Discontinuous Fields Research Articles

Transport equation and image interpolation with SBD velocity fields

We consider an extended formulation of the transport equation that remains meaningful with discontinuous velocity fields b, assuming that ( 1 , b ) is a special function of bounded deformation (SBD). We study existence, uniqueness, and continuity/stability of the presented formulation. We then apply this study to the problem of fitting to available data a space–time image subject to the optical flow constraint. Moreover, in order to carry out these studies, we refine the SBD approximation theorem of Chambolle to show the convergence of traces.

Hierarchical Partition of Unity Field Compositions (HPFC) for Optimal Design in the Presence of Cracks

In this paper, the theory of Hierarchical Partition of Unity Field Compositions (HPFC) is applied for the analysis of cracks and for the optimal design in the presence of cracks. In the HPFC procedure, the geometrical, material and behavioral fields defined over primitive domains are composed through Boolean operations similar to the Constructive Solid Geometry Procedure of CAD. Here, cracks are modeled as compositions of a continuous field with a discontinuous enrichment field. Convergence of the composed fields is ensured by developing constructions that obey the partition of unity property. Non-Uniform Rational B-Splines (NURBS) which are a common choice for modeling curves and surfaces in CAD systems are used to discretize the geometry, material, and behavioral felds (local approximations) over the primitive regions and to describe crack shapes as well as the discontinuous behavioral feld corresponding to the crack. Following several validation examples, crack propagation simulations without remeshing, and example problems aimed at determining the optimal location and shape of defense holes in the presence of fracture constraints are demonstrated.

Magnetic cloud boundary layer of 9 November 2004 and its associated space weather effects

On 9 November 2004, the WIND spacecraft detected a magnetic cloud boundary layer (MCBL) during the interval from 19:07 UT to 20:30 UT. Within the MCBL, there is intense southward magnetic field and the dynamic pressure is rather high, which makes it much geoeffective. Twenty‐three minutes later, the MCBL arrived on the magnetopause. An intense geomagnetic storm main phase was driven by the sustaining strong southward magnetic field within the MCBL. During the passage of the MCBL, a typical magnetospheric substorm was triggered. The substorm onset was synthetically identified by the aurora breakup, magnetic dipolarization, dispersionless particle injection, Pi2 pulsation, and the polar bay onset. The substorm triggering is related to the special magnetic and plasma structure within the MCBL. The MCBL accompanying adjacent sheath region formed a dynamic pressure enhancement region, which strongly compressed the magnetosphere and even pushed the magnetopause into the geosynchronous orbit so that two dayside spacecraft GOES‐10 and GOES‐12 were directly exposed in the magnetosheath for a long interval during the passage of the MCBL. In terms of Shue et al. (1998) model, the closest subsolar standoff distance even reached 5.1 RE during the passage of the MCBL. It can be inferred that the strong dynamic pressure and the strong discontinuities within the MCBL determine the intense compression effect. In addition, a very intense geomagnetically induced current (GIC) event was directly caused by the MCBL. Similar to this case, majority of MCBLs are dynamic pressure enhancement regions, and there are strong southward magnetic field and several strong discontinuities inside these regions, which can potentially drive large‐scale magnetospheric activities. In this paper, we take a case study to discuss the magnetospheric activities and the space weather effects caused by MCBLs.

Spatially mixed crops to control the stratified dispersal of airborne fungal diseases

