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.

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.

Investigation of granite failure precursor under axial load using modified LSTM framework

Granite is usually composed of quartz,biotite,feldspar,and cracks,and the variation characteristics of these components could reflect the deformation and failure process of rock well.Taking granite as an example,the video camera was used to record the deformation and failure process of rock.The distribution of meso-components in video images was then identified.The meso-components of rock failure precursors were also discussed.Moreover,a modified LSTM(long short-term memory method)based on SSA(sparrow search algorithm)was proposed to estimate the change of meso-components of rock failure precursor.It shows that the initiation and expansion of cracks are mainly caused by feldspar and quartz fracture,and when the quartz and feldspar exit the stress framework,rock failure occurs;the second large increase of crack area and the second large decrease of quartz or feldspar area may be used as a precursor of rock failure;the precursor time of rock failure based on meso-scopic components is about 4 s earlier than that observed by the naked eye;the modified LSTM network has the strongest estimation ability for quartz area change,followed by feldspar and biotite,and has the worst estimation ability for cracks;when using the modified LSTM network to predict the precursors of rock instability and failure,quartz and feldspar could be given priority.The results presented herein may provide reference in the investigation of rock failure mechanism.

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.

Modified Augmented Lagrange Multiplier Methods for Large-Scale Chemical Process Optimization

Chemical process optimization can be described as large-scale nonlinear constrained minimization. The modified augmented Lagrange multiplier methods (MALMM) for large-scale nonlinear constrained minimization are studied in this paper. The Lagrange function contains the penalty terms on equality and inequality constraints and the methods can be applied to solve a series of bound constrained sub-problems instead of a series of unconstrained sub-problems. The steps of the methods are examined in full detail. Numerical experiments are made for a variety of problems, from small to very large-scale, which show the stability and effectiveness of the methods in large-scale problems.

Propagation of traveling wave solutions to the Vakhnenko-Parkes dynamical equation via modified mathematical methods

In this paper, we investigate some new traveling wave solutions to Vakhnenko-Parkes equation via three modified mathematical methods. The derived solutions have been obtained including periodic and solitons solutions in the form of trigonometric, hyperbolic, and rational function solutions. The graphical representations of some solutions by assigning particular values to the parameters under prescribed conditions in each solutions and comparing of solutions with those gained by other authors indicate that these employed techniques are more effective, efficient and applicable mathematical tools for solving nonlinear problems in applied science.

Dispersive propagation of optical solitions and solitary wave solutions of Kundu-Eckhaus dynamical equation via modified mathematical method

In this research work,we constructed the optical soliton solutions of nonlinear complex Kundu-Eckhaus(KE)equation with the help of modified mathematical method.We obtained the solutions in the form of dark solitons,bright solitons and combined dark-bright solitons,travelling wave and periodic wave solutions with general coefficients.In our knowledge earlier reported results of the KE equation with specific coefficients.These obtained solutions are more useful in the development of optical fibers,dynamics of solitons,dynamics of adiabatic parameters,dynamics of fluid,problems of biomedical,industrial phenomena and many other branches.All calculations show that this technique is more powerful,effective,straightforward,and fruitfulness to study analytically other higher-order nonlinear complex PDEs involves in mathematical physics,quantum physics,Geo physics,fluid mechanics,hydrodynamics,mathematical biology,field of engineering and many other physical sciences.

Application of the Modified Adomian Decomposition Method on a Mathematical Model of COVID-19

In this study, we constructed and analysed a mathematical model of COVID-19 in order to comprehend the transmission dynamics of the disease. The reproduction number (RC) was calculated via the next generation matrix method. We also used the Lyaponuv method to show the global stability of both the disease free and endemic equilibrium points. The results showed that the disease-free equilibrium point is globally asymptotically stable if RC RC > 1. We further used the Adomian decomposition method and the modified Adomian decomposition method to obtain the solutions of the model. Numerical analysis of the model was done using Sagemath 9.0 software.

A modified single edge V-notched beam method for evaluating surface fracture toughness of thermal barrier coatings

The surface fracture toughness is an important mechanical parameter for studying the failure behavior of air plasma sprayed(APS)thermal barrier coatings(TBCs).As APS TBCs are typical multilayer porous ceramic materials,the direct applications of the traditional single edge notched beam(SENB)method that ignores those typical structural characters may cause errors.To measure the surface fracture toughness more accurately,the effects of multilayer and porous characters on the fracture toughness of APS TBCs should be considered.In this paper,a modified single edge V-notched beam(MSEVNB)method with typical structural characters is developed.According to the finite element analysis(FEA),the geometry factor of the multilayer structure is recalculated.Owing to the narrower V-notches,a more accurate critical fracture stress is obtained.Based on the Griffith energy balance,the reduction of the crack surface caused by micro-defects is corrected.The MSEVNB method can measure the surface fracture toughness more accurately than the SENB method.

