The use of near-front ray expansions in deformation dynamics

A method is proposed for the constructing approximate solutions of problems in impact deformation dynamics in the form of a ray expansion of the solution at the strain discontinuity surface. The fundamental difference in the proposed approximate method is the fact that, when constructing the ray expansions of the solutions, account is taken of the regularities in the change in the curvatures of the surfaces of strain discontinuities and the divergence of the rays. The main qualitative features of the method are illustrated using the example of a one-dimensional cylindrical shock wave.     

On the weak discontinuity surfaces and tapes of equation systems in fluid mechanics

The weak discontinuity surfaces for a system of quasi-linear differential equations of higher order are developed. The classification of equation systems in fluid mechanics is based on the propagative weak discontinuity surfaces. Types of equations for different flow models are discussed. The conclusion is as follows:(a) For incompressible nonviscous flow, incompressible viscous flow and compressible viscous flow, the types of equations are all parabolic in the unsteady case and elliptic in the steady case.(b) For compressible nonviscous flow, the type of equations is hyperbolic in the unsteady case or steady supersonic case, and the type is elliptic in the steady subsonic case.     

模糊数四则运算的交点-间断点法

对模糊数的四则运算提出了一种交点-间断点法:将模糊数的加法、减法和除法运算,化为求两个函数在某区间上其交点处的值;当交点不存在且两函数在该区间上的间断点为有限个时,将其运算化为求一函数在这些间断点处及区间的两端点处的最大值;将模糊数的乘法运算,化为用交点法或间断点法分别在两个区间上求出相应的值,然后取它们中最大者。     

Computation of symbolic dynamics for one-dimensional maps

In this paper we design and implement rigorous algorithms for computing symbolic dynamics for piecewise-monotone-continuous maps of the interval. The algorithms are based on computing forwards and backwards approximations of the boundary, discontinuity and critical points. We explain how to handle the discontinuities in the symbolic dynamics which occur when the computed partition element boundaries are not disjoint. The method is applied to compute the symbolic dynamics and entropy bounds for the return map of the singular limit of a switching system with hysteresis and the forced Van der Pol equation.     

Surfaces of discontinuity and waves in an anisotropic plasma with generalized polytropic equations of state

Summary In this paper, using the theory of the surfaces of discontinuity, we study the non-linear, three-dimensional wave propagation in an anisotropic plasma with generalized polytropic equations of state. A detailed account of the surfaces of weak discontinuity is given, whether they are wavefronts or material surfaces. The specific subjects treated in the paper are indicated by the titles of the Sections. Entrata in Redazione il 28 settembre 1978.     

Determination of Discontinuities of a Function in a Domain with Refraction from Its Attenuated Ray Transform

Some results of numerical investigations are presented for the problem of determination of discontinuities of an unknown function that has the meaning of the internal source distribution and is given in a domain with absorption and refraction, on using the attenuated ray transform of the function. The refraction and the absorption coefficient are assumed to be given. The behavior of the available and newly constructed discontinuity indicator operators is investigated in some numerical tests. Some modification of discontinuity indicators was carried out for the purpose of applying them in the model of refractive tomography with absorption. Numerical methods are applied to investigate the possibility of using the operators of this kind for solving the problem of finding the discontinuities of a function from its attenuated ray transform; the degree is investigated of the influence on the recovery quality of such factors as the level of the noise introduced into the generated data, the parameters of metrics, the magnitude and variation of the absorption coefficient.     

Effect of discontinuity in density and conductivity on the hydromagnetic stability of a two-dimensional jet

The hydromagnetic stability of an incompressible, inviscid, and two-dimensional jet in the presence of a discontinuity in density and conductivity across the jet surfaces is studied against small and large wavelength disturbances. In the equilibrium state the magnetic field is uniform and aligned with the direction of jet velocity. Study of several different cases shows that discontinuity in density or conductivity or in the product of density and conductivity across the jet surfaces has a stabilizing effect on the jet.     

On calculating the second term of the series of the ray method for the vector of longitudinal displacements in isotropic nonhomogeneous elastic media

This paper contiues the series of papers devoted to designing algorithms for computing the second and subsequent terms of the ray series for the vector of longitudinal displacements in isotropic nonhomogeneous elastic media. The method proposed in this paper essentially differs from the previous ones. It is based on expanding the amplitudes, the eikonal, and the given problem parameters into power series with respect to the coordinates transversal to the direction of propagation of the waves considered. The methods for computing the correction term of the ray expansion for the vector of longitudinal displacements are compared. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 218, 1994, pp. 25–43.     

