Bound-Preserving Discontinuous Galerkin Methods with Modified Patankar Time Integrations for Chemical Reacting Flows

In this paper,we develop bound-preserving discontinuous Galerkin(DG)methods for chemical reactive flows.There are several difficulties in constructing suitable numerical schemes.First of all,the density and internal energy are positive,and the mass fraction of each species is between 0 and 1.Second,due to the rapid reaction rate,the system may contain stiff sources,and the strong-stability-preserving explicit Runge-Kutta method may result in limited time-step sizes.To obtain physically relevant numerical approximations,we apply the bound-preserving technique to the DG methods.Though traditional positivity-preserving techniques can successfully yield positive density,internal energy,and mass fractions,they may not enforce the upper bound 1 of the mass fractions.To solve this problem,we need to(i)make sure the numerical fluxes in the equations of the mass fractions are consistent with that in the equation of the density;(ii)choose conservative time integrations,such that the summation of the mass fractions is preserved.With the above two conditions,the positive mass fractions have summation 1,and then,they are all between 0 and 1.For time discretization,we apply the modified Runge-Kutta/multi-step Patankar methods,which are explicit for the flux while implicit for the source.Such methods can handle stiff sources with relatively large time steps,preserve the positivity of the target variables,and keep the summation of the mass fractions to be 1.Finally,it is not straightforward to combine the bound-preserving DG methods and the Patankar time integrations.The positivity-preserving technique for DG methods requires positive numerical approximations at the cell interfaces,while Patankar methods can keep the positivity of the pre-selected point values of the target variables.To match the degree of freedom,we use polynomials on rectangular meshes for problems in two space dimensions.To evolve in time,we first read the polynomials at the Gaussian points.Then,suitable slope limiters can be applied to enforce the positivity of the solutions at those points,which can be preserved by the Patankar methods,leading to positive updated numerical cell averages.In addition,we use another slope limiter to get positive solutions used for the bound-preserving technique for the flux.Numerical examples are given to demonstrate the good performance of the proposed schemes.

A Modified Lagrange Method for Solving Convex Quadratic Optimization Problems

In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality of the Lagrangian function with respect to the primary variables of the problem, decomposes the solution process into two independent ones, in which the primary variables are solved for independently, and then the secondary variables, which are the Lagrange multipliers, are solved for, afterward. This is an innovation that leads to solving independently two simpler systems of equations involving the primary variables only, on one hand, and the secondary ones on the other. Solutions obtained for small sized problems (as preliminary test of the method) demonstrate that the new method is generally effective in producing the required solutions.

A Fourth-Order Modified Method for the Cauchy Problem of the Modified Helmholtz Equation

This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain 0<x≤1,y∈R.The Cauchy data at x = 0 is given and the solution is then sought for the interval 0<x≤1.This problem is highly ill-posed and the solution(if it exists) does not depend continuously on the given data. In this paper,we propose a fourth-order modified method to solve the Cauchy problem. Convergence estimates are presented under the suitable choices of regularization parameters and the a priori assumption on the bounds of the exact solution.Numerical implementation is considered and the numerical examples show that our proposed method is effective and stable.

A Modified Certainty Coefficient Method(M-CF) for Debris Flow Susceptibility Assessment:A Case Study for the Wenchuan Earthquake Meizoseismal Areas

