Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.

The discontinuous Petrov-Galerkin method for one-dimensional compressible Euler equations in the Lagrangian coordinate

In this paper, a Petrov-Galerkin scheme named the Runge-Kutta control volume （RKCV） discontinuous finite ele- ment method is constructed to solve the one-dimensional compressible Euler equations in the Lagrangian coordinate. Its advantages include preservation of the local conservation and a high resolution. Compared with the Runge-Kutta discon- tinuous Galerkin （RKDG） method, the RKCV method is easier to implement. Moreover, the advantages of the RKCV and the Lagrangian methods are combined in the new method. Several numerical examples are given to illustrate the accuracy and the reliability of the algorithm.

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.

Geophysical Study: Estimation of Deposit Depth Using Gravimetric Data and Euler Method (Jalalabad Iron Mine, Kerman Province of IRAN)

Mineral exploration is done by different methods. Geophysical and geochemical studies are two powerful tools in this field. In integrated studies, the results of each study are used to determine the location of the drilling boreholes. The purpose of this study is to use field geophysics to calculate the depth of mineral reserve. The study area is located 38 km from Zarand city called Jalalabad iron mine. In this study, gravimetric data were measured and mineral depth was calculated using the Euler method. 1314 readings have been performed in this area. The rocks of the region include volcanic and sedimentary. The source of the mineralization in the area is hydrothermal processes. After gravity measuring in the region, the data were corrected, then various methods such as anomalous map remaining in levels one and two, upward expansion, first and second-degree vertical derivatives, analytical method, and analytical signal were drawn, and finally, the depth of the deposit was estimated by Euler method. As a result, the depth of the mineral deposit was calculated to be between 20 and 30 meters on average.

Energy Stable Nodal DG Methods for Maxwell’s Equations of Mixed-Order Form in Nonlinear Optical Media

In this work,we develop energy stable numerical methods to simulate electromagnetic waves propagating in optical media where the media responses include the linear Lorentz dispersion,the instantaneous nonlinear cubic Kerr response,and the nonlinear delayed Raman molecular vibrational response.Unlike the first-order PDE-ODE governing equations considered previously in Bokil et al.(J Comput Phys 350:420–452,2017)and Lyu et al.(J Sci Comput 89:1–42,2021),a model of mixed-order form is adopted here that consists of the first-order PDE part for Maxwell’s equations coupled with the second-order ODE part(i.e.,the auxiliary differential equations)modeling the linear and nonlinear dispersion in the material.The main contribution is a new numerical strategy to treat the Kerr and Raman nonlinearities to achieve provable energy stability property within a second-order temporal discretization.A nodal discontinuous Galerkin(DG)method is further applied in space for efficiently handling nonlinear terms at the algebraic level,while preserving the energy stability and achieving high-order accuracy.Indeed with d_(E)as the number of the components of the electric field,only a d_(E)×d_(E)nonlinear algebraic system needs to be solved at each interpolation node,and more importantly,all these small nonlinear systems are completely decoupled over one time step,rendering very high parallel efficiency.We evaluate the proposed schemes by comparing them with the methods in Bokil et al.(2017)and Lyu et al.(2021)(implemented in nodal form)regarding the accuracy,computational efficiency,and energy stability,by a parallel scalability study,and also through the simulations of the soliton-like wave propagation in one dimension,as well as the spatial-soliton propagation and two-beam interactions modeled by the two-dimensional transverse electric(TE)mode of the equations.

The Semi-implicit Euler Method for Stochastic Pantograph Equations with Jumps

In this paper,we present the semi-implicit Euler（SIE）numerical solution for stochastic pantograph equations with jumps and prove that the SIE approximation solution converges to the exact solution in the mean-square sense under the Local Lipschitz condition.

