Discontinuous Fields Research Articles

Spatial estimation of daily precipitation in regions with complex relief and scarce data using terrain orientation

Precipitation is the most relevant variable in the hydrological cycle which drives continental hydrologic processes. Its spatial occurrence and behavior are complex and its daily estimation is hard in poorly gauged regions where the topography is highly irregular. Several interpolation methods are available for this purpose, but their performance is quite uncertain. This study develops a spatial interpolation method for daily precipitation that considers both spatial discontinuities and the influence of topography. The method first identifies the precipitation occurrence in each grid-cell as a function of measurements in surrounding rain gauges, and then uses daily elevation vs. precipitation linear regressions throughout the grid-cells where precipitation occurrence is identified. These regressions are classified according to the terrain orientation with respect to the prevailing wind direction. The method was evaluated using categorical statistics that quantify the skill to identify the precipitation occurrence/non-occurrence, and goodness-of-fit statistics to evaluate the error and efficiency. The methodology was compared against inverse distance weighted and simple regression methods, which were implemented considering both continuous and discontinuous precipitation fields. The new method better simulates the occurrence of precipitation, whereas traditional methods applied without considering the spatial discontinuity of precipitation tend to overestimate the frequency of the rainfall events, and thus the magnitude of precipitation at the basin scale. When spatial discontinuity is considered, traditional methods improve their performance and are comparable to the proposed method. Overall, the new method increases the number of days in which elevation vs. precipitation linear regression can be used, thus improving the spatial representation of precipitation in areas with complex relief.

Discontinuities in the magnetic field near Enceladus

AbstractThe plasma interaction of Saturn's icy moon Enceladus generates a hemisphere coupling current system that directly connects the giant planet's northern and southern polar magnetosphere. Based on Cassini magnetometer observations from all 20 targeted Enceladus flybys between 2004 and 2014, we study the magnetic field discontinuities associated with these hemisphere coupling currents. We identify a total number of 11 events during which the magnetic field was discontinuous at the surface of the Enceladus flux tube (defined by the bundle of magnetic field lines tangential to the solid body of the moon). A minimum variance analysis is applied to calculate the surface normals of these discontinuities. In agreement with theoretical expectations, the normals are found to be perpendicular to the surface of the Enceladus flux tube. The variation of the hemisphere coupling currents with Enceladean longitude leaves a clear imprint in the strengths of the observed magnetic field jumps as well.

Comparing faceted and smoothed tool surface descriptions in sheet metal forming simulation

This study deals with different tool surface description methods used in the finite element analysis of sheet metal forming processes. The description of arbitrarily-shaped tool surfaces using the traditional linear finite elements is compared with two distinct smooth surface description approaches: (i) Bézier patches obtained from the Computer-Aided Design model and (ii) smoothing the finite element mesh using Nagata patches. The contact search algorithm is presented for each approach, exploiting its special features in order to ensure an accurate and efficient contact detection. The influence of the tool modelling accuracy on the numerical results is analysed using two sheet forming examples, the unconstrained cylindrical bending and the reverse deep drawing of a cylindrical cup. Smoothing the contact surfaces with Nagata patches allows creating more accurate tool models, both in terms of shape and normal vectors, when compared with the conventional linear finite element mesh. The computational efficiency is evaluated in this study through the total number of increments and the required CPU time. The mesh refinement in the faceted description approach is not effective in terms of computational efficiency due to large discontinuities in the normal vector field across facets, even when adopting fine meshes.

Improving profile of multi-electrode Fresnel-type liquid crystal lens utilizing local control of pretilt angle

A multi-electrode driving Fresnel liquid crystal lenticular lens (MeD-Fresnel type LC lens) for switchable 2D/3D liquid crystal displays has been studied. The demonstrated lens which has fine-patterned electrodes only on one substrate, minimizing lens error by mis-assembly between top and bottom substrate, and uses a discontinuous non-uniform electric field with local control of surface pretilt angle by polymer stabilization technique, exhibits excellent lens profile compared with conventional lens without local control of pretilt angle.

