A new shock/discontinuity detector

A new effective shock/discontinuity detector has been developed in this work. The detector has two steps. The first step is to check the ratio of the truncation errors on the coarse and fine grids, and the second step is to check the local ratio of the left- and right-hand slopes. The currently popular shock/discontinuity detectors such as Harten's, Jameson's and WENO can detect shock, but mistake high-frequency waves and critical points as shock and then damp the physically important high-frequency waves. Preliminary results show that the new shock/discontinuity detector is very delicate and can detect all shocks including strong, weak and oblique shocks or discontinuity in function and first-, second- and third-order derivatives without artificial case-related constants, but never mistake high-frequency waves and critical points or expansion waves as shock. This will overcome the bottleneck problem with numerical simulation for the shock–boundary layer interaction, shock–acoustic interaction, image process, porous media flow, multiple phase flow, detonation wave, and anywhere the high-frequency waves are important, but discontinuity exists and is mixed with high-frequency waves. After detecting the shock, we can then use one-side high-order scheme for shocks and high-order central compact scheme for the smooth parts if the shock is appropriately located. Then a high-order universal subroutine for the finite difference method is developed, which can be used for any finite difference code for accurate numerical derivatives.     

A general scheme for discontinuity detection

This paper is concerned with the initial segmentation step of locating the dominant discontinuities in a variety of image ensembles. Using vector representation for the image ensembles, we model the image vectors over a small neighborhood as an outcome of a linear transformation operating on the image vector at the center of the neighborhood. It is shown that such a modelling leads to a two-stage discontinuity detection process. During the first state, a measure of similarity is computed between the neighboring image vectors using the normalized inner product. These similarity values are then combined through two masks to obtain a measure of the discontinuity present at each location. Such a general formulation makes the proposed discontinuity detector capable of handling a variety of images.     

An embedded strong discontinuity model for cracking of plain concrete

A numerical model formulated within the framework of a nonsymmetric strong discontinuity approach for fracture simulations of plain concrete is presented. The model is based on the fixed crack concept and makes use of the concept of the elements with embedded discontinuities. Discontinuity segments of individual elements are considered to form a C⁰-continuous path. Enforcement of continuity of the crack path across adjacent elements is established by a partial domain crack tracking algorithm. Orientation of individual crack segments is derived from a nonlocal strain field. The capabilities of the model are shown by means of numerical examples.     

A treatment of discontinuities for nonlinear systems with linearly degenerate fields

In this paper we propose an artificial compression technique to avoid the numerical diffusion that standard numerical methods present in contact discontinuities. The main idea is to replace contact discontinuities by shocks. For nonlinear 1D systems we replace locally linearly degenerate fields by genuinely nonlinear fields, in such a way the solution does not vary. We apply this technique to a family of numerical schemes and we deduce that this can be seen as a discretization of the system modified by a new term, when we are in a jump of a contact discontinuity. We have also extended this technique for the multidimensional case. We prove by applying the artificial compression technique that the numerical scheme is stable under the same CFL condition. We also present different numerical schemes: Sod’s problem for 1D Euler equations, transport of a discontinuity, a stationary contact discontinuity and in the multidimensional case the transversal transport of two different geometries. We observe that in all cases the numerical diffusion is reduced.     

2D-discontinuity detection from scattered data

We describe a numerical approach for the detection of discontinuities of a two dimensional function distorted by noise. This problem arises in many applications as computer vision, geology, signal processing. The method we propose is based on the two-dimensional continuous wavelet transform and follows partially the ideas developed in [2], [6] and [8]. It is well-known that the wavelet transform modulus maxima locate the discontinuity points and the sharp variation points as well. Here we propose a statistical test which, for a suitable scale value, allows us to decide if a wavelet transform modulus maximum corresponds to a function value discontinuity. Then we provide an algorithm to detect the discontinuity curves fromscattered and noisy data.     

A front tracking method on unstructured grids

A numerical method is developed for tracking discontinuities which is integrated in a generalized finite-volume solution framework for systems of conservation laws on unstructured grids of arbitrary element type. The location, geometry and the movement of the discontinuities are described by a local level set method on a restricted, dynamic definition range. Special algorithms based on least square methods are developed for handling the transport and renormalization of the level set function within the restricted range. An additional error correction is employed to minimize topological errors of the tracked front geometry. The jump conditions at the front are updated by one-sided extrapolation which define the local front velocity and the Riemann problem. A flux separation concept enables the treatment of the discontinuity within the finite-volume concept. The front tracking method is demonstrated by a number of computational examples for shock wave problems.     

Efficient integration over discontinuities for differential-algebraic systems

In recent years, hybrid (discrete/continuous) dynamic systems that exhibit coupled continuous and discrete behavior have attracted much attention. Many engineering problems can be formulated as ODE/DAEs which can be solved by numerical methods. In process design, dynamic optimization and simulation, however, many systems of interest experience significant discontinuities during transients. This paper describes some simple strategies for discontinuity detecting and handling in DAE embedded systems. The described algorithm supports flexible representation of state conditions in propositional logic. By making full use of the discontinuity functions, both efficiency of integration and effectiveness of discontinuity detection are achieved. Considerable care is taken to handle problems that can arise during mode transitions. We outline the implementation in a general-purpose solver DASPKE for differential-algebraic equations containing discontinuities, and present some numerical experiments which illustrate its effectiveness.     

