能量泛函及Euler-Lagrange方程在图像降噪中的应用研究

研究了自适应分数阶偏微分方程修正模型的能量泛函及Euler-Lagrange方程。首先,定义了自适应分数阶偏微分方程修正模型的能量泛函,其中包含未知函数和拉格朗日乘子的集合。然后,通过求解能量泛函的极值方程,推导出了Euler-Lagrange方程。最后,讨论了Euler-Lagrange方程在自适应分数阶偏微分方程修正模型中的应用。

THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX

We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux,more specifically,for pressureless flow on the left and polytropic flow on the right separated by a discontinuity x=x(t).We prove that this problem admits global Radon measure solutions for all kinds of initial data.The over-compressing condition on the discontinuity x=x(t)is not enough to ensure the uniqueness of the solution.However,there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve x=x(t)+0,in addition to the full adhesion condition on its left-side.As an application,we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas.In particular,we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas.This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field.

Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.

LaNets:Hybrid Lagrange Neural Networks for Solving Partial Differential Equations

We propose new hybrid Lagrange neural networks called LaNets to predict the numerical solutions of partial differential equations.That is,we embed Lagrange interpolation and small sample learning into deep neural network frameworks.Concretely,we first perform Lagrange interpolation in front of the deep feedforward neural network.The Lagrange basis function has a neat structure and a strong expression ability,which is suitable to be a preprocessing tool for pre-fitting and feature extraction.Second,we introduce small sample learning into training,which is beneficial to guide themodel to be corrected quickly.Taking advantages of the theoretical support of traditional numerical method and the efficient allocation of modern machine learning,LaNets achieve higher predictive accuracy compared to the state-of-the-artwork.The stability and accuracy of the proposed algorithmare demonstrated through a series of classical numerical examples,including one-dimensional Burgers equation,onedimensional carburizing diffusion equations,two-dimensional Helmholtz equation and two-dimensional Burgers equation.Experimental results validate the robustness,effectiveness and flexibility of the proposed algorithm.

CONSTRAINT VIOLATION STABILIZATION OF EULER-LAGRANGE EQUATIONS WITH NON-HOLONOMIC CONSTRAINTS

Two constraint violation stabilization methods are presented to solve the Euler Lagrange equations of motion of a multibody system with nonholonomic constraints. Compared to the previous works, the newly devised methods can deal with more complicated problems such as those with nonholonomic constraints or redundant constraints, and save the computation time. Finally a numerical simulation of a multibody system is conducted by using the methods given in this paper.

The Hamiltonian Canonical Form for Euler-Lagrange Equations

Based on the theory of calculus of variation, some suffcient conditions are given for some Euler-Lagrangcequations to be equivalently represented by finite or even infinite many Hamiltonian canonical equations. Meanwhile,some further applications for equations such as the KdV equation, MKdV equation, the general linear Euler Lagrangeequation and the cylindric shell equations are given.

On the Homotopy Analysis Method and Optimal Value of the Convergence Control Parameter: Solution of Euler-Lagrange Equation

This paper presents, an efficient approach for solving Euler-Lagrange Equation which arises from calculus of variations. Homotopy analysis method to find an approximate solution of variational problems is proposed. An optimal value of the convergence control parameter is given through the square residual error. By minimizing the the square residual error, the optimal convergence-control parameters can be obtained. It is showed that the homotopy analysis method was valid and feasible to the study of variational problems.

Fractional Euler Lagrange Equations for Irregular Lagrangian with Holonomic Constraints

In this paper the fractional Euler Lagrange equations for irregular Lagrangian with holonomic constraints have been presented. The equations of motion are obtained using fractional Euler Lagrange equations in a similar manner to the usual mechanics. The results of fractional calculus reduce to those obtained from classical calculus (the standard Euler Lagrange equations) when γ→0 and α, βare equal unity only. Two problems are considered to demonstrate the application of the formalism.