Intraspecific crop diversification is thought to be a possible solution to the disease susceptibility of monocultured crops. We modelled the stratified dispersal of an airborne pathogen population in order to identify the spatial patterns of cultivar mixtures that could slow epidemic spread driven by dual dispersal mechanisms acting over both short and long distances. We developed a model to simulate the propagation of a fungal disease in a 2D field, including a reaction-diffusion model for short-distance disease dispersal, and a stochastic model for long-distance dispersal. The model was fitted to data for the spatio-temporal spread of faba bean rust (caused by Uromyces viciae- fabae) through a discontinuous field. The model was used to compare the effectiveness of eight different planting patterns of cultivar mixtures against a disease spread by short-distance and stratified dispersal. Our combined modelling approach provides a reasonably good fit with the observed data for the spread of faba bean rust. Similar predictive power could be expected for the management of resource-mediated invasions by other airborne fungi. If a disease spreads by short-distance dispersal, random mixtures can be used to slow the epidemic spread, since their spatial irregularity creates a natural barrier to the progression of a smooth epidemic wave. In the context of stratified dispersal, heterogeneous patterns should be used that include a minimum distance between susceptible units, which decreases the probability of infection by long-distance spore dispersal. We provide a simple framework for modelling the stratified dispersal of disease in a diversified crop. The model suggests that the spatial arrangement of components in cultivar mixtures has to accord with the dispersal characteristics of the pathogen in order to increase the efficiency of diversification strategies in agro-ecosystems and forestry. It can be applied in low input agriculture to manage pathogen invasion by intercropping and cultivar mixtures, and to design sustainable systems of land use.

YIELD DESIGN HOMOGENIZATION METHOD FOR COMPACTION OF MONOSIZED SPHERICAL POWDERS

The compaction of a package of monosized spherical solid grains by rate-independent plasticity deformation is examined in this paper through the use of both yield design homogenization method and finite element simulation. Both modes of compaction, isostatic and closed die, are considered. In this study, the arrangement of powder consists of hexagonal array of identical spherical grains touching each other in its initial state. During the compaction process the response of the powder compacts is monitored in terms of behaviors of appropriate representative unit cells subject to axisymmetrical loading conditions. The kinematic approach of the yield design homogenization method has been used to determine external estimates of macroscopic strength criteria of powders at various stages of compaction. The obtained upper bound estimates are based on consideration of discontinuous incompressible velocity fields satisfying conditions of homogeneous strain rate. The shapes and sizes of the macroscopic yield surfaces are determined at various stages of compaction and it has been found that they depend upon the loading history as well as the relative density of the compact. Finite element simulations similar to those of Ogbonna N. and Fleck N. A. [1995] "Compaction of an array of spherical particles," Acta Metall. Mater.43(2), 603–620. have also been performed in order to (i) obtain the deformation modes as well as the evolution of the deformation mechanism of the powder compact during the whole process of compaction; (ii) derive the evolution of contact sizes between adjacent grains; (iii) examine the dependence of the macroscopic yield surface upon the degree of compaction, using the "yield probing technique" Gurson, A. L. and Yuan, D. W. [1995] A Material Model for a Ceramic Powder Based on Ultrasound, TRS Bend Bar, and Axisymmetric Triaxial Compression Test Results (ASME, New York), pp. 57–68, and (iv) validate, to some extent, the results provided by the kinematic approach.

Approximation of Loop Subdivision Surfaces for Fast Rendering

This paper describes an approach to the approximation of Loop subdivision surfaces for real-time rendering. The approach consists of two phases, which separately construct the approximation geometry and the normal field of a subdivision surface. It first exploits quartic triangular Bézier patches to approximate the geometry of the subdivision surface by interpolating a grid of sampled points. To remedy the artifact of discontinuity of normal fields between adjacent patches, a continuous normal field is then reconstructed by approximating the tangent vector fields of the subdivision surfaces with quartic triangular Bézier patches. For regular triangles, the approach reproduces the associated subdivision patches, quartic three-directional box splines.

Non local damage model

ABSTRACT The present contribution aims at providing a closer insight on boundary effects in non local damage modelling. From micromechanics, we show that on a boundary interaction stress components normal to the surface should vanish. These interaction stresses are at the origin of non locality and therefore the material response of points located on the boundary should be partially local. Then, we discuss a tentative modification of the classical non local damage model aimed at accounting for this effect due to existing boundaries and also boundaries that arise from crack propagation. One-dimensional computations show that the profiles of damage are quite different compared to those obtained with the original formulation. The region in which damage is equal to 1 is small. The modified model performs better at complete failure, with a consistent description of discontinuity of the displacement field after failure.

