LONG-TERM RIGOROUS NUMERICAL INTEGRATION OF NAVIER-STOKES EQUATION BY NEWTON-GMRES ITERATION

The recent result of an orbit continuation algorithm has provided a rigorous method for long-term numerical integration of an orbit on the unstable manifold of a periodic solution.This algorithm is matrix-free and employs a combination of the Newton-Raphson method and the Krylov subspace method.Moreover,the algorithm adopts a multiple shooting method to address the problem of orbital instability due to long-term numerical integration.The algorithm is described through computing the extension of unstable manifold of a recomputed Nagata′s lowerbranch steady solution of plane Couette flow,which is an example of an exact coherent state that has recently been studied in subcritical transition to turbulence.

A numerical case study on the long-term seismic assessment of reinforced concrete tunnels in corrosive environments

The paper investigates the long-term seismic behaviour of an underground reinforced concrete(RC)metro tunnel in Santiago,Chile,considering the combined effects of chloride-induced corrosion and cumulative,low-amplitude seismic shaking on the structure’s performance.The soil-tunnel response is evaluated with the aid of transient,nonlinear finite element analysis using a two-dimensional(2D)plane strain numerical model that adopts advanced nonlinear models for the simulation of soil and concrete plasticity and the dynamic stiffness behaviour.The effects of corrosion deterioration are demonstrated in terms of time-dependent loss of rebar area and cover concrete stiffness and strength.The study illustrates the influence of ageing and repeated seismic shaking on lining deformation,crack development,and the modal characteristics of the intact and degrading systems.The results indicate that multiple lowamplitude events drive the non-degrading RC tunnel beyond its elastic regime without significant structural response consequences.A noticeable impact of corrosion deterioration on the structure’s seismic performance is revealed,increasing with the number and intensity of earthquake events.Two different tunnel embedment depths are comparatively assessed.The analyses demonstrate larger coseismic section convergence in the case of the deeper tunnel,yet a less pronounced effect of ageing and successive seismic loading compared to the shallow section,which is evident in the RC lining cracks at the end of shaking.

Monitoring Study of Long-Term Land Subsidence during Subway Operation in High-Density Urban Areas Based on DInSAR-GPS-GIS Technology and Numerical Simulation

During subway operation,various factors will cause long-term land subsidence,such as the vibration subsidence of foundation soil caused by train vibration load,incomplete consolidation deformation of foundation soil during tunnel construction,dense buildings and structures in the vicinity of the tunnel,and changes in water level in the stratum where the tunnel is located.The monitoring of long-term land subsidence during subway operation in high-density urban areas differs from that in low-density urban construction areas.The former is the gathering point of the entire urban population.There are many complex buildings around the project,busy road traffic,high pedestrian flow,and less vegetation cover.Several existing items requiremonitoring.However,monitoring distance is long,and providing early warning is difficult.This study uses the 2.8 km operation line between Wulin Square station and Ding’an Road station of Hangzhou Subway Line 1 as an example to propose the integrated method of DInSAR-GPS-GIS technology and the key algorithm for long-term land subsidence deformation.Then,it selects multiscene image data to analyze long-termland subsidence of high-density urban areas during subway operation.Results show that long-term land subsidence caused by the operation of Wulin Square station to Ding’an Road station of Hangzhou Subway Line 1 is small,with maximumsubsidence of 30.64 mm,and minimumsubsidence of 11.45 mm,and average subsidence ranging from 19.27 to 21.33 mm.And FLAC3D software was used to verify the monitoring situation,using the geological conditions of the soil in the study area and the tunnel profile to simulate the settlement under vehicle load,and the simulation results tended to be consistent with the monitoring situation.

