Discontinuous Fields Research Articles

The Mutual Influence of Disk-Shaped Cracks in Three-Dimensional Elastic Space

The mutual influence of three-dimensional disk-shaped cracks located in parallel planes of an elastic medium is studied. The medium is under tensile stress directed perpendicular to the crack planes. The cracks are modeled by mathematical cuts of the continuous medium with the possibility of a strong discontinuity of the displacement field on the faces of the cut. The discontinuous displacement method is used to solve the problem numerically.

Unfitted finite element for optimal control problem of the temperature in composite media with contact resistance

This paper presents a numerical method for the optimal control problem governed by the heat diffusion equation inside a composite medium. The contact resistance at the interface of constitute materials allows for jumps of the temperature field. The derivation process of the Karush-Kuhn-Tucher system is given by the formal Lagrange method. Due to the discontinuity of the temperature field, the standard linear finite element method cannot achieve optimal convergence when the uniform mesh is used. Therefore, the unfitted finite element method is applied to discrete the state equation required in the variational discretization approach. Optimal error estimates in the broken H1-norm and L2-norm for the control, state, and adjoint state are derived. Some numerical examples are provided to confirm the theoretical results.

Enriched degree of freedom locking in crack analysis using the extended finite element method

In this article, the enriched degree of freedom locking that can occur in a crack analysis with the extended finite element method is described. The discontinuous displacement field formulated by the enriched degree of freedom in the extended finite element method does not activate due to the enriched degree of freedom locking. Using the phantom node method, the occurrence of locking when two adjacent elements are simultaneously cracked in a loading step was verified. Two adjacent cracks can be determined to have developed simultaneously when an analysis model reveals a relatively uniform stress distribution on two adjacent elements. Numerical examples of a simply tensioned bar and a reinforced concrete beam are presented to demonstrate the erroneous analysis result due to the enriched degree of freedom locking. As a simple method to circumvent the enriched degree of freedom locking, the tensile strength of the neighboring elements was slightly increased in the numerical examples, and the effectiveness of the method was demonstrated. The proposed method is simple and easy for practicing engineers, and it can be easily applied to the three-dimensional crack propagation analysis.

AN IMPROVED SPH ALGORITHM FOR LARGE DENSITY RATIOS MULTIPHASE FLOWS BASED ON RIEMANN SOLUTION

The discontinuities of physical fields (such as density, viscosity and so on) exist in the different phase interface of multiphase flow problems, and the numerical simulation method using the traditional SPH model is liable to cause spurious~oscillations in the pressure and velocity field at the interface, which is a big problem in the application of multiphase flows. An improved SPH model based on Riemann solution on dealing with the abrupt physical quantities of multiphase flows with large density radios is presented. Using the advantage of Riemann solution in dealing with the contact discontinuity problems, we introduce it into the SPH multiphase flow model. For the sake of accurately calculating the physical viscosity of multiphase fluid and decreasing the Riemann dissipation, the SPH momentum equation of the Riemann solution form is improved. In the new model, we combine the Adami fixed particle wall-boundary with the one-sided Riemann problem to impose solid boundary of the SPH multiphase flow, and consider the influence of the surface tension on the small-scale interface. The new model without adding any artificial viscosity, artificial dissipation and non-physical treatment technology can simulate the real physical viscousity and the physical evolution process of multiphase flow problems. In order to verify the ability of the improved model in dealing with the multiphase flow problems with the discontinuous interface and the convergence of the model particle spacing, firstly the squared droplet oscillating problem is simulated under different discrete particle spacing. Afterwards, the multiphase flow problems of the Rayleigh-Taylor instability, the single bubble buoyancy and the double bubble buoyancy are simulated. The interface is clearly capture and the results are good in agreement with the literature, which proved that the improved multiphase flow SPH model can stably and effectively deal with the problems of the multiphase flow large with density ratio and large viscosity ratio.