In the meizoseismal areas hit by the China Wenchuan earthquake on May 12, 2008, the disasterprone environment has changed dramatically, making the susceptibility assessment of debris flow more complex and uncertain. After the earthquake, debris flow hazards occurred frequently and effective susceptibility assessment of debris flow has become extremely important. Shenxi gully in Du Jiangyan city, located in the meizoseismal areas, was selected as the study area. Based on the research of disaster-prone environment and the main factors controlling debris flow, the susceptibility zonations of debris flow were mapped using factor weight method(FW), certainty coefficient method(CF) and geomorphic information entropy method(GI). Through comparative analysis, the study showed that these three methods underestimated susceptible degree of debris flow when used in the meizoseismal areas of Wenchuan earthquake. In order to solve this problem, this paper developed a modified certainty coefficient method(M-CF) to reflect the impact of rich loose materials on the susceptible degree of debris flow. In the modified method, the distribution and area of loose materials were obtained by field investigations and postearthquake remote sensing image, and four data sets, namely, lithology, elevation, slop and aspect, wereused to calculate the CF values. The result of M-CF method is in agreement with field investigations and the accuracy of the method is satisfied. The method has a wide application to the susceptibility assessment of debris flow in the earthquake stricken areas.

An Accurate Numerical Solution for the Modified Equal Width Wave Equation Using the Fourier Pseudo-Spectral Method

In this study, the numerical solution for the Modified Equal Width Wave (MEW) equation is presented using Fourier spectral method that use to discretize the space variable and Leap-frog method scheme for time dependence. Test problems including the single soliton wave motion, interaction of two solitary waves and interaction of three solitary waves will use to validate the proposed method. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Finally, a Maxwellian initial condition pulse is then studied. The L2 and L∞ error norms are computed to study the accuracy and the simplicity of the presented method.

A Novel Method of the Preparation ofChemically Modified Electrodes

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Numerical Treatment of Initial Value Problems of Nonlinear Ordinary Differential Equations by Duan-Rach-Wazwaz Modified Adomian Decomposition Method

We employ the Duan-Rach-Wazwaz modified Adomian decomposition method for solving initial value problems for the systems of nonlinear ordinary differential equations numerically. In order to confirm practicality, robustness and reliability of the method, we compare the results from the modified Adomian decomposition method with those from the MATHEMATICA solutions and also from the fourth-order Runge Kutta method solutions in some cases. Furthermore, we apply Padé approximants technique to improve the solutions of the modified decomposition method whenever the exact solutions exist.

A MODIFIED METHOD FOR THE PREPARATION OF GOETHITE

Synthetic goethite(α-FeOOH) was prepared in the laboratory using a modified version of Brauer's method.The process of steam-crystallization was prolonged and impurities were removed by dialysis.The products have been characterised by Surface Area Analyser and Scanning Electron Microscope, and proved to have larger Specific Surface Area than that in literature.

Solving Different Types of Differential Equations Using Modified and New Modified Adomian Decomposition Methods

The Modified Adomian Decomposition Method (MADM) is presented. A number of problems are solved to show the efficiency of the method. Further, a new solution scheme for solving boundary value problems with Neumann conditions is proposed. The scheme is based on the modified Adomian decomposition method and the inverse linear operator theorem. Several differential equations with Neumann boundary conditions are solved to demonstrate the high accuracy and efficiency of the proposed scheme.

PARALLEL ADAPTIVELY MODIFIED CHARACTERISTIC BASIS FUNCTION METHOD BASED ON STATIC LOAD BALANCE

Characteristic Basis Function Method (CBFM) is a novel approach for analyzing the ElectroMagnetic (EM) scattering from electrically large objects. Based on dividing the studied object into small blocks, the CBFM is suitable for parallel computing. In this paper, a static load balance parallel method is presented by combining Message Passing Interface (MPI) with Adaptively Modified CBFM (AMCBFM). In this method, the object geometry is partitioned into distinct blocks, and the serial number of blocks is sent to related nodes according to a certain rule. Every node only needs to calculate the information on local blocks. The obtained results confirm the accuracy and efficiency of the proposed method in speeding up solving large electrical scale problems.

A modified soft docking method for proteins and peptides

A modified two-stage soft-docking procedure was developed for the theoretic researches on the recognition of protein-protein or protein-peptide complexes. Some systems have been used to test our program and the results are encouraging.