A linear nonconforming finite element method for Maxwell’s equations in two dimensions. Part I: Frequency domain

We suggest a linear nonconforming triangular element for Maxwell’s equations and test it in the context of the vector Helmholtz equation. The element uses discontinuous normal fields and tangential fields with continuity at the midpoint of the element sides, an approximation related to the Crouzeix–Raviart element for Stokes. The element is stabilized using the jump of the tangential fields, giving us a free parameter to decide. We give dispersion relations for different stability parameters and give some numerical examples, where the results converge quadratically with the mesh size for problems with smooth boundaries. The proposed element is free from spurious solutions and, for cavity eigenvalue problems, the eigenfrequencies that correspond to well-resolved eigenmodes are reproduced with the correct multiplicity.

Particle dynamics inside shocks in Hamilton–Jacobi equations

The characteristic curves of a Hamilton-Jacobi equation can be seen as action-minimizing trajectories of fluid particles. For non-smooth 'viscosity' solutions, which give rise to discontinuous velocity fields, this description is usually pursued only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we show that, for any convex Hamiltonian, there exists a uniquely defined canonical global non-smooth coalescing flow that extends particle trajectories and determines the dynamics inside shocks. We also provide a variational description of the corresponding effective velocity field inside shocks, and discuss the relation to the 'dissipative anomaly' in the limit of vanishing viscosity.

Modeling of large deformation – Large sliding contact via the penalty X-FEM technique

Abstract In this paper, an extended finite element method is employed to simulate the presence of discontinuities caused by the contact surface. In X-FEM, the need for mesh adaption to interface is neglected and the process is accomplished by partitioning the domain with some triangular sub-elements whose Gauss points are used for integration of the elements. The modified level set technique and the Heaviside enrichment function are employed to approximate the discontinuous displacement field of elements located on the contact surface. The penalty method is used to impose the contact constraints and establish the non-penetration condition. An efficient numerical algorithm is employed to model the large deformation contact behavior based on the node-to-segment technique. Finally, numerical examples are presented to demonstrate the accuracy and capability of proposed X-FEM technique in modeling of large deformation – large sliding contact problems.

Discontinuous velocity domain splitting in limit analysis

We present a new limit analysis method, originated in Ionescu and Lachand-Robert (2005) and called discontinuous velocity domain splitting method (DVDS). DVDS is a mesh free method which focuses on the strain localization and completely neglect the bulk deformations. It considers the kinematic variational principle on a special class of virtual velocity fields to get an upper bound of the limit load. To construct this class of virtual velocity fields, the rigid-plastic body is splinted into simple connected sub-domains and on each such sub-domain a rigid motion is associated. The discontinuous collapse flow velocity field results in localized deformations only, located at the boundary of the sub-domains. In the numerical applications of the DVDS method, we introduce a numerical technique based on a level set description of the partition of the rigid-plastic body and on genetic minimization algorithms. For the anti-plane flow of a von Mises material, DVDS is exact in solving the limit analysis problem: the collapse flow is a rigid motion of a sub-domain. The associated deformation rate is localized on the smooth boundary of the moving sub-domain representing the fracture surface. In the case of in-plane deformation of pressure insensitive materials, the internal boundaries of the sub-domains are parts of circles or straight lines, tangent to the collapse velocity jumps. In this case, DVDS reduces to the block decomposition method, which was intensively used to get analytical upper bounds of the limit loads. When applied to the two notched tensile problem of a von Mises material, DVDS gives excellent results with a low computational cost. Furthermore, DVDS was applied to model collapse in pressure sensitive plastic materials. Illustrative examples for homogenous and heterogeneous Coulomb and Cam-Clay materials shows that DVDS gives excellent prediction of limit loads and on the collapse flow.