Dynamic instability of damped composite plates with embedded delaminations

A layerwise finite element formulation based on B-spline function is used to study dynamic instability of damped laminated composite plates with embedded delaminations in this article. The out-of-plane variation of in-plane displacement components are interpolated by Linear Lagrange interpolation function and the in-plane variation of in-plane and out-of-plane displacement fields are interpolated by B-spline functions. Delamination between layers of composite plate is modeled by jump discontinuities in displacement fields using Heaviside functions. In the delaminated region, virtual springs are added to prevent interpenetration of lower and upper sublaminates. The local buckling of thinner sublaminate is observed when delamination is placed near the top/bottom interface. The dynamic instability regions are computed by Bolotin's method. The effects of delamination lengths and its position on the natural frequencies, mode shapes, buckling loads and dynamic instability regions of the composite plates with various boundary conditions are examined. Both uniform and non-uniform periodic edge loading in the form of concentrated and partial edge loading are considered. Furthermore, the effect of damping on the dynamic instability regions of composite plate is studied. From the linear and nonlinear responses in the stable and unstable regions, various characteristics such as beats, its dependence on forcing frequency and damping are observed.

Generalized Boundary Conditions in Surface Electromagnetics: Fundamental Theorems and Surface Characterizations

Generalized boundary conditions (GBCs) for electromagnetic surfaces are investigated in this paper, which can be used to analytically characterize field discontinuities across two-dimensional surfaces. First, five representative features are described to categorize various surface problems which require proper GBC-based characterization procedures. Next, the generalized boundary conditions are discussed in details, in association with impedance boundary conditions, generalized sheet transition conditions, and surface scattering coefficients. Then an extraction method for surface susceptibilities, the characteristic parameters of surfaces in GBCs, are proposed and validated. Finally, to illustrate the applications of GBCs, two representative surface problems are demonstrated, namely, isolated-aperture surface analysis and surface-wave mode characterization. For the isolated-aperture surface, a characterization procedure is derived based on GBCs and Babinet’s principle, and the accuracy is validated through comparison with full-wave simulations. For surface wave analysis, a characterization procedure is also developed from GBCs, and the feasibility is verified by numerical examples as well.

Nonlinear Dynamic Modeling and Analysis of Self-Oscillating H-Bridge Parallel Resonant Converter Under Zero Current Switching Control: Unveiling Coexistence of Attractors

This paper deals with the global dynamical analysis of an H-bridge parallel resonant converter under a zero current switching control. Due to the discontinuity of the vector field in this system, sliding dynamics may take place. Here, the sliding set is found to be an escaping region. Different tools are combined for studying the stability of oscillations of the system. The desired crossing limit cycles are computed by solving their initial value problem and their stability analysis is performed using Floquet theory. The resulting monodromy matrix reveals that these cycles are created according to a smooth cyclic-fold bifurcation. Under parameter variation, an unstable symmetric crossing limit cycle undergoes a crossing-sliding bifurcation leading to the creation of a symmetric unstable sliding limit cycle. Finally, this limit cycle undergoes a double homoclinic connection giving rise to two different unstable asymmetric sliding limit cycles. The analysis is performed using a piecewise-smooth dynamical model of a Filippov type. Sliding limit cycles divide the state plane in three basins of attraction, and hence, different steady-state solutions may coexist which may lead the system to start-up problems. Numerical simulations corroborate the theoretical predictions, which have been experimentally validated.

Linear-slip discrete fracture network model and multiscale seismic wave simulation

We propose a method to design a discrete fracture network (DFN) model for seismic wave simulations that eliminates complex meshing requirements and poor mesh design. An ant-tracking volume is generated from the original seismic to capture the global fracture geometry. A mesh structure is specified prior to adding fractures. Fractures are then specified on the mesh in the form of small fracture patches following the trend of ant-tracking signals in the previously defined mesh. In the proposed method, the implementation of the DFN model is thus independent of the mesh structure. Also, fracture clusters in the final DFN model are composed of multiple small patches, and each small patch is represented by the linear slip model in the form of a discontinuity of the displacement field. We also performed wave simulations to validate the proposed method of DFN model building, and investigated the wave behavior in the given fracture structure. For efficient implementation of wave modeling, we employed the generalized multiscale finite element method to discretize the elastic wave equation with explicit fractures. The wave simulation results show that a small fracture segment could be ignored when the wavelength is long enough; however, fracture clusters can induce large displacement discontinuities and wave scattering, and these effects become obvious as we apply high source frequency.