Response Sensitivity Analysis of the Dynamic Milling Process Based on the Numerical Integration Method

As one of the bases of gradient-based optimization algorithms, sensitivity analysis is usually required to calculate the derivatives of the system response with respect to the machining parameters. The most widely used approaches for sensitivity analysis are based on time-consuming numerical methods, such as finite difference methods. This paper presents a semi-analytical method for calculation of the sensitivity of the stability boundary in milling. After transforming the delay-differential equation with time-periodic coefficients governing the dynamic milling process into the integral form, the Floquet transition matrix is constructed by using the numerical integration method. Then, the analytical expressions of derivatives of the Floquet transition matrix with respect to the machining parameters are obtained. Thereafter, the classical analytical expression of the sensitivity of matrix eigenvalues is employed to calculate the sensitivity of the stability lobe diagram. The two-degree-of-freedom milling example illustrates the accuracy and efficiency of the proposed method. Compared with the existing methods, the unique merit of the proposed method is that it can be used for analytically computing the sensitivity of the stability boundary in milling, without employing any finite difference methods. Therefore, the high accuracy and high efficiency are both achieved. The proposed method can serve as an effective tool for machining parameter optimization and uncertainty analysis in high-speed milling.

Efficiency analysis of numerical integrations for finite element substructure in real-time hybrid simulation

Finite element（FE） is a powerful tool and has been applied by investigators to real-time hybrid simulations（RTHSs）. This study focuses on the computational efficiency, including the computational time and accuracy, of numerical integrations in solving FE numerical substructure in RTHSs. First, sparse matrix storage schemes are adopted to decrease the computational time of FE numerical substructure. In this way, the task execution time（TET） decreases such that the scale of the numerical substructure model increases. Subsequently, several commonly used explicit numerical integration algorithms, including the central difference method（CDM）, the Newmark explicit method, the Chang method and the Gui-λ method, are comprehensively compared to evaluate their computational time in solving FE numerical substructure. CDM is better than the other explicit integration algorithms when the damping matrix is diagonal, while the Gui-λ（λ = 4） method is advantageous when the damping matrix is non-diagonal. Finally, the effect of time delay on the computational accuracy of RTHSs is investigated by simulating structure-foundation systems. Simulation results show that the influences of time delay on the displacement response become obvious with the mass ratio increasing, and delay compensation methods may reduce the relative error of the displacement peak value to less than 5% even under the large time-step and large time delay.

Consistency and Stability Issues in the Numerical Integration of the First and Second Order Initial Value Problem

In this note we consider some basic, yet unusual, issues pertaining to the accuracy and stability of numerical integration methods to follow the solution of first order and second order initial value problems (IVP). Included are remarks on multiple solutions, multi-step methods, effect of initial value perturbations, as well as slowing and advancing the computed motion in second order problems.

Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method

In this paper a new ODE numerical integration method was successfully applied to solving nonlinear equations. The method is of same simplicity as fixed point iteration, but the efficiency has been significantly improved, so it is especially suitable for large scale systems. For Brown’s equations, an existing article reported that when the dimension of the equation N = 40, the subroutines they used could not give a solution, as compared with our method, we can easily solve this equation even when N = 100. Other two large equations have the dimension of N = 1000, all the existing available methods have great difficulties to handle them, however, our method proposed in this paper can deal with those tough equations without any difficulties. The sigularity and choosing initial values problems were also mentioned in this paper.

AN INTEGRATION METHOD WITH FITTING CUBIC SPLINE FUNCTIONS TO A NUMERICAL MODEL OF 2ND-ORDER SPACE-TIME DIFFERENTIAL REMAINDER——FOR AN IDEAL GLOBAL SIMULATION CASE WITH PRIMITIVE ATMOSPHERIC EQUATIONS