基于能量整形方案实现具有通讯时滞欠驱动Euler-Lagrange网络的一致性

本文主要基于能量整形方案研究具有通讯时滞网络化欠驱动Euler-Lagrange(EL)系统的一致性问题,通过利用阻尼注入和互连分配的无源控制(PBC)技术,在有向连通网络拓扑下提出了一个简单的分布式协议,来实现在无引导者和有引导者-跟随者两种情形下欠驱动EL网络的一致性.本文提出的一致性能量整形方案的主要特点是有机地整合了系统欠驱动和驱动部分以及控制器三部分能量作为整个系统的总能量,这个总能量被利用作为一个合适的Lyapunov函数,它能够充分确保网络化欠驱动EL系统达到所期望的分布式一致性.最后,通过由欠驱动EL网络所描述柔性关节机械臂系统的数值模拟,来分析通讯时滞对一致性的效应和验证所提出控制算法的正确性.

Numerical Solution of Euler-Bernoulli Beam Equation by Using Barycentric Lagrange Interpolation Collocation Method

Euler-Bernoulli beam equation is very important that can be applied in the field of mechanics, science and technology. Some authors have put forward many different numerical methods, but the precision is not enough high. In this paper, we will illustrate the high-precision numerical method to solve Euler-Bernoulli beam equation. Three numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by our method indicate new algorithm has the following advantages: small computational work, fast convergence speed and high precision.

基于LAGRANGE方程的深水钻井隔水管–水下井口系统动力分析

为了探究大质量、大刚度防喷器组(blowout preventers,BOPs)对深水钻井隔水管系统动态相应预测精度的影响,根据细长钻井隔水管与刚性防喷器组的结构特点,提出两者刚柔耦合的概念,采用能量法推导隔水管–防喷器组–水下井口系统的动能和势能,采用LAGRANGE方法建立耦合系统动力学理论模型,采用科学计算软件和Newmark-β直接积分法对动力学模型进行数值计算。以南海某深水钻井隔水管为例,开展基于耦合动力学模型的隔水管系统动态响应分析。结果表明,采用本文理论模型得到的隔水管不同位置的节点位移、单元弯矩、上部和下部挠性接头转角时程曲线、整体侧向位移包络线和弯矩包络线等与ABAQUS仿真结果均吻合良好,最大误差为8.8%。此方法可为深水钻井隔水管和水下井口系统动态分析提供参考。

Radon Measure Solutions to Riemann Problems for Isentropic Compressible Euler Equations of Polytropic Gases

We solve the Riemann problems for isentropic compressible Euler equations of polytropic gases in the class of Radon measures,and the solutions admit the concentration of mass.It is found that under the requirement of satisfying the over-compressing entropy condition:(i)there is a unique delta shock solution,corresponding to the case that has two strong classical Lax shocks;(ii)for the initial data that the classical Riemann solution contains a shock wave and a rarefaction wave,or two shocks with one being weak,there are infinitely many solutions,each consists of a delta shock and a rarefaction wave;(iii)there are no delta shocks for the case that the classical entropy weak solutions consist only of rarefaction waves.These solutions are self-similar.Furthermore,for the generalized Riemann problem with mass concentrated initially at the discontinuous point of initial data,there always exists a unique delta shock for at least a short time.It could be prolonged to a global solution.Not all the solutions are self-similar due to the initial velocity of the concentrated point-mass(particle).Whether the delta shock solutions constructed satisfy the over-compressing entropy condition is clarified.This is the first result on the construction of singular measure solutions to the compressible Euler system of polytropic gases,that is strictly hyperbolic,and whose characteristics are both genuinely nonlinear.We also discuss possible physical interpretations and applications of these new solutions.

A Modified Lagrange Method for Solving Convex Quadratic Optimization Problems

In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality of the Lagrangian function with respect to the primary variables of the problem, decomposes the solution process into two independent ones, in which the primary variables are solved for independently, and then the secondary variables, which are the Lagrange multipliers, are solved for, afterward. This is an innovation that leads to solving independently two simpler systems of equations involving the primary variables only, on one hand, and the secondary ones on the other. Solutions obtained for small sized problems (as preliminary test of the method) demonstrate that the new method is generally effective in producing the required solutions.