Solutions of Zhiber-Shabat and Related Equations Using a Modified tanh-coth Function Method

The modified tanh-coth function method is used to obtain new exact travelling wave solutions for Zhiber-Shabat equation and the related equations: Liouville equation, sinh-Gordon equation, Dodd-Bullough-Mikhailov equation, and Tzitzeica-Dodd-Bullough equation. Exact travelling wave solutions for each equation are derived and expressed in terms of hyperbolic functions, trigonometric functions and rational functions. The modified tanh-coth function method is easy to execute, brief, efficient, and can be used to solve many other nonlinear evolution equations.

Zirconium Modified Aluminide Coatings Obtained by the CVD and PVD Methods

The paper presents the comparison of the structures of the zirconium modified aluminide coatings deposited on pure nickel by the CVD and PVD methods. In the CVD process, zirconium was deposited from the ZrCl3 gas phase at the 1000°C. Zirconium thin layer (1 or 7 μm thick) and aluminum thin layer (1.0, 0.7 or 0.5 μm thick) were deposited by the EB-PVD method. Deposition velocity was about 1 ?m/min. The layers obtained by the Electron Beam Evaporation method were subjected to diffusion treatment for 2 h in the argon atmosphere. The obtained coatings were examined by the use of an optical microscope (microstructure and coating thickness) a scanning electron microscope (chemical composition on the cross-section of the modified aluminide coating) and XRD phase analysis. Microstructures and phase compositions of coatings obtained by different methods differ significantly. NiAl(Zr), Ni3Al and Ni(Al) phases were found in the CVD aluminide coatings, whereas Ni5Zr, Ni7Zr2 and γNi(Al,Zr) were observed in coatings obtained by the PVD method. The results indicate that the microstructure of the coating is strongly influenced by the method of manufacturing.

Research Progress of Agricultural Genetically Modified Crop Safety Evaluation

The commercial cultivation of genetically modified(GM)crops has eased the global food crisis and brought considerable economic and social benefits to countries.Because of the potential safety problems,it is necessary to make clear the molecular genetic characteristics,edible safety,planting,processing,and other aspects of the safety evaluation of GM crops.The safety problems existing in the cultivation of GM crops,safety evaluation and detection of GM crops were introduced in this paper,which provided the basis for safety evaluation and effective supervision of GM crops and their products.Commercial cultivation and reasonable supervision based on safety evaluation have far-reaching significance for ensuring consumer safety,enhancing the credibility of the national political system and enhancing citizens'confidence in the safety of GM crop products for consumption.

Grouping response method for equivalent static wind loads based on a modified LRC method

Wind loading is one of the most important loads for controlling the design of large-span roof structures. Equivalent static wind loads, which can generally aim at determining a specific response, are widely used by structural designers. A method for equivalent static wind loads applicable to multi-responses is proposed in this paper. A modified load- response-correlation （LRC） method corresponding to a particular peak response is presented, and the similarity algorithm implemented for the group response is described. The main idea of the algorithm is that two responses can be put into one group if the value of one response is close to that of the other response, when the structure is subjected to equivalent static wind loads aiming at the other response. Based on the modified LRC, the grouping response method is put forward to construct equivalent static wind loading. This technique can simultaneously reproduce peak responses for some grouped responses. To verify its computational accuracy, the method is applied to an actual large-span roof structure. Calculation results show that when the similarity of responses in the same group is high, equivalent static wind loads with high accuracy and reasonable magnitude of equivalent static wind distribution can be achieved.

Density of Ni-Al Alloys in Liquid and Solid-Liquid Coexistence State Measured by a Modified Pycnometric Method

The density of Ni-AI alloys in both liquid state and solid-liquid coexistence state was measured with a modified pycnometric method. It was found that the density of Ni-AI alloys decreases with increasing temperature and Al concentration in the alloys. The molar volume of liquid Ni-AI binary alloys increases with the increase of temperature and Al concentration. The partial molar volume of Al in Ni-AI binary alloy was calculated approximately. The molar volume of liquid Ni-AI alloy determined in the present work shows a negative deviation from the ideal linear molar volume.

