The purpose of this paper is to report the feasibility of constructing high-order rogue waves with controllable fission and asymmetry for high-dimensional nonlinear evolution equations.Such a nonlinear model considere...The purpose of this paper is to report the feasibility of constructing high-order rogue waves with controllable fission and asymmetry for high-dimensional nonlinear evolution equations.Such a nonlinear model considered in this paper as the concrete example is the(3+1)-dimensional generalized Boussinesq(gB)equation,and the corresponding method is Zhaqilao’s symbolic computation approach containing two embedded parameters.It is indicated by the(3+1)-dimensional gB equation that the embedded parameters can not only control the center of the first-order rogue wave,but also control the number of the wave peaks split from higher-order rogue waves and the asymmetry of higher-order rogue waves about the coordinate axes.The main novelty of this paper is that the obtained results and findings can provide useful supplements to the method used and the controllability of higher-order rogue waves.展开更多
Both numerical simulation and theoretical analysis of seawater intrusion in coastal regions are of great theoretical importance in environmental sciences. The mathematical model can be described as a problem of the in...Both numerical simulation and theoretical analysis of seawater intrusion in coastal regions are of great theoretical importance in environmental sciences. The mathematical model can be described as a problem of the initial boundary values for a system of 3-dimensional nonlinear parabolic partial differential equations, one being the pressure flow equation and the other is the concentration convection-dispersion equation of the salt contained. For a generic case of a 3-dimensional bounded region, a backward-difference time-stepping scheme is defined. It approximates the pressure by the standard Galerkin procedure and the concentration by a Galerkin method of charederistics, where calculus of variations, theory of prior estimates and techniques are made use of Optimal order estimates in H1 norm are derived for the errors in the approximate solution.展开更多
The (180)<sup>3</sup> third-order mixed sensitivities of the leakage response of a polyethylene-reflected plutonium (PERP) experimental benchmark with respect to the benchmark’s 180 microscopic total cros...The (180)<sup>3</sup> third-order mixed sensitivities of the leakage response of a polyethylene-reflected plutonium (PERP) experimental benchmark with respect to the benchmark’s 180 microscopic total cross sections have been computed in accompanying works [1] [2]. This work quantifies the contributions of these (180)<sup>3</sup> third-order mixed sensitivities to the PERP benchmark’s leakage response distribution moments (expected value, variance and skewness) and compares these contributions to those stemming from the corresponding first- and second-order sensitivities of the PERP benchmark’s leakage response with respect to the total cross sections. The numerical results obtained in this work reveal that the importance of the 3<sup>rd</sup>-order sensitivities can surpass the importance of the 1<sup>st</sup>- and 2<sup>nd</sup>-order sensitivities when the parameters’ uncertainties increase. In particular, for a uniform standard deviation of 10% of the microscopic total cross sections, the 3<sup>rd</sup>-order sensitivities contribute 80% to the response variance, whereas the contribution stemming from the 1st- and 2nd-order sensitivities amount only to 2% and 18%, respectively. Consequently, neglecting the 3<sup>rd</sup>-order sensitivities could cause a very large non-conservative error by under-reporting the response variance by a factor of 506%. The results obtained in this work also indicate that the effects of the 3<sup>rd</sup>-order sensitivities are to reduce the response’s skewness in parameter space, rendering the distribution of the leakage response more symmetric about its expected value. The results obtained in this work are the first such results ever published in reactor physics. Since correlations among the group-averaged microscopic total cross sections are not available, only the effects of typical standard deviations for these cross sections could be considered. Due to this lack of correlations among the cross sections, the effects of the <em>mixed</em> 3<sup>rd</sup>-order sensiti展开更多
