A Filtered Milne-Simpson ODE Method with Enhanced Stability Properties

The Milne-Simpson method is a two-step implicit linear multistep method for the numerical solution of ODEs that obtains the theoretically highest order of convergence for such a method. The stability region of the method is only an interval on the imaginary axis and the method is classified as weakly stable which causes non-physical oscillations to appear in numerical solutions. For this reason, the method is seldom used in applications. This work examines filtering techniques that improve the stability properties of the Milne-Simpson method while retaining its fourth-order convergence rate. The resulting filtered Milne-Simpson method is attractive as a method of lines integrator of linear time-dependent partial differential equations.

A SPECTRAL METHOD FOR A WEAKLY SINGULAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION WITH PANTOGRAPH DELAY

In this paper,a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay,which contains a weakly singular kernel.We use a function transformation and a variable transformation to change the equation into a new Volterra integral equation defined on the standard interval[-1,1],so that the Jacobi orthogonal polynomial theory can be applied conveniently.In order to obtain high order accuracy for the approximation,the integral term in the resulting equation is approximated by Jacobi spectral quadrature rules.In the end,we provide a rigorous error analysis for the proposed method.The spectral rate of convergence for the proposed method is established in both the L^(∞)-norm and the weighted L^(2)-norm.

GENERALIZED JACOBI SPECTRAL GALERKIN METHOD FOR FRACTIONAL-ORDER VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH WEAKLY SINGULAR KERNELS

For fractional Volterra integro-differential equations(FVIDEs)with weakly singular kernels,this paper proposes a generalized Jacobi spectral Galerkin method.The basis functions for the provided method are selected generalized Jacobi functions(GJFs),which can be utilized as natural basis functions of spectral methods for weakly singular FVIDEs when appropriately constructed.The developed method's spectral rate of convergence is determined using the L^(∞)-norm and the weighted L^(2)-norm.Numerical results indicate the usefulness of the proposed method.

Study on the critical stress threshold of weakly cemented sandstone damage based on the renormalization group method

During the microstructural analysis of weakly cemented sandstone,the granule components and ductile structural parts of the sandstone are typically generalized.Considering the contact between granules in the microstructure of weakly cemented sandstone,three basic units can be determined:regular tetrahedra,regular hexahedra,and regular octahedra.Renormalization group models with granule-and pore-centered weakly cemented sandstone were established,and,according to the renormalization group transformation rule,the critical stress threshold of damage was calculated.The results show that the renormalization model using regular octahedra as the basic units has the highest critical stress threshold.The threshold obtained by iterative calculations of the granule-centered model is smaller than that obtained by the pore-centered model.The granule-centered calculation provides the lower limit(18.12%),and the pore-centered model provides the upper limit(36.36%).Within this range,the weakly cemented sandstone is in a phase-like critical state.That is,the state of granule aggregation transforms from continuous to discrete.In the relative stress range of 18.12%-36.36%,the weakly cemented sandstone exhibits an increased proportion of high-frequency signals(by 83.3%)and a decreased proportion of low-frequency signals(by 23.6%).The renormalization calculation results for weakly cemented sandstone explain the high-low frequency conversion of acoustic emission signals during loading.The research reported in this paper has important significance for elucidating the damage mechanism of weakly cemented sandstone.

Weak Galerkin Finite Element Method for the Unsteady Stokes Equation

The Weak Galerkin (WG) finite element method for the unsteady Stokes equations in the primary velocity-pressure formulation is introduced in this paper. Optimal-order error estimates are established for the corresponding numerical approximation in an H1 norm for the velocity, and L2 norm for both the velocity and the pressure by use of the Stokes projection.

Sliding surface searching method for slopes containing a potential weak structural surface

Weak structural surface is one of the key factors controlling the stability of slopes. The stability of rock slopes is in general concerned with set of discontinuities. However, in soft rocks, failure can occur along surfaces approaching to a circular failure surface. To better understand the position of potential sliding surface, a new method called simplex-finite stochastic tracking method is proposed. This method basically divides sliding surface into two parts： one is described by smooth curve obtained by random searching, the other one is po｜yline formed by the weak structural surface. Single or multiple sliding surfaces can be considered, and consequently several types of combined sliding surfaces can be simu- lated. The paper will adopt the arc-polyline to simulate potential sliding surface and analyze the searching process of sliding surface. Accordingly, software for slope stability analysis using this method was developed and applied in real cases. The results show that, using simplex-finite stochastic tracking method, it is possible to locate the position of a potential sliding surface in the slope.

