场地-城市相互作用(site-city interaction,SCI)效应会显著改变场地地震波场分布及建筑反应,基于SCI效应理论计算研究方法的发展现状,发挥谱元(spectral element,SE)法可快速高效求解三维地震波场传播和多自由度(multi-degree of freedo...场地-城市相互作用(site-city interaction,SCI)效应会显著改变场地地震波场分布及建筑反应,基于SCI效应理论计算研究方法的发展现状,发挥谱元(spectral element,SE)法可快速高效求解三维地震波场传播和多自由度(multi-degree of freedom,MDOF)模型计算量小且可同时模拟大量建筑的优势,同时,结合频率波数域(frequency wave number analysis,FK)方法,以等效地震荷载的方式施加地震波场,建立了FK-SE-MDOF耦合方法,实现了SE-MDOF耦合模型中多种波型(P波、SV波和SH波)的斜入射输入,解决了当前三维SCI效应研究方法中未能同时考虑建筑非线性、频谱特性、地震波波型及入射角度影响的问题。首先对方法原理进行了介绍;然后,通过与振动台试验的对比,验证了方法的正确性;进而,采用该方法建立理想场地-城市建筑群相互作用耦合模型,主要探讨了入射角度和地震波波型对SCI效应的影响,得到了一些有益结论。该方法较为真实地反映SCI效应影响的同时,可反映建筑基础轮廓对地震波场的影响,适用于需考虑建筑轮廓信息的社区尺度SCI效应研究,可为城市规划、抗震设计、风险评估以及震后救援等工作提供定量指导。展开更多
Four key stress thresholds exist in the compression process of rocks,i.e.,crack closure stress(σ_(cc)),crack initiation stress(σ_(ci)),crack damage stress(σ_(cd))and compressive strength(σ_(c)).The quantitative id...Four key stress thresholds exist in the compression process of rocks,i.e.,crack closure stress(σ_(cc)),crack initiation stress(σ_(ci)),crack damage stress(σ_(cd))and compressive strength(σ_(c)).The quantitative identifications of the first three stress thresholds are of great significance for characterizing the microcrack growth and damage evolution of rocks under compression.In this paper,a new method based on damage constitutive model is proposed to quantitatively measure the stress thresholds of rocks.Firstly,two different damage constitutive models were constructed based on acoustic emission(AE)counts and Weibull distribution function considering the compaction stages of the rock and the bearing capacity of the damage element.Then,the accumulative AE counts method(ACLM),AE count rate method(CRM)and constitutive model method(CMM)were introduced to determine the stress thresholds of rocks.Finally,the stress thresholds of 9 different rocks were identified by ACLM,CRM,and CMM.The results show that the theoretical stress−strain curves obtained from the two damage constitutive models are in good agreement with that of the experimental data,and the differences between the two damage constitutive models mainly come from the evolutionary differences of the damage variables.The results of the stress thresholds identified by the CMM are in good agreement with those identified by the AE methods,i.e.,ACLM and CRM.Therefore,the proposed CMM can be used to determine the stress thresholds of rocks.展开更多
In order to study the dynamic responses in the microstructures of the pavement structure, the multi-scale modeling subjected to moving load is analyzed using the discrete element method (DEM). The macro-scale discre...In order to study the dynamic responses in the microstructures of the pavement structure, the multi-scale modeling subjected to moving load is analyzed using the discrete element method (DEM). The macro-scale discrete element model of the flexible pavement structure is established. The stress and strain at the bottom of the asphalt concrete layer under moving load are calculated. The DEM model is validated through comparison between DEM predictions and the results from the classical program. Based on the validated macro-scale DEM model, the distribution and the volumetric fraction of coarse aggregate, mastics and air voids at the bottom of the asphalt layer are modeled, and then the multi-scale model is constructed. The dynamic response in the microstructures of the multi-scale model are calculated and compared with the results from the macro model. The influence of mastic stiffness on the distribution of dynamic response in the microstructures is also analyzed. Results show that the average values and the variation coefficient of the tensile stress at the aggregate-mastic interface are far more than those within the mastics. The dynamic response including stress and strain distributes non-uniformly in both mastics and the interface. An increase in mastic stiffness tends to a uniform distribution of tensile stress in asphalt concrete.展开更多
The location of model errors in a stiffness matrix by using test data has been investigated by the others.The present paper deals with the problem of updating stiffness elements in the erroneous areas. Firstly,a model...The location of model errors in a stiffness matrix by using test data has been investigated by the others.The present paper deals with the problem of updating stiffness elements in the erroneous areas. Firstly,a model that bears relation to erroneous elements only is derived.This model is termed local errors model,which reduces orders and computational loads compared with global stiffness matrix. Secondly,an inverse eigenvalue method is used to update model errors. The results of a numerical experiment demonstrate that the method is quite effective.展开更多
Frequency-domain airborne electromagnetics is a proven geophysical exploration method.Presently,the interpretation is mainly based on resistivity-depth imaging and onedimensional layered inversion;nevertheless,it is d...Frequency-domain airborne electromagnetics is a proven geophysical exploration method.Presently,the interpretation is mainly based on resistivity-depth imaging and onedimensional layered inversion;nevertheless,it is difficult to obtain satisfactory results for two- or three-dimensional complex earth structures using 1D methods.3D forward modeling and inversion can be used but are hampered by computational limitations because of the large number of data.Thus,we developed a 2.5D frequency-domain airborne electromagnetic forward modeling and inversion algorithm.To eliminate the source singularities in the numerical simulations,we split the fields into primary and secondary fields.The primary fields are calculated using homogeneous or layered models with analytical solutions,and the secondary(scattered) fields are solved by the finite-element method.The linear system of equations is solved by using the large-scale sparse matrix parallel direct solver,which greatly improves the computational efficiency.The inversion algorithm was based on damping leastsquares and singular value decomposition and combined the pseudo forward modeling and reciprocity principle to compute the Jacobian matrix.Synthetic and field data were used to test the effectiveness of the proposed method.展开更多