This work extends to fourth-order previously published work on developing the adjoint sensitivity and uncertainty analysis of the numerical model of a <u>p</u>oly<u>e</u>thylene-<u>r</...This work extends to fourth-order previously published work on developing the adjoint sensitivity and uncertainty analysis of the numerical model of a <u>p</u>oly<u>e</u>thylene-<u>r</u>eflected <u>p</u>lutonium (acronym: PERP) OECD/NEA reactor physics benchmark. The PERP benchmark comprises 7477 imprecisely known (uncertain) model parameters which have nonzero values. These parameters are as follows: 180 microscopic total cross sections;7101 microscopic scattering sections;60 microscopic fission cross sections;60 parameters that characterize the average number of neutrons per fission;60 parameters that characterize the fission spectrum;10 parameters that characterize the fission source;and 6 parameters that characterize the isotope number densities. Previous works have used the adjoint sensitivity analysis methodology to compute exactly and efficiently all of the 7477 first-order and 27,956,503 second-order sensitivities of the PERP benchmark’s leakage response to all of the benchmark’s uncertain parameters. These works showed that largest response sensitivities involve the total microscopic cross sections, which motivated the recent computation of all of the (180)<sup>3</sup> third-order sensitivities of the PERP leakage response with respect to these total microscopic cross sections. It turned out that some of these 3<sup>rd</sup>-order cross sections were far larger than the corresponding 2<sup>nd</sup>-order ones, thereby having the largest impact on the uncertainties induced in the PERP benchmark’s response. This finding has motivated the development of the original 4<sup>th</sup>-order formulas presented in this work, which are valid not only for the PERP benchmark but can also be used for computing the 4<sup>th</sup>-order sensitivities of response of any nuclear system involving fissionable material and internal or external neutron sources. Subsequent works will use the adjoint-based mathematical expressions obtained in this work to compute exactly and efficiently the numerical values of the larges展开更多
This work presents the results of the exact computation of (180)<sup>3</sup> = 5,832,000 third-order mixed sensitivities of the leakage response of a polyethylene-reflected plutonium (PERP) experimental be...This work presents the results of the exact computation of (180)<sup>3</sup> = 5,832,000 third-order mixed sensitivities of the leakage response of a polyethylene-reflected plutonium (PERP) experimental benchmark with respect to the benchmark’s 180 microscopic total cross sections. This computation was made possible by applying the Third-Order Adjoint Sensitivity Analysis Methodology developed by Cacuci. The numerical results obtained in this work revealed that many of the 3<sup>rd</sup>-order sensitivities are significantly larger than their corresponding 1<sup>st</sup>- and 2<sup>nd</sup>-order ones, which is contrary to the widely held belief that higher-order sensitivities are all much smaller and hence less important than the first-order ones, for reactor physics systems. In particular, the largest 3<sup>rd</sup>-order relative sensitivity is the mixed sensitivity <img src="Edit_754b8437-dfdf-487d-af68-c78c637e6d4e.png" width="180" height="24" alt="" />of the PERP leakage response with respect to the lowest energy-group (30) total cross sections of <sup>1</sup>H (“isotope 6”) and <sup>239</sup>Pu (“isotope 1”). These two isotopes are shown in this work to be the two most important parameters affecting the PERP benchmark’s leakage response. By comparison, the largest 1<sup>st</sup>-order sensitivity is that of the PERP leakage response with respect to the lowest energy-group total cross section of isotope <sup>1</sup>H, having the value <img src="Edit_a5cfcc11-6a99-41ee-b844-a5ee84b454b3.png" width="100" height="24" alt="" />, while the largest 2<sup>nd</sup>-order sensitivity is <img src="Edit_05166a2b-97f7-43f1-98ff-b21368c00228.png" width="120" height="22" alt="" />. The 3<sup>rd</sup>-order sensitivity analysis presented in this work is the first ever such analysis in the field of reactor physics. The consequences of the results presented in this work on the uncertainty analysis of the PERP benchmark’s leakage response will be presented in a subsequent work.展开更多
This work presents the mathematical framework of the “Fifth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (5<sup>th</sup>-CASAM-N),” which generalizes and extends all...This work presents the mathematical framework of the “Fifth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (5<sup>th</sup>-CASAM-N),” which generalizes and extends all of the previous works performed to date on this subject. The 5<sup>th</sup>-CASAM-N enables the exact and efficient computation of all sensitivities, up to and including fifth-order, of model responses to uncertain model parameters and uncertain boundaries of the system’s domain of definition, thus enabling, inter alia, the quantification of uncertainties stemming from manufacturing tolerances. The 5<sup>th</sup>-CASAM-N provides a fundamental step towards overcoming the curse of dimensionality in sensitivity and uncertainty analysis.展开更多