Correlation method estimation of the modulation signal in the weak equivalence principle test

In a test of the weak equivalence principle （WEP） with a rotating torsion pendulum, it is important to estimate the amplitude of the modulation signal with high precision. We use a torsional filter to remove the free oscillation signal and employ the correlation method to estimate the amplitude of the modulation signal. The data analysis of an experiment shows that the uncertainties of amplitude components of the modulation signal obtained by the correlation method are in agreement with those due to white noise. The power spectral density of the modulation signal obtained by the correlation method is about one order higher than the thermal noise limit. It indicates that the correlation method is an effective way to estimate the amplitude of the modulation signal and it is instructive to conduct a high-accuracy WEP test.

Weakness Ranking Method for Subsystems of Heavy-Duty Machine Tools Based on FMECA Information

Heavy-duty machine tools are composed of many subsystems with different functions,and their reliability is governed by the reliabilities of these subsystems.It is important to rank the weaknesses of subsystems and identify the weakest subsystem to optimize products and improve their reliabilities.However,traditional ranking methods based on failure mode effect and critical analysis(FMECA)does not consider the complex maintenance of products.Herein,a weakness ranking method for the subsystems of heavy-duty machine tools is proposed based on generalized FMECA information.In this method,eight reliability indexes,including maintainability and maintenance cost,are considered in the generalized FMECA information.Subsequently,the cognition best worst method is used to calculate the weight of each screened index,and the weaknesses of the subsystems are ranked using a technique for order preference by similarity to an ideal solution.Finally,based on the failure data collected from certain domestic heavy-duty horizontal lathes,the weakness ranking result of the subsystems is obtained to verify the effectiveness of the proposed method.An improved weakness ranking method that can comprehensively analyze and identify weak subsystems is proposed herein for designing and improving the reliability of complex electromechanical products.

Weakly Singular Symmetric Galerkin Boundary Element Method for Fracture Analysis of Three-Dimensional Structures Considering Rotational Inertia and Gravitational Forces

The Symmetric Galerkin Boundary Element Method is advantageous for the linear elastic fracture and crackgrowth analysis of solid structures,because only boundary and crack-surface elements are needed.However,for engineering structures subjected to body forces such as rotational inertia and gravitational loads,additional domain integral terms in the Galerkin boundary integral equation will necessitate meshing of the interior of the domain.In this study,weakly-singular SGBEM for fracture analysis of three-dimensional structures considering rotational inertia and gravitational forces are developed.By using divergence theorem or alternatively the radial integration method,the domain integral terms caused by body forces are transformed into boundary integrals.And due to the weak singularity of the formulated boundary integral equations,a simple Gauss-Legendre quadrature with a few integral points is sufficient for numerically evaluating the SGBEM equations.Some numerical examples are presented to verify this approach and results are compared with benchmark solutions.

A LEAST SQUARE BASED WEAK GALERKIN FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC EQUATIONS IN NON-DIVERGENCE FORM

This article is devoted to establishing a least square based weak Galerkin method for second order elliptic equations in non-divergence form using a discrete weak Hessian operator.Naturally,the resulting linear system is symmetric and positive definite,and thus the algorithm is easy to implement and analyze.Convergence analysis in the H2 equivalent norm is established on an arbitrary shape regular polygonal mesh.A superconvergence result is proved when the coefficient matrix is constant or piecewise constant.Numerical examples are performed which not only verify the theoretical results but also reveal some unexpected superconvergence phenomena.

LOCAL CONVERGENCE OF INEXACT NEWTON-LIKE METHOD UNDER WEAK LIPSCHITZ CONDITIONS

The paper develops the local convergence of Inexact Newton-Like Method(INLM)for approximating solutions of nonlinear equations in Banach space setting.We employ weak Lipschitz and center-weak Lipschitz conditions to perform the error analysis.The obtained results compare favorably with earlier ones such as[7,13,14,18,19].A numerical example is also provided.

Algebraic structure and Poisson method for a weakly nonholonomic system

The algebraic structure and the Poisson method for a weakly nonholonomic system are studied.The differential equations of motion of the system can be written in a contravariant algebra form and its algebraic structure is discussed.The Poisson theory for the systems which possess Lie algebra structure is generalized to the weakly nonholonomic system.An example is given to illustrate the application of the result.

