Metrological atomic force microscopes(Met.AFMs)with built-in interferometers are one of the main workhorses for versatile dimensional nanometrology.The interferometric nonlinearity error,particularly the high-order(i....Metrological atomic force microscopes(Met.AFMs)with built-in interferometers are one of the main workhorses for versatile dimensional nanometrology.The interferometric nonlinearity error,particularly the high-order(i.e.,3rd-and 4th-order)nonlinearity errors,is a dominant error source for further improving their metrology performance,which cannot be corrected using the conventional Heydemann correction method.To solve this problem,two new methods were developed.One uses a capacitive sensor embedded in the Met.AFM,and the other applies an external physical artifact with a flat surface.Both methods can be applied very conveniently and can effectively reduce the nonlinearity error.In this paper,the propagation of the(residual)nonlinearity error in step height calibrations is examined.Finally,the performance of the improved tool is verified in the calibration of a highly demanding industrial sample.For the measurements performed at 25 different positions and repeated six times,the standard deviation of the total 150 measured values is 0.08 nm,which includes the contributions from the reproducibility of the metrology tool and sample inhomogeneity.This research has significantly improved our dimensional nanometrology service.For instance,the extended measurement uncertainty(k=2)is reduced from 1.0 to 0.3 nm for the step height or etching depth calibrations.展开更多
A grating eddy current displacement sensor(GECDS) can be used in a watertight electronic transducer to realize long range displacement or position measurement with high accuracy in difficult industry conditions.The pa...A grating eddy current displacement sensor(GECDS) can be used in a watertight electronic transducer to realize long range displacement or position measurement with high accuracy in difficult industry conditions.The parameters optimization of the sensor is essential for economic and efficient production.This paper proposes a method to combine an artificial neural network(ANN) and a genetic algorithm(GA) for the sensor parameters optimization.A neural network model is developed to map the complex relationship between design parameters and the nonlinearity error of the GECDS,and then a GA is used in the optimization process to determine the design parameter values,resulting in a desired minimal nonlinearity error of about 0.11%.The calculated nonlinearity error is 0.25%.These results show that the proposed method performs well for the parameters optimization of the GECDS.展开更多
Extended range （10-30 d） heavy rain forecasting is difficult but performs an important function in disaster prevention and mitigation. In this paper, a nonlinear cross prediction error （NCPE） algorithm that combin...Extended range （10-30 d） heavy rain forecasting is difficult but performs an important function in disaster prevention and mitigation. In this paper, a nonlinear cross prediction error （NCPE） algorithm that combines nonlinear dynamics and statistical methods is proposed. The method is based on phase space reconstruction of chaotic single-variable time series of precipitable water and is tested in 100 global cases of heavy rain. First, nonlinear relative dynamic error for local attractor pairs is calculated at different stages of the heavy rain process, after which the local change characteristics of the attractors are analyzed. Second, the eigen-peak is defined as a prediction indicator based on an error threshold of about 1.5, and is then used to analyze the forecasting validity period. The results reveal that the prediction indicator features regarded as eigenpeaks for heavy rain extreme weather are all reflected consistently, without failure, based on the NCPE model; the prediction validity periods for 1-2 d, 3-9 d and 10-30 d are 4, 22 and 74 cases, respectively, without false alarm or omission. The NCPE model developed allows accurate forecasting of heavy rain over an extended range of 10-30 d and has the potential to be used to explore the mechanisms involved in the development of heavy rain according to a segmentation scale. This novel method provides new insights into extended range forecasting and atmospheric predictability, and also allows the creation of multi-variable chaotic extreme weather prediction models based on high spatiotemporal resolution data.展开更多
Real time remaining useful life(RUL) prediction based on condition monitoring is an essential part in condition based maintenance(CBM). In the current methods about the real time RUL prediction of the nonlinear degrad...Real time remaining useful life(RUL) prediction based on condition monitoring is an essential part in condition based maintenance(CBM). In the current methods about the real time RUL prediction of the nonlinear degradation process, the measurement error is not considered and forecasting uncertainty is large. Therefore, an approximate analytical RUL distribution in a closed-form of a nonlinear Wiener based degradation process with measurement errors was proposed. The maximum likelihood estimation approach was used to estimate the unknown fixed parameters in the proposed model. When the newly observed data are available, the random parameter is updated by the Bayesian method to make the estimation adapt to the item's individual characteristic and reduce the uncertainty of the estimation. The simulation results show that considering measurement errors in the degradation process can significantly improve the accuracy of real time RUL prediction.展开更多
With the observational wind data and the Zebiak-Cane model, the impact of Madden-Iulian Oscillation （MJO） as external forcing on El Nino-Southern Oscillation （ENSO） predictability is studied. The observational dat...With the observational wind data and the Zebiak-Cane model, the impact of Madden-Iulian Oscillation （MJO） as external forcing on El Nino-Southern Oscillation （ENSO） predictability is studied. The observational data are analyzed with Continuous Wavelet Transform （CWT） and then used to extract MJO signals, which are added into the model to get a new model. After the Conditional Nonlinear Optimal Perturbation （CNOP） method has been used, the initial errors which can evolve into maximum prediction error, model errors and their join errors are gained and then the Nifio 3 indices and spatial structures of three kinds of errors are investigated. The results mainly show that the observational MJO has little impact on the maximum prediction error of ENSO events and the initial error affects much greater than model error caused by MJO forcing. These demonstrate that the initial error might be the main error source that produces uncertainty in ENSO prediction, which could provide a theoretical foundation for the adaptive data assimilation of the ENSO forecast and contribute to the ENSO target observation.展开更多
