A unifying formulation of the discrete and continuum approximations for embedded discontinuities

A methodology for the numerical implementation of embedded discontinuities into the finite element method is developed. This is applicable for the discrete and continuum approximations of discontinuities. The variational formulation of the problem of a solid with discontinuities is established for both approximations, yielding the equations used in this methodology. Three sets of equations are obtained by applying this methodology; all are suitable to be numerically implemented. To show the application potential of this method, the numerical simulation of the formation and propagation of a discontinuity in a concrete specimen is carried out and the results are compared with those from the physical experiment, demonstrating the adequacy of the methodology and its corresponding implementations to model discontinuities. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

On the localization of discontinuities of the first kind for a function of bounded variation

Methods of the localization (detection of positions) of discontinuities of the first kind for a univariate function of bounded variation are constructed and investigated. Instead of an exact function, its approximation in L ₂(?∞,+∞) and the error level are known. We divide the discontinuities into two sets, one of which contains discontinuities with the absolute value of the jump greater than some positive Δ^min; the other set contains discontinuities satisfying a smallness condition for the value of the jump. It is required to find the number of discontinuities in the former set and localize them using the approximately given function and the error level. Since the problem is ill-posed, regularizing algorithms should be used for its solution. Under additional conditions on the exact function, we construct regular methods for the localization of discontinuities and obtain estimates for the accuracy of localization and for the separability threshold, which is another important characteristic of the method. The (order) optimality of the constructed methods on the classes of functions with singularities is established.     

Local edge detectors using a sigmoidal transformation for piecewise smooth data

For piecewise smooth data, edges can be recognized by jump discontinuities in the data. Successful edge detection is essential in digital signal processing as the most relevant information is often observed near the edges in each segmented region. In this paper, using the concentration property of existing local edge detectors and the clustering property of sigmoidal transformations, we provide enhanced edge detectors which diminish the oscillations of the local detector near jump discontinuities as well as highly improve rate of convergence away from the discontinuities. Numerical results of some examples illustrate efficiency of the presented method.     

Space–time discontinuous Galerkin finite element method for shallow water flows

A space–time discontinuous Galerkin (DG) finite element method is presented for the shallow water equations over varying bottom topography. The method results in nonlinear equations per element, which are solved locally by establishing the element communication with a numerical HLLC flux. To deal with spurious oscillations around discontinuities, we employ a dissipation operator only around discontinuities using Krivodonova's discontinuity detector. The numerical scheme is verified by comparing numerical and exact solutions, and validated against a laboratory experiment involving flow through a contraction. We conclude that the method is second order accurate in both space and time for linear polynomials.     

Front tracking for hyperbolic systems

A method of front tracking yields a zero diffusion calculation of fluid interface discontinuities. The method is applied to the problem of petroleum reservoir simulation. Unstable interfaces with multiple fingers can be resolved by this method even on a coarse grid.     

An $h$-Adaptive Runge-Kutta Discontinuous Galerkin Method for Hamilton-Jacobi Equations

In [35, 36], we presented an $h$-adaptive Runge-Kutta discontinuous Galerkin method using troubled-cell indicators for solving hyperbolic conservation laws. A tree data structure (binary tree in one dimension and quadtree in two dimensions) is used to aid storage and neighbor finding. Mesh adaptation is achieved by refining the troubled cells and coarsening the untroubled "children". Extensive numerical tests indicate that the proposed $h$-adaptive method is capable of saving the computational cost and enhancing the resolution near the discontinuities. In this paper, we apply this $h$-adaptive method to solve Hamilton-Jacobi equations, with an objective of enhancing the resolution near the discontinuities of the solution derivatives. One- and two-dimensional numerical examples are shown to illustrate the capability of the method.     