A NUMERICAL METHOD FOR FRACTIONAL INTEGRAL WITH APPLICATIONS

A new numerical method for the fractional integral that only stores part history data is presented, and its discretization error is estimated. The method can be used to solve the integro_differential equation including fractional integral or fractional derivative in a long history. The difficulty of storing all history data is overcome and the error can be controlled. As application,motion equations governing the dynamical behavior of a viscoelastic Timoshenko beam with fractional derivative constitutive relation are given. The dynamical response of the beam subjected to a periodic excitation is studied by using the separation variables method. Then the new numerical method is used to solve a class of weakly singular Volterra integro_differential equations which are applied to describe the dynamical behavior of viscoelastic beams with fractional derivative constitutive relations. The analytical and unmerical results are compared. It is found that they are very close.

QUANTITATIVE ANALYSIS OF VERY WEAK ACID AND BASE BY pH-FIXED TITRATION METHOD

pH-fixed titration method for the determination of weak acids and bases has been studied in this paper.It is not necessary to know the ionization constant of weak acid or base and the concentration of titrant. This method had been applied to determine phenol,4-aminoantipyrine and glycine,whose ionization constants range from 10^(-10)to 10^(-12).The results were satisfactory.

A Modified Weak Galerkin Finite Element Method for the Biharmonic Equation on Polytopal Meshes

A modified weak Galerkin(MWG)finite element method is developed for solving the biharmonic equation.This method uses the same finite element space as that of the discontinuous Galerkin method,the space of discontinuous polynomials on polytopal meshes.But its formulation is simple,symmetric,positive definite,and parameter independent,without any of six inter-element face-integral terms in the formulation of the discontinuous Galerkin method.Optimal order error estimates in a discrete H2 norm are established for the corresponding finite element solutions.Error estimates in the L^(2)norm are also derived with a sub-optimal order of convergence for the lowest-order element and an optimal order of convergence for all high-order of elements.The numerical results are presented to confirm the theory of convergence.

A Uniformly Convergent Numerical Method Using Weak Formulation for Singularly Perturbed Differential Equations

A numerical method using weak formulation is proposed to solve singularly perturbed differential equations. The numerical method is applied to both linear and nonlinear perturbation problems. A linear differential equation is solved using its weak formulation with a test space composed of exponential functions matching boundary layers. A nonlinear singular perturbation problem is converted into a system of linear differentiation equations. Then each linear differential equation is solved iteratively. The uniform convergence, which is independent of the singular perturbation parameter, is numerically verified.

Temporally Modulated Phase Retrieval Method for Weak Temporal Phase Measurement of Laser Pulses

We propose a simple iterative algorithm based on a temporally movable phase modulation process to retrieve the weak temporal phase of laser pulses. This unambiguous method can be used to achieve a high accuracy and to simultaneously measure the weak temporal phase and temporal profile of pulses, which are almost transform- limited. A detailed analysis shows that this iterative method has valuable potential applications in the charac- terization of pulses with weak temporal phase.

New Method for the Determination of Ionization Constants of Polyprotic Weak Acid

A new method for the determination of ionization constants of polyprotic weak acids is presented. Based on dissociation equilibrium,mass balance and charge balance, the mathematic model is established and the non linear least squares Gauss Newton method is applied to numerically solve the model equations. In order to get the concentration of hydrogen ion, the Debye Hückel equation is used to calculate its activity coefficient. The ionization constants of H 2SO 3 and H 2C 2O 4 obtained by this method are in good agreement with the literature values.

The Chromatic Point-spread Function of Weak Lensing Measurement in the Chinese Space Station Survey Telescope

Weak gravitational lensing is a powerful tool in modern cosmology.To accurately measure the weak lensing signal,one has to control the systematic bias on a small level.One of the most difficult problems is how to correct the smearing effect of the Point-Spread Function(PSF)on the shape of the galaxies.The chromaticity of PSF for a broad-band observation can lead to new subtle effects.Since the PSF is wavelength-dependent and the spectrum energy distributions between stars and galaxies are different,the effective PSF measured from the star images will be different from those that smear the galaxies.Such a bias is called color bias.We estimate it in the optical bands of the Chinese Space Station Survey Telescope from simulated PSFs,and show the dependence on the color and redshift of the galaxies.Moreover,due to the spatial variation of spectra over the galaxy image,another higher-order bias exists:color gradient bias.Our results show that both color bias and color gradient bias are generally below 0.1%in CSST.Only for small-size galaxies,one needs to be careful about the color gradient bias in the weak lensing analysis using CSST data.