Discontinuous Fields Research Articles

The Linear Peridinamic Model in Elasticity Theory

The peridinamics theory, proposed by Silling in 2000, is a nonlocal theory of continuum mechanics based on an integro-differential equation without spatial derivatives and may cope with discontinuous displacement fields commonly occurring in fracture mechanics. Instead of spatial differential operators, integration over differences of the displacement field is used to describe the existing, possibly nonlinear, forces between particles of the solid body. Beside an overview of the peridynamics modelling and its application, results concerning the mathematical solution of the governing equation, which is a partial integro-differential equation with second-order time derivative, are presented. In this paper we consider well-posed Cauchy problem for the singular periodic integro-differential equation of peridinamics. In case of two-dimensional space the existence and uniqueness of solution are proved in Sobolev spaces. In multidimensional space the unique solvability of the problem in logarithmic scale of Hilbert spaces is proved.

Upsetting calculation of edge on plate during their assembly

The relations for calculating of the deformation and force upsetting modes of edge on the plate (sheet) during their assembly are proposed. The state of viscoplasticity of the material under isothermal heating is assumed. The discontinuous fi eld of travel speeds is used. The calculation of the loss of continuity of the workpiece material is given.

A proper integral for the electric field at the surface of a conducting sphere

Abstract In a recent paper, I have shown that the electric field at the surface of a charged conducting sphere evaluates to half the field discontinuity across the surface of the sphere. This exact result has been found by solving an improper integral, which has been criticized by Assad in a very recent paper in this journal [6]. In this note, I present a simpler approach that yields a proper integral for that field. This integral also evaluates to exactly half the field discontinuity, contrarily to Assad's objections.

On the Correct Determination of Flow of a Discontinuous Solenoidal Vector Field

We prove inequalities connecting a flow through the (n−1)-dimensional surface S of a smooth solenoidal vector field with its Lp(U)-norm (U is an n-dimensional domain that contains S ). On the basis of these inequalities, we propose a correct definition of the flow through the surface S of a discontinuous solenoidal vector field f ∈ Lp(U) (or, more precisely, of the class of vector fields that are equal almost everywhere with respect to the Lebesgue measure).

Review and analysis of heat source models for additive manufacturing

As additive manufacturing (AM) becomes a viable manufacturing solution, demand for an accurate thermo-structural model of the process increases. Iteratively correcting discrepancies between the CAD model and additively manufactured product through trial and error can be an expensive and time-consuming process, taking up to several hours to build and costing up to tens of thousands of dollars Lindgren et al. (Addit Manuf 12:144–158, 2016). A numerical model reduces manufacturing cost and time considerably by predicting discrepancies that will arise due to the complex thermal history induced by the AM process, thus reducing the need for iterative manufacturing. An important part of any additive manufacturing model is the heat source model. The heat source model is a mathematical function which represents how much of a heat source’s power actually goes into heating the powdered metal and how this heat is distributed across the heat-affected zone (HAZ). This paper provides a review and analysis of heat source models in the AM literature to date in order to alleviate some of the confusion and provide emerging researchers in the field with perspective on the issue. Both two-dimensional surface models and three dimensional volumetric models are explored. Next, an analysis of the models was performed and presented in an effort to validate their physical accuracy and mathematical usability. This analysis consisted of checking for sensible boundary conditions and ensuring that energy conservation is upheld. In surface models, the TEM00 model is a classic representation of the Gaussian power distribution of most heat sources used in AM. Researchers interested in simply modeling the heat distribution, without accounting for any other phenomena that intervene in the heat transfer process (such as molten pool dynamics) will find the TEM00 model suitable. The literature also shows cases where the TEM00 model has been modified to have a sharper radial gradient, and these modifications can be suitable for high-powered heat sources. For volumetric models, Goldak’s ellipsoidal model (Metall Trans B 15(2)299–305, 1984) remains a straightforward and accurate model that is physically sound and applicable to a variety of cases. The Gaussian cone model presented by Rogeon et al. [48] also performs well, meeting all the required physical and mathematical restrictions. This model’s linearly decaying penetration is better suited for high-energy applications. The non-Gaussian cone proposed by Tsirkas et al. (J Mater Process Technol 134(1):59–69, 2003) imposes inaccurate boundary conditions and violates the first law of thermodynamics, and is thus deemed an inadequate model. Other novel models have been introduced in recent years, most notably the line model and the elongated ellipsoidal model presented by Irwin and Michaleris (J Manuf Sci Eng 138(11):111004, 2016). Both of these models are based on Goldak’s ellipsoidal model and attempt to maintain the accuracy of that model while allowing for fewer time steps and requiring less computational resources. These models appear to function well and can be used effectively in some applications, but could benefit from further study and validation. Care must be taken to ensure that the parameters used with these models do not result in averaging errors or a discontinuous thermal field. These tools must be used carefully with a thorough understanding of the underlying mathematics.