This work extends to third-order previously published work on developing the adjoint sensitivity and uncertainty analysis of the numerical model of a <u>p</u>oly<u>e</u>thylene-<u>r</u...This work extends to third-order previously published work on developing the adjoint sensitivity and uncertainty analysis of the numerical model of a <u>p</u>oly<u>e</u>thylene-<u>r</u>eflected <u>p</u>lutonium (acronym: PERP) OECD/NEA reactor physics benchmark. The PERP benchmark comprises 21,976 imprecisely known (uncertain) model parameters. Previous works have used the adjoint sensitivity analysis methodology to compute exactly and efficiently all of the 21,976 first-order and (21,976)<sup>2</sup> second-order sensitivities of the PERP benchmark’s leakage response to all of the benchmark’s uncertain parameters, showing that the largest and most consequential 1<sup>st</sup>- and 2<sup>nd</sup>-order response sensitivities are with respect to the total microscopic cross sections. These results have motivated extending the previous adjoint-based derivations to third-order, leading to the derivation, in this work, of the exact mathematical expressions of the (180)<sup>3</sup> third-order sensitivities of the PERP leakage response with respect to these total microscopic cross sections. The formulas derived in this work are valid not only for the PERP benchmark but can also be used for computing the 3<sup>rd</sup>-order sensitivities of the leakage response of any nuclear system involving fissionable material and internal or external neutron sources. Subsequent works will use the adjoint-based mathematical expressions obtained in this work to compute exactly and efficiently the numerical values of these (180)<sup>3</sup> third-order sensitivities (which turned out to be very large and consequential) and use them for a third-order uncertainty analysis of the PERP benchmark’s leakage response.展开更多
A modified hydrophilic penta-bismuth hepta-oxide nitrate （Bi5O7NO3） surface was synthesized via a precipitation method using TiO2 and Ag as modified agents. The synthesized product was characterized by different ana...A modified hydrophilic penta-bismuth hepta-oxide nitrate （Bi5O7NO3） surface was synthesized via a precipitation method using TiO2 and Ag as modified agents. The synthesized product was characterized by different analytical techniques. The removal efficiency was evaluated using mono- and di-sulphonated azo dyes as model pollutants. Different kinetic, isotherm and diffusion models were chosen to describe the adsorption process. X-ray photoelectron spectroscopy （XPS） results revealed no noticeable differences in the chemical states of modified adsorbent when compared to pure BisOTNO3; however, the presence of hydrophilic centres such as TiO2 and Ag developed positively charged surface groups and improved its adsorption performance to a wide range of azo dyes. Dyes removal was found to be a function of adsorbent dosage, initial dye concentration, solution pH and temperature. The reduction of Langmuir 1,2-mixed order kinetics to the second or first-order kinetics could be successfully used to describe the adsorption of dyes onto the modified adsorbent. Mass transfer can be described by intra-particle diffusion at a certain stage, but it was not the rate limiting step that controlled the adsorption process. Homogenous behavior of adsorbent surface can be explored by applying Langmuir isotherm to fit the adsorption data.展开更多
基金supported by Liaoning Bai Qian Wan Talents Program of China(LRS2020[78])the Natural Science Foundation of Education Department of Liaoning Province of China(LJ2020002)the National Natural Science Foundation of China(11547005)
文摘The purpose of this paper is to report the feasibility of constructing high-order rogue waves with controllable fission and asymmetry for high-dimensional nonlinear evolution equations.Such a nonlinear model considered in this paper as the concrete example is the(3+1)-dimensional generalized Boussinesq(gB)equation,and the corresponding method is Zhaqilao’s symbolic computation approach containing two embedded parameters.It is indicated by the(3+1)-dimensional gB equation that the embedded parameters can not only control the center of the first-order rogue wave,but also control the number of the wave peaks split from higher-order rogue waves and the asymmetry of higher-order rogue waves about the coordinate axes.The main novelty of this paper is that the obtained results and findings can provide useful supplements to the method used and the controllability of higher-order rogue waves.
文摘Both numerical simulation and theoretical analysis of seawater intrusion in coastal regions are of great theoretical importance in environmental sciences. The mathematical model can be described as a problem of the initial boundary values for a system of 3-dimensional nonlinear parabolic partial differential equations, one being the pressure flow equation and the other is the concentration convection-dispersion equation of the salt contained. For a generic case of a 3-dimensional bounded region, a backward-difference time-stepping scheme is defined. It approximates the pressure by the standard Galerkin procedure and the concentration by a Galerkin method of charederistics, where calculus of variations, theory of prior estimates and techniques are made use of Optimal order estimates in H1 norm are derived for the errors in the approximate solution.