Detecting derivative discontinuity locations in piecewise continuous functions from Fourier spectral data

We present a method that uses Fourier spectral data to locate jump discontinuities in the first derivatives of functions that are continuous with piecewise smooth derivatives. Since Fourier spectral methods yield strong oscillations near jump discontinuities, it is often difficult to distinguish true discontinuities from artificial oscillations. In this paper we show that by incorporating a local difference method into the global derivative jump function approximation, we can reduce oscillations near the derivative jump discontinuities without losing the ability to locate them. We also present an algorithm that successfully locates both simple and derivative jump discontinuities. This work was partially supported by NSF grants CNS 0324957 and DMS 0510813, and NIH grant EB 02553301 (AG).     

A semi-analytical model for dynamic analysis of non-uniform plates

Dynamic properties of the plate structures can be enhanced by introducing discontinuities of different kinds such as using surface-bonded discrete patches or spatially varying the stiffness and mass properties of the plate. Fast and reliable design of such complex structures requires efficient and accurate modeling tools. In this study, a novel semi-analytical model is developed for the dynamic analysis of plates having discrete and/or continuous non-uniformities. Two-dimensional Heaviside unit step functions are utilized to represent the discontinuities. Different from existing numerical methods based on Heaviside functions, a numerical technique is proposed for modeling the discontinuities that are not necessarily aligned with the plate axes. The governing equations are derived using Hamilton's principle and Rayleigh–Ritz method is used for determining the modal variables. The surface-bonded patches are used to demonstrate discrete non-uniformities where variable-stiffness laminates are selected to represent continuous non-uniform structures. Natural frequencies and mode shapes obtained using the proposed method are validated with finite element analyses and the existing results from the literature. The results show that the developed model performs accurately and efficiently.     

Approximating the jump discontinuities of a function by its Fourier-Jacobi coefficients

In the present paper we generalize Eckhoff's method, i.e., the method for approximating the locations of discontinuities and the associated jumps of a piecewise smooth function by means of its Fourier-Chebyshev coefficients.

A new method enables us to approximate the locations of discontinuities and the associated jumps of a discontinuous function, which belongs to a restricted class of the piecewise smooth functions, by means of its Fourier-Jacobi coefficients for arbitrary indices. Approximations to the locations of discontinuities and the associated jumps are found as solutions of algebraic equations. It is shown as well that the locations of discontinuities and the associated jumps are recovered exactly for piecewise constant functions with a finite number of discontinuities.

In addition, we study the accuracy of the approximations and present some numerical examples.

     

A HIGHLY ACCURATE NUMERICAL METHOD FOR FLOW PROBLEMS WITH INTERACTIONS OF DISCONTINUITIES

1. IntroductionCompared with the widely used shock capturing methods for the compressible flows, theshock fitting methods have the main advantage of accuracy. The shock fitting methods areusually very accurate wherever it can be applied. Several kinds of shock fitting methods havebeen developed il1 the last tl1ree decades. Glimrn and his coworkers have worked extensively onthe front tracking methods and applied them to many complicated problems with shocks andother types of singularities [1]-[…     

A General Numerical Method for Solving Riemann Problems

In this paper a general numerical method for solving Riemann problems is discussed. It can be used to solve the Riemann problems of various hyperbolic systems of differential equations with two independent variables. The problem of reflection of discontinuities from external boundaries can also be solved by this general numerical method.     

微分方程右函数间断及其处理方法

§1.引言 现代电气、化工、生物工程和航天飞行器、航空飞行器、核反应工程等都是复杂的系统工程.在这类工程的科学研究和工程设计中,经常需要用常微分方程组的初值问题来建立系统运动行为的数学模型.为了获得研究和设计的数据,需要进行科学工程计算和动     

Dynamically adapted grids for interacting discontinuous solutions

Further development of the dynamic adaptation method for gas dynamics problems that describe multiple interactions of shock waves, rarefaction waves, and contact discontinuities is considered. Using the Woodward-Colella problem and a nonuniformly accelerating piston as examples, the efficiency of the proposed method is demonstrated for the gas dynamics problems with shock wave and contact discontinuity tracking. The grid points are distributed under the control of the diffusion approximation. The choice of the diffusion coefficient for obtaining both quasi-uniform and strongly nonuniform grids for each subdomain of the solution is validated. The interaction between discontinuities is resolved using the Riemann problem for an arbitrary discontinuity. Application of the dynamic adaptation method to the Woodward-Colella problem made it possible to obtain a solution on a grid consisting of 420 cells that is almost identical to the solution obtained using the WENO5m method on a grid consisting of 12 800 cells. In the problem for a nonuniformly accelerating piston, a proper choice of the diffusion coefficient in the transformation functions makes it possible to generate strongly nonuniform grids, which are used to simulate the interaction of a series of shock waves using shock wave and contact discontinuity tracking.     