The forecasting efficiency under different selected regions by Pattern Informatics Method and seismic potential estimation in the North-South Seismic Zone

In 2022,four earthquakes with M_(S)≥6.0 including the Menyuan M_(S)6.9 and Luding M_(S)6.8 earthquakes occurred in the North-South Seismic Zone(NSSZ),which demonstrated high and strong seismicity.Pattern Informatics(PI)method,as an effective long and medium term earthquake forecasting method,has been applied to the strong earthquake forecasting in Chinese mainland and results have shown the positive performance.The earthquake catalog with magnitude above M_(S)3.0 since 1970 provided by China Earthquake Networks Center was employed in this study and the Receiver Operating Characteristic(ROC)method was applied to test the forecasting efficiency of the PI method in each selected region related to the North-South Seismic Zone systematically.Based on this,we selected the area with the best ROC testing result and analyzed the evolution process of the PI hotspot map reflecting the small seismic activity pattern prior to the Menyuan M_(S)6.9 and Luding M_(S)6.8 earthquakes.A“forward”forecast for the area was carried out to assess seismic risk.The study shows the following.1)PI forecasting has higher forecasting efficiency in the selected study region where the difference of seismicity in any place of the region is smaller.2)In areas with smaller differences of seismicity,the activity pattern of small earthquakes prior to the Menyuan M_(S)6.9 and Luding M_(S)6.8 earthquakes can be obtained by analyzing the spatio-temporal evolution process of the PI hotspot map.3)The hotspot evolution in and around the southern Tazang fault in the study area is similar to that prior to the strong earthquakes,which suggests the possible seismic hazard in the future.This study could provide some ideas to the seismic hazard assessment in other regions with high seismicity,such as Japan,Californi,Turkey,and Indonesia.

2021年9月16日四川泸县M_(S)6.0地震前地磁扰动异常特征分析

选取四川地区3个地磁扰动台站的观测数据,利用基于脉冲幅度法的地磁垂直强度极化方法,对2021年9月16日四川泸县M_(S)6.0地震前的地磁扰动异常演化特征进行分析。研究结果表明:(1)在中强地震前,台站的垂直强度极化值和超阈值极化值频次逐日变化均会出现高值异常,其中频次异常呈现先增强后降低,再增强-降低的形态;(2)发生异常变化的多个台站距离震中300 km范围内,时间上属准同步变化;(3)地震多发生在异常出现的一个月后,频次异常值往往是研究时段的最高值;(4)机理上,这些异常演化形态和特征是由岩石的破裂特性决定的,一定程度上反映了地震的孕育过程;(5)这种震前的地磁扰动异常与外源场无关。

Analysis of regular and chaotic dynamics of the Euler-Bernoulli beams using finite difference and finite element methods

Chaotic vibrations of flexible non-linear Euler-Bernoulli beams subjected to harmonic load and with various boundary conditions（symmetric and non-symmetric）are studied in this work.Reliability of the obtained results is verified by the finite difference method（FDM）and the finite element method（FEM）with the Bubnov-Galerkin approximation for various boundary conditions and various dynamic regimes（regular and non-regular）.The influence of boundary conditions on the Euler-Bernoulli beams dynamics is studied mainly,dynamic behavior vs.control parameters { ωp,q0 } is reported,and scenarios of the system transition into chaos are illustrated.

^(125)I的俄歇电子和氚的β射线细胞S值及DNA损伤模拟研究

氚的辐射生物效应一直是研究热点,特别是在福岛核电站排放核废水后成为公众关注的焦点。氚极易进入人体并参与机体代谢活动,衰变放出的β射线对人体具有极大的危害,而靶向放射治疗则利用摄入的俄歇电子发射体能够进入癌细胞,并放出俄歇电子造成DNA严重损伤这一特性,达到有效杀死癌细胞的目的。为探究β射线在微纳米尺度的辐射生物效应,本研究利用清华大学研发的纳剂量学程序NASIC模拟计算了^(125)I和氚在不同尺度细胞内的细胞S值以及造成的DNA双链断裂(DSB)产额,深入探讨了其DNA损伤机制。结果表明,由于研究方法和粒子截面的不同,细胞S值的计算结果存在偏差,尤其对于S(N←Cy),相对偏差可达26.35%。此外,相较于^(125)I或氚均匀分布在细胞核内,与DNA结合时造成的DSB产额显著增加:^(125)I结合到DNA时的DSB产额是均匀分布时的2倍;而氚结合到DNA中使得DSB产额从1.31×10^(-11)Gy^(-1)·Da^(-1)增加至1.46×10^(-11)Gy^(-1)·Da^(-1)。本研究深入探究了微纳米尺度下放射性核素的损伤作用效应,为精确医疗和环境放射性污染风险评估提供重要的理论基础。

