二维浅水波方程的间断周期解和间断孤立波解讨论

本文讨论地球流体运动浅水波模式中间断周期解与间断孤立波解.在系统的非平衡点即奇点附近考虑轨线性质时,我们发现只要引入广义解的概念(分片光滑连续解),就会产生间断周期解并得到了间断周期解的条件.当系统在退化的过程中,发现系统此时会产生间断的孤立波解,与此同时其它物理量也产生了间断.这里我们发现,一般认为在超高速情况下解会产生间断,然而在非超高速时也会产生间断现象.本文讨论了上述一系列问题得到了间断解的解析解表达式,并把这一事实与飑线的实例进行比较,得到了不少类似之处.     

连续半鞅随机微分方程解的性质

本文在扩散系数可退化可间断的情况下,讨论了一维连续半鞅型随机微分方程解的非合流性质和强比较定理。     

一维随机微分方程强解的某些结果

利用局部时及估计得到了连续半鞅型随机微分方程强解比较定理与存在唯一性定理,方程的系数可以退化可以间断。     

Sobolev方程的时间间断Galerkin有限元方法

引入Sobolev方程的等价积分方程,构造Sobolev方程的新的时间间断Galerkin有限元格式.该格式不仅保持有限元解在时间剖分点处的间断特性,而且避免了传统时空有限元格式中跳跃项的出现,从而降低了格式理论分析和数值模拟的复杂性.证明了Sobolev方程的时间间断而空间连续的时空有限元解的稳定性、存在唯一性、L2...     

一类具有转点的右端不连续奇摄动边值问题

研究了一类具有转点的右端不连续二阶半线性奇摄动边值问题解的渐近性.首先,在间断处将原问题分为左右两个问题,通过修正左问题退化问题的正则化方程,提高了左问题渐近解的精度,并利用Nagumo定理证明了左问题光滑解的存在性.其次,证明了右问题具有空间对照结构的解,并通过在间断点的光滑缝接,得到了原问题的渐近解.最后,通过一个算例验证了结果的正确性.     

磁流体力学方程组的激波生成

研究磁流体横向流动的一维模型,在解的强间断出现后流场的性质。利用迭代法具体构造了该方程组的强间断—激波以及问题的熵解。同时,利用激波的性质,给出了各物理参量在爆破点附近的奇性估计。     

一类半线性对称双曲组的间断初值问题

本文证明了一类半线性对称双曲组的间断初值问题局部强解的存在性，同时得到解的光滑性，推广了Metivier及其他人关于严格双曲组的结果．     

带径向接管的圆柱壳端部受力矩作用时的薄壳理论解

给出可适用于大开孔率 ( ρ0 ≤ 0 .8)的带径向接管的圆柱壳受端部力矩作用时的薄壳理论解 .采用修正的Morley方程代替前人使用的Donnell扁壳方程 ,在主壳展开面极坐标 (α ,β)系中求解开孔圆柱壳齐次解 ,既保持了薄壳理论的精度量级 ,又不受开孔率的限制 ;利用Goldenveizer的位移函数圆柱壳方程 ,在支管展开面主坐标 ( ζ ,θ)系中求得非平齐端头支管的齐次解 .主壳与支管交界处的边界位移和边界力分别由 (α ,β)、( ζ ,θ)系转换到总体柱坐标 ( ρ,θ ,z)系后均为θ的周期函数 ,因而可分别展成Fourier级数 ,各谐的Fourier系数可利用数值积分获得 ,再由连续条件得到整体结构的薄壳理论解 .经前人的实验和三维有限元计算结果的检验 ,证明解是可靠的 .     

合成展开法求解圆薄板大挠度问题

本文用合成展开摄动法,把外场解和内层解结合起来,求解圆薄板大挠度问题.本文把Hencky的薄膜解当作外场解的一级近似解,并求出了外场解的二级近似解.利用边界内层坐标,求得了相应的各级内层解,即边界层解.本文采用最大位移和板厚之比的倒数作为小参数,所得结果大大改进了1948年作者所得的结果.     

对流扩散方程的间断时空有限元方法的误差估计

研究了一类线性对流扩散方程的间断时空有限元方法,即空间连续,时间允许间断的时空有限元方法.将有限元方法和有限差分方法相结合,在每一时间层上充分利用Lagrange插值多项式在Radau点处的特性,给出了有限元解的最优阶L∞(L2)模误差估计.     

