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.     

An approximation result for free discontinuity functionals by means of non‐local energies

We approximate, in the sense of Γ‐convergence, free discontinuity functionals with linear growth by a sequence of non‐local integral functionals depending on the average of the gradient on small balls. The result extends to a higher dimension what is already proved in (Ann. Mat. Pura Appl. 2007; 186 (4): 722–744), where there is the proof of the general one‐dimensional case, and in (ESAIM Control Optim. Calc. Var. 2007; 13 (1):135–162), where the n‐dimensional case with ?=Id is treated. Moreover, we investigate whether it is possible to approximate a given free discontinuity functional by means of non‐local energies. Copyright © 2008 John Wiley & Sons, Ltd.     

Modified kinetic flux vector splitting schemes for compressible flows

We investigate the traditional kinetic flux vector splitting (KFVS) and BGK schemes for the compressible Euler equations. First, based on a careful study of the behavior of the discrete physical variables across the contact discontinuity, we analyze quantitatively the mechanism of inducing spurious oscillations of the velocity and pressure in the vicinity of the contact discontinuity for the first-order KFVS and BGK schemes. Then, with the help of this analysis, we propose a first-order modified KFVS (MKFVS) scheme which is oscillation-free in the vicinity of the contact discontinuity, provided certain consistent conditions are satisfied. Moreover, by using piecewise linear reconstruction and van Leer’s limiter, the first-order MKFVS scheme is extended to a second-order one, consequently, a nonoscillatory second-order MKFVS scheme is constructed. Finally, by combing the MKFVS schemes with the γ-model, we successfully extend the MKFVS schemes to multi-flows, and propose therefore a first- and second-order MKFVS schemes for multi-fluid computations, which are nonoscillatory across fluid interfaces. A number of numerical examples presented in this paper validate the theoretic analysis and demonstrate the good performance of the MKFVS schemes in simulation of contact discontinuities for both single- and multi-fluids.     

Forced Burgers Equation in an Unbounded Domain

The inviscid Burgers equation with random and spatially smooth forcing is considered in the limit when the size of the system tends to infinity. For the one-dimensional problem, it is shown both theoretically and numerically that many of the features of the space-periodic case carry over to infinite domains as intermediate time asymptotics. In particular, for large time T we introduce the concept of T-global shocks replacing the notion of main shock which was considered earlier in the periodic case (1997, E et al., Phys. Rev. Lett. 78, 1904). In the case of spatially extended systems these objects are no anymore global. They can be defined only for a given time scale and their spatial density behaves as (T)T ^–2/3 for large T. The probability density function p(A) of the age A of shocks behaves asymptotically as A ^–5/3. We also suggest a simple statistical model for the dynamics and interaction of shocks and discuss an analogy with the problem of distribution of instability islands for a simple first-order stochastic differential equation.     

Modification of Lytle’s theory: Interpretation of the extended fine structure associated with LIII absorption discontinuity of some ytterbium systems

In Lytle’s theory for the extended fine structure in x-ray absorption spectra, the potential at the boundary of the ‘equivalent sphere’ around the absorbing atom, having volume equal to that of the Wigner-Seitz cell is considered to be infinite. It has been observed that Lytle’s theory is applicable only in the case of metals and metallic systems. In the present paper the extended fine structure associated with the L_III absorption spectra of some systems of ytterbium is interpreted on the basis of Lytle’s model, modified by using a finite potential instead of an infinite one at the boundary of the equivalent sphere. The values of this potential are estimated for eight systems of ytterbium. It has been shown that there exists a correlation between the potentials and covalency of the compounds.     

Transformation of hanging discontinuities into vortex systems in a stratified flow behind a cylinder

The mechanisms of formation of hanging discontinuities, vortex dipoles, and vortex arrays in the wave wake behind a cylinder moving at a constant velocity in a stratified fluid are investigated using various schlieren methods. The existence of discontinuities is attributable to the distortion of the internal-wave phase pattern in the shear flow and to the varying stratification and subsequent interaction of the waves with the appearing nonuniformities. Hanging discontinuities and vortex systems are low-velocity analogs of shock waves. An analysis of the internal-wave pattern indicates that the values of the normal velocity component differ on the upper and lower edges of the discontinuities. A regime diagram for flows of this kind is given.     

Modeling quasi-static crack growth with the extended finite element method Part I: Computer implementation

The extended finite element method (X-FEM) is a numerical method for modeling strong (displacement) as well as weak (strain) discontinuities within a standard finite element framework. In the X-FEM, special functions are added to the finite element approximation using the framework of partition of unity. For crack modeling in isotropic linear elasticity, a discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are used to account for the crack. This enables the domain to be modeled by finite elements without explicitly meshing the crack surfaces, and hence quasi-static crack propagation simulations can be carried out without remeshing. In this paper, we discuss some of the key issues in the X-FEM and describe its implementation within a general-purpose finite element code. The finite element program Dynaflow™ is considered in this study and the implementation for modeling 2-d cracks in isotropic and bimaterial media is described. In particular, the array-allocation for enriched degrees of freedom, use of geometric-based queries for carrying out nodal enrichment and mesh partitioning, and the assembly procedure for the discrete equations are presented. We place particular emphasis on the design of a computer code to enable the modeling of discontinuous phenomena within a finite element framework.     

On a new centered strategy to control the accuracy of weighted essentially non oscillatory algorithm for conservation laws close to discontinuities

In this paper we try to redesign an algorithm analyzed in previous works by Amat et al. in order to obtain a centered strategy aimed to approximate the solution of linear and nonlinear systems of conservation laws. We perform the analysis of the new algorithm in order to explain why the new strategy works close to shocks in the solution of conservation laws. The objective is to force the use of the most centered stencil of those used whenever the data are affected by the presence of a high gradient.     

Generalized Modal Scattering Matrix of Bandpass Filters Using Ridged Waveguides

Simple and accurate integral equation formulation using a three dimensional finite element (3DFEM) to determine the propagation of microwave passive structures is presented. This technique is applied to solve the scattering problems due to various forms of irises in lossless rectangular waveguide.The analysis of these components is highly dependent of the numbers of modes considered in the neighbourhood of the discontinuity. The theoretical model is compared with measurements and Mode Matching Method (MMM). Results are found to be good agreement.