Some comparisons and analyses of time or space discontinuous Galerkin methods applied to elastic wave propagation in anisotropic and heterogeneous media

This research work presents some comparisons and analyses of the time discontinuous space–time Galerkin method and the space discontinuous Galerkin method applied to elastic wave propagation in anisotropic and heterogeneous media. Mechanism of both methods to ensure their stability using time or space discontinuities of unknown fields is analyzed and compared. The most general case of anisotropic and heterogeneous media with physical interfaces of discontinuous material properties is considered, especially for the space discontinuous Galerkin method. A new stability result is proved. Numerical applications to different elastic media, more particularly polycrystalline materials containing a large number of physical interfaces, are also presented to confirm theoretical analyses.

Consistent discretization of higher-order interface models for thin layers and elastic material surfaces, enabled by isogeometric cut-cell methods

Many interface formulations, e.g. based on asymptotic thin interphase models or material surface theories, involve higher-order differential operators and discontinuous solution fields. In this article, we are taking first steps towards a variationally consistent discretization framework that naturally accommodates these two challenges by synergistically combining recent developments in isogeometric analysis and cut-cell finite element methods. Its basis is the mixed variational formulation of the elastic interface problem that provides access to jumps in displacements and stresses for incorporating general interface conditions. Upon discretization with smooth splines, derivatives of arbitrary order can be consistently evaluated, while cut-cell meshes enable discontinuous solutions at potentially complex interfaces. We demonstrate via numerical tests for three specific nontrivial interfaces (two regimes of the Benveniste–Miloh classification of thin layers and the Gurtin–Murdoch material surface model) that our framework is geometrically flexible and provides optimal higher-order accuracy in the bulk and at the interface.

Structural shape optimization with meshless method and swarm-intelligence based optimization

This paper presents a distinctive numerical approach for shape optimization by coupling meshless method with stochastic swarm intelligence based optimization technique for two dimensional linear elastic problems. Element free Galerkin method has been used here for structural analysis to circumvent frequently encountered issues with traditional grid-based technique like FEM in shape optimization such as heavy reliance on quality mesh for accurate solutions needing remeshing due to initial mesh distortion in case of large shape changes, discontinuous secondary field variables across element boundaries needing post-processing techniques and mesh optimization to minimize computational errors. Another distinguishing feature of present work is deployment of gradient-free particle swarm optimization technique for obtaining near optimal solution in shape optimization which eradicates computational efforts and errors associated with sensitivity computation. In this work, for design boundary representation Akima spline has been used due to its better stability against outlier points during shape optimization which generates natural looking boundaries. The performance of proposed technique is validated through numerical examples of shape optimization with behavior constraints on displacement and stress. To demonstrate effectiveness of present technique, results obtained through the proposed technique are compared with other techniques of past literature. To ensure acceptable levels of solution accuracy during shape optimization, h-refinement for initial problem geometry has been carried out as some shapes generated during the optimization process may have low field node density.

Design With SADSF Method and Analyses of Elastic Properties of Torsion-Loaded Double-Tee Section With Torsional Box

The work presents the results of preliminary strength design of a thin-walled structure based on double-tee section loaded with a torsion moment. One of the solutions to this problem is considered, in which the torsional box is introduced in the central part. Then, one constructs a series of solution variants that differ in the torsional box length. In the design one uses the method of statically admissible discontinuous stress fields (SADSF) assuming the condition of equalized equivalent stress in the limit state. The work is complemented with elastic FEM analyses of one of the solution variants. Using this example, one shows good load-carrying properties of structures designed with the SADSF method, and proves that they could be several times better than the properties of structures designed with traditional or intuitive ways.