Strong discontinuities in partially saturated poroplastic solids

This paper considers the analysis and numerical simulation of strong discontinuities in partially saturated solids. The goal is to study observed localized failures in such media like shear bands and similar. The developments consider a fully coupled partially saturated elastoplastic model for the (continuum) bulk response of the solid formulated in effective stresses, identifying the necessary mathematical conditions for the appearance of strong discontinuities (that is, discontinuities in the displacement field leading to singular strains) as well as the proper treatment for the fields characterizing the flow of the different fluid phases, namely, the fluid contents of these phases and their individual pore pressures. The geometrically linear range of infinitesimal strains is considered. These developments allow the formulation of multiphase cohesive laws along the strong discontinuity, capturing in this way the coupled localized dissipation observed in the aforementioned failures. Furthermore, the paper also presents the formulation of enhanced finite elements capturing all these discontinuous solutions in general unstructured meshes. In particular, the finite elements capture the strong discontinuity through the proper enhancements of the discrete element strains, allowing for a complete local resolution of these effects. This results in a particularly efficient computational approach, easily accommodated in an existing finite element code. Different representative numerical simulations are presented illustrating the performance of the proposed formulation, as well as its use in practical applications like the modeling of the excavation of tunnels in variably saturated media.

On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types

We consider a simplified model for two-phase flows in one-dimensional heterogeneous porous media made of two different rocks.We focus on the effects induced by the discontinuity of thecapillarity field at interface. We first consider a model withcapillarity forces within the rocks, stating an existence/uniquenessresult. Then we look for the asymptotic problem for vanishingcapillarity within the rocks, remaining only on the interface. Weshow that either the solution to the asymptotic problem is theoptimal entropy solution to a scalar conservation law withdiscontinuous flux, or it admits a non-classical shock at theinterface modeling oil-trapping.

A Magma Accretion Model for the Formation of Oceanic Lithosphere: Implications for Global Heat Loss

A magma accretion model of oceanic lithosphere is proposed and its implications for understanding its thermal field examined. The new model (designated Variable Basal Accretion—VBA) assumes existence of lateral variations in magma accretion rates and temperatures at the boundary zone between the lithosphere and the asthenosphere. However, unlike the previous thermal models of the lithosphere, the ratio of advection to conduction heat transfer is considered a space dependent variable. The results of VBA model simulations reveal that the thickness of the young lithosphere increases with distance from the ridge axis, at rates faster than those predicted by Half-Space Cooling models. Another noteworthy feature of the new model is its ability to account for the main features in the thermal behavior of oceanic lithosphere. The improved fits to bathymetry have been achieved for the entire age range and without the need to invoke the ad-hoc hypothesis of large-scale hydrothermal circulation. Also, use of VBA model does not lead to artificial discontinuities in the temperature field of the lithosphere, as is the case with GDH (Global Depth Heat Flow) reference models. The results suggest that estimates of global heat loss need to be downsized by at least 25%.

A monotone nonlinear finite volume method for advection–diffusion equations on unstructured polyhedral meshes in 3D

We present a new monotone finite volume method for the advection–diffusion equation with a full anisotropic discontinuous diffusion tensor and a discontinuous advection field on 3D conformal polyhedral meshes. The proposed method is based on a nonlinear flux approximation both for diffusive and advective fluxes and guarantees solution non-negativity. The approximation of the diffusive flux uses the nonlinear two-point stencil described in [Danilov and Vassilevski, Russ. Numer. Anal. Math. Modelling 24: 207–227, 2009]. Approximation of the advective flux is based on the second-order upwind method with a specially designed minimal nonlinear correction [Lipnikov, Svyatskiy, and Vassilevski, J. Comp. Phys. 229: 4017–4032, 2010]. The second-order convergence rate and monotonicity are verified with numerical experiments.

On the analysis of heterogeneous fluids with jumps in the viscosity using a discontinuous pressure field

Heterogeneous incompressible fluid flows with jumps in the viscous properties are solved with the particle finite element method using continuous and discontinuous pressure fields. We show the importance of using discontinuous pressure fields to avoid errors in the incompressibility condition near the interface.