Weak discontinuity annihilation in solidification modelling

Discontinuous behaviour provides substantial obstacles to the efficient application of mesh based numerical techniques. Accounting for strong discontinuities is presently of particular interest to the finite element research community with for example the development of cohesive and enriched elements to cater for material separation. Although strong discontinuities are of importance, of equal if not of greater interest and the focus in this paper, are weak discontinuities, which are present at any material change. A recent innovation for accounting for weak discontinuities has been the discovery of non-physical variables which are founded and defined using transport equations.This paper is concerned with the application of the non-physical approach to solidification modelling in the presence of more than one material discontinuity. A typical feature of the enthalpy-temperature response in solidification is discontinuities at phase transition temperatures as a consequence of phase change and latent heat release. In these circumstances, depending on the conditions that prevail, an element in a finite element mesh can have more than one discontinuity present.Presented in the paper is a methodology that can cater for multiple discontinuities. The non-physical approach permits the precise removal of weak discontinuities arising in the governing transport equations. In order to facilitate the application of the approach the finite element equations are presented in the form of weighted transport equations. The method utilises a non-physical form of enthalpy that possesses a remarkable source distribution like property at a discontinuity. It is demonstrated in the paper that it is through this property that multiple discontinuities can be exactly removed from an element so facilitating the use of continuous approximations.The new methodology is applied to a range of simple problems to provide an in-depth treatment and for ease of understanding to demonstrate the methods remarkable accuracy and stability.     

Discontinuity meshing for accurate radiosity

An algorithm for compactly and accurately capturing the illumination of a diffuse polyhedral environment caused by an area light source is presented. The algorithm constructs a discontinuity mesh that explicitly represents discontinuities in the radiance function as boundaries between mesh elements. A piecewise quadratic interpolant is used to approximate the radiance function, preserving the discontinuities associated with the edges in the mesh     

Interactions due to body forces in generalized thermo-elasticity III

The generalized thermo-elasticity theory III is employed to study thermo-elastic interactions in a homogeneous isotropic unbounded solid due to distributed continuous and instantaneous body forces. The solutions are derived by using a Laplace transform on time and then a Fourier transform on space. It is found that the interactions consist of a wave part traveling with the speed of the dilatational wave and a diffusive part. For continuous body forces, both temperature and deformation are continuous at the elastic dilatational wave front, while the stress suffers finite discontinuity at this location. For instantaneous body forces, both deformation and temperature suffer finite discontinuities at the elastic wave front, while stress exhibits delta function discontinuity resulting from the Dirac delta function at this location. All the fields suffer exponential attenuation at the elastic wave front and the attenuation is influenced by thermo-elastic coupling and thermal diffusivity of the medium. The results achieved in the present analysis are compared to those obtained by using generalized thermo-elasticity theory II without energy dissipation and other generalized theories. Lastly, numerical results applicable to a copper-like material are presented in order to illustrate the analytical result.     

Structure from Motion: Beyond the Epipolar Constraint

The classic approach to structure from motion entails a clear separation between motion estimation and structure estimation and between two-dimensional (2D) and three-dimensional (3D) information. For the recovery of the rigid transformation between different views only 2D image measurements are used. To have available enough information, most existing techniques are based on the intermediate computation of optical flow which, however, poses a problem at the locations of depth discontinuities. If we knew where depth discontinuities were, we could (using a multitude of approaches based on smoothness constraints) accurately estimate flow values for image patches corresponding to smooth scene patches; but to know the discontinuities requires solving the structure from motion problem first. This paper introduces a novel approach to structure from motion which addresses the processes of smoothing, 3D motion and structure estimation in a synergistic manner. It provides an algorithm for estimating the transformation between two views obtained by either a calibrated or uncalibrated camera. The results of the estimation are then utilized to perform a reconstruction of the scene from a short sequence of images.The technique is based on constraints on image derivatives which involve the 3D motion and shape of the scene, leading to a geometric and statistical estimation problem. The interaction between 3D motion and shape allows us to estimate the 3D motion while at the same time segmenting the scene. If we use a wrong 3D motion estimate to compute depth, we obtain a distorted version of the depth function. The distortion, however, is such that the worse the motion estimate, the more likely we are to obtain depth estimates that vary locally more than the correct ones. Since local variability of depth is due either to the existence of a discontinuity or to a wrong 3D motion estimate, being able to differentiate between these two cases provides the correct motion, which yields the least varying estimated depth as well as the image locations of scene discontinuities. We analyze the new constraints, show their relationship to the minimization of the epipolar constraint, and present experimental results using real image sequences that indicate the robustness of the method.     

