EXTENSION TO THE DIRECT METHODS AND APPLICATIONS TO THE GENERALIZED BURGERS EQUATION

A generalization of the direct method of Clarkson and Kruskal for finding similarity reductions of partial differential equations with arbitrary functions is found and discussed for the generalized Burgers equation. The corresponding reductions and the exact solutions due to the methods of the ordinary differential equations are then given by the methods. The results given here answer partially an open problem proposed by Clarkson, that is how to develop the direct method to seek symmetry reductions of nonlinear PDEs with arbitrary functions.

Superconvergence of Direct Discontinuous Galerkin Methods:Eigen-structure Analysis Based on Fourier Approach

This paper investigates superconvergence properties of the direct discontinuous Galerkin(DDG)method with interface corrections and the symmetric DDG method for diffusion equations.We apply the Fourier analysis technique to symbolically compute eigenvalues and eigenvectors of the amplification matrices for both DDG methods with different coefficient settings in the numerical fluxes.Based on the eigen-structure analysis,we carry out error estimates of the DDG solutions,which can be decomposed into three parts:(i)dissipation errors of the physically relevant eigenvalue,which grow linearly with the time and are of order 2k for P^(k)(k=2,3)approximations;(ii)projection error from a special projection of the exact solution,which is decreasing over the time and is related to the eigenvector corresponding to the physically relevant eigenvalue;(iii)dissipative errors of non-physically relevant eigenvalues,which decay exponentially with respect to the spatial mesh sizeΔx.We observe that the errors are sensitive to the choice of the numerical flux coefficient for even degree P^(2)approximations,but are not for odd degree P^(3)approximations.Numerical experiments are provided to verify the theoretical results.

Alternating Direction Finite Volume Element Methods for Three-Dimensional Parabolic Equations

This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes, one is analogous to Douglas finite difference scheme with second-order splitting error, the other two schemes have third-order splitting error, and the last one is an extended LOD scheme. The L2 norm and H1 semi-norm error estimates are obtained for the first scheme and second one, respectively. Finally, two numerical examples are provided to illustrate the efficiency and accuracy of the methods.

Investigation of a cyanine dye assay for the evaluation of the biocompatibility of magnesium alloys by direct and indirect methods

Magnesium and its alloys are promising candidates for a new generation of biodegradable metals in orthopaedic applications due to their excellent biocompatibility,biodegradability,and mechanical properties that are similar to natural bone.However,direct in vitro assessment of these materials in the presence of cells is complicated by degradation products from the alloy that lead to a false positive for the most commonly used cell adhesion and cell proliferation assays.In this paper,a cyanine dye was used to quantitatively evaluate the in vitro biocompatibility of a Mg AZ31 alloy by both direct and indirect methods.The cytotoxicity of the corrosion products was evaluated via an indirect method;a 25%decrease in cell viability compared to control samples was observed.Moreover,direct assessment of cell adhesion and proliferation showed a statistically significant increase in cell number at the surface after 72 h.In addition,the degradation rate and surface characteristics of the Mg AZ31 alloy were evaluated for both direct and indirect tests.The degradation rate was unaffected by the presence of cells while evidence of an increase in calcium phosphate deposition on the magnesium alloy surface in the presence of cells was observed.This study demonstrates that a cyanine dye based assay provides a more accurate assessment of the overall in vitro biocompatibility of biodegradable metals than the more commonly used assays reported in the literature to date.

The efficiency of direct integration methods in elastic contact-impact problems

A comparison of direct integration methods is madeand their efficiency is investigated for impact problems.New-mark,Wilson-θ,Central Difference and Houbolt Methodsare used as direct integration methods.Impact analysisincludes that of elastic and large deformation based uponupdated Lagrangian including buckling check.The resultsshow that the direct integration methods give differentresults in different contact-impact cases.

The Direct Discontinuous Galerkin Methods with Implicit-Explicit Runge-Kutta Time Marching for Linear Convection-Diffusion Problems

In this paper,a fully discrete stability analysis is carried out for the direct discontinuous Galerkin(DDG)methods coupled with Runge-Kutta-type implicit-explicit time marching,for solving one-dimensional linear convection-diffusion problems.In the spatial discretization,both the original DDG methods and the refined DDG methods with interface corrections are considered.In the time discretization,the convection term is treated explicitly and the diffusion term implicitly.By the energy method,we show that the corresponding fully discrete schemes are unconditionally stable,in the sense that the time-step is only required to be upper bounded by a constant which is independent of the mesh size h.Opti-mal error estimate is also obtained by the aid of a special global projection.Numerical experiments are given to verify the stability and accuracy of the proposed schemes.

The Error Estimates of Direct Discontinuous Galerkin Methods Based on Upwind-Baised Fluxes

In this paper, we study the error estimates for direct discontinuous Galerkin methods based on the upwind-biased fluxes. We use a newly global projection to obtain the optimal error estimates. The numerical experiments imply that L2 norms error estimates can reach to order k + 1 by using time discretization methods.