A numerical approach for the simulation of ductile fracture with embedded discontinuities

In the present study, a computational approach for the numerical simulation of ductile fracture within the framework of the finite element method is proposed. In the developed macroscopic formulation, the inelastic behavior in the bulk of the material is described by the finite elasto‐plastic material model proposed in [4]. The failure process is modeled by introducing discontinuities when a special local fracture criterion is satisfied. The discontinuities are incorporated via special triangular finite elements with embedded interfaces following the line of [2]. Finally, the numerical procedure is evaluated for a twodimensional representative test problem. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

EXISTENCE OF NONNEGATIVE SOLUTIONS FOR DEGENERATE ECOLOGICAL MODELS

In this paper,nonnegative solutions for the degenerate elliptic systems are considered.First,nonnegative solutions for scalar equation with spatial discontinuities are studied. Then themethod developed for scalar equation is applied to study elliptic systems. At last,the existence criteria of nonnegative solutions of elliptic systems are given.     

Discontinuous enrichment in finite elements with a partition of unity method

A technique is presented to model arbitrary discontinuities in the finite element framework by locally enriching a displacement-based approximation through a partition of unity method. This technique allows discontinuities to be represented independently of element boundaries. The method is applied to fracture mechanics, in which crack discontinuities are represented using both a jump function and the asymptotic near-tip fields. As specific examples, we consider cracks and crack growth in two-dimensional elasticity and Mindlin–Reissner plates. A domain form of the J-integral is also derived to extract the moment intensity factors. The accuracy and utility of the method is also discussed.     

Iterative reproducing kernel method for nonlinear oscillator with discontinuity

In this paper, iterative reproducing kernel method is applied to obtain the analytical approximate solution of a nonlinear oscillator with discontinuities. The solution obtained by using the method takes the form of a convergent series with easily computable components. An illustrative example is given to demonstrate the effectiveness of the present method. The results obtained using the scheme presented here show that the numerical scheme is very effective and convenient for the nonlinear oscillator with discontinuities.     

Localization of subsets of discontinuity points of a noisy function

We consider an ill-posed problem of localization of discontinuities of the first kind of a one-dimensional function, when knowing only its approximation and the error level δ in the metric of L ₂(?∞,+∞). We propose a new statement of the problem when all discontinuities are divisible into subsets, and the localization takes place for subsets of discontinuities. Assuming additionally that all discontinuities in each subset have jumps of one sign, we construct a new regular method that allows to determine the number of subsets of discontinuities, to approximate their boundaries, and to estimate the approximation accuracy.     

Regularizing algorithms for detecting discontinuities in ill-posed problems

The problem of detecting singularities (discontinuities of the first kind) of a noisy function in L ₂ is considered. A wide class of regularizing algorithms that can detect discontinuities is constructed. New estimates of accuracy of determining the location of discontinuities are obtained and their optimality in terms of order with respect to the error level δ is proved for some classes of functions with isolated singularities. New upper bounds for the singularity separation threshold are obtained.     

A method for improving the performance of the WENO5 scheme near discontinuities

WENO5 uses a convex combination of the polynomials reconstructed on the three stencils of ENO3 in order to achieve higher accuracy on smooth profiles. However, in some cases WENO5 generates oscillations or smears near discontinuities due to the time scheme used. Here, we present a method to reduce those oscillations without damping and this yields a sharper approximation. Our technique uses smoothness indicators to identify severe shocks and switches from WENO5 to ENO3. Numerical tests show that the behaviour of WENO5 is improved near discontinuities while preserving high accuracy on smooth profiles.