Two important extensions of a technique to perform a nonlinear error propagation analysis for an explicit pseudodynamic algorithm （Chang, 2003） are presented. One extends the stability study from a given time step t...Two important extensions of a technique to perform a nonlinear error propagation analysis for an explicit pseudodynamic algorithm （Chang, 2003） are presented. One extends the stability study from a given time step to a complete step-by-step integration procedure. It is analytically proven that ensuring stability conditions in each time step leads to a stable computation of the entire step-by-step integration procedure. The other extension shows that the nonlinear error propagation results, which are derived for a nonlinear single degree of freedom （SDOF） system, can be applied to a nonlinear multiple degree of freedom （MDOF） system. This application is dependent upon the determination of the natural frequencies of the system in each time step, since all the numerical properties and error propagation properties in the time step are closely related to these frequencies. The results are derived from the step degree of nonlinearity. An instantaneous degree of nonlinearity is introduced to replace the step degree of nonlinearity and is shown to be easier to use in practice. The extensions can be also applied to the results derived from a SDOF system based on the instantaneous degree of nonlinearity, and hence a time step might be appropriately chosen to perform a pseudodynamic test prior to testing.展开更多
With the wide application of laser in the field of skin plastic surgery, micro laser galvanometer scanner has made great progress in this field with its portability. However, the measurement method used to measure the...With the wide application of laser in the field of skin plastic surgery, micro laser galvanometer scanner has made great progress in this field with its portability. However, the measurement method used to measure the deflection angle of laser galvanometer in the narrow space of scanner with high precision remains to be studied. In this paper, an angle measurement method based on magnetic field is proposed, and the effect of the shapes of permanent magnets(PMs) on the measurement is studied by theoretical and experimental study under the condition that the maximum available space for the PMs is a 10 mm side cube. An angle measuring experimental device is set up, and the contrast experiment is carried out with different PMs which are the same as simulation. The experimental results show that cylindrical PM is more suitable than other PMs, which is consistent with the simulation results, and the maximum nonlinearity error is 0.562%. This method has the advantages of small volume,non-contact measurement, small moment of inertia, good dynamic response and no external excitation for the PMs, so it has a broad application prospect in micro laser galvanometer scanner.展开更多
It has been well studied that the γ-function explicit method can be effective in providing favorable numerical dissipation for linear elastic systems. However, its performance for nonlinear systems is unclear due to ...It has been well studied that the γ-function explicit method can be effective in providing favorable numerical dissipation for linear elastic systems. However, its performance for nonlinear systems is unclear due to a lack of analytical evaluation techniques. Thus, a novel technique is proposed herein to evaluate its efficiency for application to nonlinear systems by introducing two parameters to describe the stiffness change. As a result, the numerical properties and error propagation characteristics of the γ-function explicit method for the pseudodynamic testing of a nonlinear system are analytically assessed. It is found that the upper stability limit decreases as the step degree of nonlinearity increases; and it increases as the current degree of nonlinearity increases. It is also shown that this integration method provides favorable numerical dissipation not only for linear elastic systems but also for nonlinear systems. Furthermore, error propagation analysis reveals that the numerical dissipation can effectively suppress the severe error propagation of high frequency modes while the low frequency responses are almost unaffected for both linear elastic and nonlinear systems.展开更多
This study investigated <i>Bacillus subtilis</i> ATCC13952 as an adsorbent for arsenic in groundwater. Batch experiments were used to determine the effect of contact time, adsorbent dose, arsenic (III) con...This study investigated <i>Bacillus subtilis</i> ATCC13952 as an adsorbent for arsenic in groundwater. Batch experiments were used to determine the effect of contact time, adsorbent dose, arsenic (III) concentration, pH, and temperature on the process. The percentage of arsenic (III) removed was high at a contact time of four days, 3.0 mL of <i>Bacillus subtilis</i> ATCC13952, pH 8 and temperature of 35°C. The kinetics of the process showed the Elovich kinetics model as the best fit for the process. This indicates that arsenic removal was by chemisorption. The analysis of the nonlinear equilibrium isotherms and the error functions showed the Langmuir isotherm as best fit for the process. Mechanistic study of the process indicated bulk diffusion to be the rate-determining step. Thermodynamically, the process was favourable, spontaneous and feasible. When the community water samples were treated with the <i>Bacillus subtilis</i> ATCC13952 at the optimum contact time, adsorbent dose, pH and temperature, 99.96% - 99.97% of arsenic was removed across all sampling points within the studied communities. Hence, the results show that <i>Bacillus subtilis</i> ATCC13952 is an efficient adsorbent for arsenic in aqueous systems and the organism appears to hold the key to purging the environment of arsenic contamination.展开更多