FIXED/PREASSIGNED-TIME SYNCHRONIZATION OF QUATERNION-VALUED NEURAL NETWORKS INVOLVING DELAYS AND DISCONTINUOUS ACTIVATIONS: A DIRECT APPROACH

The fixed-time synchronization and preassigned-time synchronization are investigated for a class of quaternion-valued neural networks with time-varying delays and discontinuous activation functions. Unlike previous efforts that employed separation analysis and the real-valued control design, based on the quaternion-valued signum function and several related properties, a direct analytical method is proposed here and the quaternion-valued controllers are designed in order to discuss the fixed-time synchronization for the relevant quaternion-valued neural networks. In addition, the preassigned-time synchronization is investigated based on a quaternion-valued control design, where the synchronization time is preassigned and the control gains are finite. Compared with existing results, the direct method without separation developed in this article is beneficial in terms of simplifying theoretical analysis, and the proposed quaternion-valued control schemes are simpler and more effective than the traditional design, which adds four real-valued controllers. Finally, two numerical examples are given in order to support the theoretical results.

OPTIMIZATION OF ADAPTIVE DIRECT METHOD FOR APPROXIMATE SOLUTION OF INTEGRAL EQUATIONS OF SEVERAL VARIABLES

This paper determines the exact error order on optimization of adaptive direct methods of approximate solution of the class of Fredholm integral equations of the second kind with kernel belonging to the anisotropic Sobolev classes, and also gives an optimal algorithm.

Radial solution of the Logarithmic Laplacian system

The paper generalizes the direct method of moving planes to the Logarithmic Laplacian system.Firstly,some key ingredients of the method are discussed,for example,Narrow region principle and Decay at infinity.Then,the radial symmetry of the solution of the Logarithmic Laplacian system is obtained.

RSDM:A Powerful Direct Method to Predict the Asymptotic Cyclic Behavior of Elastoplastic Structures

Mechanical engineering structures and structural components are often subjected to cyclic thermomechanical loading which stresses their material beyond its elastic limits well inside the inelastic regime.Depending on the level of loading inelastic strains may lead either to failure,due to low cycle fatigue or ratcheting,or to safety,through elastic shakedown.Thus,it is important to estimate the asymptotic stress state of such structures.This state may be determined by cumbersome incremental time-stepping calculations.Direct methods,alternatively,have big computational advantages as they focus on the characteristics of these states and try to establish them,in a direct way,right from the beginning of the calculations.Among the very few such general-purpose direct methods,a powerful direct method which has been called RSDM has appeared in the literature.The method may directly predict any asymptotic state when the exact time history of the loading is known.The advantage of the method is due to the fact that it addresses the physics of the asymptotic cycle and exploits the cyclic nature of its expected residual stress distribution.Based on RSDM a method for the shakedown analysis of structures,called RSDM-S has also been developed.Despite most direct methods for shakedown,RSDM-S does not need an optimization algorithm for its implementation.Both RSDM and RSDM-S may be implemented in any Finite Element Code.A thorough review of both these methods,together with examples of implementation are presented herein.

New expression of bimodal phase distributions in direct-method phasing of protein single-wavelength anomalous diffraction data

One of the essential points of the direct-method single-wavelength anomalous diffraction （SAD） phasing for proteins is to express the bimodal SAD phase distribution by the sum of two Gaussian functions peaked respectively at φh″＋｜△φh｜ and φh″-｜△φh｜. The probability for △φh being positive （P＋） can be derived based on the Cochran distribution in direct methods. Hence the SAD phase ambiguity can be resolved by multiplying the Gaussian function peaked at φh″＋｜△φh｜ with P＋ and multiplying the Gaussian function peaked at φh″-｜△φh｜ with P_ （=1- P＋）. The direct-method SAD h phasing has been proved powerful in breaking SAD phase ambiguities, in particular when anomalous-scattering signals are weak. However, the approximation of bimodal phase distributions by the sum of two Gaussian functions introduces considerable errors. In this paper we show that a much better approximation can be achieved by replacing the two Gaussian functions with two von Mises distributions. Test results showed that this leads to significant improvement on the efficiency of direct-method SAD-phasing.

Fully Distributed Learning for Deep Random Vector Functional-Link Networks

In the contemporary era, the proliferation of information technology has led to an unprecedented surge in data generation, with this data being dispersed across a multitude of mobile devices. Facing these situations and the training of deep learning model that needs great computing power support, the distributed algorithm that can carry out multi-party joint modeling has attracted everyone’s attention. The distributed training mode relieves the huge pressure of centralized model on computer computing power and communication. However, most distributed algorithms currently work in a master-slave mode, often including a central server for coordination, which to some extent will cause communication pressure, data leakage, privacy violations and other issues. To solve these problems, a decentralized fully distributed algorithm based on deep random weight neural network is proposed. The algorithm decomposes the original objective function into several sub-problems under consistency constraints, combines the decentralized average consensus (DAC) and alternating direction method of multipliers (ADMM), and achieves the goal of joint modeling and training through local calculation and communication of each node. Finally, we compare the proposed decentralized algorithm with several centralized deep neural networks with random weights, and experimental results demonstrate the effectiveness of the proposed algorithm.

