissue sheet movement is first dipolar, and next quadrupolar. In vivo air puff tonometry reveals a simple visco-elastic behaviour of the living material. The jump from a dipolar to a quadrupolar flow changes the topology of the early morphogenetic field which is observed towards a complex vortex winding with a trail (the eye cups and brain folds). The hydrodynamical model accounts for the discontinuity of the vector field at the moment of collision of the left and right halves of the embryo, at a quantitative level. This suggests a possible mechanism for the morphogenesis of the head of amniotes, as compared to cephalochordates and anamniotes.

Study on vortex structure near wind-driven air-water interfaces

A one-fluid numerical model for two-phase air-water flows is developed using finite volume scheme and free surface capturing method. Proper numerical dissipation introduced by advection scheme assures computational stability when dealing with the discontinuous density field and singularities in source terms. Furthermore, numerical dissipation implicitly represents the subgrid-scale effects instead of using an eddy-viscosity model in turbulence calculations. As exploratory research in this field, generation and evolution of wind-driven waves are simulated by proposed model and vortex structure in air side is studied based on the numerical results. Features of vortex structure, relations between vortex structure and Reynolds stress and characteristics of streamwise vortex transport are analyzed through comparison with channel flow.

Discretization errors and free surface stabilization in the finite difference and marker-in-cell method for applied geodynamics: A numerical study

The finite difference–marker-in-cell (FD-MIC) method is a popular method in thermomechanical modeling in geodynamics. Although no systematic study has investigated the numerical properties of the method, numerous applications have shown its robustness and flexibility for the study of large viscous deformations. The model setups used in geodynamics often involve large smooth variations of viscosity (e.g., temperature-dependent viscosity) as well large discontinuous variations in material properties (e.g., material interfaces). Establishing the numerical properties of the FD-MIC and showing that the scheme is convergent adds relevance to the applications studies that employ this method. In this study, we numerically investigate the discretization errors and order of accuracy of the velocity and pressure solution obtained from the FD-MIC scheme using two-dimensional analytic solutions. We show that, depending on which type of boundary condition is used, the FD-MIC scheme is a second-order accurate in space as long as the viscosity field is constant or smooth (i.e., continuous). With the introduction of a discontinuous viscosity field characterized by a viscosity jump (η*) within the control volume, the scheme becomes first-order accurate. We observed that the transition from second-order to first-order accuracy will occur with only a small increase in the viscosity contrast (η* ≈ 5). We have employed two methods for projecting the material properties from the Lagrangian markers onto the Eulerian nodes. The methods are based on the size of the interpolation volume (4-cell, 1-cell). The use of a more local interpolation scheme (1-cell) decreases the absolute velocity and pressure discretization errors. We also introduce a stabilization algorithm that damps the potential oscillations that may arise from quasi free surface calculations in numerical codes that employ the strong form of the Stokes equations. This correction term is of particular interest for topographic modeling, since the surface of the Earth is generally represented by a free surface. Including the stabilization enables physically meaningful solutions to be obtained from our simulations, even in cases where the time step value exceeds the isostatic relaxation time. We show that including the stabilization algorithm in our FD stencil does not affect the convergence properties of our scheme. In order to verify our approach, we performed time-dependent simulations of free surface Rayleigh-Taylor instability.

Estimation of slipping organ motion by registration with direction-dependent regularization

Accurate estimation of respiratory motion is essential for many applications in medical 4D imaging, for example for radiotherapy of thoracic and abdominal tumors. It is usually done by non-linear registration of image scans at different states of the breathing cycle but without further modeling of specific physiological motion properties. In this context, the accurate computation of respiration-driven lung motion is especially challenging because this organ is sliding along the surrounding tissue during the breathing cycle, leading to discontinuities in the motion field. Without considering this property in the registration model, common intensity-based algorithms cause incorrect estimation along the object boundaries. In this paper, we present a model for incorporating slipping motion in image registration. Extending the common diffusion registration by distinguishing between normal- and tangential-directed motion, we are able to estimate slipping motion at the organ boundaries while preventing gaps and ensuring smooth motion fields inside and outside. We further present an algorithm for a fully automatic detection of discontinuities in the motion field, which does not rely on a prior segmentation of the organ. We evaluate the approach for the estimation of lung motion based on 23 inspiration/expiration pairs of thoracic CT images. The results show a visually more plausible motion estimation. Moreover, the target registration error is quantified using manually defined landmarks and a significant improvement over the standard diffusion regularization is shown.