In this paper,the forecasting equations of a 2nd-order space-time differential remainder are deduced from the Navier-Stokes primitive equations and Eulerian operator by Taylor-series expansion.Here we introduce a cubic spline numerical model(Spline Model for short),which is with a quasi-Lagrangian time-split integration scheme of fitting cubic spline/bicubic surface to all physical variable fields in the atmospheric equations on spherical discrete latitude-longitude mesh.A new algorithm of"fitting cubic spline—time step integration—fitting cubic spline—……"is developed to determine their first-and2nd-order derivatives and their upstream points for time discrete integral to the governing equations in Spline Model.And the cubic spline function and its mathematical polarities are also discussed to understand the Spline Model’s mathematical foundation of numerical analysis.It is pointed out that the Spline Model has mathematical laws of"convergence"of the cubic spline functions contracting to the original functions as well as its 1st-order and 2nd-order derivatives.The"optimality"of the 2nd-order derivative of the cubic spline functions is optimal approximation to that of the original functions.In addition,a Hermite bicubic patch is equivalent to operate on a grid for a 2nd-order derivative variable field.Besides,the slopes and curvatures of a central difference are identified respectively,with a smoothing coefficient of 1/3,three-point smoothing of that of a cubic spline.Then the slopes and curvatures of a central difference are calculated from the smoothing coefficient 1/3 and three-point smoothing of that of a cubic spline,respectively.Furthermore,a global simulation case of adiabatic,non-frictional and"incompressible"model atmosphere is shown with the quasi-Lagrangian time integration by using a global Spline Model,whose initial condition comes from the NCEP reanalysis data,along with quasi-uniform latitude-longitude grids and the so-called"shallow atmosphere"Navier-Stokes primitive equations in the spherical coordinates.The Spline Model,which adopted the Navier-Stokes primitive equations and quasi-Lagrangian time-split integration scheme,provides an initial ideal case of global atmospheric circulation.In addition,considering the essentially non-linear atmospheric motions,the Spline Model could judge reasonably well simple points of any smoothed variable field according to its fitting spline curvatures that must conform to its physical interpretation.

Numerical Integration Method in Analysis of Wire Antennas

The numerical evaluation of an integral is a frequently encountered problem in antenna analysis. A special Gauss Christoffel quadrature formula for nonclassical weight function is constructed for solving the pseudo singular integration problem arising from the analysis of thin wire antennas. High integration accuracy is obtained at comparable low computation cost by the quadrature formula constructed. This integration method can be also used in other electromagnetic integral equation problems.

Behavior of the Numerical Integration Error

In this work, we consider different numerical methods for the approximation of definite integrals. The three basic methods used here are the Midpoint, the Trapezoidal, and Simpson’s rules. We trace the behavior of the error when we refine the mesh and show that Richardson’s extrapolation improves the rate of convergence of the basic methods when the integrands are sufficiently differentiable many times. However, Richardson’s extrapolation does not work when we approximate improper integrals or even proper integrals from functions without smooth derivatives. In order to save computational resources, we construct an adaptive recursive procedure. We also show that there is a lower limit to the error during computations with floating point arithmetic.

The Numerical Integration of Discrete Functions on a Triangular Element

With the application of Hammer integral formulas of a continuous function on a triangular element, the numerical integral formulas of some discrete functions on the element are derived by means of decomposition and recombination of base functions. Hammer integral formulas are the special examples of those of the paper.

Solving Large Scale Unconstrained Minimization Problems by a New ODE Numerical Integration Method

In reference [1], for large scale nonlinear equations , a new ODE solving method was given. This paper is a continuous work. Here has gradient structure i.e. , is a scalar function. The eigenvalues of the Jacobian of;or the Hessian of , are all real number. So the new method is very suitable for this structure. For quadratic function the convergence was proved and the spectral radius of iteration matrix was given and compared with traditional method. Examples show for large scale problems (dimension ) the new method is very efficient.

PARALLEL INFORMATION-BASED COMPLEXITY OF NUMERICAL INTEGRATION ON SOBOLEV CLASS W_q^s(Ω)

This paper deals with the parallel information-based complexity of numerical integration on Sobolev class. We obtain tight bounds on the complexity, considered as a function of two variables simultaneously:the number of processors, the rquired precision. This result seems to be new even in serial case.

