Augmented Lagrangian Methods for p-Harmonic Flows with the Generalized Penalization Terms and Application to Image Processing

In this paper,we propose a generalized penalization technique and a convex constraint minimization approach for the p-harmonic flow problem following the ideas in[Kang&March,IEEE T.Image Process.,16(2007),2251–2261].We use fast algorithms to solve the subproblems,such as the dual projection methods,primal-dual methods and augmented Lagrangian methods.With a special penalization term,some special algorithms are presented.Numerical experiments are given to demonstrate the performance of the proposed methods.We successfully show that our algorithms are effective and efficient due to two reasons:the solver for subproblem is fast in essence and there is no need to solve the subproblem accurately(even 2 inner iterations of the subproblem are enough).It is also observed that better PSNR values are produced using the new algorithms.

Augmented Lagrangian Methods for Convex Matrix Optimization Problems

In this paper,we provide some gentle introductions to the recent advance in augmented Lagrangian methods for solving large-scale convex matrix optimization problems(cMOP).Specifically,we reviewed two types of sufficient conditions for ensuring the quadratic growth conditions of a class of constrained convex matrix optimization problems regularized by nonsmooth spectral functions.Under a mild quadratic growth condition on the dual of cMOP,we further discussed the R-superlinear convergence of the Karush-Kuhn-Tucker(KKT)residuals of the sequence generated by the augmented Lagrangian methods(ALM)for solving convex matrix optimization problems.Implementation details of the ALM for solving core convex matrix optimization problems are also provided.

耦合拉格朗日-欧拉方法及其在海洋工程中的应用

Combining the strengths of Lagrangian and Eulerian descriptions,the coupled Lagrangian–Eulerian methods play an increasingly important role in various subjects.This work reviews their development and application in ocean engineering.Initially,we briefly outline the advantages and disadvantages of the Lagrangian and Eulerian descriptions and the main characteristics of the coupled Lagrangian–Eulerian approach.Then,following the developmental trajectory of these methods,the fundamental formulations and the frameworks of various approaches,including the arbitrary Lagrangian–Eulerian finite element method,the particle-in-cell method,the material point method,and the recently developed Lagrangian–Eulerian stabilized collocation method,are detailedly reviewed.In addition,the article reviews the research progress of these methods with applications in ocean hydrodynamics,focusing on free surface flows,numerical wave generation,wave overturning and breaking,interactions between waves and coastal structures,fluid–rigid body interactions,fluid–elastic body interactions,multiphase flow problems and visualization of ocean flows,etc.Furthermore,the latest research advancements in the numerical stability,accuracy,efficiency,and consistency of the coupled Lagrangian–Eulerian particle methods are reviewed;these advancements enable efficient and highly accurate simulation of complicated multiphysics problems in ocean and coastal engineering.By building on these works,the current challenges and future directions of the hybrid Lagrangian–Eulerian particle methods are summarized.

Hybrid N-order Lagrangian Interpolation Eulerian-Lagrangian Method for Salinity Calculation

The Eulerian？Lagrangian method（ELM） has been used by many ocean models as the solution of the advection equation,but the numerical error caused by interpolation imposes restriction on its accuracy.In the present study,hybrid N-order Lagrangian interpolation ELM（Li ELM） is put forward in which the N-order Lagrangian interpolation is used at first,then the lower order Lagrangian interpolation is applied in the points where the interpolation results are abnormally higher or lower.The calculation results of a step-shaped salinity advection model are analyzed,which show that higher order（N=3？8） Li ELM can reduce the mean numerical error of salinity calculation,but the numerical oscillation error is still significant.Even number order Li ELM makes larger numerical oscillation error than its adjacent odd number order Li ELM.Hybrid N-order Li ELM can remove numerical oscillation,and it significantly reduces the mean numerical error when N is even and the current is in fixed direction,while it makes less effect on mean numerical error when N is odd or the current direction changes periodically.Hybrid odd number order Li ELM makes less mean numerical error than its adjacent even number order Li ELM when the current is in the fixed direction,while the mean numerical error decreases as N increases when the current direction changes periodically,so odd number of N may be better for application.Among various types of Hybrid N-order Li ELM,the scheme reducing N-order directly to 1st-order may be the optimal for synthetic selection of accuracy and computational efficiency.

