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A Review on Sources,Extractions and Analysis Methods of a Sustainable Biomaterial:Tannins 被引量:2
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作者 Antonio Pizzi Marie-Pierre Laborie Zeki Candan 《Journal of Renewable Materials》 EI CAS 2024年第3期397-425,共29页
Condensed and hydrolysable tannins are non-toxic natural polyphenols that are a commercial commodity industrialized for tanning hides to obtain leather and for a growing number of other industrial applications mainly ... Condensed and hydrolysable tannins are non-toxic natural polyphenols that are a commercial commodity industrialized for tanning hides to obtain leather and for a growing number of other industrial applications mainly to substitute petroleum-based products.They are a definite class of sustainable materials of the forestry industry.They have been in operation for hundreds of years to manufacture leather and now for a growing number of applications in a variety of other industries,such as wood adhesives,metal coating,pharmaceutical/medical applications and several others.This review presents the main sources,either already or potentially commercial of this forestry by-materials,their industrial and laboratory extraction systems,their systems of analysis with their advantages and drawbacks,be these methods so simple to even appear primitive but nonetheless of proven effectiveness,or very modern and instrumental.It constitutes a basic but essential summary of what is necessary to know of these sustainable materials.In doing so,the review highlights some of the main challenges that remain to be addressed to deliver the quality and economics of tannin supply necessary to fulfill the industrial production requirements for some materials-based uses. 展开更多
关键词 TANNINS FLAVONOIDS SOURCES extraction methods analysis methods
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Overview of Precious Metal Content Analysis Methods in Automotive Catalytic Converter 被引量:1
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作者 Xianpeng Yang Chengbao Xie +4 位作者 Ning Liu Xin Du Suqing Wang Huasheng Jiang Zhengang Zhang 《Journal of Materials Science and Chemical Engineering》 2023年第11期7-14,共8页
With the increasing awareness of environmental protection, people’s concern of pollution issues arising. Vehicles, as the most important means of transportation, its exhaust emission has received considerable attenti... With the increasing awareness of environmental protection, people’s concern of pollution issues arising. Vehicles, as the most important means of transportation, its exhaust emission has received considerable attention. The catalytic converter is able to purify harmful substances in exhaust gas. The absolute content of precious metals in the catalytic converter dominates the exhaust gas purification effect. Accurate detection of precious metal content is of great significance for controlling the cost of catalysts, ensuring catalytic performance and recovering precious metals from spent catalysts. We herein summarized several instruments for precious metals content exploration, such as X-ray fluorescence spectrometer (XRF), atomic absorption spectrometer (AAS), inductively coupled plasma emission spectrometer (ICP) and spectrophotometer. In this thesis, the feasibility of using various devices for characterizing precious metal content in catalytic converters is analyzed and their strengths or weaknesses are elaborated. 展开更多
关键词 Catalytic Converter Precious Metal Content analysis methods
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Comprehensive Analysis Method of Slope Stability Based on the Limit Equilibrium and Finite Element Methods and Its Application
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作者 Yajun Wang Yifeng Li Jinzhou Chen 《Open Journal of Civil Engineering》 2023年第4期555-571,共17页
To study the safety and stability of large slopes, taking the right side slope of the Yuxi’an tunnel of the Yuchu Expressway Bridge in Yunnan Province as an example, limit equilibrium and finite element analysis were... To study the safety and stability of large slopes, taking the right side slope of the Yuxi’an tunnel of the Yuchu Expressway Bridge in Yunnan Province as an example, limit equilibrium and finite element analysis were applied to engineering examples to calculate the stability coefficient of the slope before and after excavation in the natural state. After comparative analysis, it was concluded that the former had a clear mechanical model and concept, which could quickly provide stability results;the latter could accurately determine the sliding surface of the slope and simulate the stress state changes of the rock and soil mass. The stability coefficients calculated by the two methods were within the stable range, but their values were different. On this basis, combined with the calculation principles, advantages and disadvantages of the two methods, a comprehensive analysis method of slope stability based on the limit equilibrium and finite element methods was proposed, and the rationality of the stability coefficient calculated by this method was judged for a slope case. 展开更多
关键词 Slope Body Excavation Mechanical Model Sliding Surface Coefficient of Stability Calculation Principle Comprehensive analysis method
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Sensitivity and inverse analysis methods for parameter intervals