Although the step degree of nonlinearity has been introduced to conduct basic analysis and error propagation analysis for the pseudodynamic testing of nonlinear systems, it cannot be reliably used to select an appropr...Although the step degree of nonlinearity has been introduced to conduct basic analysis and error propagation analysis for the pseudodynamic testing of nonlinear systems, it cannot be reliably used to select an appropriate time step before performing a pseudodynamic test. Therefore, a novel parameter of instantaneous degree of nonlinearity is introduced to monitor the stiffness change at the end of a time step, and can thus be used to evaluate numerical and error propagation properties for nonlinear systems. Based on these properties, it is possible to select an appropriate time step to conduct a pseudodynamic test in advance. This possibility is very important in pseudodynamic testing, since the use of an arbitrary time step might lead to unreliable results or even destroy the test specimen. In this paper, guidelines are proposed for choosing an appropriate time step for accurate integration of nonlinear systems. These guidelines require estimation of the maximum instantaneous degree of nonlinearity and the solution of the initial natural frequency. The Newmark explicit method is chosen for this study. All the analytical results and the guidelines proposed are thoroughly confirmed with numerical examples.展开更多
High-precision piezoresistive pressure sensors play a significant role in aerospace,automotive,and other fields.Nonlinear error is the key factor that restricts the improvement of the sensor precision.A mathematical m...High-precision piezoresistive pressure sensors play a significant role in aerospace,automotive,and other fields.Nonlinear error is the key factor that restricts the improvement of the sensor precision.A mathematical model for evaluating the sensor’s nonlinear error is established,based on which a piezoresistor sensitivity matching method is proposed to suppress the nonlinear error.By adjusting the piezoresistors'structure and position on the sensing membrane,four piezoresistors with equal sensitivity are obtained,and theoretical quasi-zero nonlinear error is achieved.To verify the design,sensor prototypes are fabricated utilizing the MEMS technology.After sensor packaging,a cylindrical absolute pressure sensor featuring a 4 mm diameter with a range from 0 to 100 kPa is acquired.The experimental results demonstrate the excellent performance of the proposed sensor,which indicates a nonlinear error as low as±0.004%FS.Besides,the proposed sensor has a sensitivity of 1.6810 mV/kPa,a hysteresis of 0.025%,a repeatability of 0.015%,a zero drift of 0.03%FS,and a 3 dB frequency from 0 to 121.82 kHz.Moreover,the prototype is tested in the Mach 4 wind tunnel,and the measurement error between the proposed sensor and the true pressure is±0.98%.This paper provides key sensing technology for high-precision surface pressure analysis of aircraft.展开更多
In nonlinear error growth dynamics,the initial error cannot be accurately determined,and the forecast error,which is also uncertain,can be considered to be a random variable.Entropy in information theory is a natural ...In nonlinear error growth dynamics,the initial error cannot be accurately determined,and the forecast error,which is also uncertain,can be considered to be a random variable.Entropy in information theory is a natural measure of the uncertainty of a random variable associated with a probability distribution.This paper effectively combines statistical information theory and nonlinear error growth dynamics,and introduces some fundamental concepts of entropy in information theory for nonlinear error growth dynamics.Entropy based on nonlinear error can be divided into time entropy and space entropy,which are used to estimate the predictabilities of the whole dynamical system and each of its variables.This is not only applicable for investigating the dependence between any two variables of a multivariable system,but also for measuring the influence of each variable on the predictability of the whole system.Taking the Lorenz system as an example,the entropy of nonlinear error is applied to estimate predictability.The time and space entropies are used to investigate the spatial distribution of predictability of the whole Lorenz system.The results show that when moving around two chaotic attractors or near the edge of system space,a Lorenz system with lower sensitivity to the initial field behaves with higher predictability and a longer predictability limit.The example analysis of predictability of the Lorenz system demonstrates that the predictability estimated by the entropy of nonlinear error is feasible and effective,especially for estimation of predictability of the whole system.This provides a theoretical foundation for further work in estimating real atmospheric multivariable joint predictability.展开更多
To estimate atmospheric predictability for multivariable system, based on information theory in nonlinear error growth dynamics, a quantitative method is introduced in this paper using multivariable joint predictabili...To estimate atmospheric predictability for multivariable system, based on information theory in nonlinear error growth dynamics, a quantitative method is introduced in this paper using multivariable joint predictability limit(MJPL) and corresponding single variable predictability limit(SVPL). The predictability limit, obtained from the evolutions of nonlinear error entropy and climatological state entropy, is not only used to measure the predictability of dynamical system with the constant climatological state entropy, but also appropriate to the case of climatological state entropy changed with time. With the help of daily NCEP-NCAR reanalysis data, by using a method of local dynamical analog, the nonlinear error entropy, climatological state entropy, and predictability limit are obtained, and the SVPLs and MJPL of the winter 500-hPa temperature field, zonal wind field and meridional wind field are also investigated. The results show that atmospheric predictability is well associated with the analytical variable. For single variable predictability, there exists a big difference for the three variables, with the higher predictability found for the temperature field and zonal wind field and the lower predictability for the meridional wind field. As seen from their spatial distributions, the SVPLs of the three variables appear to have a property of zonal distribution, especially for the meridional wind field, which has three zonal belts with low predictability and four zonal belts with high predictability. For multivariable joint predictability, the MJPL of multivariable system with the three variables is not a simple mean or linear combination of its SVPLs. It presents an obvious regional difference characteristic. Different regions have different results. In some regions, the MJPL is among its SVPLs. However, in other regions, the MJPL is less than its all SVPLs.展开更多
