Formation of a jet and vortices behind a shock wave reflected from a concave body

A jet and vortices have been observed when a plane shock wave reflects from a concave body in a shock tube. If the cavity is deep enough then two reflected shocks appear near its edges. Air, carbon tetrafluoride (CF) and dichlorodifluoromethane (CClF) were chosen as test gases. The flow was visualized with the aid of a conventional shadow technique. Pressure measurements at the body surface were also obtained. Numerical studies have been conducted using a two-dimensional inviscid model. There is a good qualitative agreement between the experimental and numerical results. Received 8 February 1996 / Accepted 30 June 1997     

Deformation waves in thermoelastic media and the concept of internal variables

Summary Nonlinear thermoelastic continua with Fourier's type of heat conduction illustrate complex systems composed of internal and reaction-diffusion subsystems. An evolution equation is derived for the observable variable (deformation) when temperature effects are viewed as internal processes. This approach, which pertains to the so-called instantaneous wave analysis, shows that thermal losses accompanying the deformation wave correspond to the low-frequency approximation.     

Burgers方程的小波精细积分算法

求解偏微分方程的常用方法包括有限差分法、有限元法等。近年来,小波分析在偏微分方程数值求解中的应用已引起很多学者的关注,例如采用Daubechies小波或shannon小波构造的小波配置方法已经取得较好的结果。钟万勰院士提出的偏微分方程的子域精细积分方法是一种半解析方法,方法简单,精度高。将小波方法和精细积分方法相结合应用于偏微分方程的数值求解中将有利于提高算法的精度和稳定性,为此本文以Burgers方程为例,提出了一种求解一维非线性抛物型偏微分方程的小波精积分方法。该方法用拟小波配点法对空间域进行离散,建立起对时间的常微分方程组,然后采用精细时程积分方法对该方程组求解。数值计算结果表明,该方法同其它方法相比,具有计算格式简单,数值稳定性和精度较高的优点。     

Nonlinear Boundary Control of the Generalized Burgers Equation

In this paper, the adaptive and non-adaptive stabilization of the generalized Burgers equation by nonlinear boundary control are analyzed. For the non-adaptive case, we show that the controlled system is exponentially stable in L². As for the adaptive case, we present a novel and elegant approach to show the L² regulation of the solution of the generalized Burgers system. Numerical results supporting and reinforcing the analytical ones of both the controlled and uncontrolled system for the non-adaptive and adaptive cases are presented using the Chebychev collocation method with backward Euler method as a temporal scheme.     

A two-step back–forth algorithm for the solution of the unsteady adjoint equations

A new algorithm for the solution of the unsteady adjoint equations is proposed in this article, aiming at overcoming the excessive computational cost and memory requirements of the conventional adjoint approach for the optimisation of unsteady problems in computational mechanics. The total cost is equal to four times the cost of the unsteady state solution, which is twice the cost of the conventional backward-in-time adjoint calculation but the memory requirements are very small, equivalent to those of a steady-state problem, while stability is acceptable. The proposed algorithm is validated in the case of the 1D unsteady Burgers equation with non-smooth source terms.     

Numerical simulation of silencers

The problem of attenuating the noise from weapons firing is studied experimentally and numerically. As a possible method of attenuating the noise significantly, a silencer with no internal baffles is attached to the M242 cannon. The internal pressures inside the muffler are measured. The near-field overpressures outside the muffler at various polar angles are also measured. A numerical simulation of the flow through the muffler is performed, using Harten's shock-capturing method to solve the Euler equations of ideal compressible flow. The numerical simulation yields a detailed picture of the flow field as displayed by the pressure and Mach contours. Pressure–time curves at selected locations are obtained and compared with experimental data. There is good agreement, except that the numerical simulation generates more vigorous oscillations.     