An efficient approach for multilayered beams and wide plates with imperfect interfaces and delaminations

In multilayered systems with interfacial imperfections, such as imperfect bonding of the layers or delaminations, or where the plies are separated by thin interlayers allowing relative motion, the displacement field is highly discontinuous in the thickness, with a characteristic zig-zag pattern and interfacial jumps. Stresses also have large variations, especially for highly deformable interlayers or fully debonded layers. These effects cannot be captured using classical, first- or higher-order single-layer theories. A mechanical model is formulated for the solution of multilayered beams/wide plates with an arbitrary number of imperfect interfaces/delaminations loaded dynamically. The formulation is in the framework of the discrete-layer approach and the interfaces are described through affine traction laws which relate interfacial tractions and relative displacements. Homogenization and variational techniques define novel equilibrium equations depending on only four generalized displacement functions. Comparison with 2D elasticity solutions shows that complex discontinuous fields in thick, highly-anisotropic plates with an arbitrary number of sliding-interfaces are accurately predicted. The approach extends the range of problems which can be solved analytically compared to discrete-layer models where the unknowns depend on the number of layers/interfaces. The affine traction laws describe arbitrary branches of piecewise linear functions approximating nonlinear traction laws to represent different interfacial mechanisms.

Formation of magnetic discontinuities through viscous relaxation

According to Parker's magnetostatic theorem, tangential discontinuities in magnetic field, or current sheets (CSs), are generally unavoidable in an equilibrium magnetofluid with infinite electrical conductivity and complex magnetic topology. These CSs are due to a failure of a magnetic field in achieving force-balance everywhere and preserving its topology while remaining in a spatially continuous state. A recent work [Kumar, Bhattacharyya, and Smolarkiewicz, Phys. Plasmas 20, 112903 (2013)] demonstrated this CS formation utilizing numerical simulations in terms of the vector magnetic field. The magnetohydrodynamic simulations presented here complement the above work by demonstrating CS formation by employing a novel approach of describing the magnetofluid evolution in terms of magnetic flux surfaces instead of the vector magnetic field. The magnetic flux surfaces being the possible sites on which CSs develop, this approach provides a direct visualization of the CS formation, helpful in understanding the governing dynamics. The simulations confirm development of tangential discontinuities through a favorable contortion of magnetic flux surfaces, as the magnetofluid undergoes a topology-preserving viscous relaxation from an initial non-equilibrium state with twisted magnetic field. A crucial finding of this work is in its demonstration of CS formation at spatial locations away from the magnetic nulls.

The morphing method as a flexible tool for adaptive local/non-local simulation of static fracture

We introduce a framework that adapts local and non-local continuum models to simulate static fracture problems. Non-local models based on the peridynamic theory are promising for the simulation of fracture, as they allow discontinuities in the displacement field. However, they remain computationally expensive. As an alternative, we develop an adaptive coupling technique based on the morphing method to restrict the non-local model adaptively during the evolution of the fracture. The rest of the structure is described by local continuum mechanics. We conduct all simulations in three dimensions, using the relevant discretization scheme in each domain, i.e., the discontinuous Galerkin finite element method in the peridynamic domain and the continuous finite element method in the local continuum mechanics domain.

Weak discontinuity in porous media: an enriched EFG method for fully coupled layered porous media

SUMMARY In this paper, a series of multimaterial benchmark problems in saturated and partially saturated two-phase and three-phase deforming porous media are addressed. To solve the process of fluid flow in partially saturated porous media, a fully coupled three-phase formulation is developed on the basis of available experimental relations for updating saturation and permeabilities during the analysis. The well-known element free Galerkin mesh-free method is adopted. The partition of unity property of MLS shape functions allows for the field variables to be extrinsically enriched by appropriate functions that introduce existing discontinuities in the solution field. Enrichment of the main unknowns including solid displacement, water phase pressure, and gas phase pressure are accounted for, and a suitable enrichment strategy for different discontinuity types are discussed. In the case of weak discontinuity, the enrichment technique previously used by Krongauz and Belytschko [Int. J. Numer. Meth. Engng., 1998; 41:1215–1233] is selected. As these functions possess discontinuity in their first derivatives, they can be used for modeling material interfaces, generating only minor oscillations in derivative fields (strain and pressure gradients for multiphase porous media), as opposed to unenriched and constrained mesh-free methods. Different problems of multimaterial poro-elasticity including fully saturated, partially saturated one, and two-phase flows under the assumption of fully coupled extended formulation of Biot are examined. As a further development, problems involved with both material interface and impermeable discontinuities, where no fluid exchange is permitted across the discontinuity, are considered and numerically discussed. Copyright © 2014 John Wiley & Sons, Ltd.