A Preconditioned Gridless Method for Solving Euler Equations at Low Mach Numbers

A preconditioned gridless method is developed for solving the Euler equations at low Mach numbers.The preconditioned system in a conservation form is obtained by multiplying apreconditioning matrix of the type of Weiss and Smith to the time derivative of the Euler equations,which are discretized using agridless technique wherein the physical domain is distributed by clouds of points.The implementation of the preconditioned gridless method is mainly based on the frame of the traditional gridless method without preconditioning,which may fail to converge for low Mach number simulations.Therefore,the modifications corresponding to the affected terms of preconditioning are mainly addressed.The numerical results show that the preconditioned gridless method still functions for compressible transonic flow simulations and additionally,for nearly incompressible flow simulations at low Mach numbers as well.The paper ends with the nearly incompressible flow over a multi-element airfoil,which demonstrates the ability of the method presented for treating flows over complicated geometries.

Sloshing simulation of standing wave with time-independent finite difference method for Euler equations

The numerical solutions of standing waves for Euler equations with the nonlinear free surface boundary condition in a two-dimensional （2D） tank are studied. The irregular tank is mapped onto a fixed square domain through proper mapping functions. A staggered mesh system is employed in a 2D tank to calculate the elevation of the transient fluid. A time-independent finite difference method, which is developed by Bang- fuh Chen, is used to solve the Euler equations for incompressible and inviscid fluids. The numerical results agree well with the analytic solutions and previously published results. The sloshing profiles of surge and heave motion with initial standing waves are presented. The results show very clear nonlinear and beating phenomena.

Variational iteration method for solving compressible Euler equations

This paper applies the variational iteration method to obtain approximate analytic solutions of compressible Euler equations in gas dynamics. This method is based on the use of Lagrange multiplier for identification of optimal values of parameters in a functional. Using this method, a rapid convergent sequence is produced which converges to the exact solutions of the problem. Numerical results and comparison with other two numerical solutions verify that this method is very convenient and efficient.

Research on the Application of Montessori Education Method in Cognitive Training of Patients with Alzheimer’s Disease

Objective:To study the application of the Montessori education method in cognitive training in patients with Alzheimer’s disease(AD).Methods:40 cases of senile dementia patients who were admitted to our hospital from January 2022 to January 2023 were selected and randomly divided into an intervention group and a control group according to the single and double number table method,with 20 cases in each group.The intervention group used the Montessori education method,the principle of which was to implement individualized health interventions based on the individual conditions of the patients,for a period of 6 months;the control group was given conventional treatment and nursing of the disease.The Mini-Mental State Examination(MMSE)was used to compare the effects of the two groups of patients before and after health intervention and conduct statistical analysis.Results:The score of the intervention group was higher than that of the control group,and there was a statistical difference between the two(P<0.05).Conclusion:Implementing the Montessori education method for diagnosed Alzheimer’s patients can effectively improve their cognitive function and delay the progress of further dementia.

Input-to-state stability of Euler-Maruyama method for stochastic delay control systems

This paper develops the mean-square exponential input-to-state stability（exp-ISS） of the Euler-Maruyama（EM） method for stochastic delay control systems（SDCSs）.The definition of mean-square exp-ISS of numerical methods is established.The conditions of the exact and EM method for an SDCS with the property of mean-square exp-ISS are obtained without involving control Lyapunov functions or functional.Under the global Lipschitz coefficients and mean-square continuous measurable inputs,it is proved that the mean-square exp-ISS of an SDCS holds if and only if that of the EM method is preserved for a sufficiently small step size.The proposed results are evaluated by using numerical experiments to show their effectiveness.

