High-Order Decoupled and Bound Preserving Local Discontinuous Galerkin Methods for a Class of Chemotaxis Models

In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-dependent problems.We use the convex splitting method,the variant energy quadratization method,and the scalar auxiliary variable method coupled with the LDG method to construct first-order temporal accurate schemes based on the gradient flow structure of the models.These semi-implicit schemes are decoupled,energy stable,and can be extended to high accuracy schemes using the semi-implicit spectral deferred correction method.Many bound preserving DG discretizations are only worked on explicit time integration methods and are difficult to get high-order accuracy.To overcome these difficulties,we use the Lagrange multipliers to enforce the implicit or semi-implicit LDG schemes to satisfy the bound constraints at each time step.This bound preserving limiter results in the Karush-Kuhn-Tucker condition,which can be solved by an efficient active set semi-smooth Newton method.Various numerical experiments illustrate the high-order accuracy and the effect of bound preserving.

Local singularity and S–A methods for analyzing ore-producing anomalies in the Jianbiannongchang area of Heilongjiang,China

The Heilongjiang Jianbiannongchang area is located at the confluence of the Great and Lesser Xing’an Ranges.This area has a complex magmatic and tectonic evolutionary history that has resulted in a complex and diverse geological background for mineralization.In this study,isometric logarithmic ratio(ILR)transformations of Au,Cu,Pb,Zn,and Sb contents were performed in the1:50,000 soil geochemical data of the Jianbiannongchang area.Robust principal component analysis(RPCA)was conducted based on ILR transformation.The local singularity and spectrum-area(S-A)methods were used to extract information on mineralogic anomalies.The results showed that:(1)the transformed data eliminated the influence of the original data closure effect,and the PC1and PC2 information obtained by applying RPCA reflected ore-producing element anomalies dominated by Au and Cu.(2)The local singularity method can enhance the information of the local strong and weak slow anomalies.After performing local singularity analysis on PC1 and PC2,the obtained local anomalies reflected the local singularity spatial anomaly patterns related to Cu and Au mineralization in this area,which is an effective method for trapping ore-producing anomalies.(3)Furthermore,the composite anomaly decomposition of PC1 and PC2 was performed using the S-A method,and the screened anomalous and background fields reflect the ore-producing anomalies related to Cu and Au mineralization.This information is in agreement with known Cu and Au mineralization.(4)The geochemical anomalies with mineralization potential were obtained outside the known mineralization sites by integrating the information of oreproducing anomalies extracted by the local singularity and S-A methods,providing the theoretical basis and exploration direction for future exploration in the study area.

A MESHLESS LOCAL PETROV-GALERKIN METHOD FOR GEOMETRICALLY NONLINEAR PROBLEMS

Nonlinear formulations of the meshless local Petrov-Galerkin （MLPG） method are presented for geometrically nonlinear problems. The method requires no mesh in computation and therefore avoids mesh distortion difficulties in the large deformation analysis. The essential boundary conditions in the present formulation axe imposed by a penalty method. An incremental and iterative solution procedure is used to solve geometrically nonlinear problems. Several examples are presented to demonstrate the effectiveness of the method in geometrically nonlinear problems analysis. Numerical results show that the MLPG method is an effective one and that the values of the unknown variable are quite accurate.

NUMERICAL ANALYSIS OF MINDLIN SHELL BY MESHLESS LOCAL PETROV-GALERKIN METHOD

The objectives of this study are to employ the meshless local Petrov-Galerkin method （MLPGM） to solve three-dimensional shell problems. The computational accuracy of MLPGM for shell problems is affected by many factors, including the dimension of compact support domain, the dimension of quadrture domain, the number of integral cells and the number of Gauss points. These factors＇ sensitivity analysis is to adopt the Taguchi experimental design technology and point out the dimension of the quadrature domain with the largest influence on the computational accuracy of the present MLPGM for shells and give out the optimum combination of these factors. A few examples are given to verify the reliability and good convergence of MLPGM for shell problems compared to the theoretical or the finite element results.

LOCAL PETROV-GALERKIN METHOD FOR A THIN PLATE

The meshless local Petrov_Galerkin (MLPG) method for solving the bending problem of the thin plate were presented and discussed. The method used the moving least_squares approximation to interpolate the solution variables, and employed a local symmetric weak form. The present method was a truly meshless one as it did not need a finite element or boundary element mesh, either for purpose of interpolation of the solution, or for the integration of the energy. All integrals could be easily evaluated over regularly shaped domains (in general, spheres in three_dimensional problems) and their boundaries. The essential boundary conditions were enforced by the penalty method. Several numerical examples were presented to illustrate the implementation and performance of the present method. The numerical examples presented show that high accuracy can be achieved for arbitrary grid geometries for clamped and simply_supported edge conditions. No post processing procedure is required to computer the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.