Accurate evaluation of stress intensity factors using dual interpolation boundary face method

Accurate computation of stress intensity factors for two-dimensional cracks by a dual interpolation boundary face method (DiBFM) is presented in this paper. Dual interpolation method combines traditional element polynomial interpolation and moving least-squares approximation. Dual interpolation elements are obtained by adding virtual nodes on the boundary of traditional discontinuous elements, and they unify traditional continuous and discontinuous elements. With the dual interpolation elements, both the continuous and discontinuous fields can be approximated, and the accuracy of interpolation increases by two orders compared with the corresponding discontinuous elements. In addition, a new singular element based on the dual interpolation method for modeling displacement fields around the crack tip is proposed in this paper. The stress intensity factors are extracted from the relative displacements between the crack surfaces. Using the DiBFM combined with the proposed singular element, accurate results of the stress intensity factors can be obtained. Numerical examples have demonstrated the accuracy and efficiency of the proposed method.

Estimation of Errors in Stress Distributions Computed in Finite Element Simulations of Polycrystals

The accuracy of the stresses predicted from crystal plasticity-based finite element formulation depends on estimation and control of the errors associated with the discretization. In the current work, the errors in the stress distribution are estimated in virtual polycrystalline samples of α-phase titanium (hexagonal close-packed phase of Ti–6Al–4V). To estimate the error, the stress field, which does not possess inter-element continuity, is smoothed over a grain using an $$L_2$$ projection, thereby providing continuous stress distributions with inter-element continuity. The differences between the continuous (smooth) and discontinuous (raw) stress fields are calculated at individual Gauss quadrature points and used to estimate errors for corresponding elements and grains. Error estimations are performed for a Voronoi-tessellated microstructure, an equiaxed microstructure, and two microstructures with varying grain sizes for tensile loading extending into the fully plastic regime ($$\approx $$ 5% extension). Magnitudes of the errors are found to depend on microstructural characteristics, particularly the shape and size of grains. Samples having variations in grain size or having less spherical grains exhibited higher errors than samples with uniformly sized, equiaxed grains, with the size variations having a more pronounced effect. Errors correlate with proximity to grain boundaries at small (elastic) strains and with deformation-induced features (deformation bands) at large strains.

Validation of an improved contact method for multi-material Eulerian hydrocodes in three-dimensions

Realistic and accurate modeling of contact for problems involving large deformations and severe distortions presents a host of computational challenges. Due to their natural description of surfaces, Lagrangian finite element methods are traditionally used for problems involving sliding contact. However, problems such as those involving ballistic penetrations, blast-structure interactions, and vehicular crash dynamics, can result in elements developing large aspect ratios, twisting, or even inverting. For this reason, Eulerian, and by extension Arbitrary Lagrangian-Eulerian (ALE), methods have become popular. However, additional complexities arise when these methods permit multiple materials to occupy a single finite element.Multi-material Eulerian formulations in computational structural mechanics are traditionally approached using mixed-element thermodynamic and constitutive models. These traditional approaches treat discontinuous pressure and stress fields that exist in elements with material interfaces by using a single approximated pressure and stress field. However, this approximation often has little basis in the physics taking place at the contact boundary and can easily lead to unphysical behavior. This work presents a significant departure from traditional Eulerian contact models by solving the conservation equations separately for each material within each computational element and then imposing inequality constraints associated with contact to the solutions for each material with the appropriate tractions included. The advantages of this method have been demonstrated with several computational examples. These examples are limited to contact between two materials. Future work will examine contact between three or more materials within an element. This work concludes by drawing a comparison between the method put forth in this work and traditional treatment of multi-material contact in Eulerian methods.

Monitoring of Low-Level Wind Shear by Ground-based 3D Lidar for Increased Flight Safety, Protection of Human Lives and Health.