The dynamic response of moored crane-ship is studied. Governing equations for the dynamic response of a crane-ship coupled with the pendulum motion of the payload are derived based on Lagrange’s equations. The boom i...The dynamic response of moored crane-ship is studied. Governing equations for the dynamic response of a crane-ship coupled with the pendulum motion of the payload are derived based on Lagrange’s equations. The boom is modeled based on finite element method, while the payload is modeled as a planar pendulum of point mass. The dynamic response was studied using numerical method. The calculation results show that the large-amplitude responses occur at wave periods near the natural period of the payload. Load swing angle is smaller for crane-ship with flexible boom, in comparison with rigid boom. The ship surge mo- tions have large vibrations for crane-ship with flexible boom, which were not observed for a rigid boom. The analysis identifies the significance of key parameters and reveals how the system design can be adjusted to avoid critical conditions.展开更多
The virtual laminated element method (VLEM) can resolve structural shap e optimization problems with a new method. According to the characteristics of V LEM , only some characterized layer thickness values need be def...The virtual laminated element method (VLEM) can resolve structural shap e optimization problems with a new method. According to the characteristics of V LEM , only some characterized layer thickness values need be defined as design v ariables instead of boundary node coordinates or some other parameters determini ng the system boundary. One of the important features of this method is that it is not necessary to regenerate the FE(finite element) grid during the optimizati on process so as to avoid optimization failures resulting from some distortion grid elements. Th e thickness distribution in thin plate optimization problems in other studies be fore is of stepped shape. However, in this paper, a continuous thickness distrib ution can be obtained after optimization using VLEM, and is more reasonable. Fur thermore, an approximate reanalysis method named ″behavior model technique″ ca n be used to reduce the amount of structural reanalysis. Some typical examples are offered to prove the effectiveness and practicality of the proposed method.展开更多
A three-dimensional finite element analysis of process-induced residual stress in resin transfer molding (RTM) process is presented. The finite element method (FEM) was employed to solve the coupled equations involved...A three-dimensional finite element analysis of process-induced residual stress in resin transfer molding (RTM) process is presented. The finite element method (FEM) was employed to solve the coupled equations involved in the transient heat transfer and the cure kinetics of the resin, and the distributions of internal temperature and cure degree of the composite at any instant time were obtained. The self-consistent field micro-mechanics model was used to predict the cure-dependent mechanical properties of the composites. Thermal expansion and cure shrinkage were included in the analysis. The thermo-elastic mechanical governing equations were solved using the incremental stress-strain relationship based FEM and the residual stress development was predicted. The present results were validated by the comparisons with the pertinent literature. The numerical example of a half cylinder was presented. The results show that it is necessary to carry out the three-dimensional analysis due to the complex distributions of temperatures, cure degrees and process-induced stress for thick parts, which can be predicted at any point within composite structures in the present analysis.展开更多
There are various analytical, empirical and numerical methods to calculate groundwater inflow into tun- nels excavated in rocky media. Analytical methods have been widely applied in prediction of groundwa- ter inflow ...There are various analytical, empirical and numerical methods to calculate groundwater inflow into tun- nels excavated in rocky media. Analytical methods have been widely applied in prediction of groundwa- ter inflow to tunnels due to their simplicity and practical base theory. Investigations show that the real amount of water infiltrating into jointed tunnels is much less than calculated amount using analytical methods and obtained results are very dependent on tunnel's geometry and environmental situations. In this study, using multiple regression analysis, a new empirical model for estimation of groundwater seepage into circular tunnels was introduced. Our data was acquired from field surveys and laboratory analysis of core samples. New regression variables were defined after perusing single and two variables relationship between groundwater seepage and other variables. Finally, an appropriate model for estima- tion of leakage was obtained using the stepwise algorithm. Statistics like R, R2, R2e and the histogram of residual values in the model represent a good reputation and fitness for this model to estimate the groundwater seepage into tunnels. The new experimental model was used for the test data and results were satisfactory. Therefore, multiple regression analysis is an effective and efficient way to estimate the groundwater seeoage into tunnels.展开更多