Electronic Structure and Geminate Pair Energetics at Organic–Organic Interfaces: The Case of Pentacene/C60 Heterojunctions

AbstractOrganic semiconductors are characterized by localized states whose energies are predominantly determined by electrostatic interactions with their immediate molecular environment. As a result, the details of the energy landscape at heterojunctions between different organic semiconductors cannot simply be deduced from those of the individual semiconductors, and they have so far remained largely unexplored. Here, microelectrostatic computations are performed to clarify the nature of the electronic structure and geminate pair energetics at the pentacene/C60 interface, as archetype for an interface between a donor molecule and a fullerene electron acceptor. The size and orientation of the molecular quadrupole moments, determined by material choice, crystal orientation, and thermodynamic growth parameters of the semiconductors, dominate the interface energetics. Not only do quadrupoles produce direct electrostatic interactions with charge carriers, but, in addition, the discontinuity of the quadrupole field at the interface induces permanent interface dipoles. That discontinuity is particularly striking for an interface with C60 molecules, which by virtue of their symmetry possess no quadrupole. Consequently, at a pentacene/C60 interface, both the vacuum‐level shift and geminate pair dissociation critically depend on the orientation of the pentacene π‐system relative to the adjacent C60.

Development of a quadratic element formulation based on the X‐FEM and NURBS surfaces

AbstractThe FE‐simulation of inhomogeneous structures such as composite materials, biological tissues or foams requires the generation of respective finite element meshes. The Extended Finite Element Method (X‐FEM) allows for an efficient modelling technique where the element boundary needs not to be aligned to the interfaces. In this paper, a new plane element based on the X‐FEM and quadratic shape functions will be presented. This element allows for the description of curved material interfaces inside the element domain. The computation of the element stiffness matrix requires a numerical integration process which accounts for the discontinuous fields. In the case of a curved interface, the triangulation of the element domain as applied for linear element formulations is inappropriate. Hence, Non‐Uniform Rational B‐Spline (NURBS) surfaces are used to evaluate the integrals numerically. Finally, the results of different examples will show the properties and the quality of this element formulation. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Generalized finite element enrichment functions for discontinuous gradient fields

AbstractA general GFEM/XFEM formulation is presented to solve two‐dimensional problems characterized by C0 continuity with gradient jumps along discrete lines, such as those found in the thermal and structural analysis of heterogeneous materials or in line load problems in homogeneous media. The new enrichment functions presented in this paper allow solving problems with multiple intersecting discontinuity lines, such as those found at triple junctions in polycrystalline materials and in actively cooled microvascular materials with complex embedded networks. We show how the introduction of enrichment functions yields accurate finite element solutions with meshes that do not conform to the geometry of the discontinuity lines. The use of the proposed enrichments in both linear and quadratic approximations is investigated, as well as their combination with interface enrichment functions available in the literature. Through a detailed convergence study, we demonstrate that quadratic approximations do not require any correction to the method to recover optimal convergence rates and that they perform better than linear approximations for the same number of degrees of freedom in the solution of these types of problems. In the linear case, the effectiveness of correction functions proposed in the literature is also investigated. Copyright © 2009 John Wiley & Sons, Ltd.

Isotachophoresis of nucleic acids in agarose gel rods

A new method of electrophoresis (isotachophoresis in agarose gel rods) in which nucleic acid molecules are not separated but, oppositely, are brought together into one band, was elaborated. Heterogeneous in size DNA and RNA polymers present in a few milliliters of a solution at so low concentration that their isolation by other methods is hardly attainable and fraught with losses are brought together into one visible narrow band when put in a discontinuous electric field. Polynucleotides migrate in dilute (0.1%) semifluid agarose gel that permits easy quantitative isolation of the band of interest. Resulting DNA can be used directly in PCR. The suggested method for isolation of micro amounts of nucleic acids from dilute solutions can be applied to forensic and clinical research and cancer gene diagnostics by the analysis of fragmented circulating DNA from bodily fluids.