文摘The (180)<sup>3</sup> third-order mixed sensitivities of the leakage response of a polyethylene-reflected plutonium (PERP) experimental benchmark with respect to the benchmark’s 180 microscopic total cross sections have been computed in accompanying works [1] [2]. This work quantifies the contributions of these (180)<sup>3</sup> third-order mixed sensitivities to the PERP benchmark’s leakage response distribution moments (expected value, variance and skewness) and compares these contributions to those stemming from the corresponding first- and second-order sensitivities of the PERP benchmark’s leakage response with respect to the total cross sections. The numerical results obtained in this work reveal that the importance of the 3<sup>rd</sup>-order sensitivities can surpass the importance of the 1<sup>st</sup>- and 2<sup>nd</sup>-order sensitivities when the parameters’ uncertainties increase. In particular, for a uniform standard deviation of 10% of the microscopic total cross sections, the 3<sup>rd</sup>-order sensitivities contribute 80% to the response variance, whereas the contribution stemming from the 1st- and 2nd-order sensitivities amount only to 2% and 18%, respectively. Consequently, neglecting the 3<sup>rd</sup>-order sensitivities could cause a very large non-conservative error by under-reporting the response variance by a factor of 506%. The results obtained in this work also indicate that the effects of the 3<sup>rd</sup>-order sensitivities are to reduce the response’s skewness in parameter space, rendering the distribution of the leakage response more symmetric about its expected value. The results obtained in this work are the first such results ever published in reactor physics. Since correlations among the group-averaged microscopic total cross sections are not available, only the effects of typical standard deviations for these cross sections could be considered. Due to this lack of correlations among the cross sections, the effects of the <em>mixed</em> 3<sup>rd</sup>-order sensiti
文摘This work extends to fourth-order previously published work on developing the adjoint sensitivity and uncertainty analysis of the numerical model of a <u>p</u>oly<u>e</u>thylene-<u>r</u>eflected <u>p</u>lutonium (acronym: PERP) OECD/NEA reactor physics benchmark. The PERP benchmark comprises 7477 imprecisely known (uncertain) model parameters which have nonzero values. These parameters are as follows: 180 microscopic total cross sections;7101 microscopic scattering sections;60 microscopic fission cross sections;60 parameters that characterize the average number of neutrons per fission;60 parameters that characterize the fission spectrum;10 parameters that characterize the fission source;and 6 parameters that characterize the isotope number densities. Previous works have used the adjoint sensitivity analysis methodology to compute exactly and efficiently all of the 7477 first-order and 27,956,503 second-order sensitivities of the PERP benchmark’s leakage response to all of the benchmark’s uncertain parameters. These works showed that largest response sensitivities involve the total microscopic cross sections, which motivated the recent computation of all of the (180)<sup>3</sup> third-order sensitivities of the PERP leakage response with respect to these total microscopic cross sections. It turned out that some of these 3<sup>rd</sup>-order cross sections were far larger than the corresponding 2<sup>nd</sup>-order ones, thereby having the largest impact on the uncertainties induced in the PERP benchmark’s response. This finding has motivated the development of the original 4<sup>th</sup>-order formulas presented in this work, which are valid not only for the PERP benchmark but can also be used for computing the 4<sup>th</sup>-order sensitivities of response of any nuclear system involving fissionable material and internal or external neutron sources. Subsequent works will use the adjoint-based mathematical expressions obtained in this work to compute exactly and efficiently the numerical values of the larges
文摘This work presents the results of the exact computation of (180)<sup>3</sup> = 5,832,000 third-order mixed sensitivities of the leakage response of a polyethylene-reflected plutonium (PERP) experimental benchmark with respect to the benchmark’s 180 microscopic total cross sections. This computation was made possible by applying the Third-Order Adjoint Sensitivity Analysis Methodology developed by Cacuci. The numerical results obtained in this work revealed that many of the 3<sup>rd</sup>-order sensitivities are significantly larger than their corresponding 1<sup>st</sup>- and 2<sup>nd</sup>-order ones, which is contrary to the widely held belief that higher-order sensitivities are all much smaller and hence less important than the first-order ones, for reactor physics systems. In particular, the largest 3<sup>rd</sup>-order relative sensitivity is the mixed sensitivity <img src="Edit_754b8437-dfdf-487d-af68-c78c637e6d4e.png" width="180" height="24" alt="" />of the PERP leakage response with respect to the lowest energy-group (30) total cross sections of <sup>1</sup>H (“isotope 6”) and <sup>239</sup>Pu (“isotope 1”). These two isotopes are shown in this work to be the two most important parameters affecting the PERP benchmark’s leakage response. By comparison, the largest 1<sup>st</sup>-order sensitivity is that of the PERP leakage response with respect to the lowest energy-group total cross section of isotope <sup>1</sup>H, having the value <img src="Edit_a5cfcc11-6a99-41ee-b844-a5ee84b454b3.png" width="100" height="24" alt="" />, while the largest 2<sup>nd</sup>-order sensitivity is <img src="Edit_05166a2b-97f7-43f1-98ff-b21368c00228.png" width="120" height="22" alt="" />. The 3<sup>rd</sup>-order sensitivity analysis presented in this work is the first ever such analysis in the field of reactor physics. The consequences of the results presented in this work on the uncertainty analysis of the PERP benchmark’s leakage response will be presented in a subsequent work.