Numerical Integration for DAEs of Multibody System Dynamics

During the simulation of constrained multibody system,numerical integration is important for solving the Euler-Lagrange equation of multibody system dynamics,which is usually a Differential-Algebraic Equations(DAEs).Using the discrete Hamilton principle,discrete EulerLagrangian equation is obtained first based on Lagrange Interpolation.Then the Romberg,Gauss integral is used to solve the DAEs.At last,numerical results are compared by using Euler method,Runge-Kutta method,Romberg method and Gauss method for a double pendulum system.

NEW NUMERICAL METHOD FOR VOLTERRA INTEGRAL EQUATION OF THE SECOND KIND IN PIEZOELASTIC DYNAMIC PROBLEMS

The elastodynamic problems of piezoelectric hollow cylinders and spheres under radial deformation can be transformed into a second kind Volterra integral equation about a function with respect to time, which greatly simplifies the solving procedure for such elastodynamic problems. Meanwhile, it becomes very important to find a way to solve the second kind Volterra integral equation effectively and quickly. By using an interpolation function to approximate the unknown function, two new recursive formulae were derived, based on which numerical solution can be obtained step by step. The present method can provide accurate numerical results efficiently. It is also very stable for long time calculating.

Multistage Numerical Picard Iteration Methods for Nonlinear Volterra Integral Equations of the Second Kind

Using the Picard iteration method and treating the involved integration by numerical quadrature formulas, we propose a numerical scheme for the second kind nonlinear Volterra integral equations. For enlarging the convergence region of the Picard iteration method, multistage algorithm is devised. We also introduce an algorithm for problems with some singularities at the limits of integration including fractional integral equations. Numerical tests verify the validity of the proposed schemes.

The Numerical Solutions of Systems of Nonlinear Integral Equations with the Spline Functions

The main goal of this work is to develop an effective technique for solving nonlinear systems of Volterra integral equations. The main tools are the cardinal spline functions on small compact supports. We solve a system of algebra equations to approximate the solution of the system of integral equations. Since the matrix for the algebraic system is nearly triangular, It is relatively painless to solve for the unknowns and an approximation of the original solution with high precision is accomplished. In order to enhance the accuracy, several cardinal splines are employed in the paper. Our schemes were compared with other techniques proposed in recent papers and the advantage of our method was exhibited with several numerical examples.

On Trigonometric Numerical Integrator for Solving First Order Ordinary Differential Equation

In this paper, we used an interpolation function with strong trigonometric components to derive a numerical integrator that can be used for solving first order initial value problems in ordinary differential equation. This numerical integrator has been tested for desirable qualities like stability, convergence and consistency. The discrete models have been used for a numerical experiment which makes us conclude that the schemes are suitable for the solution of first order ordinary differential equation.

Numerical investigation on damping coefficient of the integral squeeze film damper

The elimination of rotor vibration is usually achieved by applying additional damping to the system.Squeeze film dampers are widely used in various aerospace and turbine equipments.The research is carried out on flow characteristics in the integral squeeze film dampers(ISFDs).The dynamic response to the operation condition is investigated through the computational fluid dynamics(CFD) model of ISFD.Due to the large pressure loss at the oil inlet,the oil film force only changes slightly with the increase of oil supply pressure,and the damping increases slightly.The vibration amplitude only affects the film force,but has no effect on the damping.The oil film force and damping show an upward tendency with the decrease of thickness of the end seal clearance.

RECURSIVE PARAMETER ESTIMATION OF LINEAR DELAYED SYSTEM VIA NUMERICAL INTEGRAL OPERATION

In this paper,the problem of Parameter estimation in linear delayedsystems from sampled data is treated.Using numerical integral operation(NIO),anidentification model which is parametrized directly in the linear delayed system pa-rameters is got.With the least square(LS)method or the instrumental variable(Ⅳ)method,the recursive algorithm of parameter estimation is given.The result ofthe illustrated example shows that this algorithm is simple,rapid and accurrate.