Exact Traveling Wave Solutions to Phi-4 Equation and Joseph-Egri (TRLW) Equation and Calogro-Degasperis (CD) Equation by Modified (G'/G2)-Expansion Method

In this study, we will introduce the modified (G'/G2)-expansion method to explore some of the exact traveling wave solutions of some nonlinear partial differential equations namely, Phi-4 equation, Joseph-Egri (TRLW) equation, and Calogro-Degasperis (CD) equation. As a result, we have obtained solutions for the equations expressed in terms of trigonometric, hyperbolic and rational functions. Moreover, some selected solutions are plotted using some specific values for the parameters.

A Method Suitable for Extracting Genomic DNA from Animal and Plant——Modified CTAB Method

[Objective] The study aimed to introduce a rapid and effective method that is suitable for extracting genomic DNA from animal and plant. [ Method ] The genomic DNAs were extracted from tender leaves of 24 peanut cuhivars and from the liver, lung and kidney of white mouse through the specifically modified CTAB method. The DNAs were run on agarose gel, next detected by DNA/Protein analyzer. Finally PCR amplification was conducted to detect the quality of DNAs extracted using the modified CTAB method. [ Result] The clear and orderly bands were observed in gel detection, and the values of OD200/OD200 for DNAs extracted via modified CTAB method were between 1.77 - 1.83. The DNAs performed well in PCR amplification. [ Conclusion] The DNAs extracted by modified CTAB method could satisfy the requirement of PCR amplification.

Modified CTAB Method for Extracting Genomic DNA from Wheat Leaf

ObjectiveThe aim was to seek for a rapid DNA minipreparation method from wheat leaf. MethodThe total DNA of wheat leaf was extracted using CTAB, SDS and boiling water, separately, with some modifications. Integrity and purity of nucleic acids were detected through agarose gel electrophoresis, ultraviolet absorption and PCR. ResultThe DNA extracted by the modified CTAB method had high quality and purity, and was not degraded. Two hundreds of DNA samples could be extracted each workday by per capita using this method; and the PCR detection of wheat transgenic plants showed that amplified bands of target gene were clear, without false-positive, and the test results were satisfactory. The DNA purity and concentration extracted by modified SDS method were not as good as that extracted by modified CTAB method, but it also met the DNA requirements of major molecular research. The DNA quantity extracted by modified boiling method was small and there were a lot of impurities in it, PCR detection of this DNA showed no amplified band. ConclusionModified CTAB method is a simple and rapid method for DNA minipreparation from wheat leaf, and was suitable for PCR amplification and other molecular biology researches.

Modified embedded-atom interatomic potential for Co-W and Al-W systems

A semi-empirical interatomic potential formalism,the second-nearest-neighbor modified embedded-atom method(2NN MEAM),has been applied to obtaining interatomic potentials for the Co-W and Al-W binary system using previously developed MEAM potentials of Co,Al and W.The potential parameters were determined by fitting the experimental data on the enthalpy of formation,lattice parameter,melting point and elastic constants.The present potentials generally reproduce the fundamental physical properties of the Co-W and Al-W systems accurately.The lattice parameters,the enthalpy of formation,the thermal stability and the elastic constants match well with experiment and the first-principles results.The enthalpy of mixing and the enthalpy of formation and mixing of liquid are in good agreement with CALPHAD calculations.The potentials can be easily combined with already-developed MEAM potentials for binary cobalt systems and can be used to describe Co-Al-W-based multicomponent alloys,especially for interfacial properties.

MODIFIED STOCHASTIC EXTRAGRADIENT METHODS FOR STOCHASTIC VARIATIONAL INEQUALITY

In this paper,we consider two kinds of extragradient methods to solve the pseudo-monotone stochastic variational inequality problem.First,we present the modified stochastic extragradient method with constant step-size(MSEGMC)and prove the convergence of it.With the strong pseudo-monotone operator and the exponentially growing sample sequences,we establish the R-linear convergence rate in terms of the mean natural residual and the oracle complexity O(1/ǫ).Second,we propose a modified stochastic extragradient method with adaptive step-size(MSEGMA).In addition,the step-size of MSEGMA does not depend on the Lipschitz constant and without any line-search procedure.Finally,we use some numerical experiments to verify the effectiveness of the two algorithms.