Slow–fast systems on algebraic varieties bordering piecewise-smooth dynamical systems

This article extends results contained in Buzzi et al. (2006) [4] , Llibre et al. (2007, 2008) [12] , [13] concerning the dynamics of non-smooth systems. In those papers a piecewise C k discontinuous vector field Z on R n is considered when the discontinuities are concentrated on a codimension one submanifold. In this paper our aim is to study the dynamics of a discontinuous system when its discontinuity set belongs to a general class of algebraic sets. In order to do this we first consider F : U → R a polynomial function defined on the open subset U ⊂ R n . The set F − 1 ( 0 ) divides U into subdomains U 1 , U 2 , … , U k , with border F − 1 ( 0 ) . These subdomains provide a Whitney stratification on U . We consider Z i : U i → R n smooth vector fields and we get Z = ( Z 1 , … , Z k ) a discontinuous vector field with discontinuities in F − 1 ( 0 ) . Our approach combines several techniques such as ε -regularization process, blowing-up method and singular perturbation theory. Recall that an approximation of a discontinuous vector field Z by a one parameter family of continuous vector fields is called an ε -regularization of Z (see Sotomayor and Teixeira, 1996 [18] ; Llibre and Teixeira, 1997 [15] ). Systems as discussed in this paper turn out to be relevant for problems in control theory (Minorsky, 1969 [16] ), in systems with hysteresis (Seidman, 2006 [17] ) and in mechanical systems with impacts (di Bernardo et al., 2008 [5] ).

Monopole and dipole layers in curved spacetimes: Formalism and examples

The discontinuities of electromagnetic test fields generated by general layers of electric and magnetic monopoles and dipoles are investigated in general curved spacetimes. The equivalence of electric currents and magnetic dipoles is discussed. The results are used to describe exact "Schwarzschild disk" solutions endowed with such sources. The resulting distributions of charge and dipole densities on the disks are corroborated using the membrane paradigm.

Beyond analog gravity: the case of exceptional dynamics

We show that it is possible to go beyond the propagation aspects usually contemplated in the analog models of gravity. Until now, the emergence of a metric appears in the description of excitations around a given background solution or in the study of field discontinuities in the geometrical optics regime. We now overcome some limitations of the above analogies and exhibit the form of a Lagrangian that describes the dynamics of a self-interacting field φ as an interaction between φ and its associated effective metric In other words, the nonlinear equation of motion of the field may be interpreted as the gravitational influence on φ by its own effective metric which, in our scheme, becomes an active partner of the dynamics of φ.

Complete Tensor Field Topology on 2D Triangulated Manifolds embedded in 3D

AbstractThis paper is concerned with the extraction of the surface topology of tensor fields on 2D triangulated manifolds embedded in 3D. In scientific visualization topology is a meaningful instrument to get a hold on the structure of a given dataset. Due to the discontinuity of tensor fields on a piecewise planar domain, standard topology extraction methods result in an incomplete topological skeleton. In particular with regard to the high computational costs of the extraction this is not satisfactory. This paper provides a method for topology extraction of tensor fields that leads to complete results. The core idea is to include the locations of discontinuity into the topological analysis. For this purpose the model of continuous transition bridges is introduced, which allows to capture the entire topology on the discontinuous field. The proposed method is applied to piecewise linear three‐dimensional tensor fields defined on the vertices of the triangulation and for piecewise constant two or three‐dimensional tensor fields given per triangle, e.g. rate of strain tensors of piecewise linear flow fields.