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作者 Guojian Shao Jingbo Su 《Journal of Rock Mechanics and Geotechnical Engineering》 SCIE 2010年第3期274-280,共7页
This paper proposes a sensitivity analysis method for engineering parameters using interval analyses.This method substantially extends the application of interval analysis method.In this scheme,parameter intervals and... This paper proposes a sensitivity analysis method for engineering parameters using interval analyses.This method substantially extends the application of interval analysis method.In this scheme,parameter intervals and decision-making target intervals are determined using the interval analysis method.As an example,an inverse analysis method for uncertainty is presented.The intervals of unknown parameters can be obtained by sampling measured data.Even for limited measured data,robust results can also be obtained with the inverse analysis method,which can be intuitively evaluated by the uncertainty expressed in terms of an interval.For complex nonlinear problems,an iteratively optimized inverse analysis model is proposed.In a given set of loose parameter intervals,all the unknown parameter intervals that satisfy the measured information can be obtained by an iteratively optimized inverse analysis model.The influences of measured precisions and the number of parameters on the results of the inverse analysis are evaluated.Finally,the uniqueness of the interval inverse analysis method is discussed. 展开更多
关键词 interval analysis method sensitivity analysis reversible inverse analysis method iteratively optimized inverse analysis method
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A Research on Competitiveness of Guangxi City——Based on System Clustering Method and Principal Component Analysis Method
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作者 FAN Chang-ke WU Yu 《Asian Agricultural Research》 2010年第2期13-16,共4页
A total of 10 indices of regional economic development in Guangxi are selected.According to the relevant economic data,regional economic development in Guangxi is analyzed by using System Clustering Method and Princip... A total of 10 indices of regional economic development in Guangxi are selected.According to the relevant economic data,regional economic development in Guangxi is analyzed by using System Clustering Method and Principal Component Analysis Method.Result shows that System Clustering Method and Principal Component Analysis Method have revealed similar results analysis of economic development level.Overall economic strength of Guangxi is weak and Nanning has relatively high scores of factors due to its advantage of the political,economic and cultural center.Comprehensive scores of other regions are all lower than 1,which has big gap with the development of Nanning.Overall development strategy points out that Guangxi should accelerate the construction of the Ring Northern Bay Economic Zone,create a strong logistics system having strategic significance to national development,use the unique location advantage and rely on the modern transportation system to establish a logistics center and business center connecting the hinterland and the Asean Market.Based on the problems of unbalanced regional economic development in Guangxi,we should speed up the development of service industry in Nanning,construct the circular economy system of industrial city,and accelerate the industrialization process of tourism city in order to realize balanced development of regional economy in Guangxi,China. 展开更多
关键词 Clustering analysis method Factor analysis method Economic development level Economic strength China
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SIMPLIFIED ANALYSIS METHOD FOR ULTIMATE LOAD CAPACITY OF WEB PLATES OF BOX GIRDERS
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作者 韩庆华 程万海 +1 位作者 尹越 刘锡良 《Transactions of Tianjin University》 EI CAS 2000年第1期32-35,共4页
The prosperous post buckling load capacity of web plates of box girders can be used.In this article,the post buckling behaviour of web plates of box girders under different loading conditions is theoretically analyz... The prosperous post buckling load capacity of web plates of box girders can be used.In this article,the post buckling behaviour of web plates of box girders under different loading conditions is theoretically analyzed and on the basis of domestic and overseas design codes of steel structures,the corresponding simplified analysis methods are put forward for the engineering design or code revision.It is proved that the simplified methods are safe,efficient and practicable through the comparison between several results. 展开更多
关键词 post buckling behaviour ultimate load capacity simplified analysis method web plates of box girders
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Determination of Fe and Zn contents and distributions in natural sphalerite/marmatite by various analysis methods 被引量:7
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作者 Hao LAI Jiu-shuai DENG +2 位作者 Zi-long LIU Shu-ming WEN Ling-yun HUANG 《Transactions of Nonferrous Metals Society of China》 SCIE EI CAS CSCD 2020年第5期1364-1374,共11页