In this paper,three different modeling ranges were selected in the structural analysis for a hydropower house.The analysis was carried out using ABAQUS 6.6.The modeling range has a remarkable effect on finite element ...In this paper,three different modeling ranges were selected in the structural analysis for a hydropower house.The analysis was carried out using ABAQUS 6.6.The modeling range has a remarkable effect on finite element method(FEM) calculation result at the middle position of typical cross-sections where the concrete is relatively thin,and at the region close to turbine floor.If the ventilation barrel,floor slabs and columns above turbine floor are excluded from FEM model,the maximum rise difference of pedestal structure increases by about 24% compared with that of the whole model.It is indicated that different modeling ranges indeed affect FEM calculation result,and the structure above turbine floor in the FEM model should be included.展开更多
We consider a statically determinate structural truss problem where all of the physical model parameters are uncertain: not just the material values and applied loads, but also the positions of the nodes are assumed ...We consider a statically determinate structural truss problem where all of the physical model parameters are uncertain: not just the material values and applied loads, but also the positions of the nodes are assumed to be inexact but bounded and are represented by intervals. Such uncertainty may typically arise from imprecision during the process of manufacturing or construction, or round-off errors. In this case the application of the finite element method results in a system of linear equations with numerous interval parameters which cannot be solved conventionally. Applying a suitable variable substitution, an iteration method for the solution of a parametric system of linear equations is firstly employed to obtain initial bounds on the node displacements. Thereafter, an interval tightening (pruning) technique is applied, firstly on the element forces and secondly on the node displacements, in order to obtain tight guaranteed enclosures for the interval solutions for the forces and displacements.展开更多
In order to verify the validity of finite element numerical simulation method for asphalt mixture, which consists of aggregates, mastic (where mastic is a kind of fine mixture composed of asphalt binder mixed with fi...In order to verify the validity of finite element numerical simulation method for asphalt mixture, which consists of aggregates, mastic (where mastic is a kind of fine mixture composed of asphalt binder mixed with fines and fine aggregates) and air voids, based on three-dimensional (3D) heterogeneous specimen, X-ray computerized tomography (X-ray CT) was used to scan the asphalt specimens to obtain the real internal microstrnctures of asphalt mixture. CT images were reconstructed to build up 3D digital specimen, and the viscoelastic properties of mastic were described with Burgers model The uniaxial creep numerical simulations of three different levels of aggregate gradation were conducted. The simulation results agree well with the experimental results. The numerical simulation of asphalt mixture incorporated with real 3D microstructure based on finite element method is a promising application to conduct research of asphalt concrete. Additionally, this method can increase the mechanistic understanding of global viscoelastic properties of asphalt mixtures by linking the real 3D microstructure.展开更多
Recently, solutions to inverse problems have been required in various engineering fields. The neural network inversion method has been studied as one of the neural network-based solutions. On the other hand, the exten...Recently, solutions to inverse problems have been required in various engineering fields. The neural network inversion method has been studied as one of the neural network-based solutions. On the other hand, the extension of the neural network to a higher-dimensional domain, e.g., complex-value or quaternion, has been proposed, and a number of higher-dimensional neural network models have been proposed. Using the quatemion, we have the advantage of expressing 3D (three-dimensional) object attitudes easily. In the quaternion domain, we can define inverse problems where the cause and the result are expressed by the quaternion. In this paper, we extend the neural network inversion method to the quatemion domain. Further, we provide the results of the computer experiments to demonstrate the process and effectiveness of our method.展开更多
文摘场地-城市相互作用(site-city interaction,SCI)效应会显著改变场地地震波场分布及建筑反应,基于SCI效应理论计算研究方法的发展现状,发挥谱元(spectral element,SE)法可快速高效求解三维地震波场传播和多自由度(multi-degree of freedom,MDOF)模型计算量小且可同时模拟大量建筑的优势,同时,结合频率波数域(frequency wave number analysis,FK)方法,以等效地震荷载的方式施加地震波场,建立了FK-SE-MDOF耦合方法,实现了SE-MDOF耦合模型中多种波型(P波、SV波和SH波)的斜入射输入,解决了当前三维SCI效应研究方法中未能同时考虑建筑非线性、频谱特性、地震波波型及入射角度影响的问题。首先对方法原理进行了介绍;然后,通过与振动台试验的对比,验证了方法的正确性;进而,采用该方法建立理想场地-城市建筑群相互作用耦合模型,主要探讨了入射角度和地震波波型对SCI效应的影响,得到了一些有益结论。该方法较为真实地反映SCI效应影响的同时,可反映建筑基础轮廓对地震波场的影响,适用于需考虑建筑轮廓信息的社区尺度SCI效应研究,可为城市规划、抗震设计、风险评估以及震后救援等工作提供定量指导。