Phase Change Heat Transfer Simulation for Boiling Bubbles Arising from a Vapor Film by the VOSET Method

This article presents a numerical method directed towards the simulation of flows with changes of phase. The volume-of-fluid level set (VOSET) method, which is a new interface capturing method and combines the advantages of both volume-of-fluid (VOF) and level set methods, is used for interface tracking. A difficulty occurs for the problems studied here: the discontinuous velocity field due to the difference between mass-weighted velocity and volume weighted velocity caused by the phase change at the interface. In this article, some special treatment is made to overcome this difficulty. The VOSET method and the developed treatment for the difference between mass-weighted and volume-weighted velocities are adopted to simulate a one-dimensional Stefan problem, two-dimensional horizontal film boiling, and horizontal film boiling of water at near critical pressure. The predicted results in both Nusselt number and flow patterns are agreeable with experimental results available in the literature.

Limit cycles and invariant cylinders for a class of continuous and discontinuous vector field in dimention 2n

The subject of this paper concerns with the bifurcation of limit cycles and invariant cylinders from a global center of a linear differential system in dimension 2n perturbed inside a class of continuous and discontinuous piecewise linear differential systems. Our main results show that at most one limit cycle and at most one invariant cylinder can bifurcate using the expansion of the displacement function up to first order with respect to a small parameter. This upper bound is reached. For proving these results we use the averaging theory in a form where the differentiability of the system is not needed.

AES for multiscale localization modeling in granular media

This work presents a multiscale strong discontinuity approach to tackle key challenges in modeling localization behavior in granular media: accommodation of discontinuities in the kinematic fields, and direct linkage to the underlying grain-scale information. Assumed enhanced strain (AES) concepts are borrowed to enhance elements for post-localization analysis, but are reformulated within a recently-proposed hierarchical multiscale computational framework. Unlike classical AES methods, where material properties are usually constants or assumed to evolve with some arbitrary phenomenological laws, this framework provides a bridge to extract evolutions of key material parameters, such as friction and dilatancy, based on grain scale computational or experimental data. More importantly, the phenomenological softening modulus typically used in AES methods is no longer required. Numerical examples of plane strain compression tests are presented to illustrate the applicability of this method and to analyze its numerical performance.

Adaptive non‐conformal mesh refinement and extended finite element method for viscous flow inside complex moving geometries

SUMMARYTwo different techniques to analyze non‐Newtonian viscous flow in complex geometries with internal moving parts and narrow gaps are compared. The first technique is a non‐conforming mesh refinement approach based on the fictitious domain method (FDM), and the second one is the extended finite element method (XFEM). The refinement technique uses one fixed reference mesh, and to impose continuity across non‐conforming regions, constraints using Lagrangian multipliers are used. The size of elements locally in the high shear rate regions is reduced to increase accuracy. FDM is shown to have limitations; therefore, XFEM is applied to decouple the fluid from the internal moving rigid bodies. In XFEM, the discontinuous field variables are captured by using virtual degrees of freedom that serve as enrichment and by applying special integration over the intersected elements. The accuracy of the two methods is demonstrated by direct comparison with results of a boundary‐fitted mesh applied to a two‐dimensional cross section of a twin‐screw extruder. Compared with non‐conforming FDM, XFEM shows a considerable improvement in accuracy around the rigid body, especially in the narrow gap regions. Copyright © 2011 John Wiley & Sons, Ltd.

Shear band systems in plane strain extension: analytical solution and comparison with experimental results

Shear banding represents a local failure mechanism of a soil structure as a response to shear loading. In soil structures of different spatial scales systems of regularly spaced shear bands can be observed as a consequence of extensional loading. The phenomenon of single shear bands, defined as thin zones of localized deformation with a discontinuity of the strain field at its boundaries, is well understood. Inside the shear band the material undergoes inelastic strain softening accompanied by shearing and dilation, whereas the material outside the shear band unloads accompanied by elastic contraction in extension tests. Despite numerous experimental and numerical investigations, the physical mechanisms and parameters determining the spacing of parallel shear bands remained unknown. The paper in hand presents an analytical solution for the spacing of the shear bands and a comparison with a large base of experimental data gained from 1g and ng (geotechnical centrifuge) model experiments. The analytical solution is based on the assumption that the elastic energy rate in the unloaded zone between the shear bands tends to a minimum value. The spacing was calculated as the energetically preferred solution for a broad range of cohesive-frictional granular materials. The dependency of the calculated spacing on initial and boundary conditions as well as on material parameters was found to be in good agreement with the experimental results.