The contents of Fe and Zn in natural sphalerite samples were determined by chemical titration and spectroscopic techniques(portable X-ray fluorescence(P-XRF) spectrometry, electron probe microanalysis with energy disp... The contents of Fe and Zn in natural sphalerite samples were determined by chemical titration and spectroscopic techniques(portable X-ray fluorescence(P-XRF) spectrometry, electron probe microanalysis with energy dispersive spectroscopy(EPMA-EDS), electron probe microanalysis with wavelength dispersive spectroscopy(EPMA-WDS), and time-of-flight secondary ion mass spectrometry(To F-SIMS)). Besides, the distribution of Fe and Zn in sphalerite samples was analyzed by imaging EPMA-WDS and imaging To F-SIMS. The results show that Fe and Zn contents determined by each spectroscopic technique have good linearity with those determined by chemical titration(R^2>0.77), and the R^2 values of Fe are generally greater than those of Zn. The imaging analysis results revealed that Fe and Zn are not uniformly distributed in the sphalerite. 展开更多
关键词 SPHALERITE MARMATITE FE Zn analysis methods imaging
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Species distribution of polymeric aluminium ferrum——timed complexation colorimetric analysis method of Al-Fe-Ferron 被引量:8
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作者 Hu, YY Tu, CQ Wu, HH 《Journal of Environmental Sciences》 SCIE EI CAS CSCD 2001年第4期418-421,共4页
The effects of the calorimetric buffer solutions were investigated while the two colorimetric reactions of AI-ferron complex and Fe-ferron complex occurred individually, and the effects of the testing wavelength and t... The effects of the calorimetric buffer solutions were investigated while the two colorimetric reactions of AI-ferron complex and Fe-ferron complex occurred individually, and the effects of the testing wavelength and the pH of the solutions were also investigated. A timed complexatian colorimetric analysis method of Al-Fe-ferron in view of the total concentration of {AI + Fe} was then established to determine the species distribution of polymeric Al-Fe. The testing wavelength was recommended at 362 net and the testing pH value was 5. With a comparison of the ratios of n(Al)/n(Fe), the standard adsorption curves of the polymeric Al-Fe solutions were derived from the experimental results. Furthermore, the solutions' composition were carious in both the molar n(Al)/n(Fe) ratios, i.e. 0/0, 5/5, 9/1 and 0/10, and the concentrations associated with the total ( Al + Fe which ranged from 10(-5) to 10(-4) mol/L.. 展开更多
关键词 polymeric aluminum-ferrum species distribution timed complexation colarimetric analysis method Al-Fe-ferron
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Slope stability analysis under seismic load by vector sum analysis method 被引量:15
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作者 Mingwei Guo Xiurun Ge Shuilin Wang 《Journal of Rock Mechanics and Geotechnical Engineering》 SCIE 2011年第3期282-288,共7页
The vibration characteristics and dynamic responses of rock and soil under seismic load can be estimated with dynamic finite element method (DFEM). Combining with the DFEM, the vector sum analysis method (VSAM) is... The vibration characteristics and dynamic responses of rock and soil under seismic load can be estimated with dynamic finite element method (DFEM). Combining with the DFEM, the vector sum analysis method (VSAM) is employed in seismic stability analysis of a slope in this paper. Different from other conventional methods, the VSAM is proposed based on the vector characteristic of force and current stress state of the slope. The dynamic stress state of the slope at any moment under seismic load can he obtained by the DFEM, thus the factor of safety of the slope at any moment during earthquake can be easily obtained with the VSAM in consideration of the DFEM. Then, the global stability of the slope can be estimated on the basis of time-history curve of factor of safety and reliability theory. The VSAM is applied to a homogeneous slope under seismic load. The factor of safety of the slope is 1.30 under gravity only and the dynamic factor of safety under seismic load is 1.21. The calculating results show that the dynamic characteristics and stability state of the slope with input ground motion can be actually analyzed. It is believed that the VSAM is a feasible and practical approach to estimate the dynamic stability of slopes under seismic load. 展开更多
关键词 slope stability vector sum analysis method (VSAM) seismic load dynamic finite element method (DFEM)
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Interval analysis method and convex models for impulsive response of structures with uncertain-but-bounded external loads 被引量:7
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作者 Zhiping Qiu Xiaojun Wang 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2006年第3期265-276,共12页
Two non-probabilistic, set-theoretical methods for determining the maximum and minimum impulsive responses of structures to uncertain-but-bounded impulses are presented. They are, respectively, based on the theories o... Two non-probabilistic, set-theoretical methods for determining the maximum and minimum impulsive responses of structures to uncertain-but-bounded impulses are presented. They are, respectively, based on the theories of interval mathematics and convex models. The uncertain-but-bounded impulses are assumed to be a convex set, hyper-rectangle or ellipsoid. For the two non-probabilistic methods, less prior information is required about the uncertain nature of impulses than the probabilistic model. Comparisons between the interval analysis method and the convex model, which are developed as an anti-optimization problem of finding the least favorable impulsive response and the most favorable impulsive response, are made through mathematical analyses and numerical calculations. The results of this study indicate that under the condition of the interval vector being determined from an ellipsoid containing the uncertain impulses, the width of the impulsive responses predicted by the interval analysis method is larger than that by the convex model; under the condition of the ellipsoid being determined from an interval vector containing the uncertain impulses, the width of the interval impulsive responses obtained by the interval analysis method is smaller than that by the convex model. 展开更多