西风带阻塞高压形成及崩溃动力机制的Euler-Lagrange理论解释

本文主要采用Euler-Lagrange理论,讨论正压剪切流的切变参数对阻塞高压形成的影响。研究表明:存在一个临界参数,当切变参数的绝对值超过这个临界参数,正压剪切流将分歧出阻塞流型,这意味着阻塞高压的形成;反之,当切变参数的绝对值小于这个临界参数,阻塞流型将恢复到剪切流,这意味着阻塞高压的崩溃。该研究一定程度回答了阻塞流型“怎么形成”和“怎么崩溃”这一基本而关键的问题。

Lagrange equation在RLC电路中的应用

将Lagrange equation应用于RLC电路来讨论其对非力学体系的应用。首先针对一般RLC电路,利用类比的方法,得到RLC电路的Lagrange function和Lagrange equation,进而得出了求解RLC电路的一般方法。

A Fourth-Order Unstructured NURBS-Enhanced Finite Volume WENO Scheme for Steady Euler Equations in Curved Geometries

In Li and Ren(Int.J.Numer.Methods Fluids 70:742–763,2012),a high-order k-exact WENO finite volume scheme based on secondary reconstructions was proposed to solve the two-dimensional time-dependent Euler equations in a polygonal domain,in which the high-order numerical accuracy and the oscillations-free property can be achieved.In this paper,the method is extended to solve steady state problems imposed in a curved physical domain.The numerical framework consists of a Newton type finite volume method to linearize the nonlinear governing equations,and a geometrical multigrid method to solve the derived linear system.To achieve high-order non-oscillatory numerical solutions,the classical k-exact reconstruction with k=3 and the efficient secondary reconstructions are used to perform the WENO reconstruction for the conservative variables.The non-uniform rational B-splines(NURBS)curve is used to provide an exact or a high-order representation of the curved wall boundary.Furthermore,an enlarged reconstruction patch is constructed for every element of mesh to significantly improve the convergence to steady state.A variety of numerical examples are presented to show the effectiveness and robustness of the proposed method.

Conical Sonic-Supersonic Solutions for the 3-D Steady Full Euler Equations

This paper concerns the sonic-supersonic structures of the transonic crossflow generated by the steady supersonic flow past an infinite cone of arbitrary cross section.Under the conical assumption,the three-dimensional(3-D)steady Euler equations can be projected onto the unit sphere and the state of fluid can be characterized by the polar and azimuthal angles.Given a segment smooth curve as a conical-sonic line in the polar-azimuthal angle plane,we construct a classical conical-supersonic solution near the curve under some reasonable assumptions.To overcome the difficulty caused by the parabolic degeneracy,we apply the characteristic decomposition technique to transform the Euler equations into a new degenerate hyperbolic system in a partial hodograph plane.The singular terms are isolated from the highly nonlinear complicated system and then can be handled successfully.We establish a smooth local solution to the new system in a suitable weighted metric space and then express the solution in terms of the original variables.

Blowup of Solutions to the Non-Isentropic Compressible Euler Equations with Time-Dependent Damping and Vacuum

This paper mainly studies the blowup phenomenon of solutions to the compressible Euler equations with general time-dependent damping for non-isentropic fluids in two and three space dimensions. When the initial data is assumed to be radially symmetric and the initial density contains vacuum, we obtain that classical solution, especially the density, will blow up on finite time. The results also reveal that damping can really delay the singularity formation.

滑模控制下多Euler-Lagrange系统预设时间二分一致性

本文研究了具有外部干扰的多Euler-Lagrange系统预设时间滑模二分一致性问题.通过设计分布式控制协议,使得具有动态领导者的多Euler-Lagrange系统在预设时间内到达滑模面,从而进一步在另一预设时间内实现二分一致性.本文采用的终端滑模控制方法可以在克服外部干扰的基础上保证系统在预设时间内实现期望的控制目标.最后,通过数值仿真验证了理论结果的可行性和有效性.

网络化Euler-Lagrange系统的分布式编队机动控制

研究了网络化Euler-Lagrange系统自适应编队机动控制问题.针对参数不确定的Euler-Lagrange系统,利用滑模控制方法提出了一种自适应编队机动控制算法.基于Lyapunov稳定性理论,证明了闭环系统的稳定性.该算法的显著特点是通过引入一种特殊的有向网络拓扑来描述智能体之间的通信交互行为,使得系统中跟随者在不需要知道或估计时变机动参数的情况下,能够实现编队的方向、平移、形状的连续改变.最后对提出的自适应编队机动控制算法进行数值模拟以验证该控制方案的有效性.