Asymmetries of the StokesVprofiles observed by HINODE SOT/SP in the quiet Sun

We present the first classification of SOT/SP circular polarization measurements with the aim of highlighting exhaustively the whole variety of Stokes V shapes emerging from the quiet Sun. k-means is used to classify HINODE SOT/SP Stokes V profiles observed in the quiet Sun network and internetwork (IN). We analyze a 302 x 162 square arcsec field-of-view (FOV) which can be considered a complete sample of quiet Sun measurements performed at at the disk center with 0.32 arcsec angular resolution and 0.001 polarimetric sensitivity. Such a classification allows us to divide the whole dataset in classes, with each class represented by a cluster profile, i.e., the average of the profiles in the class. The set of 35 cluster profiles derived from the analysis completely characterizes SOT/SP quiet Sun measurements. The separation between network and IN profile shapes is evident - classes in the network are not present in the IN, and vice versa. Asymmetric profiles are approximatively 93 % of the total number of profiles. Among these, approximatively 34 % of the profiles are strongly asymmetric profiles, and they can be divided in three families: blue-lobe, red-lobe, and Q-like profiles. The blue-lobe profiles tend to be associated with upflows (granules), whereas the red-lobe and Q-like ones appear in downflows (intergranular lanes). Such profiles need to be interpreted considering model atmospheres different from a uniformly magnetized Milne-Eddington (ME) atmosphere, i.e., characterized by gradients and/or discontinuities in the magnetic field and velocity along the line-of-sight (LOS).

Passive scalar structures in supersonic turbulence

We conduct a systematic numerical study of passive scalar structures in supersonic turbulent flows. We find that the degree of intermittency in the scalar structures increases only slightly as the flow changes from transonic to highly supersonic, while the velocity structures become significantly more intermittent. This difference is due to the absence of shocklike discontinuities in the scalar field. The structure functions of the scalar field are well described by the intermittency model of She and Lévêque [Phys. Rev. Lett 72, 336 (1994)], and the most intense scalar structures are found to be sheetlike at all Mach numbers.

Freehand HDR Imaging of Moving Scenes with Simultaneous Resolution Enhancement

AbstractDespite their high popularity, common high dynamic range (HDR) methods are still limited in their practical applicability: They assume that the input images are perfectly aligned, which is often violated in practise. Our paper does not only free the user from this unrealistic limitation, but even turns the missing alignment into an advantage: By exploiting the multiple exposures, we can create a super‐resolution image. The alignment step is performed by a modern energy‐based optic flow approach that takes into account the varying exposure conditions. Moreover, it produces dense displacement fields with subpixel precision. As a consequence, our approach can handle arbitrary complex motion patterns, caused by severe camera shake and moving objects. Additionally, it benefits from several advantages over existing strategies: (i) It is robust under outliers (noise, occlusions, saturation problems) and allows for sharp discontinuities in the displacement field. (ii) The alignment step neither requires camera calibration nor knowledge of the exposure times. (iii) It can be efficiently implemented on CPU and GPU architectures. After the alignment is performed, we use the obtained subpixel accurate displacement fields as input for an energy‐based, joint super‐resolution and HDR (SR‐HDR) approach. It introduces robust data terms and anisotropic smoothness terms in the SR‐HDR literature. Our experiments with challenging real world data demonstrate that these novelties are pivotal for the favourable performance of our approach.

Estimation of geophysical parameters from ambient noise correlation.

The passive fathometer correlates the ambient noise recorded by elements of an array to infer the depth of seabed reflecting layers. This includes two distinct processes: the cross-correlation of two elements separated by a finite distance and the autocorrelation of a single element. A model is presented for the correlation of ambient noise in the presence of reflecting boundaries with one sensor, a pair of sensors, and an array of sensors. The results obtained from experimental data from pressure and particle-velocity (vector) sensors are presented. The presence and location of reflecting boundaries can be inferred when the amplitude of the incident noise field varies smoothly across all directions of incidence. In the presence of discontinuities in the incident noise field, spurious peaks appear in the noise correlation which can be attenuated relative to the reflection peaks with use of adaptive array processing (minimum variance distortionless response). The reflection coefficient of the reflecting boundary affects the height of the reflection peaks in the noise correlation and in the ocean this can be used to infer seabed properties such as density and sound speed.