Low-level wind shear, i.e., sudden changes in wind speed and/or wind direction up to altitudes of 1600 ft (500 m) above-ground is a hazardous meteorological phenomenon in aviation. It may radically change the aerodynamic circumstances of the flight, particularly during landing and take-off and consequently, it may threaten human lives and the health of passengers, people at the airport and its surrounding areas. The Bratislava Airport, the site of this case study, is one of the few airports worldwide and the first in Central Europe that is equipped with a Doppler lidar system, a perspective remote sensing tool for detecting low-level wind shear. The main objective of this paper was to assess the weather events collected over a period of one year with the occurrences of low-level wind shear situations, such as vertical discontinuities in the wind field, frontal passages and gust fronts to increase the level of flight safety and protect human lives and health. The lidar data were processed by a computer algorithm with the main focus on potential wind shear alerts and microburst alerts, guided by the recommendations of the International Civil Aviation Organisation. In parallel, the selected weather events were analyzed by the nearby located meteorological radar to utilize the strengths of both approaches. Additionally, an evaluation of the lidar capability to scan dynamics of aerosol content above the airport is presented.

Quiet Time Structured Pc1 Waves Generated During Transient Foreshock

AbstractWe study two events of structured Pc1 waves (pearls) observed by CARISMA magnetometers during a period of quiet magnetic (Kp ~1) and solar wind conditions on 1 August 2008. Bursts of proton precipitations and electromagnetic ion‐cyclotron waves were simultaneously observed by NOAA/POES and GOES satellites at dayside. Apparent correspondence was found between ground pearl pulsations and magnetospheric compressions observed by GOES and THEMIS satellites. We have found the Pc1 wave splitting into very close subfrequencies (within 0.1 Hz) that can be explained by weak magnetospheric compressions. The compressions are caused by pressure pulses originated from transient foreshock and interplanetary magnetic field discontinuities observed upstream of the bow shock by the THEMIS‐C probe. Thus, the observed pearls are compression‐related Pc1 waves generated under transient foreshock conditions.

WENO interpolation for Lagrangian particles in highly compressible flow regimes

Lagrangian particles can be used either to analyze different complex flows, as massless tracers, or to model multiphase flows, as particles with mass. As particles can have arbitrary positions within the computational domain, interpolation of different flow quantities from the Eulerian grid is essential. Low-order centered interpolation schemes generally do not provide a sufficient level of accuracy for many flow configurations of interest. Thus, typically, high-order centered interpolation schemes are utilized. The current study demonstrates that for highly compressible non-reacting and reacting flow regimes, in which discontinuities in the flow field, e.g., shock waves arise, centered interpolation schemes tend to smear the shock and high order schemes also produce numerical oscillations. It is shown that this problem can be remedied by using Weighted-Essentially-Non-Oscillatory (WENO) interpolation schemes. Extensive numerical tests are performed in order to demonstrate this, comparing the performance of WENO-3 and WENO-5 interpolation schemes with a variety of centered interpolation schemes. The first test case involves specially designed steady state two-dimensional vortex flows. For a smooth flow, WENO schemes provide comparable accuracy to other high-order centered schemes. For flow fields with discontinuities, WENO-3 and WENO-5 interpolation schemes can decrease the interpolation error by more than two and three orders of magnitude, respectively, in comparison with centered schemes. Solution accuracy is further studied with a normal shock wave test. It is demonstrated that centered schemes tend to smear the shock, whereas, WENO schemes capture it in a much sharper manner. Moreover, high-order centered schemes tend to oscillate in the vicinity of the discontinuity leading to non-physical results, such as negative absolute pressure values. The same trends are retained for an unsteady three-dimensional spherical blast wave, where the shock is smeared across several grid cells, which is typical for shock-capturing flow solvers. Finally, the benefits of WENO schemes for the Lagrangian-particle tracking analysis of highly compressible reactive flows are explored by comparing various Lagrangian particle trajectories and joint probability density functions (PDFs) for a two-dimensional cellular detonation. The results indicate that high-order centered schemes lead to unphysical results, whereas WENO schemes can provide high-order interpolation that is free of severe numerical oscillations and non-physical artifacts, which is critical for the proper analysis of the flow.

Characteristics of stick-slip oscillations in dry friction backward whirl of piecewise smooth rotor/stator rubbing systems

Abstract Self-excited dry friction backward whirl, or shortly denoted as DFBW, are the most destructive rotor/stator rubbing response. From the point of view of sliding bifurcations, this paper first unveils the characteristics of stick-slip oscillations exhibited in the response of self-excited dry friction backward whirl. By studying a four-dimensional piecewise smooth rotor/stator rubbing system, which has a switching manifold defined by the zero relative velocity on the rubbing point, three types of sliding regions on the curved hypersurface of the switching manifold are identified according to the characters of the two discontinuous vector fields near it. The influence of the system parameters on the stick-slip transitions in DFBW is also explored, which shows the interplay of the different parameters and reveals the phenomenon observed experimentally. On basis of the relation between the existence boundaries of DFBW and the characters of sliding motions on it, a new way, from the theory of non-smooth dynamics, to analytically derive the existence condition of the self-excited dry friction backward whirl (DFBW) is presented and coincident with the existing formula in the literature. The work provides a deeper insight into the more detailed response characteristics in the self-excited dry friction backward whirl of rotor/stator rubbing systems.