Stabilized extended finite elements for the approximation of saddle point problems with unfitted interfaces

We address a two-phase Stokes problem, namely the coupling of two fluids with different kinematic viscosities. The domain is crossed by an interface corresponding to the surface separating the two fluids. We observe that the interface conditions allow the pressure and the velocity gradients to be discontinuous across the interface. The eXtended Finite Element Method (XFEM) is applied to accommodate the weak discontinuity of the velocity field across the interface and the jump in pressure on computational meshes that do not fit the interface. Numerical evidence shows that the discrete pressure approximation may be unstable in the neighborhood of the interface, even though the spatial approximation is based on inf-sup stable finite elements. It means that XFEM enrichment locally violates the satisfaction of the stability condition for mixed problems. For this reason, resorting to pressure stabilization techniques in the region of elements cut by the unfitted interface is mandatory. In alternative, we consider the application of stabilized equal order pressure/velocity XFEM discretizations and we analyze their approximation properties. On one side, this strategy increases the flexibility on the choice of velocity and pressure approximation spaces. On the other side, symmetric pressure stabilization operators, such as local pressure projection methods or the Brezzi–Pitkaranta scheme, seem to be effective to cure the additional source of instability arising from the XFEM approximation. We will show that these operators can be applied either locally, namely only in proximity of the interface, or globally, that is on the whole domain when combined with equal order approximations. After analyzing the stability, approximation properties and the conditioning of the scheme, numerical results on benchmark cases will be discussed, in order to thoroughly compare the performance of different variants of the method.

Three-dimensional finite elements with embedded strong discontinuities to model failure in electromechanical coupled materials

This paper presents new finite elements with embedded strong discontinuities to model failure in three dimensional electromechanical coupled materials. Following the strong discontinuity approach for plane electromechanical problems, the coupled boundary value problem is decomposed into a continuous global part and into a discontinuous local part where strong discontinuities in the displacement field and electric potential are introduced. Those are incorporated into general three-dimensional brick finite elements through nine mechanical separation modes and three new electrical separation modes. All the local enhanced parameters related to those modes can be statically condensed out on the element level, yielding a computationally efficient framework to model failure in electromechanical coupled materials. Impermeable electric boundary conditions are assumed along the strong discontinuities. Their initiation and orientation is detected through a configurational force driven failure criterion. A marching cubes based crack propagation concept is used to obtain smooth failure surfaces in the three dimensional problems of interest. Several representative numerical simulations are included and compared with experimental results of failure in piezoelectric ceramics to outline the performance of the new finite elements.

A multiscale framework for localizing microstructures towards the onset of macroscopic discontinuity

This paper presents a multiscale computational homogenization model for the post localization behavior of a macroscale domain crossed by a cohesive discontinuity emanating from microstructural damage. The stress---strain and the cohesive macroscopic responses are obtained incorporating the underlying microstructure, in which the damage evolution results in the formation of a strain localization band. The macro structural kinematics entails a discontinuous displacement field and a non-uniform deformation field across the discontinuity. Novel scale transitions are formulated to provide a consistent coupling to the continuous microscale kinematics. From the solution of the micromechanical boundary value problem, the macroscale stress responses at both sides of the discontinuity are recovered, providing automatically the cohesive tractions at the interface. The effective displacement jump and deformation field discontinuity are derived from the same microscale analysis. This contribution focusses on scale transition relations and on the solution procedure at the microlevel; the highlights of the approach are demonstrated on microscale numerical examples. Coupled two-scale solution strategy will be presented in a subsequent paper.