A Novel Localized Meshless Method for Solving Transient Heat Conduction Problems in Complicated Domains

This paper first attempts to solve the transient heat conduction problem by combining the recently proposed local knot method(LKM)with the dual reciprocity method(DRM).Firstly,the temporal derivative is discretized by a finite difference scheme,and thus the governing equation of transient heat transfer is transformed into a non-homogeneous modified Helmholtz equation.Secondly,the solution of the non-homogeneous modified Helmholtz equation is decomposed into a particular solution and a homogeneous solution.And then,the DRM and LKM are used to solve the particular solution of the non-homogeneous equation and the homogeneous solution of the modified Helmholtz equation,respectively.The LKM is a recently proposed local radial basis function collocationmethod with themerits of being simple,accurate,and free ofmesh and integration.Compared with the traditional domain-type and boundary-type schemes,the present coupling algorithm could be treated as a really good alternative for the analysis of transient heat conduction on high-dimensional and complicated domains.Numerical experiments,including two-and three-dimensional heat transfer models,demonstrated the effectiveness and accuracy of the new methodology.

The complex variable meshless local Petrov-Galerkin method of solving two-dimensional potential problems

Based on the complex variable moving least-square（CVMLS） approximation and a local symmetric weak form,the complex variable meshless local Petrov-Galerkin（CVMLPG） method of solving two-dimensional potential problems is presented in this paper.In the present formulation,the trial function of a two-dimensional problem is formed with a one-dimensional basis function.The number of unknown coefficients in the trial function of the CVMLS approximation is less than that in the trial function of the moving least-square（MLS） approximation.The essential boundary conditions are imposed by the penalty method.The main advantage of this approach over the conventional meshless local Petrov-Galerkin（MLPG） method is its computational efficiency.Several numerical examples are presented to illustrate the implementation and performance of the present CVMLPG method.

Quasi Ellipsoid Gear Surface Reconstruction Based on Meshless Local Petrov-Galerkin Method and Transmission Characteristic

Special transmission 3D model simulation must be based on surface discretization and reconstruction, but special transmission usually has complicated tooth shape and movement, so present software can＇t provide technical support for special transmission 3D model simulation. Currently, theoretical calculation and experimental method are difficult to exactly solve special transmission contact analysis problem. How to reduce calculation and computer memories consume and meet calculation precision is key to resolve special transmission contact analysis problem. According to 3D model simulation and surface reconstruction of quasi ellipsoid gear is difficulty, this paper employes meshless local Petrov-Galerkin （MLPG） method. In order to reduce calculation and computer memories consume, we disperse tooth mesh into finite points--sparseness points cloud or grid mesh, and then we do interpolation reconstruction in some necessary place of the 3D surface model during analysis. Moving least square method （MLSM） is employed for tooth mesh interpolation reconstruction, there are some advantages to do interpolation by means of MLSM, such as high precision, good flexibility and no require of tooth mesh discretization into units. We input the quasi ellipsoid gear reconstruction model into simulation software, we complete tooth meshing simulation. Simulation transmission ratio during meshing period was obtained, compared with theoretical transmission ratio, the result inosculate preferably. The method using curve reconstruction realizes surface reconstruction, reduce simulation calculation enormously, so special gears simulation can be realized by minitype computer. The method provides a novel solution for special transmission 3D model simulation analysis and contact analysis.

A LOCAL DISCONTINUOUS GALERKIN METHOD FOR TIME-FRACTIONAL DIFFUSION EQUATIONS

In this paper,a local discontinuous Galerkin(LDG)scheme for the time-fractional diffusion equation is proposed and analyzed.The Caputo time-fractional derivative(of orderα,with 0<α<1)is approximated by a finite difference method with an accuracy of order3-α,and the space discretization is based on the LDG method.For the finite difference method,we summarize and supplement some previous work by others,and apply it to the analysis of the convergence and stability of the proposed scheme.The optimal error estimate is obtained in the L2norm,indicating that the scheme has temporal(3-α)th-order accuracy and spatial(k+1)th-order accuracy,where k denotes the highest degree of a piecewise polynomial in discontinuous finite element space.The numerical results are also provided to verify the accuracy and efficiency of the considered scheme.