Solution to the Riemann problem in a one-dimensional magnetogasdynamic flow

The Riemann problem for a quasilinear hyperbolic system of equations governing the one-dimensional unsteady flow of an inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field, is solved approximately. This class of equations includes as a special case the Euler equations of gasdynamics. It has been observed that in contrast to the gasdynamic case, the pressure varies across the contact discontinuity. The iterative procedure is used to find the densities between the left acoustic wave and the right contact discontinuity and between the right contact discontinuity and the right acoustic wave, respectively. All other quantities follow directly throughout the (x, t)-plane, except within rarefaction waves, where an extra iterative procedure is used along with a Gaussian quadrature rule to find particle velocity; indeed, the determination of the particle velocity involves numerical integration when the magneto-acoustic wave is a rarefaction wave. Lastly, we discuss numerical examples and study the solution influenced by the magnetic field.     

Radiosity in Flatland

The radiosity method for the simulation of interreflection of light between diffuse surfaces is such a common image synthesis technique that its derivation is worthy of study. We here examine the radiosity method in a two dimensional, flatland world. It is shown that the radiosity method is a simple finite element method for the solution of the integral equation governing global illumination. These two-dimensional studies help explain the radiosity method in general and suggest a number of improvements to existing algorithms. In particular, radiosity solutions can be improved using a priori discontinuity meshing, placing mesh boundaries on discontinuities such as shadow edges. When discontinuity meshing is used along with piecewise-linear approximations instead of the current piecewise-constant approximations, the accuracy of radiosity simulations can be greatly increased.     

An improved deconvolution method for bremsstrahlung spectra from hot plasmas

The influence of tip region cross sections and the energy resolution of X-ray detectors on the deconvolution of X-ray spectra from hot plasmas into the electronic distribution function is discussed. From the discussion an improved deconvolution method is derived starting from an integral equation of the second kind. Here numerical discontinuities can be handled by a differentiation process, the reliability of which increases with increasing refinement of the measurements.     

A node-centric indirect boundary element method: three-dimensional displacement discontinuities

An indirect boundary element formulation based on unknown physical values, defined only at the nodes (vertices) of a boundary discretization of a linear elastic continuum, is introduced. As an adaptation of this general framework, a linear displacement discontinuity density distribution using a flat triangular boundary discretization is considered. A unified element integration methodology based on the continuation principle is introduced to handle regular as well as near-singular and singular integrals. The boundary functions that form the basis of the integration methodology are derived and tabulated in the appendix for linear displacement discontinuity densities. The integration of the boundary functions is performed numerically using an adaptive algorithm which ensures a specified numerical accuracy. The applications include verification examples which have closed-form analytical solutions as well as practical problems arising in rock engineering. The node-centric displacement discontinuity method is shown to be numerically efficient and robust for such problems.     

Noniterative Algorithms for Sensitivity Analysis Attacks

Sensitivity analysis attacks constitute a powerful family of watermark "removal" attacks. They exploit vulnerability in some watermarking protocols: the attacker's unlimited access to the watermark detector. This paper proposes a mathematical framework for designing sensitivity analysis attacks and focuses on additive spread-spectrum embedding schemes. The detectors under attack range in complexity from basic correlation detectors to normalized correlation detectors and maximum-likelihood (ML) detectors. The new algorithms precisely estimate and then eliminate the watermark from the watermarked signal. This is accomplished by exploiting geometric properties of the detection boundary and the information leaked by the detector. Several important extensions are presented, including the case of a partially unknown detection function, and the case of constrained detector inputs. In contrast with previous art, our algorithms are noniterative and require, at most, O(n) detection operations in order to precisely estimate the watermark, where n is the dimension of the signal. The cost of each detection operation is O(n); hence, the algorithms can be executed in quadratic time. The method is illustrated with an application to image watermarking using an ML detector based on a generalized Gaussian model for images     

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.     

A novel algorithmic framework for the numerical implementation of locally embedded strong discontinuities

In this paper, a novel algorithmic framework for the numerical modeling of locally embedded strong discontinuities suitable for the analysis of material failure such as cracking in brittle structures or shear bands in soils is proposed. Based on the enhanced assumed strain (EAS) concept, the final failure kinematics of solids, approximated by discontinuous displacement fields, are incorporated into the finite element formulation. In contrast to previous works, the discontinuous part of the deformation mapping is condensed out at the material level without employing the standard static condensation technique. As a consequence, the resulting constitutive equations are formally identical to those of standard plasticity models. Hence, the return-mapping algorithm is applied to the numerical implementation. Only slight modifications of this, by now classical, algorithm are necessary. The proposed framework is applicable to any constitutive equation characterizing the inelastic part of deformation. Referring to the yield (failure) function and the evolution equations, no special assumption has to be made. Despite the differences between the presented numerical framework and the original implementation of the strong discontinuity approach (SDA), it is shown that both finite element models are completely equivalent. The applicability of the novel algorithmic formulation and its numerical performance are investigated by means of two two-dimensional as well as by means of a fully three-dimensional numerical analysis of shear band formation.