关键词 Impulsive response Interval analysis method Convex model Uncertain-but-bounded impulse
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Nonlinear oscillations with parametric excitation solved by homotopy analysis method 被引量:5
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作者 Jianmin Wen Zhengcai Cao 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2008年第3期325-329,共5页
An analytical technique, namely the homotopy analysis method (HAM), is used to solve problems of nonlinear oscillations with parametric excitation. Unlike perturbation methods, HAM is not dependent on any small phys... An analytical technique, namely the homotopy analysis method (HAM), is used to solve problems of nonlinear oscillations with parametric excitation. Unlike perturbation methods, HAM is not dependent on any small physical parameters at all, and thus valid for both weakly and strongly nonlinear problems. In addition, HAM is different from all other analytic techniques in providing a simple way to adjust and control convergence region of the series solution by means of an auxiliary parameter h. In the present paper, a periodic analytic approximations for nonlinear oscillations with parametric excitation are obtained by using HAM, and the results are validated by numerical simulations. 展开更多
关键词 Nonlinear oscillation Parametric excitation Series solutions Homotopy analysis method
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Rutting influencing factors and prediction model for asphalt pavements based on the factor analysis method 被引量:4
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作者 Liu Gang Chen Leilei +1 位作者 Qian Zhendong Zhou Xiayang 《Journal of Southeast University(English Edition)》 EI CAS 2021年第4期421-428,共8页
To clarify the importance of various influencing factors on asphalt pavement rutting deformation and determine a screening method of model indicators,the data of the RIOHTrack full-scale track were examined using the ... To clarify the importance of various influencing factors on asphalt pavement rutting deformation and determine a screening method of model indicators,the data of the RIOHTrack full-scale track were examined using the factor analysis method(FAM).Taking the standard test pavement structure of RIOHTrack as an example,four rutting influencing factors from different aspects were determined through statistical analysis.Furthermore,the common influencing factors among the rutting influencing factors were studied based on FAM.Results show that the common factor can well characterize accumulative ESALs,center-point deflection,and temperature,besides humidity,which indicates that these three influencing factors can have an important impact on rutting.Moreover,an empirical rutting prediction model was established based on the selected influencing factors,which proved to exhibit high prediction accuracy.These analysis results demonstrate that the FAM is an effective screening method for rutting prediction model indicators,which provides a reference for the selection of independent model indicators in other rutting prediction model research when used in other areas and is of great significance for the prediction and control of rutting distress. 展开更多
关键词 asphalt pavement rutting prediction influencing factors RIOHTrack full-scale track factor analysis method
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Improving convergence of incremental harmonic balance method using homotopy analysis method 被引量:3
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作者 Yanmao Chen Jike Liu 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2009年第5期707-712,共6页
We have deduced incremental harmonic balance an iteration scheme in the (IHB) method using the harmonic balance plus the Newton-Raphson method. Since the convergence of the iteration is dependent upon the initial va... We have deduced incremental harmonic balance an iteration scheme in the (IHB) method using the harmonic balance plus the Newton-Raphson method. Since the convergence of the iteration is dependent upon the initial values in the iteration, the convergent region is greatly restricted for some cases. In this contribution, in order to enlarge the convergent region of the IHB method, we constructed the zeroth-order deformation equation using the homotopy analysis method, in which the IHB method is employed to solve the deformation equation with an embedding parameter as the active increment. Taking the Duffing and the van der Pol equations as examples, we obtained the highly accurate solutions. Importantly, the presented approach renders a convenient way to control and adjust the convergence. 展开更多
关键词 Incremental harmonic balance method Homotopy analysis method Initial value CONVERGENCE
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Application of the homotopy analysis method to nonlinear characteristics of a piezoelectric semiconductor fiber 被引量:3
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作者 Minghao ZHAO Zelong MA +1 位作者 Chunsheng LU Qiaoyun ZHANG 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2021年第5期665-676,共12页