Based on the theory of information entropy concerning nonlinear errors, the growth rules for the nonlinear errors of the Lorenz system and its predictable components are studied. The results show that the impact of th...Based on the theory of information entropy concerning nonlinear errors, the growth rules for the nonlinear errors of the Lorenz system and its predictable components are studied. The results show that the impact of the uncertainties, both in the initial error and in the system itself, needs to be considered in a quantitative estimation of the system predictability. The nonlinear error growth is related to the magnitude of the initial error, and to the spatial distribution of the initial error vectors. Even if these initial errors have the same magnitude but different directions, there are also differences in the nonlinear error growth. The predictability of nonlinear error growth is related to the error component, but not related to the ratio of these components. The component with the highest/lowest rate of enntribution does not necessarily have the greatest/least predictability. The different components have different predictabilities, and in different time periods, the different predictable components also have different predictabilities.展开更多
While the numerical solution of one-dimensional Volterra integral equations of the second kind with regular kernels is well understood, there exist no systematic studies of asymptotic error expansion for the approxima...While the numerical solution of one-dimensional Volterra integral equations of the second kind with regular kernels is well understood, there exist no systematic studies of asymptotic error expansion for the approximate solution. In this paper,we analyse the Nystrom solution of one-dimensional nonlinear Volterra integral equation of the second kind and show that approkimate solution admits an asymptotic error expansion in even powers of the step-size h, beginning with a term in h2. So that the Richardson's extrapolation can be done. This will increase the accuracy of numerical solution greatly.展开更多
The measurement of mass,or equivalently the binding energy,of exotic nuclei has reached the limits of nuclear existence,which are characterized by tiny production cross-sections and short half-lives.The isochronousmod...The measurement of mass,or equivalently the binding energy,of exotic nuclei has reached the limits of nuclear existence,which are characterized by tiny production cross-sections and short half-lives.The isochronousmode of the Spectrometer Ring at the High Intensity heavy-ion Accelerator Facility project in China(HIAF-SRing)offers the capacity for such measurements.However,many factors limit the revolution time resolution of the isochronous mode of the large acceptance HIAF-SRing.Nonlinear field errors as well as fringe fields of the wide aperture dipoles and quadrupoles strongly excite the higher-order aberrations,which negatively affect the revolution time resolution.Moreover,the transverse emittance of the beam is inversely proportional to the revolution time resolution.Their influence is investigated here,and a possible correction scheme with sextupoles and octupoles is shown.With higher-order corrections,a mass resolution of R(FWHM)=1×10^(6),corresponding to a relative revolution time ofσ(T)/T~4.9×10^(-7),is with the isochronous settingγt=1.43 within the momentum acceptance of±0.2%.展开更多
According to the specific input-output characteristics of a pressure sensor, a novel calibration algorithm is presented and a calibration system is developed to correct the nonlinear error caused by temperature. In co...According to the specific input-output characteristics of a pressure sensor, a novel calibration algorithm is presented and a calibration system is developed to correct the nonlinear error caused by temperature. In contrast to the routine BP and RBF, curve fitting based on RBF is first used to get the slope and intercept, and then the voltage-pressure curve is described. Test results show that the algorithm features fast convergence speed, strong robustness and minimum SSE （sum of squares for error）. It is proven by practical applications that this calibration system works well and the measurement precision is better than the design demands. Furthermore, this calibration system has a good real-time capability.展开更多
In this work, two-grid characteristic finite volume schemes for the nonlinear parabolic problem are considered. In our algorithms, the diffusion term is discretized by the finite volume method, while the temporal diff...In this work, two-grid characteristic finite volume schemes for the nonlinear parabolic problem are considered. In our algorithms, the diffusion term is discretized by the finite volume method, while the temporal differentiation and advection terms are treated by the characteristic scheme. Under some conditions about the coefficients and exact solution, optimal error estimates for the numerical solution are obtained. Furthermore, the two- grid characteristic finite volume methods involve solving a nonlinear equation on coarse mesh with mesh size H, a large linear problem for the Oseen two-grid characteristic finite volume method on a fine mesh with mesh size h = O（H2） or a large linear problem for the Newton two-grid characteristic finite volume method on a fine mesh with mesh size h = 0（I log hll/2H3）. These methods we studied provide the same convergence rate as that of the characteristic finite volume method, which involves solving one large nonlinear problem on a fine mesh with mesh size h. Some numerical results are presented to demonstrate the efficiency of the proposed methods.展开更多
基金Open Access funding enabled and organized by Projekt DEAL.