An efficient low-dissipative WENO filter scheme

This paper presents a simple and efficient procedure developed for tracing discontinuities in flow fields. Numerical experiments are carried out to test the new sensor coupled with the associated nonlinear WENO dissipation filter developed to suppress the numerical dissipation. The tests show that, for a problem containing shocks and vortices, the implementation of the new sensor and an optimized WENO scheme can obtain a stable and highly resolved solution. The numerical experiments demonstrated that the new filter scheme performs efficiently both in parallel and serial running for one-dimensional inviscid flow problems. Direct numerical simulation of a Mach 5 turbulent boundary layer over a flat plate was carried out to demonstrate the applicability of the scheme to the DNS practices. Copyright © 2011 John Wiley & Sons, Ltd.

2-D/3-D Switchable Display by Fresnel-Type LC Lens

We demonstrate a multi-electrode Fresnel-type 2-D/3-D switchable display with electrically controlled birefringence mode liquid crystal (LC) lenses. By applying a discontinuous nonuniform electric field, the display with this structure can form a Fresnel-type refractive-index distribution to generate 3-D images. Compared to a normal LC lens, the Fresnel-type LC lens can achieve faster switching speed for dynamic applications. It can also replace the high sag solid lens for 3-D applications with relatively small cell gap. Furthermore, a Fresnel lens with suitable refractive-index profile is proposed to reduce the crosstalk in the 3-D displays.

Influence of negatively charged plume grains and hemisphere coupling currents on the structure of Enceladus' Alfvén wings: Analytical modeling of Cassini magnetometer observations

[1] We present an analytical model of the Alfven wing system that is generated by the interaction between the plume of Enceladus and the corotating plasma in Saturn's inner magnetosphere. Our primary purpose is to explain the orientation of the magnetic field perturbations detected in Enceladus' Alfven wings by the Cassini magnetometer (MAG) instrument. Observational data from numerous close Enceladus flybys show both the Bx and By components (in Enceladus interaction coordinates: Bx, along corotation direction; By, toward or away from Saturn) in the center of the northern wing tube to possess a negative sign, whereas the opposite case of Bx and By being positive was observed within the southern wing. So far, none of the available models of Enceladus' magnetospheric interaction is able to reproduce this correlation between the directions of Bx and By. On the basis of the analytical calculations of Neubauer (1980, 1998) and Saur et al. (1999, 2007), we demonstrate that the observed orientation of the magnetic field may arise from the presence of negatively charged dust grains in the plume of Enceladus, serving as a sink for “free” magnetospheric electrons. Although the current carried by these particles does not make a noteworthy contribution to the magnetic field distortions in the interaction region, the negative charge accumulated by them needs to be accounted for in the quasi-neutrality condition of the plasma. The depletion of magnetospheric electrons within the plume is therefore far from causing only some localized perturbations of the magnetic field, but it drastically alters the nature of the interaction: we show that this process yields a reversal in the sign of the Hall conductivity, thereby giving rise to the observed field signatures. By applying a modified version of the Alfven wing model developed by Saur et al. (2007), we demonstrate that the magnetic field observations from Cassini's targeted Enceladus flybys can be understood by taking into account the influence of electron-absorbing dust grains. In contrast to what is claimed in recent literature, we therefore propose that magnetic field observations near Enceladus can be completely understood in terms of a local interaction model, i.e., that it is not necessary to consider the large-scale dynamics of the flux tubes in Saturn's magnetosphere. In addition, we provide first in situ evidence that the hemisphere coupling current system predicted by Saur et al. (2007) and the associated magnetic field discontinuities are indeed present at Enceladus. The field perturbations caused by these hemisphere coupling currents arise from the partial blockage of the Alfven wing at the nonconducting icy crust of Enceladus. This effect needs to be taken into account when interpreting Cassini MAG data from flybys that intersected the Enceladus flux tube and can only be reproduced by models that apply adequate boundary conditions to the surface of the icy moon.