小波插值方法自适应数值求解时间进化微分方程

应用小波自相关函数的插值性质,得到任意给定函数的插值小波表达式,然后对其直接求导,可以得到函数导数的表达式。导数运算不再应用差分算法,扩展了小波方法在数值求解微分方程中的应用。由于小波基函数的有限支撑特点,小波方法可以有效地处理微分方程中解的局部突变问题。通过设定小波系数阀值,实现了求解过程的自适应。本文给出了两个算例,结果表明了算法的自适应特点及其向二维空间问题推广的有效性。     

Asymptotic Solution for a Kind of Boundary Layer Problem

This paper studies the partial differential equation with a small coefficient in the highest-order item. This kind of equation is also named as boundary layer problem. The Burgers equation and modified Burgers equation are analyzed in this approach. First, these equations are transferred into the strong nonlinear ones, and then the corresponding strong nonlinear equations are solved based on the perturbation method. The results from the asymptotic method are comparable with those obtained from numerical computation. An erratum to this article is available at .     

ADAPTIVE INTERVAL WAVELET PRECISE INTEGRATION METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS

IntroductionThepreciseintegrationmethod(PIM) [1],whichwasproposedforsolvingstructuraldynamicequations.Thismethodissimplerandpossesseshigherprecision .Forlinearsteadystructuraldynamicsystems,itsnumericalresultsattheintegrationpointsarealmostequaltothatoftheexactsolutioninmachineaccuracy .InthepreciseintegrationmethodforsolvingPDEs,theequationsshouldbediscretizedinthephysicalspaceforobtainingthesystemofODEsintime ,whichisoftenexecutedbythefinitedifferencemethodorthefiniteelementmethod .Inrec…     

基于Runge-Kutta的自回归物理信息神经网络求解偏微分方程

物理信息神经网络离散时间模型(PINN-RK)是深度学习技术与龙格库塔方法相结合的产物, 在求解偏微分方程时具有非常出色的稳定性和较高的求解精度. 但是, 受到龙格库塔算法本身的限制, PINN-RK模型仅能实现单步时间预测, 且计算效率较低. 因此, 为了实现多时间步长预测和提高模型的计算效率, 提出了一种基于龙格库塔法的自回归物理信息神经网络模型(SR-PINN-RK). 该模型基于自回归时间步进机制, 改进了神经网络的训练流程和网络结构, 相比PINN-RK模型, 大幅减少了神经网络的训练参数, 提高了模型的计算效率. 此外, 在自回归机制的作用下, 该模型通过对标签数据的动态更新, 成功实现了对偏微分方程解的多时间步长预测. 为了验证文中模型的求解精度和计算效率, 分别求解了Allen-Cahn方程和Burgers方程, 并与文献中的基准解进行了对比. 结果表明, 模型预测解与基准解之间具有很高的一致性, 求解Allen-Cahn方程和Burgers方程的最大相对误差均低于0.009.     

Nonlinear waves in fluid flow through a viscoelastic tube

A system of nonlinear equations for describing the perturbations of the pressure and radius in fluid flow through a viscoelastic tube is derived. A differential relation between the pressure and the radius of a viscoelastic tube through which fluid flows is obtained. Nonlinear evolutionary equations for describing perturbations of the pressure and radius in fluid flow are derived. It is shown that the Burgers equation, the Korteweg-de Vries equation, and the nonlinear fourth-order evolutionary equation can be used for describing the pressure pulses on various scales. Exact solutions of the equations obtained are discussed. The numerical solutions described by the Burgers equation and the nonlinear fourth-order evolutionary equation are compared.     

Shock wave solution of the Boltzmann kinetic equation in a 13-moment approximation

We consider the classic problem of a one-dimensional steady shock-wave solution of the Boltzmann kinetic equation utilizing a new type of 13-moment approximation proposed by Oguchi (1997). The model, unlike previous ones, expresses the collision term in an explicit function of the molecular velocity. This enables us to examine directly the nature of the singularity of the distribution function to this particular problem caused by the vanishing molecular velocity. We can thus obtain moment integrals directly because of its explicit expression. The principal value is utilized for the moment integral to cope with the singularity, and we can have five relations for five unknown functions to be determined with respect to the coordinate x. These relations can be reduced to a first-order differential equation that is solved to provide the familiar smooth monotonic transition from the upstream supersonic state to the subsonic downstream state. Computed values of shock thickness for various shock Mach numbers agree well with existing results obtained by different methods to the certain Mach number beyond which no solution exists.Received: 17 May 2002, Accepted: 1 May 2003, Published online: 15 August 2003PACS: 51.10. + y     

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