Direct error in constitutive equation formulation for inverse problems in time harmonic elastography

Shear wave imaging techniques allow the evaluation of rigidity and viscosity of tissues locally within a material. From an inverse problem perspective, the approach is quite attractive insofar as it provides a densely sampled displacement field in the interior of the object from which to invert for material properties. We consider several challenges related to elastic wave inverse problems arising in acoustic radiation force imaging. First, we validate an axisymmetric viscoelasticity model suitable for some applications of acoustic radiation force imaging. Second, we consider reconstructing lateral displacement components from measured axial displacement components. Finally, we present a new variational formulation, the direct error in constitutive equation formulation, for inverse problems in time harmonic viscoelastic wave propagation with full-field data. The formulation relies on minimizing the error in the constitutive equation with a momentum equation constraint. Numerical results on model problems show that the formulation is capable of handling discontinuous and noisy strain fields and also converging with mesh refinement for continuous and discontinuous material property distributions. Applications to MRE and ARFI measured wave data are considered.

A combined momentum-interpolation and advection upstream splitting pressure-correction algorithm for simulation of convective and acoustic transport at all levels of Mach number

A pressure-correction algorithm is presented for compressible fluid flow regimes. It is well-suited to simulate flows at all levels of Mach number with smooth and discontinuous flow field changes, by providing a precise representation of convective transport and acoustic propagation. The co-located finite volume space discretization is used with the AUSM flux splitting. It is demonstrated that two ingredients are essential for obtaining good quality solutions: the presence of an inertia term in the face velocity expression; a velocity difference diffusive term in the face pressure expression, with a correct Mach number scaling to recover the hydrodynamic and acoustic low Mach number limits. To meet these two requirements, a new flux scheme, named MIAU, for Momentum Interpolation with Advection Upstream splitting is proposed.

A solution approach for contact problems based on the dual interpolation boundary face method

• A solution approach for contact problems based on the dual interpolation boundary face method is proposed. • The solution approach is divided into outer and inner iterations. • In the outer iteration , a contact boundary detection scheme is presented to determine the size of the contact zone. • The inner iteration is only performed for frictional contact problems to determine whether the sliding occurs. • The pressure oscillations near the contact boundary can be treated by the proposed approach. The recently proposed dual interpolation boundary face method (DiBFM) has been shown to have a much higher accuracy and improved convergence rates compared with the traditional boundary element method . In addition, the DiBFM has the ability to approximate both continuous and discontinuous fields, and this provides a way to approximate the discontinuous pressure at a contact boundary. This paper presents a solution approach for two dimensional frictionless and frictional contact problems based on the DiBFM. The solution approach is divided into outer and inner iterations. In the outer iteration, the size of the contact zone is determined. Then the elements near the contact boundary are updated to approximate the discontinuous pressure. The inner iteration is used to determine the contact state (sticking or sliding), and is only performed for frictional contact problems. To make the system of equations solvable, the contact constraints and some supplementary equations are also given. Several numerical examples demonstrate the validity and high accuracy of the proposed approach. Furthermore, due to the continuity of elements in DiBFM and the detection of the contact boundary, the pressure oscillations near the contact boundary can be treated.

On the Kinetic Nature of Solar Wind Discontinuities

AbstractThe most intense currents in the solar wind are carried by magnetic field discontinuities. The formation of such structures depends on whether they are rotational or tangential discontinuities. To distinguish between these two types, variations of plasma characteristics should be studied. We analyze data from a set of discontinuities observed by the Acceleration, Reconnection, Turbulence, and Electrodynamics of the Moon's Interaction with the Sun and Magnetospheric Multiscale missions in the near‐Earth solar wind. We show that these discontinuities have properties characteristic of both tangential and rotational discontinuities: (1) Jumps in the tangential velocity component are well correlated with jumps in the Alfven speed, and (2) Electron density and temperature vary significantly across these discontinuities. Suprathermal electron pitch angle distributions change across discontinuities, revealing the importance of electron kinetics to discontinuity structure. Investigation of the discontinuity formation needs to go beyond magnetohydrodynamics theory and highly likely involves both ion and electron kinetics.