Image reconstruction based on total-variation minimization and alternating direction method in linear scan computed tomography

Linear scan computed tomography （LCT） is of great benefit to online industrial scanning and security inspection due to its characteristics of straight-line source trajectory and high scanning speed. However, in practical applications of LCT, there are challenges to image reconstruction due to limited-angle and insufficient data. In this paper, a new reconstruction algorithm based on total-variation （TV） minimization is developed to reconstruct images from limited-angle and insufficient data in LCT. The main idea of our approach is to reformulate a TV problem as a linear equality constrained problem where the objective function is separable, and then minimize its augmented Lagrangian function by using alternating direction method （ADM） to solve subproblems. The proposed method is robust and efficient in the task of reconstruction by showing the convergence of ADM. The numerical simulations and real data reconstructions show that the proposed reconstruction method brings reasonable performance and outperforms some previous ones when applied to an LCT imaging problem.

Reconstruction of electrical capacitance tomography images based on fast linearized alternating direction method of multipliers for two-phase flow system

Electrical capacitance tomography(ECT)has been applied to two-phase flow measurement in recent years.Image reconstruction algorithms play an important role in the successful applications of ECT.To solve the ill-posed and nonlinear inverse problem of ECT image reconstruction,a new ECT image reconstruction method based on fast linearized alternating direction method of multipliers(FLADMM)is proposed in this paper.On the basis of theoretical analysis of compressed sensing(CS),the data acquisition of ECT is regarded as a linear measurement process of permittivity distribution signal of pipe section.A new measurement matrix is designed and L1 regularization method is used to convert ECT inverse problem to a convex relaxation problem which contains prior knowledge.A new fast alternating direction method of multipliers which contained linearized idea is employed to minimize the objective function.Simulation data and experimental results indicate that compared with other methods,the quality and speed of reconstructed images are markedly improved.Also,the dynamic experimental results indicate that the proposed algorithm can ful fill the real-time requirement of ECT systems in the application.

THE COLLOCATION METHODS FOR SINGULAR INTEGRAL EQUATIONS WITH CAUCHY KERNELS

This paper applies the singular integral operators, singular quadrature operators and discretization matrices associated with singular integral equations with Cauchy kernels, which are established in [1], to give a unified framework for various collocation methods of numerical solutions of singular integral equations with Cauchy kernels. Under the framework, the coincidence of the direct quadrature method and the indirect quadrature method is very simple and obvious.

Determination of stress intensity factor with direct stress approach using finite element analysis

In this article, a direct stress approach based on finite element analysis to determine the stress intensity factor is improved. Firstly, by comparing the rigorous solution against the asymptotic solution for a problem of an infinite plate embedded a central crack, we found that the stresses in a restrictive interval near the crack tip given by the rigorous solution can be used to determine the stress intensity factor, which is nearly equal to the stress intensity factor given by the asymptotic solution. Secondly, the crack problem is solved numerically by the finite element method. Depending on the modeling capability of the software, we designed an adaptive mesh model to simulate the stress singularity. Thus, the stress result in an appropriate interval near the crack tip is fairly approximated to the rigorous solution of the corresponding crack problem. Therefore, the stress intensity factor may be calculated from the stress distribution in the appropriate interval, with a high accuracy.

Combined indirect and direct method for adaptive fuzzy output feedback control of nonlinear system

A novel control method for a general class of nonlinear systems using fuzzy logic systems （FLSs） is presertted. Indirect and direct methods are combined to design the adaptive fuzzy output feedback controller and a high-gain observer is used to estimate the derivatives of the system output. The closed-loop system is proven to be semiglobally uniformly ultimately bounded. In addition, it is shown that if the approximation accuracy of the fuzzy logic system is high enough and the observer gain is chosen sufficiently large, an arbitrarily small tracking error can be achieved. Simulation results verify the effectiveness of the newly designed scheme and the theoretical discussion.

Direct discontinuous Galerkin method for the generalized Burgers-Fisher equation

In this study, we use the direct discontinuous Galerkin method to solve the generalized Burgers-Fisher equation. The method is based on the direct weak formulation of the Burgers-Fisher equation. The two adjacent cells are jointed by a numerical flux that includes the convection numerical flux and the diffusion numerical flux. We solve the ordinary differential equations arising in the direct Galerkin method by using the strong stability preserving Runge^Kutta method. Numerical results are compared with the exact solution and the other results to show the accuracy and reliability of the method.

Direct method of finding first integral of two-dimensional autonomous systems in polar coordinates

A direct method to find the first integral for two-dimensional autonomous system in polar coordinates is suggested. It is shown that if the equation of motion expressed by differential 1-forms for a given autonomous Hamiltonian system is multiplied by a set of multiplicative functions, then the general expression of the first integral can be obtained, An example is given to illustrate the application of the results.