A modified domain reduction method for numerical simulation of wave propagation in localized regions

A modified domain reduction method(MDRM) that introduces damping terms to the original DRM is presented in this paper. To verify the proposed MDRM and compare the computational accuracy of these two methods, a numerical test is designed. The numerical results of the MDRM and DRM are compared using an extended meshed model. The results show that the MDRM significantly improved the computational accuracy of the DRM. Then, the MDRM is compared with two existing conventional methods, namely Liao's transmitting boundary and viscous-spring boundary with Liu's method. The MDRM shows its great advancement in computational accuracy, stability and range of applications. This paper also discusses the influence of boundary location on computational accuracy. It can be concluded that smaller models tend to have larger errors. By introducing two dimensionless parameters, φ_1 and φ_2, the rational distance between the observation point and the MDRM boundary is suggested. When φ_1 >2 or φ_2>13, the relative PGA error can be limited to 5%. In practice, the appropriate model size can be chosen based on these two parameters to achieve desired computational accuracy.

Lateral Bearing Capacity of Modified Suction Caissons Determined by Using the Limit Equilibrium Method

The modified suction caisson(MSC) adds a short-skirted structure around the regular suction caissons to increase the lateral bearing capacity and limit the deflection. The MSC is suitable for acting as the offshore wind turbine foundation subjected to larger lateral loads compared with the imposed vertical loads. Determination of the lateral bearing capacity is a key issue for the MSC design. The formula estimating the lateral bearing capacity of the MSC was proposed in terms of the limit equilibrium method and was verified by the test results. Parametric studies on the lateral bearing capacity were also carried out. It was found that the lateral bearing capacity of the MSC increases with the increasing length and radius of the external skirt, and the lateral bearing capacity increases linearly with the increasing coefficient of subgrade reaction. The maximum lateral bearing capacity of the MSC is attained when the ratio of the radii of the internal compartment to the external skirt equals 0.82 and the ratio of the lengths of the external skirt to the internal compartment equals 0.48, provided that the steel usage of the MSC is kept constant.

Density of Liquid Ni-Mo Alloys Measured by a Modified Sessile Drop Method

The density of liquid binary Ni-Mo alloys with molybdenum concentration from 0 to 20% (mass fraction) wasmeasured by a modified sessile drop method. It has been found that the density of the liquid Ni-Mo alloys decreaseswith increasing temperature, but increases with the increase of molybdenum concentration in the alloys. The molarvolume of liquid Ni-Mo binary alloys increases with the increase of temperature and molybdenum concentration. Thepartial molar volume of molybdenum in Ni-Mo binary alloy has been approximately calculated as [13.18-2.65×10^(-3)T+(-47.94+3.10×10^(-2)T)×10^(-2)X_(Mo)]×10^(-6)m^3·mol^(-1). The molar volume of Ni-Mo alloy determined inthe present work shows a negative deviation from the ideal linear mixing molar volume.

A modified stiffness spreading method for layout optimization of truss structures

The stiffness spreading method (SSM) was initially proposed for layout optimization of truss structures,in which an artificial elastic material of low modulus is uniformly distributed in the design domain to create connections between discrete members.In this paper,a modified stiffness spreading method is proposed by replacing the artificial elastic material with auxiliary bars to connect real members of the truss structure.Since the background continuum mesh for the elastic material is no longer required,the computational cost is significantly reduced.Like SSM,the new method is advantageous in that an initial design may consist of disconnected bars allocated in the design domain,and mathematical programming methods can be applied for the efficient solution of the formulated optimization problem.A number of solution strategies are also developed to achieve more practical designs with lower computational cost.Numerical examples of both 2-D and 3-D truss structures are presented to demonstrate the feasibility,robustness and effectiveness of the proposed method.