An efficient formulation based on the Lagrangian method for contact–impact analysis of flexible multi-body system

In this paper,an efficien formulation based on the Lagrangian method is presented to investigate the contact–impact problems of f exible multi-body systems.Generally,the penalty method and the Hertz contact law are the most commonly used methods in engineering applications.However,these methods are highly dependent on various non-physical parameters,which have great effects on the simulation results.Moreover,a tremendous number of degrees of freedom in the contact–impact problems will influenc thenumericalefficien ysignificantl.Withtheconsideration of these two problems,a formulation combining the component mode synthesis method and the Lagrangian method is presented to investigate the contact–impact problems in fl xible multi-body system numerically.Meanwhile,the finit element meshing laws of the contact bodies will be studied preliminarily.A numerical example with experimental verificatio will certify the reliability of the presented formulationincontact–impactanalysis.Furthermore,aseries of numerical investigations explain how great the influenc of the finit element meshing has on the simulation results.Finally the limitations of the element size in different regions are summarized to satisfy both the accuracy and efficien y.

The discontinuous Petrov-Galerkin method for one-dimensional compressible Euler equations in the Lagrangian coordinate

In this paper, a Petrov-Galerkin scheme named the Runge-Kutta control volume （RKCV） discontinuous finite ele- ment method is constructed to solve the one-dimensional compressible Euler equations in the Lagrangian coordinate. Its advantages include preservation of the local conservation and a high resolution. Compared with the Runge-Kutta discon- tinuous Galerkin （RKDG） method, the RKCV method is easier to implement. Moreover, the advantages of the RKCV and the Lagrangian methods are combined in the new method. Several numerical examples are given to illustrate the accuracy and the reliability of the algorithm.

Numerical simulation of magnetic drug targeting with Eulerian-Lagrangian model and effect of viscosity modification due to diabetics

The targeted drug delivery and targeted drug therapy transport a drug directly to the center of the disease under various conditions and thereby treat it deliber- ately without effects on the body. This paper studies the magnetic drug targeting （MDT） technique by particle tracking in the presence of magnetic field in bifurcation vessels of a healthy person and a diabetes patient. The Lagrangian particle tracking is performed to estimate particle behavior under effects of imposed magnetic field gradients along the bifurcation. The results show that the magnetic field increases the volume fraction of particle in the target region, and the efficiency of MDT on a patient with the diabetes disease is better than a healthy person. Also, for the higher magnetic numbers, the flow in the upper branch is strongly affected by the magnetic field.

Deviation of the Lagrangian particle tracing method in the evaluation of the Southern Hemisphere annual subduction rate

The classical Lagrangian particle tracing method is widely used in the evaluation of the ocean annual subduction rate.However,our analysis indicates that in addition to neglecting the effect of mixing,there are two possible deviations in the method:one is an overestimation due to not considering that the amount of subducted water at the source location may be inadequate during the late winter of the first year when the mixed layer becomes shallow;the other one is an underestimation due to the neglect of the effective subduction caused by strong vertical pumping.Quantitative analysis shows that these two deviations mainly exist in the low-latitude subduction areas of the South Pacific and South Atlantic.The two deviations have very similar distribution areas and can partially off set each other.However,the overall deviation is still large,and the maximum relative deviation ratio can reach 50%;therefore,it cannot be ignored.

An Optimization Model for the Strip-packing Problem and Its Augmented Lagrangian Method

This paper formulates a two-dimensional strip packing problem as a non- linear programming （NLP） problem and establishes the first-order optimality conditions for the NLP problem. A numerical algorithm for solving this NLP problem is given to find exact solutions to strip-packing problems involving up to 10 items. Approximate solutions can be found for big-sized problems by decomposing the set of items into small-sized blocks of which each block adopts the proposed numerical algorithm. Numerical results show that the approximate solutions to big-sized problems obtained by this method are superior to those by NFDH, FFDH and BFDH approaches.

A LAGRANGIAN LATTICE BOLTZMANN METHOD FOR EULER EQUATIONS

A Lagrangian lattice Boltzmann method for solving Euler equations is proposed. The key step in formulating this method is the introduction of the displacement distribution function. The equilibrium distribution function consists of macroscopic Lagrangian variables at time steps n and n + 1. It is different from the standard lattice Boltzmann method. In this method the element, instead of each particle, is required to satisfy the basic law. The element is considered as one large particle, which results in simpler version than the corresponding Eulerian one, because the advection term disappears here. Our numerical examples successfully reproduce the classical results.

Pile Running in Layered Soils

This paper presents a case study on incidents of offshore pile running in layered soils.The study provides a detailed description of the seabed soil data,pile driving records,and field surveillance video observations.Three-dimensional large deformation finite element(LDFE)analyses were conducted to retrospectively analyze the incidents,considering the remoulding of seabed soil and degradation of the pile-soil interface in the LDFE modeling.By comparing the field observations with the LDFE analysis,the mechanism of pile running was discussed,with a focus on investigating the pile penetration resistance in each layer.The study revealed that pile running in layered soils primarily resulted from a significant reduction in pile base resistance when transitioning from a strong layer to an adjacent weak layer.To further investigate the pile running mechanism in layered soils,a parametric study on the strength variation of adjacent soil layers and its influence on pile base resistance was conducted.Lastly,a simplified prediction model of pile base resistance,suitable for assessing the risk of pile running in layered soils,was proposed.