文摘Metrological atomic force microscopes(Met.AFMs)with built-in interferometers are one of the main workhorses for versatile dimensional nanometrology.The interferometric nonlinearity error,particularly the high-order(i.e.,3rd-and 4th-order)nonlinearity errors,is a dominant error source for further improving their metrology performance,which cannot be corrected using the conventional Heydemann correction method.To solve this problem,two new methods were developed.One uses a capacitive sensor embedded in the Met.AFM,and the other applies an external physical artifact with a flat surface.Both methods can be applied very conveniently and can effectively reduce the nonlinearity error.In this paper,the propagation of the(residual)nonlinearity error in step height calibrations is examined.Finally,the performance of the improved tool is verified in the calibration of a highly demanding industrial sample.For the measurements performed at 25 different positions and repeated six times,the standard deviation of the total 150 measured values is 0.08 nm,which includes the contributions from the reproducibility of the metrology tool and sample inhomogeneity.This research has significantly improved our dimensional nanometrology service.For instance,the extended measurement uncertainty(k=2)is reduced from 1.0 to 0.3 nm for the step height or etching depth calibrations.
文摘A grating eddy current displacement sensor(GECDS) can be used in a watertight electronic transducer to realize long range displacement or position measurement with high accuracy in difficult industry conditions.The parameters optimization of the sensor is essential for economic and efficient production.This paper proposes a method to combine an artificial neural network(ANN) and a genetic algorithm(GA) for the sensor parameters optimization.A neural network model is developed to map the complex relationship between design parameters and the nonlinearity error of the GECDS,and then a GA is used in the optimization process to determine the design parameter values,resulting in a desired minimal nonlinearity error of about 0.11%.The calculated nonlinearity error is 0.25%.These results show that the proposed method performs well for the parameters optimization of the GECDS.
基金provided by the National Natural Science Foundation of China(Grant Nos.41275039 and 41471305)the Preeminence Youth Cultivation Project of Sichuan (Grant No.2015JQ0037)
文摘Extended range （10-30 d） heavy rain forecasting is difficult but performs an important function in disaster prevention and mitigation. In this paper, a nonlinear cross prediction error （NCPE） algorithm that combines nonlinear dynamics and statistical methods is proposed. The method is based on phase space reconstruction of chaotic single-variable time series of precipitable water and is tested in 100 global cases of heavy rain. First, nonlinear relative dynamic error for local attractor pairs is calculated at different stages of the heavy rain process, after which the local change characteristics of the attractors are analyzed. Second, the eigen-peak is defined as a prediction indicator based on an error threshold of about 1.5, and is then used to analyze the forecasting validity period. The results reveal that the prediction indicator features regarded as eigenpeaks for heavy rain extreme weather are all reflected consistently, without failure, based on the NCPE model; the prediction validity periods for 1-2 d, 3-9 d and 10-30 d are 4, 22 and 74 cases, respectively, without false alarm or omission. The NCPE model developed allows accurate forecasting of heavy rain over an extended range of 10-30 d and has the potential to be used to explore the mechanisms involved in the development of heavy rain according to a segmentation scale. This novel method provides new insights into extended range forecasting and atmospheric predictability, and also allows the creation of multi-variable chaotic extreme weather prediction models based on high spatiotemporal resolution data.
基金Projects(51475462,61374138,61370031)supported by the National Natural Science Foundation of China
文摘Real time remaining useful life(RUL) prediction based on condition monitoring is an essential part in condition based maintenance(CBM). In the current methods about the real time RUL prediction of the nonlinear degradation process, the measurement error is not considered and forecasting uncertainty is large. Therefore, an approximate analytical RUL distribution in a closed-form of a nonlinear Wiener based degradation process with measurement errors was proposed. The maximum likelihood estimation approach was used to estimate the unknown fixed parameters in the proposed model. When the newly observed data are available, the random parameter is updated by the Bayesian method to make the estimation adapt to the item's individual characteristic and reduce the uncertainty of the estimation. The simulation results show that considering measurement errors in the degradation process can significantly improve the accuracy of real time RUL prediction.