文摘This work presents the mathematical framework of the “Fifth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (5<sup>th</sup>-CASAM-N),” which generalizes and extends all of the previous works performed to date on this subject. The 5<sup>th</sup>-CASAM-N enables the exact and efficient computation of all sensitivities, up to and including fifth-order, of model responses to uncertain model parameters and uncertain boundaries of the system’s domain of definition, thus enabling, inter alia, the quantification of uncertainties stemming from manufacturing tolerances. The 5<sup>th</sup>-CASAM-N provides a fundamental step towards overcoming the curse of dimensionality in sensitivity and uncertainty analysis.
文摘This work extends to third-order previously published work on developing the adjoint sensitivity and uncertainty analysis of the numerical model of a <u>p</u>oly<u>e</u>thylene-<u>r</u>eflected <u>p</u>lutonium (acronym: PERP) OECD/NEA reactor physics benchmark. The PERP benchmark comprises 21,976 imprecisely known (uncertain) model parameters. Previous works have used the adjoint sensitivity analysis methodology to compute exactly and efficiently all of the 21,976 first-order and (21,976)<sup>2</sup> second-order sensitivities of the PERP benchmark’s leakage response to all of the benchmark’s uncertain parameters, showing that the largest and most consequential 1<sup>st</sup>- and 2<sup>nd</sup>-order response sensitivities are with respect to the total microscopic cross sections. These results have motivated extending the previous adjoint-based derivations to third-order, leading to the derivation, in this work, of the exact mathematical expressions of the (180)<sup>3</sup> third-order sensitivities of the PERP leakage response with respect to these total microscopic cross sections. The formulas derived in this work are valid not only for the PERP benchmark but can also be used for computing the 3<sup>rd</sup>-order sensitivities of the leakage response of any nuclear system involving fissionable material and internal or external neutron sources. Subsequent works will use the adjoint-based mathematical expressions obtained in this work to compute exactly and efficiently the numerical values of these (180)<sup>3</sup> third-order sensitivities (which turned out to be very large and consequential) and use them for a third-order uncertainty analysis of the PERP benchmark’s leakage response.
文摘A modified hydrophilic penta-bismuth hepta-oxide nitrate （Bi5O7NO3） surface was synthesized via a precipitation method using TiO2 and Ag as modified agents. The synthesized product was characterized by different analytical techniques. The removal efficiency was evaluated using mono- and di-sulphonated azo dyes as model pollutants. Different kinetic, isotherm and diffusion models were chosen to describe the adsorption process. X-ray photoelectron spectroscopy （XPS） results revealed no noticeable differences in the chemical states of modified adsorbent when compared to pure BisOTNO3; however, the presence of hydrophilic centres such as TiO2 and Ag developed positively charged surface groups and improved its adsorption performance to a wide range of azo dyes. Dyes removal was found to be a function of adsorbent dosage, initial dye concentration, solution pH and temperature. The reduction of Langmuir 1,2-mixed order kinetics to the second or first-order kinetics could be successfully used to describe the adsorption of dyes onto the modified adsorbent. Mass transfer can be described by intra-particle diffusion at a certain stage, but it was not the rate limiting step that controlled the adsorption process. Homogenous behavior of adsorbent surface can be explored by applying Langmuir isotherm to fit the adsorption data.