Stress-corrosion coupled damage localization induced by secondary phases in bio-degradable Mg alloys:phase-field modeling

In this study,a phase-field scheme that rigorously obeys conservation laws and irreversible thermodynamics is developed for modeling stress-corrosion coupled damage(SCCD).The coupling constitutive relationships of the deformation,phase-field damage,mass transfer,and electrostatic field are derived from the entropy inequality.The SCCD localization induced by secondary phases in Mg is numerically simulated using the implicit iterative algorithm of the self-defined finite elements.The quantitative evaluation of the SCCD of a C-ring is in good agreement with the experimental results.To capture the damage localization,a micro-galvanic corrosion domain is defined,and the buffering effect on charge migration is explored.Three cases are investigated to reveal the effect of localization on corrosion acceleration and provide guidance for the design for resistance to SCCD at the crystal scale.

Integrated multi-scale approach combining global homogenization and local refinement for multi-field analysis of high-temperature superconducting composite magnets

Second-generation high-temperature superconducting(HTS)conductors,specifically rare earth-barium-copper-oxide(REBCO)coated conductor(CC)tapes,are promising candidates for high-energy and high-field superconducting applications.With respect to epoxy-impregnated REBCO composite magnets that comprise multilayer components,the thermomechanical characteristics of each component differ considerably under extremely low temperatures and strong electromagnetic fields.Traditional numerical models include homogenized orthotropic models,which simplify overall field calculation but miss detailed multi-physics aspects,and full refinement(FR)ones that are thorough but computationally demanding.Herein,we propose an extended multi-scale approach for analyzing the multi-field characteristics of an epoxy-impregnated composite magnet assembled by HTS pancake coils.This approach combines a global homogenization(GH)scheme based on the homogenized electromagnetic T-A model,a method for solving Maxwell's equations for superconducting materials based on the current vector potential T and the magnetic field vector potential A,and a homogenized orthotropic thermoelastic model to assess the electromagnetic and thermoelastic properties at the macroscopic scale.We then identify“dangerous regions”at the macroscopic scale and obtain finer details using a local refinement(LR)scheme to capture the responses of each component material in the HTS composite tapes at the mesoscopic scale.The results of the present GH-LR multi-scale approach agree well with those of the FR scheme and the experimental data in the literature,indicating that the present approach is accurate and efficient.The proposed GH-LR multi-scale approach can serve as a valuable tool for evaluating the risk of failure in large-scale HTS composite magnets.

Stability and Time-Step Constraints of Implicit-Explicit Runge-Kutta Methods for the Linearized Korteweg-de Vries Equation

This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.

Structural Reliability Assessment by a Modified Spectral Stochastic Meshless Local Petrov-Galerkin Method

This study presents a new tool for solving stochastic boundary-value problems. This tool is created by modify the previous spectral stochastic meshless local Petrov-Galerkin method using the MLPG5 scheme. This modified spectral stochastic meshless local Petrov-Galerkin method is selectively applied to predict the structural failure probability with the uncertainty in the spatial variability of mechanical properties. Except for the MLPG5 scheme, deriving the proposed spectral stochastic meshless local Petrov-Galerkin formulation adopts generalized polynomial chaos expansions of random mechanical properties. Predicting the structural failure probability is based on the first-order reliability method. Further comparing the spectral stochastic finite element-based and meshless local Petrov-Galerkin-based predicted structural failure probabilities indicates that the proposed spectral stochastic meshless local Petrov-Galerkin method predicts the more accurate structural failure probability than the spectral stochastic finite element method does. In addition, generating spectral stochastic meshless local Petrov-Galerkin results are considerably time-saving than generating Monte-Carlo simulation results does. In conclusion, the spectral stochastic meshless local Petrov-Galerkin method serves as a time-saving tool for solving stochastic boundary-value problems sufficiently accurately.

Meshless Local Discontinuous Petrov-Galerkin Method with Application to Blasting Problems

A meshless local discontinuous Petrov-Galerkin (MLDPG) method based on the local symmetric weak form (LSWF) is presented with the application to blasting problems. The derivation is similar to that of mesh-based Runge-Kutta Discontinuous Galerkin (RKDG) method. The solutions are reproduced in a set of overlapped spherical sub-domains, and the test functions are employed from a partition of unity of the local basis functions. There is no need of any traditional non-overlapping mesh either for local approximation purpose or for Galerkin integration purpose in the presented method. The resulting MLDPG method is a meshless, stable, high-order accurate and highly parallelizable scheme which inherits both the advantages of RKDG and meshless method (MM), and it can handle the problems with extremely complicated physics and geometries easily. Three numerical examples of the one-dimensional Sod shock-tube problem, the blast-wave problem and the Woodward-Colella interacting shock wave problem are given. All the numerical results are in good agreement with the closed solutions. The higher-order MLDPG schemes can reproduce more accurate solution than the lower-order schemes.