Existence and uniqueness of low-energy weak solutions to the compressible 3D magnetohydrodynamics equations

We prove the existence and uniqueness of weak solutions of the three dimensional compressible magnetohydrodynamics (MHD) equations. We first obtain the existence of weak solutions with small L2-norm which may display codimension-one discontinuities in density, pressure, magnetic field and velocity gradient. The weak solutions we consider here exhibit just enough regularity and structure which allow us to develop uniqueness and continuous dependence theory for the compressible MHD equations. Our results generalise and extend those for the intermediate weak solutions of compressible Navier-Stokes equations.

Adaptive analysis of crack propagation in thin-shell structures via an isogeometric-meshfree moving least-squares approach

This paper reports an isogeometric-meshfree moving least-squares approach for the adaptive analysis of crack propagation in thin-shell structures within the context of linear elastic fracture mechanics. The present approach is developed based on the equivalence of the moving least-squares meshfree shape functions and the isogeometric basis functions, which provides an effective strategy of adaptive mesh refinement for isogeometric analysis (IGA) in a straightforward meshfree manner. The adaptivity of the mesh refinement is achieved by utilizing a gradient-based error estimator to identify the meshes that need to be refined by adding linear reproducing points. The Kirchhoff–Love theory is further applied in the isogeometric-meshfree moving least-squares formulation to simplify the modeling of cracked thin-shell structures by neglecting the rotational degrees of freedom. In this way, the singularity of stress fields near the crack tip and the discontinuity of displacement fields around the crack surface can be efficiently captured by the adaptive mesh refinement to generate accurate results. A series of two-dimensional static and quasi-static crack propagation problems of thin-shell structures are investigated. It is found that the adaptive refinement strategy makes the present approach achieve higher convergence rate and computational efficiency than IGA and the meshfree method. The predicted propagation paths obtained by the present approach are in good agreement with the previously reported results.

Frontal Wind Field Retrieval Based on UHF Wind Profiler Radars and S-Band Radars Network

The three-dimensional wind field (WPR3D) and the multiple WPR3D (M-WPR3D) associated with the passage of a stationary front was derived from observations made by a network of eight wind profiler radars (WPR) being operated by the Korea Meteorological Administration during the summer “Jangma” season. The effectiveness of the WPR3D was determined through numerical model analysis and wind profilers at three sites, and the accuracy of the M-WPR3D was validated by comparing the trajectory of the radiosonde. The discontinuity of the wind field near the frontal interface was clearly retrieved and the penetration of the air mass in the southern front was detected. Compared with either the wind vector of three single wind profiler or a local data assimilation and predication system, the WPR3D wind field showed a wind speed accuracy of approximately 70% at an altitude of 1.5 km and underestimated the wind speed by 0.5–1.5 m s−1. The M-WPR3D with three S-band Doppler radars successfully retrieved the backing wind field as well as the pre-Jangma-frontal jet. The results of this study showed that severe weather can be effectively analyzed using a three-dimensional wind field generated on the basis of a remote sensing network.

A high-order conservative remap for discontinuous Galerkin schemes on curvilinear polygonal meshes

A data transfer (called later remap) of physical fields between two meshes is an important step of arbitrary Lagrangian-Eulerian (ALE) simulations. This step is challenging for high-order discontinuous Galerkin schemes since the Lagrangian flow motion leads to high-order meshes with curved faces. It becomes even more challenging for unstructured polygonal meshes that do not have a polynomial map from the reference to a current cell. We propose and analyze a new framework to create remap schemes on curvilinear polygonal meshes based on the theory of virtual element projectors. We derive a conservative remap scheme that is high-order accurate in space and time. The properties of this scheme are studied numerically for smooth and discontinuous fields on unstructured quadrilateral and polygonal meshes.