Near-infrared spectropolarimetry of aδ-spot

Sunspots harboring umbrae of both magnetic polarities within a common penumbra (delta-spots) are often but not always related to flares. We present first near-infrared (NIR) observations (Fe I 1078.3 nm and Si I 1078.6 nm spectra) obtained with the Tenerife Infrared Polarimeter (TIP) at the Vacuum Tower Telescope (VTT) in Tenerife on 2012 June 17, which afford accurate and sensitive diagnostics to scrutinize the complex fields along the magnetic neutral line of a delta-spot within active region NOAA 11504. We examine the vector magnetic field, line-of-sight (LOS) velocities, and horizontal proper motions of this rather inactive delta-spot. We find a smooth transition of the magnetic vector field from the main umbra to that of opposite polarity (delta-umbra), but a discontinuity of the horizontal magnetic field at some distance from the delta-umbra on the polarity inversion line. The magnetic field decreases faster with height by a factor of two above the delta-umbra. The latter is surrounded by its own Evershed flow. The Evershed flow coming from the main umbra ends at a line dividing the spot into two parts. This line is marked by the occurrence of central emission in the Ca II 854.2 nm line. Along this line, high chromospheric LOS-velocities of both signs appear. We detect a shear flow within the horizontal flux transport velocities parallel to the dividing line.

Discontinuities in the electromagnetic field [Advertisement

Discontinuities in the electromagnetic field [Advertisement

Limit analysis and lower/upper bounds to the macroscopic criterion of Drucker–Prager materials with spheroidal voids

The paper is devoted to a numerical Limit Analysis of a hollow spheroidal model with a Drucker–Prager solid matrix, for several values of the corresponding friction angle ϕ. In the first part of this study, the static and the mixed kinematic 3D-codes recently evaluated in [1] are modified to use the geometry defined in [2] for spheroidal cavities in the context of a von Mises matrix. The results in terms of macroscopic criteria are satisfactory for low and medium values of ϕ, but not enough for ϕ=30° in the highly compressive part of the criterion. To improve these results, an original mixed approach, dedicated to the axisymmetric case, was elaborated with a specific discontinuous quadratic velocity field associated with the triangular finite element. Despite the less good conditioning inherent to the axisymmetric modelization, the resulting conic programming problem appears quite efficient, allowing one take into account numerical discretization refinements unreachable with the corresponding 3D mixed code. After a first validation in the case of spherical cavities whose exact solution is known, the final results for spheroidal voids are given for three usual values of the friction angle and two values of the cavity aspect factor.

Dispersive Media Subcell Averaging in the FDTD Method Using Corrective Surface Currents

We present a corrective subcell averaging technique that improves on the accuracy of the volume-averaged finite-difference time-domain (FDTD) method in the presence of dispersive material interfaces. The method is based on an alternative effective-medium formulation that captures field discontinuities at interfaces as electric and magnetic surface currents. In calculating the spectra of strongly dispersive Mie scatterers we demonstrate that the derived FDTD algorithm is both highly efficient and able to approximately restore second order accuracy.

A semi‐Lagrangean time‐integration approach for extended finite element methods

SUMMARYMany computational problems incorporate discontinuities that evolve in time. The eXtendend Finite Element Method (XFEM) is able to represent discontinuities sharply on fixed arbitrary meshes, but numerical difficulties arise if these discontinuities move in time. We point out that this issue is crucial for interface problems with strongly discontinuous fields on fixed grids. A method using semi‐Lagrangean techniques is proposed to adequately handle time integration based on finite difference schemes in the context of the XFEM. The basic idea is to adapt previous numerical solutions to the current interface position by tracking back virtual Lagrangean particles to their previous positions, where an appropriate solution can be extrapolated from a smooth field. Convergence properties of the proposed method in time and space are thoroughly studied for two one‐dimensional model problems. Finally, the method is applied to the particularly challenging problem of premixed combustion, where the discontinuity appears at the flame front separating the burnt from the unburnt gases. A two‐dimensional and a three‐dimensional expanding flame demonstrates that the method is sufficiently accurate to retain the properties of the overall Nitsche‐type formulation for interface problems with embedded strong discontinuities. Copyright © 2014 John Wiley & Sons, Ltd.