Solution of the generalized diffraction problem

One solves a mixed boundary-value problem for an elliptic equation with linear principal terms. On some parts of the boundary one imposes the first boundary condition and on the other parts a condition with oblique derivatives. The leading coefficients of the equation may have discontinuities of the first kind on surfaces which partition the initial domain. On the discontinuity surfaces one imposes special matching conditions for the solutions. One establishes the existence of the solution of the problem with restrictions on the boundary and on the discontinuity surfaces. One proves the existence of the second generalized derivatives of the solution in each of the subdomains.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 172–183, 1984.     

Diffraction of whispering gallery waves by a conjunction line

A nonhomogeneous elastic body with stress-free boundary is considered. The boundary consists of two smooth cylindrical surfaces with continuous tangent plane. The curvature of the boundary is assumed to have a discontinuity on the conjunction line. The behavior of two kinds of transversal surface whispering gallery waves passing through the conjunction line is studied. For a wave of the first kind, corresponding to the Dirichlet boundary condition, the displacement vector is normal to the boundary. For a wave of the second kind, corresponding to the Neumann boundary condition, the displacement vector is tangent to the boundary and normal to the ray, which is similar to the case of Love waves.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 95–109.This work was supported by the Russian Foundation for Basic Research under grant No. 96-01-00666.     

Plane wave refraction on convex and concave angles in geometric acoustics approximation

An exact analytical solution to the eikonal equation for a plane wave refracted on a boundary comprising both convex and concave obtuse angles has been built. Under the convex angle summit the solution has a line of discontinuity in the ray vector field and in the first derivatives of the first arrival times, and under the concave angle it has a cone of waves diffracted on this angle. This cone corresponds to the Keller diffraction cone in the geometric diffraction theory. The relation between the eikonal equation and the resultant Hamilton–Jacoby equation for arrival times of downward waves and the ray parameter conservation equation is investigated. Solutions to these equations coincide for pre-critical incidence angles and differ for super-critical angles. It is shown that the arrival times of maximum amplitude waves, which are of the greatest practical interest, coincide with the times calculated from the ray parameter field for the ray parameter conservation equation. The numerical algorithm proposed for calculation of these times can be used for arbitrary speed models.     

Discontinuous solutions to hyperbolic systems under operator splitting

Two-dimensional systems of linear hyperbolic equations are studied with regard to their behavior under a solution strategy that in alternate time-steps exactly solves the component one-dimensional operators. The initial data is a step function across an oblique discontinuity. The manner in which this discontinuity breaks up under repeated applications of the split operator is analyzed, and it is shown that the split solution will fail to match the true solution in any case where the two operators do not share all their eigenvectors. The special case of the fluid flow equations is analyzed in more detail, and it is shown that arbitrary initial data gives rise to “pseudo acoustic waves” and a nonphysical stationary wave. The implications of these findings for the design of high-resolution computing schemes are discussed.     

基于离散网格的流形曲面构造综述

基于离散网格的流形曲面构造技术不仅能够生成具有高阶光滑性的曲面,并且该曲面可以是任意拓扑结构的.此外,在构造流形曲面时,无需进行额外的拼接操作,克服了传统曲面造型技术在进行面片之间的拼接时,计算量增大以及曲面光滑性难以保证的难题.本文介绍了流形曲面构造的流程以及构造过程中的难点,然后将目前已有的流形曲面构造技术分为三大类:传统意义上的流形构造方法;基于规范区域的流形构造方法;基于样条曲面推广的流形构造方法.并对每一类都进行详细地分类介绍.最后,对其作一个总结以及对未来的展望.     

A minimum-stencil difference scheme for computing two-dimensional axisymmetric gas flows: Examples of pulsating flows with instabilities

A minimum-stencil difference scheme for computing two-dimensional axisymmetric gas flows is described. The scheme is explicit, conservative, and second-order accurate in space and time. The numerical results obtained for pulsating flows and contact discontinuity instabilities are discussed. The mechanisms of flow pulsation and instability generation are described.     

Distributional analysis for discontinuous fields

The jump conditions that hold across singular surfaces for the fields having step function discontinuities do not, in general, apply if these surfaces themselves carry concentrated fields. In this note, the general situation when the surfaces of discontinuity carry multilayers and deform as they propagate is discussed. Formulas are presented for the first and second derivatives for these multilayers.     