Travelling wave solutions of nonlinear conformable analytical approaches

The presented study deals with the investigation of nonlinear Bogoyavlenskii equations with conformable time-derivative which has great importance in plasma physics and non-inspectoral scattering problems.Travelling wave solutions of this nonlinear conformable model are constructed by utilizing two powerful analytical approaches,namely,the modified auxiliary equation method and the Sardar sub-equation method.Many novel soliton solutions are extracted using these methods.Furthermore,3D surface graphs,contour plots and parametric graphs are drawn to show dynamical behavior of some obtained solutions with the aid of symbolic software such as Mathematica.The constructed solutions will help to understand the dynamical framework of nonlinear Bogoyavlenskii equations in the related physical phenomena.

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.

Carbon-supported ultrafine Pt nanoparticles modified with trace amounts of cobalt as enhanced oxygen reduction reaction catalysts for proton exchange membrane fuel cells

To accelerate the kinetics of the oxygen reduction reaction(ORR)in proton exchange membrane fuel cells,ultrafine Pt nanoparticles modified with trace amounts of cobalt were fabricated and decorated on carbon black through a strategy involving modified glycol reduction and chemical etching.The obtained Pt36Co/C catalyst exhibits a much larger electrochemical surface area(ECSA)and an improved ORR electrocatalytic activity compared to commercial Pt/C.Moreover,an electrode prepared with Pt36Co/C was further evaluated under H2-air single cell test conditions,and exhibited a maximum specific power density of 10.27 W mgPt^-1,which is 1.61 times higher than that of a conventional Pt/C electrode and also competitive with most state-of-the-art Pt-based architectures.In addition,the changes in ECSA,power density,and reacting resistance during the accelerated degradation process further demonstrate the enhanced durability of the Pt36Co/C electrode.The superior performance observed in this work can be attributed to the synergy between the ultrasmall size and homogeneous distribution of catalyst nanoparticles,bimetallic ligand and electronic effects,and the dissolution of unstable Co with the rearrangement of surface structure brought about by acid etching.Furthermore,the accessible raw materials and simplified operating procedures involved in the fabrication process would result in great cost-effectiveness for practical applications of PEMFCs.

MODIFIED LAYER REMOVAL METHOD FOR MEASUREMENT OF RESIDUAL STRESS DISTRIBUTION IN THICK PRE-STRETCHED ALUMINUM PLATE

The integrated structure parts are widely used in aircraft. The distortion caused by residual stresses in thick pre-stretched aluminum plates during machining integrated parts is a common and serious problem. To predict and control the machining distortion, the residual stress distribution in the thick plate must be measured firstly. The modified removal method for measuring residual stress in thick pre-stretched aluminum plates is proposed and the stress-strain relation matrix is deduced by elasticity theory. The residual stress distribution in specimen of 7050T7451 plate is measured by using the method, and measurement results are analyzed and compared with data obtained by other methods. The method is effective to measure the residual stress.

Calculation of circulation in the South China Sea during summer of 2000 by the modified inverse method

On the basis of hydrographic data obtained in August 2000 cruise, the circulation in the South China Sea (SCS) is computed by the modified inverse method in combination with SSH data from TOPEX/ERS-2 analysis. For study of the dynamical mechanism, which causes the pattern of summer circulation in the SCS, the diagnostic model (Yuan et al. 1982. Acta Oceanologica Sinica,4(1):1-11; Yuan and Su. 1992. Numerical Computation of Physical Oceanography.474-542) is used to simulate numerically the summer circulation in the SCS. The following results have been obtained. (1) The central and southwestern SCSs are dominated mainly by anticy-clonic circulation systems. They are mainly as follows. 1) There is strong anticyclonic eddy southeast of Vietnam (W1). Its horizontal scale is about 300 km, and it extends vertically from the surface to the about 1 000 m level. 2) There are a warm eddy W2 southeast of Zhongsha Islands and the anticyclonic circulation system W3 west off the Luzon Island. 3) There is a stronger cyclonic eddy C1 between the anticyclonic eddies W1 and W2.4) A strong northward coastal jet is present near the coast of Vietnam, and separates from the coast of Vietnam at about 12° N to the northeast.(2)The northern SCS is dominated mainly by a cyclonic circulation system. There is a cyclonic circulation system near and north of Section N2. (3) The southeastern SCS is dominated mainly by the cyclonic circulation system. (4) Comparing the results of circulation in the SCS during the summer of 2000 with those during the summer of 1998, it is found that they agree qualitatively, but there is the some difference between them in quantity.This shows that the circulation in the SCS has obviously seasonal feature. (5) The dynamical mechanism which products the basic pattern of summer circulation is because the following two reasons: 1) the joint effect of the baroclinity and relief (JEBAR) is essential dynamical cause; and 2) it is next important dynamical cause that the interaction between the wind stress and bottom topography under the southerly monsoon. (6) Comparing the hydrographic structure and distribution of stream functions with the SSH data from TOPEX/ERS-2 analysis in the SCS during August of 2000, they agree qualitatively.