Numerical Simulation of Crack Growth

In recent years the X-FEM based on the partition of unity method and the strong discontinuity approach (SDA) have shown to be powerful tools to model crack growth. Both methods model the crack surface by introducing additional d.o.f.. In the X-FEM the nodes in the mesh around a crack are globally enhanced with new d.o.f. while in in the SDA the new d.o.f. are commonly introduced as internal ones. Thus the jump displacement fields are constant across elements. Therefore the d.o.f. can be condensed on element level which results in jumps in the displacement field at element edges. In this contribution the strong discontinuity approach is used approximating the displacement jump linearly across the crack length similar as e.g. in [3]. New additional nodes of the cracked elements that lie on the element edges are introduced but are not considered as internal nodes but remain global. Thus crack path continuity is automatically given. These global d.o.f. approximate the discontinuous part of the displacement field. The sum of the aforementioned part and the continuous displacement field represent the total displacement field including a possible jump. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A strong discontinuity based adaptive refinement approach for the modeling of crack branching

In the current work, the physical phenomena of dynamic fracture of brittle materials involving crack growth, acceleration and consequent branching is simulated. The numerical modeling is based on the approach where the failure in the form of cracks or shear bands is modeled by a jump in the displacement field, the so called ‘strong discontinuity’. The finite element method is employed with this strong discontinuity approach where each finite element is capable of developing a strong discontinuity locally embedded into it. The focus in this work is on branching phenomena which is modeled by an adaptive refinement method by solving a new sub-boundary value problem represented by a finite element at the growing crack tip. The sub-boundary value problem is subjected to a certain kinematic constraint on the boundary in the form of a linear deformation constraint. An accurate resolution of the state of material at the branching crack tip is achieved which results in realistic dynamic fracture simulations. A comparison of resulting numerical simulations is provided with the experiment of dynamic fracture from the literature. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Local Existence of Contact Discontinuities in Relativistic Magnetohydrodynamics

We study the free boundary problem for a contact discontinuity for the system of relativistic magnetohydrodynamics. A surface of contact discontinuity is a characteristic of this system with no flow across the discontinuity for which the pressure, the velocity and the magnetic field are continuous whereas the density, the entropy and the temperature may have a jump. For the two-dimensional case, we prove the local-in-time existence in Sobolev spaces of a unique solution of the free boundary problem provided that the Rayleigh-Taylor sign condition on the jump of the normal derivative of the pressure is satisfied at each point of the initial discontinuity.     

A singularly perturbed convection–diffusion problem in a half-plane

A singularly perturbed convection–diffusion equation with constant coefficients is considered in a half plane, with Dirichlet boundary conditions. The boundary function has a specified degree of regularity except for a jump discontinuity, or jump discontinuity in a derivative of specified order, at a point. Precise pointwise bounds for the derivatives of the solution are obtained. The bounds show both the strength of the interior layer emanating from the point of discontinuity and the blowup of the derivatives resulting from the discontinuity, and make precise the dependence of the derivatives on the singular perturbation parameter.     

Group theoretic method for analyzing interaction of a discontinuity wave with a strong shock in an ideal gas

A group theoretic method is used to obtain an exact particular solution to the system of partial differential equations, describing one-dimensional unsteady planar, cylindrically and spherically symmetric motions in an ideal gas, involving shock waves. It is interesting to remark that the exact solution obtained here is precisely the blast wave solution obtained earlier using a different method of approach. Further, the evolution of a discontinuity wave and its interaction with the strong shock are studied within the state characterized by the exact particular solution. The properties of reflected and transmitted waves and the jump in the shock acceleration are completely characterized, and certain observations are noted in respect to their contrasting behavior.     

Discontinuous Solutions of Linearized, Steady-State, Viscous, Compressible Flows

In thIs paper we prove the existence and uniqueness of the solution to a linearized, two dimensional, steady state, viscous, compressible Navier-Stokes equations in a strip. We obtain a regularity result and a new a priori estimate of the solution. We establish the discontinuity of the solution when the boundary data of pressure have a jump discontinuity. We also derive a formula for the discontinuous part and show that the remainder is smooth in the strip.     

Tracking a sharp crested wave front in hyperbolic heat transfer

In this article we investigate the numerical oscillations encountered when approximating the solution to the hyperbolic heat conduction equation. We consider a benchmark problem and show that it is not well-posed, unless a jump condition is specified. The alternative is to “smooth” the jump which leads to a sharp crested wave front, but with no discontinuity. To track the wave front we split the problem into auxiliary problems and solve these using different methods. The resulting solution is oscillation-free.     

A jump discontinuity of compressible viscous flows grazing a non-convex corner

We show existence and regularity of solution for the compressible viscous steady state Navier–Stokes system on a polygon having a grazing corner and that the density has a jump discontinuity across a curve inside the domain. There are corresponding jumps in derivatives of the velocity. The solution comes from a well-posed boundary value problem on a polygonal domain with a non-convex corner. A formula for the decay of the jump is given. The decay formula suggests that density jumps can occur in a compressible flow with a non-vanishing viscosity.     

Variational formulation of the material failure process in solids by embedded discontinuities model

This work presents a variational formulation of the material failure process, idealized as strain or displacement discontinuities, by weak, strong, or discrete embedded discontinuities into a continuum. It is shown that the solution of the proposed variational formulation may be approximated by different types of finite elements with embedded discontinuities. The developed displacement approximation of a finite element split by the discontinuity leads to a symmetric stiffness matrix, which considers not only the continuity of tractions but also the rigid body relative motions of the portions in which the element is split. The variational formulation of a continuum with more than one discontinuity in its interior is developed. It is shown that this formulation may lead to finite elements with embedded discontinuities that can be classified as displacement, force, mixed, and hybrid models. To show the effectiveness of the proposed formulation, the classical example of a bar under tension is solved using one and 2D finite element approximations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009