基金The National Natural Science Foundation of China under contract No.41405062
文摘With the observational wind data and the Zebiak-Cane model, the impact of Madden-Iulian Oscillation （MJO） as external forcing on El Nino-Southern Oscillation （ENSO） predictability is studied. The observational data are analyzed with Continuous Wavelet Transform （CWT） and then used to extract MJO signals, which are added into the model to get a new model. After the Conditional Nonlinear Optimal Perturbation （CNOP） method has been used, the initial errors which can evolve into maximum prediction error, model errors and their join errors are gained and then the Nifio 3 indices and spatial structures of three kinds of errors are investigated. The results mainly show that the observational MJO has little impact on the maximum prediction error of ENSO events and the initial error affects much greater than model error caused by MJO forcing. These demonstrate that the initial error might be the main error source that produces uncertainty in ENSO prediction, which could provide a theoretical foundation for the adaptive data assimilation of the ENSO forecast and contribute to the ENSO target observation.
基金NSC, Chinese Taipei Under Grant No. NSC-97-2221-E-027-036-MY2
文摘Two important extensions of a technique to perform a nonlinear error propagation analysis for an explicit pseudodynamic algorithm （Chang, 2003） are presented. One extends the stability study from a given time step to a complete step-by-step integration procedure. It is analytically proven that ensuring stability conditions in each time step leads to a stable computation of the entire step-by-step integration procedure. The other extension shows that the nonlinear error propagation results, which are derived for a nonlinear single degree of freedom （SDOF） system, can be applied to a nonlinear multiple degree of freedom （MDOF） system. This application is dependent upon the determination of the natural frequencies of the system in each time step, since all the numerical properties and error propagation properties in the time step are closely related to these frequencies. The results are derived from the step degree of nonlinearity. An instantaneous degree of nonlinearity is introduced to replace the step degree of nonlinearity and is shown to be easier to use in practice. The extensions can be also applied to the results derived from a SDOF system based on the instantaneous degree of nonlinearity, and hence a time step might be appropriately chosen to perform a pseudodynamic test prior to testing.
基金supported by the Aeronautical Science Foundation of China [grant numbers 20175748009]。
文摘With the wide application of laser in the field of skin plastic surgery, micro laser galvanometer scanner has made great progress in this field with its portability. However, the measurement method used to measure the deflection angle of laser galvanometer in the narrow space of scanner with high precision remains to be studied. In this paper, an angle measurement method based on magnetic field is proposed, and the effect of the shapes of permanent magnets(PMs) on the measurement is studied by theoretical and experimental study under the condition that the maximum available space for the PMs is a 10 mm side cube. An angle measuring experimental device is set up, and the contrast experiment is carried out with different PMs which are the same as simulation. The experimental results show that cylindrical PM is more suitable than other PMs, which is consistent with the simulation results, and the maximum nonlinearity error is 0.562%. This method has the advantages of small volume,non-contact measurement, small moment of inertia, good dynamic response and no external excitation for the PMs, so it has a broad application prospect in micro laser galvanometer scanner.
基金National Science Council. Chinese Taipei, Under Grant No. NSC-92-2211-E-027-015
文摘It has been well studied that the γ-function explicit method can be effective in providing favorable numerical dissipation for linear elastic systems. However, its performance for nonlinear systems is unclear due to a lack of analytical evaluation techniques. Thus, a novel technique is proposed herein to evaluate its efficiency for application to nonlinear systems by introducing two parameters to describe the stiffness change. As a result, the numerical properties and error propagation characteristics of the γ-function explicit method for the pseudodynamic testing of a nonlinear system are analytically assessed. It is found that the upper stability limit decreases as the step degree of nonlinearity increases; and it increases as the current degree of nonlinearity increases. It is also shown that this integration method provides favorable numerical dissipation not only for linear elastic systems but also for nonlinear systems. Furthermore, error propagation analysis reveals that the numerical dissipation can effectively suppress the severe error propagation of high frequency modes while the low frequency responses are almost unaffected for both linear elastic and nonlinear systems.
文摘This study investigated <i>Bacillus subtilis</i> ATCC13952 as an adsorbent for arsenic in groundwater. Batch experiments were used to determine the effect of contact time, adsorbent dose, arsenic (III) concentration, pH, and temperature on the process. The percentage of arsenic (III) removed was high at a contact time of four days, 3.0 mL of <i>Bacillus subtilis</i> ATCC13952, pH 8 and temperature of 35°C. The kinetics of the process showed the Elovich kinetics model as the best fit for the process. This indicates that arsenic removal was by chemisorption. The analysis of the nonlinear equilibrium isotherms and the error functions showed the Langmuir isotherm as best fit for the process. Mechanistic study of the process indicated bulk diffusion to be the rate-determining step. Thermodynamically, the process was favourable, spontaneous and feasible. When the community water samples were treated with the <i>Bacillus subtilis</i> ATCC13952 at the optimum contact time, adsorbent dose, pH and temperature, 99.96% - 99.97% of arsenic was removed across all sampling points within the studied communities. Hence, the results show that <i>Bacillus subtilis</i> ATCC13952 is an efficient adsorbent for arsenic in aqueous systems and the organism appears to hold the key to purging the environment of arsenic contamination.