A local pseudo arc-length method for hyperbolic conservation laws

A local pseudo arc-length method（LPALM）for solving hyperbolic conservation laws is presented in this paper.The key idea of this method comes from the original arc-length method,through which the critical points are bypassed by transforming the computational space.The method is based on local changes of physical variables to choose the discontinuous stencil and introduce the pseudo arc-length parameter,and then transform the governing equations from physical space to arc-length space.In order to solve these equations in arc-length coordinate,it is necessary to combine the velocity of mesh points in the moving mesh method,and then convert the physical variable in arclength space back to physical space.Numerical examples have proved the effectiveness and generality of the new approach for linear equation,nonlinear equation and system of equations with discontinuous initial values.Non-oscillation solution can be obtained by adjusting the parameter and the mesh refinement number for problems containing both shock and rarefaction waves.

Shift control method for the local time at descendi ngnode based on sun-synchronous orbit satellite

This article analyzes the shift factors of the descending node local time for sun-synchronous satellites and proposes a shift control method to keep the local time shift within an allowance range. It is found that the satellite orbit design and the orbit injection deviation are the causes for the initial shift velocity, whereas the atmospheric drag and the sun gravitational perturbation produce the shift acceleration. To deal with these shift factors, a shift control method is put forward, through such methods as orbit variation design, orbit altitude, and inclination keeping control. The simulation experiment and practical application have proved the effectiveness of this control method.

A STUDY ON THE WEIGHT FUNCTION OF THE MOVING LEAST SQUARE APPROXIMATION IN THE LOCAL BOUNDARY INTEGRAL EQUATION METHOD

The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the basic function and of the weight function,and is mainly determined by that of the weight function.Therefore,the weight function greatly affects the accuracy of results obtained.Different kinds of weight functions,such as the spline function, the Gauss function and so on,are proposed recently by many researchers.In the present work,the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method.The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed.Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and α in Gauss and exponential weight functions are in the range of reasonable values,respectively,and the higher the smoothness of the weight function,the better the features of the solutions.

An Efficient Method for Local Buckling Analysis of Stiffened Panels

The local buckling of stiffened panels is one of possible failure modes and concerned by engineers in the preliminary design of lightweight structures. In practice,a simplified model,i.e.,a rectangular plate with elastically restrained along its unloaded edges,is established and the Ritz method is usually employed for solutions. To use the Ritz method,however,the loaded edges of the plate are usually assumed to be simply supported. An empirical correction factor has to be used to account for clamped loaded edges. Here,a simple and efficient method,called the quadrature element method(QEM),is presented for obtaining accurate buckling behavior of rectangular plates with any combinations of boundary conditions, including the elastically restrained conditions. Different from the conventional high order finite element method(FEM),non-uniformly distributed nodes are used,and thus the method can achieve an exponential rate of convergence. Formulations are worked out in detail. A computer program is developed. Improvement of solution accuracy can be easily achieved by changing the number of element nodes in the computer program. Several numerical examples are given. Results are compared with either existing solutions or finite element data for verifications. It is shown that high solution accuracy is achieved. In addition,the proposed method and developed computer program can allow quick analysis of local buckling of stiffened panels and thus is suitable for optimization routines in the preliminary design stage.

h-ADAPTIVE ANALYSIS BASED ON MESHLESS LOCAL PETROV-G ALERKIN METHOD WITH B SPLINE WAVELET FOR PLATES AND SHELLS

Using the two-scale decomposition technique, the h-adaptive meshless local Petrov- Galerkin method for solving Mindlin plate and shell problems is presented. The scaling functions of B spline wavelet are employed as the basis of the moving least square method to construct the meshless interpolation function. Multi-resolution analysis is used to decompose the field variables into high and low scales and the high scale component can commonly represent the gradient of the solution according to inherent characteristics of wavelets. The high scale component in the present method can directly detect high gradient regions of the field variables. The developed adaptive refinement scheme has been applied to simulate actual examples, and the effectiveness of the present adaptive refinement scheme has been verified.

Local discontinuous Galerkin method for solving Burgers and coupled Burgers equations

In the current work, we extend the local discontinuous Galerkin method to a more general application system. The Burgers and coupled Burgers equations are solved by the local discontinuous Galerkin method. Numerical experiments are given to verify the efficiency and accuracy of our method. Moreover the numerical results show that the method can approximate sharp fronts accurately with minimal oscillation.