A Review of Benchmark Experiments for the Validation of Peridynamics Models

Peridynamics (PD), a non-local generalization of classical continuum mechanics (CCM) allowing for discontinuous displacement fields, provides an attractive framework for the modeling and simulation of fracture mechanics applications. However, PD introduces new model parameters, such as the so-called horizon parameter. The length scale of the horizon is a priori unknown and need to be identified. Moreover, the treatment of the boundary conditions is also problematic due to the non-local nature of PD models. It has thus become crucial to calibrate the new PD parameters and assess the model adequacy based on experimental observations. The objective of the present paper is to review and catalog available experimental setups that have been used to date for the calibration and validation of peridynamics. We have identified and analyzed a total of 39 publications that compare PD-based simulation results with experimental data. In particular, we have systematically reported, whenever possible, either the relative error or the R-squared coefficient. The best correlations were obtained in the case of experiments involving aluminum and steel materials. Experiments based on imaging techniques were also considered. However, images provide large amounts of information and their comparison with simulations is in that case far from trivial. A total of six publications have been identified and summarized that introduce numerical techniques for extracting additional attributes from peridynamics simulations in order to facilitate the comparison against image-based data.

Numerical shape optimization based on meshless method and stochastic optimization technique

This paper puts forward a newer approach for structural shape optimization by combining a meshless method (MM), i.e. element-free Galerkin (EFG) method, with swarm intelligence (SI)-based stochastic ‘zero-order’ search technique, i.e. artificial bee colony (ABC), for 2D linear elastic problems. The proposed combination is extremely beneficial in structural shape optimization because MM, when used for structural analysis in shape optimization, eliminates inherent issues of well-known grid-based numerical techniques (i.e. FEM) such as mesh distortion and subsequent remeshing while handling large shape changes, poor accuracy due to discontinuous secondary field variables across element boundaries needing costly post-processing techniques and grid optimization to minimize computational errors. Population-based stochastic optimization technique such as ABC eliminates computational burden, complexity and errors associated with design sensitivity analysis. For design boundary representation, Akima spline interpolation has been used in the present work owing to its enhanced stability and smoothness over cubic spline. The effectiveness, validity and performance of the proposed technique are established through numerical examples of cantilever beam and fillet geometry in 2D linear elasticity for shape optimization with behavior constraints on displacement and von Mises stress. For both these problems, influence of a number of design variables in shape optimization has also been investigated.

Fields of a Finite Strain Tensor in the Neighborhood of Discontinuity of the Velocity Field of Displacements under Axisymmetric Strain

The problem of the distribution fields of a finite strain tensor in the neighborhood of points of discontinuities of speeds of movements under axisymmetric strain conditions is considered. The Almansi finite strain tensor is a measure of deformation, the motion of points of discontinuities is assumed to be given from the solution of the problems strain bodies taking into account change geometry of the free surface. The relations defining fields of a tensor the finite strains are obtained by integrating the system of equations, binding components of The Almansi finite strain tensor and strain rate tensor along the trajectory of the movement of the material particles. At the same time features of the displacement velocity field are considered in the form of cross points of characteristics of indicial equations which define displacement velocity field (center of the fan of characteristics for a deformation case in axisymmetric deformation of ideal rigid-plastic bodies conditions). The limiting trajectories of the motion of particles contracting to the discontinuity point are considered.

GD3: generalized discrete defect dynamics

A mesoscale model is introduced to study the dynamics of material defects lying at interface junctions. The proposed framework couples the dynamics of discrete dislocation and disclination lines. Disclinations are expected to be natural defects at interface junctions; their presence serving the purpose of accommodating discontinuities in rotation fields at material interface junctions. Crystallography-based rules are proposed to describe the kinematics of disclination motion. A discrete-continuous couple-stress framework, in which discrete defect lines are introduced as plastic eigenstrains and eigencurvatures, is proposed to explicitly follow the dynamics of interfacial defects. The framework is then applied to study left (10bar {1}2right) twin transverse propagation and thickening in magnesium. Focusing first on the case of a twin domain, It is shown that a disclination based representation of twin domains allows for an appropriate mechanistic description of the kinematics of shear transformations. In what concerns twin thickening, the stability of defects at twin interfaces is further studied. To this end, a 3D crater lying on a twin interface is described as a dipole of disclination loops. Upon self-relaxation, it is found that out of plane motion of disclinations followed by the nucleation of twinning dislocations can be activated; thereby showing that conservative non-planar motion of disclinations can be thermodynamically favorable; mechanism that had been postulated some 50 years ago.