Based on the nonlinear constitutive equation,a piezoelectric semiconductor(PSC)fiber under axial loads and Ohmic contact boundary conditions is investigated.The analytical solutions of electromechanical fields are der... Based on the nonlinear constitutive equation,a piezoelectric semiconductor(PSC)fiber under axial loads and Ohmic contact boundary conditions is investigated.The analytical solutions of electromechanical fields are derived by the homotopy analysis method(HAM),indicating that the HAM is efficient for the nonlinear analysis of PSC fibers,along with a rapid rate of convergence.Furthermore,the nonlinear characteristics of electromechanical fields are discussed through numerical results.It is shown that the asymmetrical distribution of electromechanical fields is obvious under a symmetrical load,and the piezoelectric effect is weakened by an applied electric field.With the increase in the initial carrier concentration,the electric potential decreases,and owing to the screen-ing effect of electrons,the distribution of electromechanical fields tends to be symmetrical. 展开更多
关键词 piezoelectric semiconductor(PSC)fiber homotopy analysis method(HAM) nonlinear analysis asymmetrical distribution screening effect
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Utilizing comprehensive decision analysis methods to determine an optimal planting pattern and nitrogen application for winter oilseed rape 被引量:3
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作者 DU Ya-dan CUI Bing-jing +3 位作者 ZHANG Qian SUN Jun WANG Zhen NIU Wen-quan 《Journal of Integrative Agriculture》 SCIE CAS CSCD 2020年第9期2229-2238,共10页
Oilseed rape is one of the most important oil crops globally.Attaining the appropriate cultivation method(planting pattern and nitrogen level)is necessary to achieve high yield,quality and resource utilization efficie... Oilseed rape is one of the most important oil crops globally.Attaining the appropriate cultivation method(planting pattern and nitrogen level)is necessary to achieve high yield,quality and resource utilization efficiency.However,the optimal method for oilseed rape varies across countries and regions.The objective of the present study was to determine an appropriate cultivation method,including planting pattern and nitrogen application,for winter oilseed rape in northwestern China.Two planting patterns:ridge film mulching and furrow planting(RFMF)and flat planting(FP),and six nitrogen(N)amounts:0(N0),60(N60),120(N120),180(N180),240(N240),and 300(N300)kg N ha–1 were applied across three growing seasons(2014–2017).Three comprehensive decision analysis methods:principal component analysis,grey correlation degree analysis and the combined entropy weight and dynamic technique for order preference by similarity to ideal solution method were used to evaluate the growth and physiological indicators,nutrient uptake,yield,quality,evapotranspiration,and water use efficiency of winter oilseed rape.Planting pattern,nitrogen amount and their interaction significantly affected the indicators aforementioned.The RFMF pattern significantly increased all indicators over the FP pattern.Application of N also markedly increased all the indicators except for seed oil content,but the yield,oil production and water use efficiency were decreased when N fertilizer exceeded 180 kg N ha–1 under FP and 240 kg N ha–1 under RFFM.The evaluation results of the three comprehensive decision analysis methods indicated that RFMF planting pattern with 240 kg N ha–1 is an appropriate cultivation method for winter oilseed rape in northwestern China.These findings are of vital significance to maximize yield,optimize quality and improve resource use efficiencies of winter oilseed rape. 展开更多
关键词 appropriate cultivation method comprehensive decision analysis method northwestern China winter oilseed rape
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Homotopy Analysis Method for Solving Biological Population Model 被引量:2
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作者 A.A.M.Arafa S.Z.Rida H.Mohamed 《Communications in Theoretical Physics》 SCIE CAS CSCD 2011年第11期797-800,共4页
In this paper,the homotopy analysis method (HAM) is applied to solve generalized biological populationmodels.The fractional derivatives are described by Caputo's sense.The method introduces a significant improveme... In this paper,the homotopy analysis method (HAM) is applied to solve generalized biological populationmodels.The fractional derivatives are described by Caputo's sense.The method introduces a significant improvementin this field over existing techniques.Results obtained using the scheme presented here agree well with the analyticalsolutions and the numerical results presented in Ref.[6].However,the fundamental solutions of these equations stillexhibit useful scaling properties that make them attractive for applications. 展开更多
关键词 biological populations model fractional calculus homotopy analysis method Mittag-Leffier function
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Prediction of central bursting defects in rod extrusion process with upper bound analysis method 被引量:2
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作者 Amir PARGHAZEH Heshmatollah HAGHIGHAT 《Transactions of Nonferrous Metals Society of China》 SCIE EI CAS CSCD 2016年第11期2892-2899,共8页