On resolving singularities of piecewise-smooth discontinuous vector fields via small perturbations

A two-fold singularity is a point on a discontinuity surface of a piecewise-smooth vector field at which the vector field is tangent to the surface on both sides. Due to the double tangency, forward evolution from a two-fold is typically ambiguous. This is an especially serious issue for two-folds that are reached by the forward orbits of a non-zero measure set of initial points. However, arbitrarily small perturbations of the vector field can make forward evolution well-defined, and from an applied perspective, such perturbations may represent additional model features that enhance the realism of a piecewise-smooth mathematical model. Three physically motivated forms of perturbation: hysteresis, time-delay, and noise, are analysed individually. The purpose of this paper is to characterise the perturbed dynamics in the limit that the size of the perturbation tends to zero. This concept is applied to a two-fold in two dimensions. In each case the limit leads to a novel probabilistic notion of forward evolution from the two-fold.

Prey Switching with a Linear Preference Trade-Off

In ecology, prey switching refers to a predator's adaptive change of habitat or diet in response to prey abundance. In this paper, we study piecewise-smooth models of predator-prey interactions with a linear trade-off in a predator's prey preference. We consider optimally foraging predators and derive a model for a 1 predator-2 prey interaction with a tilted switching manifold between the two sides of discontinuous vector fields. We show that the 1 predator-2 prey system undergoes a novel adding-sliding-like (center to two-part periodic orbit; "C2PO") bifurcation in which the prey ratio transitions from constant to time-dependent. Further away from the bifurcation point, the period of the oscillating prey ratio period doubles, suggesting a possible cascade to chaos. We compare our model predictions with data and demonstrate that we successfully capture the periodicity in the ratio between the predator's preferred and alternative prey types in data on freshwater plankton. Our study suggests that it is useful to investigate prey ratio as a possible indicator of how population dynamics can be influenced by ecosystem diversity.

P1/P0+ elements for incompressible flows with discontinuous material properties

We present a 3-noded triangle and a 4-noded tetrahedra with a continuous linear velocity and a discontinuous linear pressure field formed by the sum of an unknown constant pressure field and a prescribed linear field that satisfies the steady state momentum equations for a constant body force. The elements are termed P1/P0+ as the “effective” pressure field is linear, although the unknown pressure field is piecewise constant within each element. The elements have an excellent behavior for incompressible viscous flow problems with discontinuous material properties formulated in either Eulerian or Lagrangian descriptions. The necessary numerical stabilization for dealing with the inf–sup condition imposed by the incompressibility constraint and high convective effects (in Eulerian flows) is introduced via the Finite Calculus (FIC) approach. For the sake of clarity, the element derivation is presented first for the simpler Stokes equations written in the standard Eulerian frame. The extension of the formulation to the Navier–Stokes equations written in the Eulerian and Lagrangian frameworks is straightforward and is presented in the second part of the paper.The efficiency and accuracy of the new P1/P0+ triangle is verified by solving a set of incompressible multifluid flow problems using a Lagrangian approach and a classical Eulerian description. The excellent performance of the new triangular element in terms of mass conservation and general accuracy for analysis of fluids with discontinuous material properties is highlighted.

Total Variation Filtered Demons for Improved Registration of Sliding Organs

We present a new approach to regularize the displacement field of the accelerated Demons registration algorithm. The accelerated Demons algorithm uses Gaussian smoothing to penalize oscillatory motion in the displacement fields during registration. This regularization approach is often applied and ensures a smooth deformation field. However, when registering images with discontinuities in their motion field such as from organs sliding along the chest wall, the assumption of a smooth deformation field is invalid. In this work, we propose using total variation based smoothing that is known to better retain the discontinuities in the deformation field. The proposed approach is a first step towards automatically recovering breathing induced organ motion with good accuracy.