Solid Waste Management:A MADM Approach Using Fuzzy Parameterized Possibility Single-Valued Neutrosophic Hypersoft Expert Settings

The dramatic rise in the number of people living in cities has made many environmental and social problems worse.The search for a productive method for disposing of solid waste is the most notable of these problems.Many scholars have referred to it as a fuzzy multi-attribute or multi-criteria decision-making problem using various fuzzy set-like approaches because of the inclusion of criteria and anticipated ambiguity.The goal of the current study is to use an innovative methodology to address the expected uncertainties in the problem of solid waste site selection.The characteristics(or sub-attributes)that decision-makers select and the degree of approximation they accept for various options can both be indicators of these uncertainties.To tackle these problems,a novel mathematical structure known as the fuzzy parameterized possibility single valued neutrosophic hypersoft expert set(ρˆ-set),which is initially described,is integrated with a modified version of Sanchez’s method.Following this,an intelligent algorithm is suggested.The steps of the suggested algorithm are explained with an example that explains itself.The compatibility of solid waste management sites and systems is discussed,and rankings are established along with detailed justifications for their viability.This study’s strengths lie in its application of fuzzy parameterization and possibility grading to effectively handle the uncertainties embodied in the parameters’nature and alternative approximations,respectively.It uses specific mathematical formulations to compute the fuzzy parameterized degrees and possibility grades that are missing from the prior literature.It is simpler for the decisionmakers to look at each option separately because the decision is uncertain.Comparing the computed results,it is discovered that they are consistent and dependable because of their preferred properties.

Effects of King’s Theory of Goal Attainment Combined with Teach-Back Method on Improving Patients’ Standardization Rate of Eye Drop Use

Objective:To explore the effect of King’s Theory of Goal Attainment combined with teach-back method on improving the standardization rate of eye drop use in patients.Methods:A total of 200 patients who used more than two types of eye drops in the Department of Ophthalmology of our hospital were selected as the research subjects,and were randomly divided into an observation group and a control group.The control group was given routine health education,while the observation group was given King’s Theory of Goal Attainment combined with teach-back method on the basis of the control group.The standardization rate of eye drop use,knowledge awareness rate,and nursing satisfaction were compared between the two groups before and after the intervention.Results:After the intervention,the standardization rate of eye drop use,the awareness rate of eye drop knowledge,and the nursing satisfaction of the observation group were significantly higher than those of the control group(P<0.05).Conclusion:King’s Theory of Goal Attainment combined with teach-back method can effectively improve the standardization rate of eye drop use in patients,increase their awareness of eye drop knowledge,and improve nursing satisfaction,which is worthy of clinical application.

Critical care nurses and their clinical reasoning for customizing monitor alarms:a mixed-method study

Objective:To explore the clinical rationale of critical care nurses for personalizing monitor alarms.One of the most crucial jobs assigned to critical care nurses is monitoring patients'physiological indicators and carrying out the necessary associated interventions.Successful use of equipment in the nursing practice environment will be improved by a thorough understanding of the nurse's approach to alarm configuration.Methods:A mixed-method design integrating quantitative and qualitative components was used.The sample of this study recruited a convenience sample of 60 nurses who have worked in critical care areas.This study took place at Lebanese American University Medical Center Rizk Hospital,utilizing a semi-structured interview with participants.Results:The study demonstrated the high incidence of nuisance alarms and the desensitization of critical care nurses to vital ones.According to the nurses,frequent false alarms and a shortage of staff are the 2 main causes of alarm desensitization.Age was significantly associated with the perception of Smart alarms,according to the data(P=0.03).Four interconnected themes and subcategories that reflect the clinical reasoning process for alarm customization were developed as a result of the study's qualitative component:(1)unit alarm environment;(2)nursing style;(3)motivation to customize;and(4)clinical and technological customization.Conclusions:According to this study,nurses believe that alarms are valuable.However,a qualitative analysis of the experiences revealed that customization has been severely limited since the healthcare team depends on nurses to complete these tasks independently.Additionally,a staffing shortage and lack of technical training at the start of placement have also hindered customization.

The Zhou’s Method for Solving the Euler Equidimensional Equation

In this work, we apply the Zhou’s method [1] or differential transformation method (DTM) for solving the Euler equidimensional equation. The Zhou’s method may be considered as alternative and efficient for finding the approximate solutions of initial values problems. We prove superiority of this method by applying them on the some Euler type equation, in this case of order 2 and 3 [2]. The power series solution of the reduced equation transforms into an approximate implicit solution of the original equations. The results agreed with the exact solution obtained via transformation to a constant coefficient equation.