Joint computation offloading and resource allocation in vehicular edge computing networks

Vehicular Edge Computing(VEC)is a promising technique to accommodate the computation-intensive and delaysensitive tasks through offloading the tasks to the RoadSide-Unit(RSU)equipped with edge computing servers or neighboring vehicles.Nevertheless,the limited computation resources of edge computing servers and the mobility of vehicles make the offloading policy design very challenging.In this context,through considering the potential transmission gains brought by the mobility of vehicles,we propose an efficient computation offloading and resource allocation scheme in VEC networks with two kinds of offloading modes,i.e.,Vehicle to Vehicle(V2V)and Vehicle to RSU(V2R).We define a new cost function for vehicular users by incorporating the vehicles’offloading delay,energy consumption,and expenses with a differentiated pricing strategy,as well as the transmission gain.An optimization problem is formulated to minimize the average cost of all the task vehicles under the latency and computation capacity constraints.A distributed iterative algorithm is proposed by decoupling the problem into two subproblems for the offloading mode selection and the resource allocation.Matching theorybased and Lagrangian-based algorithms are proposed to solve the two subproblems,respectively.Simulation results show the proposed algorithm achieves low complexity and significantly improves the system performance compared with three benchmark schemes.

The Rate of Convergence of Augmented Lagrangian Method for Minimax Optimization Problems with Equality Constraints

The augmented Lagrangian function and the corresponding augmented Lagrangian method are constructed for solving a class of minimax optimization problems with equality constraints.We prove that,under the linear independence constraint qualification and the second-order sufficiency optimality condition for the lower level problem and the second-order sufficiency optimality condition for the minimax problem,for a given multiplier vectorμ,the rate of convergence of the augmented Lagrangian method is linear with respect to||μu-μ^(*)||and the ratio constant is proportional to 1/c when the ratio|μ-μ^(*)||/c is small enough,where c is the penalty parameter that exceeds a threshold c_(*)>O andμ^(*)is the multiplier corresponding to a local minimizer.Moreover,we prove that the sequence of multiplier vectors generated by the augmented Lagrangian method has at least Q-linear convergence if the sequence of penalty parameters(ck)is bounded and the convergence rate is superlinear if(ck)is increasing to infinity.Finally,we use a direct way to establish the rate of convergence of the augmented Lagrangian method for the minimax problem with a quadratic objective function and linear equality constraints.

A Moving Coordinate Numerical Method for Analyses of Electromagneto-Mechanical Coupled Behavior of Structures in a Strong Magnetic Field Aiming at Application to Tokamak Structures

Analysis of the electromagneto-mechanical coupling effect contributes greatly to the high accuracy estimation of the EM load of many EM devices, such as a tokamak structure during plasma disruption. This paper presents a method for the numerical analysis of the electromagnetomechanical coupling effect on the basis of Maxwell＇s equations in the Lagrangian description and staggered load transfer scheme, which can treat the coupled behaviors of magnetic damping and magnetic stiffness effects at the same time. Codes were developed based on the ANSYS development platform and were applied to solve two typical numerical examples： the TEAM Problem 16 and dynamic behavior analysis of a shallow arch under electromagnetic force. The good consistency of numerical results and experimental data demonstrates the validity and accuracy of the proposed method and the related numerical codes.

PROBLEM OF HYDRODYNAMIC PRESSURE ON SUDDENLY STARTING VESSEL

In this paper, Lagrangian method is applied to discuss the problem of the hydrodynamic pressure on a suddenly starting vessel. The free surface profile and the coefficients of the hydrodynamic pressure im the vessel wall are obtained. And it is verified that the singularity of the pressure near the free surface is only logarithmic.

STUDY ON THE PROBLEMS OF CONTACT IN THE NUMERICAL SIMULATION OF SHEET FORMING

This study is concerned with the problems of contact in the process of numerical simulation of sheet metal forming in rigid visco-plastic shell FEM. In respect of analysis of sheet deep drawing process,for the tool model described by triangular elements, a kind of contact judging algorithm about the correlation between the node of deformed mesh and the triangular element of a tool is presented. In SPF/DB Lagrangian multiplier method is adopted to solve the contact problem between deformed meshes, and a new reliable practical dynamic contact checking algorithm is presented. As computation examples, the simulation results of metal sheet deep drawing and SPF/DB are introduced in this paper.