The prediction of central bursting defects in the rod extrusion process through conical dies using the upper bound analysisis investigated. A kinematically admissible velocity field, including the radial and angular v... The prediction of central bursting defects in the rod extrusion process through conical dies using the upper bound analysisis investigated. A kinematically admissible velocity field, including the radial and angular velocity components, is proposed. A newcriterion is presented to predict the occurrence of the central bursting defects. Parameter bobt, which represents the risk probability ofcracking, is proposed. It is calculated using the shape of the boundary at the entrance by minimizing the total power dissipationduring the extrusion process. When bobt is equal to or greater than bcr, central bursting occurs. Furthermore, the quantitativerelationships between central bursting defects and process parameters (semi die angle, reduction in area and frictional factor) arestudied. The results show that the central bursting defects are affected primarily by the reduction in area and the friction factor. Thepresented criterion is verified by comparing with the FEM simulation data and the results of the published paper. 展开更多
关键词 central bursting defect upper bound analysis method rod extrusion process
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Solitary Solution of Discrete mKdV Equation by Homotopy Analysis Method 被引量:1
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作者 WANG Zhen ZOU Li ZHANG Hong-Qing 《Communications in Theoretical Physics》 SCIE CAS CSCD 2008年第6期1373-1378,共6页
In this paper, we apply homotopy analysis method to solve discrete mKdV equation and successfully obtain the bell-shaped solitary solution to mKdV equation. Comparison between our solution and the exact solution shows... In this paper, we apply homotopy analysis method to solve discrete mKdV equation and successfully obtain the bell-shaped solitary solution to mKdV equation. Comparison between our solution and the exact solution shows that homotopy analysis method is effective and validity in solving hybrid nonlinear problems, including solitary solution of difference-differential equation. 展开更多
关键词 differential-difference equation Homotopy analysis method homotopy-Pade approximation discrete mKdV equation SOLITON
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Primary resonance of multiple degree-of-freedom dynamic systems with strong non-linearity using the homotopy analysis method 被引量:1
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作者 原培新 李永强 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2010年第10期1293-1304,共12页
A homotopy analysis method(HAM)is presented for the primary resonance of multiple degree-of-freedom systems with strong non-linearity excited by harmonic forces.The validity of the HAM is independent of the existenc... A homotopy analysis method(HAM)is presented for the primary resonance of multiple degree-of-freedom systems with strong non-linearity excited by harmonic forces.The validity of the HAM is independent of the existence of small parameters in the considered equation.The HAM provides a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter.Two examples are presented to show that the HAM solutions agree well with the results of the modified Linstedt-Poincar'e method and the incremental harmonic balance method. 展开更多
关键词 homotopy analysis method primary resonance series solution strong nonlinearity MULTI-DEGREE-OF-FREEDOM
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Approximate Solutions of Primary Resonance for Forced Duffing Equation by Means of the Homotopy Analysis Method 被引量:1
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作者 YUAN Peixin LI Yongqiang 《Chinese Journal of Mechanical Engineering》 SCIE EI CAS CSCD 2011年第3期501-506,共6页
Nonlinear dynamic equation is a common engineering model.There is not precise analytical solution for most of nonlinear differential equations.These nonlinear differential equations should be solved by using approxima... Nonlinear dynamic equation is a common engineering model.There is not precise analytical solution for most of nonlinear differential equations.These nonlinear differential equations should be solved by using approximate methods.Classical perturbation methods such as LP method,KBM method,multi-scale method and the averaging method on weakly nonlinear vibration system is effective,while the strongly nonlinear system is difficult to apply.Approximate solutions of primary resonance for forced Duffing equation is investigated by means of homotopy analysis method (HAM).Different from other approximate computational method,the HAM is totally independent of small physical parameters,and thus is suitable for most nonlinear problems.The HAM provides a great freedom to choose base functions of solution series,so that a nonlinear problem may be approximated more effectively.The HAM provides us a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter and the auxiliary function.Therefore,HAM not only may solve the weakly non-linear problems but also may be suitable for the strong non-linear problem.Through the approximate solution of forced Duffing equation with cubic non-linearity,the HAM and fourth order Runge-Kutta method of numerical solution were compared,the results show that the HAM not only can solve the steady state solution,but also can calculate the unsteady state solution,and has the good computational accuracy. 展开更多
关键词 homotopy analysis method(HAM) series solution forced Duffing equation numerical solution
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