Finite element analysis of pressure vessel using beam on elastic foundation analysis

In this study, we first applied the variation principle to derive a new finite element method (FEM) based on the theory of beam on elastic foundation using line element. The derived FEM was then applied to solve, for the first time, the pressure vessel problems with uniform thickness. Our FEM results, obtained even by using only one line element, agreed exactly with the available closed-form solution, confirming the validity and computing efficiency of our finite element formulation. Moreover, we have applied our new FEM to solve pressure vessel problems with non-uniform thickness where no exact analytical solution is known to exist. The distributions of discontinuity stress in the cylindrical part were obtained. We found that shear force and bending moment were indeed discontinuous at the geometrically discontinuous juncture, due to the bending rigidity and elastic constant change by the non-uniform thickness. Finally, the case of discontinuity stresses in a bimetallic joint was also studied. The locations of maximum shear force and bending moment were found to be affected by the bending rigidity of the material.     

Analysis of arbitrarily shaped planar cracks in two-dimensional hexagonal quasicrystals with thermal effects. Part I: Theoretical solutions

An extended displacement discontinuity (EDD) boundary integral equation method is proposed for analysis of arbitrarily shaped planar cracks in two-dimensional (2D) hexagonal quasicrystals (QCs) with thermal effects. The EDDs include the phonon and phason displacement discontinuities and the temperature discontinuity on the crack surface. Green's functions for unit point EDDs in an infinite three-dimensional medium of 2D hexagonal QC are derived using the Hankel transform method. Based on the Green's functions and the superposition theorem, the EDD boundary integral equations for an arbitrarily shaped planar crack in an infinite 2D hexagonal QC body are established. Using the EDD boundary integral equation method, the asymptotic behavior along the crack front is studied and the classical singular index of 1/2 is obtained at the crack edge. The extended stress intensity factors are expressed in terms of the EDDs across crack surfaces. Finally, the energy release rate is obtained using the definitions of the stress intensity factors.     

Optimization Problem Coupled with Differential Equations: A Numerical Algorithm Mixing an Interior-Point Method and Event Detection

The numerical analysis of a dynamic constrained optimization problem is presented. It consists of a global minimization problem that is coupled with a system of ordinary differential equations. The activation and the deactivation of inequality constraints induce discontinuity points in the time evolution. A numerical method based on an operator splitting scheme and a fixed point algorithm is advocated. The ordinary differential equations are approximated by the Crank-Nicolson scheme, while a primal-dual interior-point method with warm-starts is used to solve the minimization problem. The computation of the discontinuity points is based on geometric arguments, extrapolation polynomials and sensitivity analysis. Second order convergence of the method is proved when an inequality constraint is activated. Numerical results for atmospheric particles confirm the theoretical investigations.     

On quasi‐local inversion of spiral CT data

Let be a function supported in an open set , which is located strictly inside the spiral trajectory of the X‐ray source. Proposed is an algorithm for computing a function which retains all the sharp features of . More precisely, if , then . The algorithm is quasi‐local in the sense that for each reconstruction point one uses the cone beam data corresponding to X‐ray sources located close to . The detector array is two‐dimensional, but of limited extent along the axis of the spiral. Stability of the algorithm is studied and results of numerical testing on the simulated data are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

A boundary element analysis for stress intensity factors of multiple circular arc cracks in a plane elasticity plate

In this paper, a numerical approach for analyzing interacting multiple cracks in infinite linear elastic media is presented. By extending Bueckner’s principle suited for a crack to a general system containing multiple interacting cracks, the original problem is divided into a homogeneous problem (the one without cracks) subjected to remote loads and a multiple crack problem in an unloaded body with applied tractions on the crack surfaces. Thus, the results in terms of the stress intensity factors (SIFs) can be obtained by considering the latter problem, which is analyzed easily by means of the displacement discontinuity method with crack-tip elements proposed recently by the author. Test examples are given to illustrate that the numerical approach is very accurate for analyzing interacting multiple cracks in an infinite linear elastic media under remote uniform stresses. In addition, the displacement discontinuity method with crack-tip elements is used to analyze a multiple crack problem in a finite plate. It is found that the boundary element method is also very accurate for investigating interacting multiple cracks in a finite plate. Specially, a generalization of Bueckner’s principle and the displacement discontinuity method with crack-tip elements are used to analyze multiple circular arc crack problems in infinite plate in tension (including: Two Collinear Circular Arc Cracks, Three Collinear Circular Arc Cracks, Two Parallel Circular Arc Cracks, Three Parallel Circular Arc Cracks and Two Circular Arc Cracks) in a plane elasticity plate. Many results are given.