In Quest of Calibration Density and Consistency in Hydrologic Modeling: Distributed Parameter Calibration against Streamflow Characteristics

AbstractConventional basin‐by‐basin approaches to calibrate hydrologic models are limited to gauged basins and typically result in spatially discontinuous parameter fields. Moreover, the consequent low calibration density in space falls seriously behind the need from present‐day applications like high resolution river hydrodynamic modeling. In this study we calibrated three key parameters of the Variable Infiltration Capacity (VIC) model at every 1/8° grid‐cell using machine learning‐based maps of four streamflow characteristics for the conterminous United States (CONUS), with a total of 52,663 grid‐cells. This new calibration approach, as an alternative to parameter regionalization, applied to ungauged regions too. A key difference made here is that we tried to regionalize physical variables (streamflow characteristics) instead of model parameters whose behavior may often be less well understood. The resulting parameter fields no longer presented any spatial discontinuities and the patterns corresponded well with climate characteristics, such as aridity and runoff ratio. The calibrated parameters were evaluated against observed streamflow from 704/648 (calibration/validation period) small‐to‐medium‐sized catchments used to derive the streamflow characteristics, 3941/3809 (calibration/validation period) small‐to‐medium‐sized catchments not used to derive the streamflow characteristics as well as five large basins. Comparisons indicated marked improvements in bias and Nash‐Sutcliffe efficiency. Model performance was still poor in arid and semiarid regions, which is mostly due to both model structural and forcing deficiencies. Although the performance gain was limited by the relative small number of parameters to calibrate, the study and results here served as a proof‐of‐concept for a new promising approach for fine‐scale hydrologic model calibrations.

A General Solution to the Formation Control Problem Without Collisions for First-Order Multi-Agent Systems

SUMMARYIn this paper, a general solution to the formation control problem without collisions for first-order multi-agent systems is proposed. The case of an arbitrary number of mobile agents on a plane with saturated input velocity is analysed. Besides, conditions on the communication graph among agents are relaxed to the only requirement of containing a directed spanning tree. This general approach is an extended result from the simpler case of combinations of cyclic pursuit communication graphs. The proposed solution to this problem is designed in two steps. First, the asymptotic convergence in the absence of collisions is ensured. After this, the non-collision problem is faced by analysing the most general possible geometrical scenario which can lead to collision among agents. Discontinuous vector fields with unstable counterclockwise focus behaviour are applied by every agent in order to repel each other. Numerical simulations illustrate the performance of the proposed scheme.

Mathematical analysis of discontinuities in the flow field of gas in a cylindrical pipe using diffusion MRI

In this study, Magnetic Resonance Imaging (MRI) is used to detect partial and total blockage of hydrogen gas in a cylindrical pipe. Diffusion Magnetic Resonance (DMR) equation is solved analytically for flow of fluid in a radially symmetric cylindrical pipe. Appropriate boundary conditions were imposed and the radial axis varied to depict partial and total blockage in the pipe. The results show that for free fluid flow, the magnetization is between 0.004 and 0.005. For partial blockage, the magnetization reduces (signal loss) in value to 0.00001 and for total blockage it is zero (0). This method is a viable alternative to other methods of detecting blockage in fluid pipelines in oil and gas industry due to its noninvasive analysis of flow in fluid. The MRI model also registers signal in its first few seconds or microseconds. The analysis can also be useful in process industries where different network of pipes are used or machines use cylindrical pipes or tubes in transporting materials especially when there is a partial or total blockage at any point in the network.Keywords: Bloch NMR Equations, DMR equation, Cylindrical pipe, Magnetization

On the forward modelling of three-dimensional magnetotelluric data using a radial-basis-function-based mesh-free method

SUMMARY The investigation of using a novel radial-basis-function-based mesh-free method for forward modelling magnetotelluric data is presented. The mesh-free method, which can be termed as radial-basis-function-based finite difference (RBF-FD), uses only a cloud of unconnected points to obtain the numerical solution throughout the computational domain. Unlike mesh-based numerical methods (e.g. grid-based finite difference, finite volume and finite element), the mesh-free method has the unique feature that the discretization of the conductivity model can be decoupled from the discretization used for numerical computation, thus avoiding traditional expensive mesh generation and allowing complicated geometries of the model be easily represented. To accelerate the computation, unstructured point discretization with local refinements is employed. Maxwell’s equations in the frequency domain are re-formulated using $\mathbf {A}$-ψ potentials in conjunction with the Coulomb gauge condition, and are solved numerically with a direct solver to obtain magnetotelluric responses. A major obstacle in applying common mesh-free methods in modelling geophysical electromagnetic data is that they are incapable of reproducing discontinuous fields such as the discontinuous electric field over conductivity jumps, causing spurious solutions. The occurrence of spurious, or non-physical, solutions when applying standard mesh-free methods is removed here by proposing a novel mixed scheme of the RBF-FD and a Galerkin-type weak-form treatment in discretizing the equations. The RBF-FD is applied to the points in uniform conductivity regions, whereas the weak-form treatment is introduced to points located on the interfaces separating different homogeneous conductivity regions. The effectiveness of the proposed mesh-free method is validated with two numerical examples of modelling the magnetotelluric responses over 3-D conductivity models.