基金supported by the NSC,Chinese Taipei,Under Grant No.NSC-95-2221-E-027-099
文摘Although the step degree of nonlinearity has been introduced to conduct basic analysis and error propagation analysis for the pseudodynamic testing of nonlinear systems, it cannot be reliably used to select an appropriate time step before performing a pseudodynamic test. Therefore, a novel parameter of instantaneous degree of nonlinearity is introduced to monitor the stiffness change at the end of a time step, and can thus be used to evaluate numerical and error propagation properties for nonlinear systems. Based on these properties, it is possible to select an appropriate time step to conduct a pseudodynamic test in advance. This possibility is very important in pseudodynamic testing, since the use of an arbitrary time step might lead to unreliable results or even destroy the test specimen. In this paper, guidelines are proposed for choosing an appropriate time step for accurate integration of nonlinear systems. These guidelines require estimation of the maximum instantaneous degree of nonlinearity and the solution of the initial natural frequency. The Newmark explicit method is chosen for this study. All the analytical results and the guidelines proposed are thoroughly confirmed with numerical examples.
基金supported in part by the National Natural Science Foundation of China(Grant No.52205602,No.52075455 and 62204264)the Fundamental Research Funds for the Central Universities(Grant No.D5000240226).
文摘High-precision piezoresistive pressure sensors play a significant role in aerospace,automotive,and other fields.Nonlinear error is the key factor that restricts the improvement of the sensor precision.A mathematical model for evaluating the sensor’s nonlinear error is established,based on which a piezoresistor sensitivity matching method is proposed to suppress the nonlinear error.By adjusting the piezoresistors'structure and position on the sensing membrane,four piezoresistors with equal sensitivity are obtained,and theoretical quasi-zero nonlinear error is achieved.To verify the design,sensor prototypes are fabricated utilizing the MEMS technology.After sensor packaging,a cylindrical absolute pressure sensor featuring a 4 mm diameter with a range from 0 to 100 kPa is acquired.The experimental results demonstrate the excellent performance of the proposed sensor,which indicates a nonlinear error as low as±0.004%FS.Besides,the proposed sensor has a sensitivity of 1.6810 mV/kPa,a hysteresis of 0.025%,a repeatability of 0.015%,a zero drift of 0.03%FS,and a 3 dB frequency from 0 to 121.82 kHz.Moreover,the prototype is tested in the Mach 4 wind tunnel,and the measurement error between the proposed sensor and the true pressure is±0.98%.This paper provides key sensing technology for high-precision surface pressure analysis of aircraft.
基金supported by National Natural Science Foundation of China (Grant No. 40975031)
文摘In nonlinear error growth dynamics,the initial error cannot be accurately determined,and the forecast error,which is also uncertain,can be considered to be a random variable.Entropy in information theory is a natural measure of the uncertainty of a random variable associated with a probability distribution.This paper effectively combines statistical information theory and nonlinear error growth dynamics,and introduces some fundamental concepts of entropy in information theory for nonlinear error growth dynamics.Entropy based on nonlinear error can be divided into time entropy and space entropy,which are used to estimate the predictabilities of the whole dynamical system and each of its variables.This is not only applicable for investigating the dependence between any two variables of a multivariable system,but also for measuring the influence of each variable on the predictability of the whole system.Taking the Lorenz system as an example,the entropy of nonlinear error is applied to estimate predictability.The time and space entropies are used to investigate the spatial distribution of predictability of the whole Lorenz system.The results show that when moving around two chaotic attractors or near the edge of system space,a Lorenz system with lower sensitivity to the initial field behaves with higher predictability and a longer predictability limit.The example analysis of predictability of the Lorenz system demonstrates that the predictability estimated by the entropy of nonlinear error is feasible and effective,especially for estimation of predictability of the whole system.This provides a theoretical foundation for further work in estimating real atmospheric multivariable joint predictability.
基金supported by the National Natural Science Foundation of China (Grant No. 41375063)
文摘To estimate atmospheric predictability for multivariable system, based on information theory in nonlinear error growth dynamics, a quantitative method is introduced in this paper using multivariable joint predictability limit(MJPL) and corresponding single variable predictability limit(SVPL). The predictability limit, obtained from the evolutions of nonlinear error entropy and climatological state entropy, is not only used to measure the predictability of dynamical system with the constant climatological state entropy, but also appropriate to the case of climatological state entropy changed with time. With the help of daily NCEP-NCAR reanalysis data, by using a method of local dynamical analog, the nonlinear error entropy, climatological state entropy, and predictability limit are obtained, and the SVPLs and MJPL of the winter 500-hPa temperature field, zonal wind field and meridional wind field are also investigated. The results show that atmospheric predictability is well associated with the analytical variable. For single variable predictability, there exists a big difference for the three variables, with the higher predictability found for the temperature field and zonal wind field and the lower predictability for the meridional wind field. As seen from their spatial distributions, the SVPLs of the three variables appear to have a property of zonal distribution, especially for the meridional wind field, which has three zonal belts with low predictability and four zonal belts with high predictability. For multivariable joint predictability, the MJPL of multivariable system with the three variables is not a simple mean or linear combination of its SVPLs. It presents an obvious regional difference characteristic. Different regions have different results. In some regions, the MJPL is among its SVPLs. However, in other regions, the MJPL is less than its all SVPLs.