拉格朗日无网格方法中用于拉普拉斯算子离散的二次SFDI研究

In the Lagrangian meshless(particle)methods,such as the smoothed particle hydrodynamics(SPH),moving particle semi-implicit(MPS)method and meshless local Petrov-Galerkin method based on Rankine source solution(MLPG_R),the Laplacian discretisation is often required in order to solve the governing equations and/or estimate physical quantities(such as the viscous stresses).In some meshless applications,the Laplacians are also needed as stabilisation operators to enhance the pressure calculation.The particles in the Lagrangian methods move following the material velocity,yielding a disordered(random)particle distribution even though they may be distributed uniformly in the initial state.Different schemes have been developed for a direct estimation of second derivatives using finite difference,kernel integrations and weighted/moving least square method.Some of the schemes suffer from a poor convergent rate.Some have a better convergent rate but require inversions of high order matrices,yielding high computational costs.This paper presents a quadric semi-analytical finite-difference interpolation(QSFDI)scheme,which can achieve the same degree of the convergent rate as the best schemes available to date but requires the inversion of significant lower-order matrices,i.e.3×3 for 3D cases,compared with 6×6 or 10×10 in the schemes with the best convergent rate.Systematic patch tests have been carried out for either estimating the Laplacian of given functions or solving Poisson’s equations.The convergence,accuracy and robustness of the present schemes are compared with the existing schemes.It will show that the present scheme requires considerably less computational time to achieve the same accuracy as the best schemes available in literatures,particularly for estimating the Laplacian of given functions.

Two-phase micro-and macro-time scales in particle-laden turbulent channel flows

The micro- and macro-time scales in two-phase turbulent channel flows are investigated using the direct nu- merical simulation and the Lagrangian particle trajectory methods for the fluid- and the particle-phases, respectively. Lagrangian and Eulerian time scales of both phases are cal- culated using velocity correlation functions. Due to flow anisotropy, micro-time scales are not the same with the theo- retical estimations in large Reynolds number （isotropic） tur- bulence. Lagrangian macro-time scales of particle-phase and of fluid-phase seen by particles are both dependent on particle Stokes number. The fluid-phase Lagrangian inte- gral time scales increase with distance from the wall, longer than those time scales seen by particles. The Eulerian inte- gral macro-time scales increase in near-wall regions but de- crease in out-layer regions. The moving Eulerian time scales are also investigated and compared with Lagrangian integral time scales, and in good agreement with previous measure- ments and numerical predictions. For the fluid particles the micro Eulerian time scales are longer than the Lagrangian ones in the near wall regions, while away from the walls the micro Lagrangian time scales are longer. The Lagrangian integral time scales are longer than the Eulerian ones. The results are useful for further understanding two-phase flow physics and especially for constructing accurate prediction models of inertial particle dispersion.

On the stress–strain states of cellular materials under high loading rates

A virtual Taylor impact of cellular materials is analyzed with a wave propagation technique, i.e. the Lagrangian analysis method, of which the main advantage is that no pre-assumed constitutive relationship is required. Time histories of particle velocity, local strain, and stress profiles are calculated to present the local stress-strain history curves, from which the dynamic stress-strain states are obtained. The present results reveal that the dynamic-rigid-plastic hardening （D-R-PH） material model introduced in a previous study of our group is in good agreement with the dynamic stress-strain states under high loading rates obtained by the Lagrangian analysis method. It directly reflects the effectiveness and feasibility of the D-R-PH material model for the cellular materials under high loading rates.

IMAGE SUPER-RESOLUTION RECONSTRUCTION BY HUBER REGULARIZATION AND TAILORED FINITE POINT METHOD

In this paper,we propose using the tailored finite point method(TFPM)to solve the resulting parabolic or elliptic equations when minimizing the Huber regularization based image super-resolution model using the augmented Lagrangian method(ALM).The Hu-ber regularization based image super-resolution model can ameliorate the staircase for restored images.TFPM employs the method of weighted residuals with collocation tech-nique,which helps get more accurate approximate solutions to the equations and reserve more details in restored images.We compare the new schemes with the Marquina-Osher model,the image super-resolution convolutional neural network(SRCNN)and the classical interpolation methods:bilinear interpolation,nearest-neighbor interpolation and bicubic interpolation.Numerical experiments are presented to demonstrate that with the new schemes the quality of the super-resolution images has been improved.Besides these,the existence of the minimizer of the Huber regularization based image super-resolution model and the convergence of the proposed algorithm are also established in this paper.