基金supported by the National Natural Science Foundation of China(Grant No.41375063)
文摘Based on the theory of information entropy concerning nonlinear errors, the growth rules for the nonlinear errors of the Lorenz system and its predictable components are studied. The results show that the impact of the uncertainties, both in the initial error and in the system itself, needs to be considered in a quantitative estimation of the system predictability. The nonlinear error growth is related to the magnitude of the initial error, and to the spatial distribution of the initial error vectors. Even if these initial errors have the same magnitude but different directions, there are also differences in the nonlinear error growth. The predictability of nonlinear error growth is related to the error component, but not related to the ratio of these components. The component with the highest/lowest rate of enntribution does not necessarily have the greatest/least predictability. The different components have different predictabilities, and in different time periods, the different predictable components also have different predictabilities.
文摘While the numerical solution of one-dimensional Volterra integral equations of the second kind with regular kernels is well understood, there exist no systematic studies of asymptotic error expansion for the approximate solution. In this paper,we analyse the Nystrom solution of one-dimensional nonlinear Volterra integral equation of the second kind and show that approkimate solution admits an asymptotic error expansion in even powers of the step-size h, beginning with a term in h2. So that the Richardson's extrapolation can be done. This will increase the accuracy of numerical solution greatly.
基金the the National Nature Science Foundation of China(12105333,12135017,12121005,11975280,12205340,12322507,12305126,12305151)。
文摘The measurement of mass,or equivalently the binding energy,of exotic nuclei has reached the limits of nuclear existence,which are characterized by tiny production cross-sections and short half-lives.The isochronousmode of the Spectrometer Ring at the High Intensity heavy-ion Accelerator Facility project in China(HIAF-SRing)offers the capacity for such measurements.However,many factors limit the revolution time resolution of the isochronous mode of the large acceptance HIAF-SRing.Nonlinear field errors as well as fringe fields of the wide aperture dipoles and quadrupoles strongly excite the higher-order aberrations,which negatively affect the revolution time resolution.Moreover,the transverse emittance of the beam is inversely proportional to the revolution time resolution.Their influence is investigated here,and a possible correction scheme with sextupoles and octupoles is shown.With higher-order corrections,a mass resolution of R(FWHM)=1×10^(6),corresponding to a relative revolution time ofσ(T)/T~4.9×10^(-7),is with the isochronous settingγt=1.43 within the momentum acceptance of±0.2%.
基金Project supported by the National Natural Science Foundation of China(No.61275081)
文摘According to the specific input-output characteristics of a pressure sensor, a novel calibration algorithm is presented and a calibration system is developed to correct the nonlinear error caused by temperature. In contrast to the routine BP and RBF, curve fitting based on RBF is first used to get the slope and intercept, and then the voltage-pressure curve is described. Test results show that the algorithm features fast convergence speed, strong robustness and minimum SSE （sum of squares for error）. It is proven by practical applications that this calibration system works well and the measurement precision is better than the design demands. Furthermore, this calibration system has a good real-time capability.
基金Acknowledgments. The work was supported by the Natural Science Foundation of China （No.11126117）, CAPES and CNPq of Brazil, and the Doctor Fund of Henan Polytechnic Univer- sity （B2012-098）. The author is very grateful to Professor JinYun Yuan for his kind invitation to visit the Universidade Federal do Paran, Brazil.
文摘In this work, two-grid characteristic finite volume schemes for the nonlinear parabolic problem are considered. In our algorithms, the diffusion term is discretized by the finite volume method, while the temporal differentiation and advection terms are treated by the characteristic scheme. Under some conditions about the coefficients and exact solution, optimal error estimates for the numerical solution are obtained. Furthermore, the two- grid characteristic finite volume methods involve solving a nonlinear equation on coarse mesh with mesh size H, a large linear problem for the Oseen two-grid characteristic finite volume method on a fine mesh with mesh size h = O（H2） or a large linear problem for the Newton two-grid characteristic finite volume method on a fine mesh with mesh size h = 0（I log hll/2H3）. These methods we studied provide the same convergence rate as that of the characteristic finite volume method, which involves solving one large nonlinear problem on a fine mesh with mesh size h. Some numerical results are presented to demonstrate the efficiency of the proposed methods.