基金Projects(2021RC3007,2020RC3090)supported by the Science and Technology Innovation Program of Hunan Province,ChinaProjects(52374150,52174099)supported by the National Natural Science Foundation of China。
文摘Four key stress thresholds exist in the compression process of rocks,i.e.,crack closure stress(σ_(cc)),crack initiation stress(σ_(ci)),crack damage stress(σ_(cd))and compressive strength(σ_(c)).The quantitative identifications of the first three stress thresholds are of great significance for characterizing the microcrack growth and damage evolution of rocks under compression.In this paper,a new method based on damage constitutive model is proposed to quantitatively measure the stress thresholds of rocks.Firstly,two different damage constitutive models were constructed based on acoustic emission(AE)counts and Weibull distribution function considering the compaction stages of the rock and the bearing capacity of the damage element.Then,the accumulative AE counts method(ACLM),AE count rate method(CRM)and constitutive model method(CMM)were introduced to determine the stress thresholds of rocks.Finally,the stress thresholds of 9 different rocks were identified by ACLM,CRM,and CMM.The results show that the theoretical stress−strain curves obtained from the two damage constitutive models are in good agreement with that of the experimental data,and the differences between the two damage constitutive models mainly come from the evolutionary differences of the damage variables.The results of the stress thresholds identified by the CMM are in good agreement with those identified by the AE methods,i.e.,ACLM and CRM.Therefore,the proposed CMM can be used to determine the stress thresholds of rocks.
基金The National Natural Science Foundation of China (No.51208178,51108157)China Postdoctoral Science Foundation (No.2012M520991)
文摘In order to study the dynamic responses in the microstructures of the pavement structure, the multi-scale modeling subjected to moving load is analyzed using the discrete element method (DEM). The macro-scale discrete element model of the flexible pavement structure is established. The stress and strain at the bottom of the asphalt concrete layer under moving load are calculated. The DEM model is validated through comparison between DEM predictions and the results from the classical program. Based on the validated macro-scale DEM model, the distribution and the volumetric fraction of coarse aggregate, mastics and air voids at the bottom of the asphalt layer are modeled, and then the multi-scale model is constructed. The dynamic response in the microstructures of the multi-scale model are calculated and compared with the results from the macro model. The influence of mastic stiffness on the distribution of dynamic response in the microstructures is also analyzed. Results show that the average values and the variation coefficient of the tensile stress at the aggregate-mastic interface are far more than those within the mastics. The dynamic response including stress and strain distributes non-uniformly in both mastics and the interface. An increase in mastic stiffness tends to a uniform distribution of tensile stress in asphalt concrete.
文摘The location of model errors in a stiffness matrix by using test data has been investigated by the others.The present paper deals with the problem of updating stiffness elements in the erroneous areas. Firstly,a model that bears relation to erroneous elements only is derived.This model is termed local errors model,which reduces orders and computational loads compared with global stiffness matrix. Secondly,an inverse eigenvalue method is used to update model errors. The results of a numerical experiment demonstrate that the method is quite effective.
基金supported by the Doctoral Fund Project of the Ministry of Education(No.20130061110060 class tutors)the National Natural Science Foundation of China(No.41504083)National Basic Research Program of China(973Program)(No.2013CB429805)
文摘Frequency-domain airborne electromagnetics is a proven geophysical exploration method.Presently,the interpretation is mainly based on resistivity-depth imaging and onedimensional layered inversion;nevertheless,it is difficult to obtain satisfactory results for two- or three-dimensional complex earth structures using 1D methods.3D forward modeling and inversion can be used but are hampered by computational limitations because of the large number of data.Thus,we developed a 2.5D frequency-domain airborne electromagnetic forward modeling and inversion algorithm.To eliminate the source singularities in the numerical simulations,we split the fields into primary and secondary fields.The primary fields are calculated using homogeneous or layered models with analytical solutions,and the secondary(scattered) fields are solved by the finite-element method.The linear system of equations is solved by using the large-scale sparse matrix parallel direct solver,which greatly improves the computational efficiency.The inversion algorithm was based on damping leastsquares and singular value decomposition and combined the pseudo forward modeling and reciprocity principle to compute the Jacobian matrix.Synthetic and field data were used to test the effectiveness of the proposed method.
基金Project supported by the National Natural Science Foundation of China (No. 50675077)the Research Fund for the Doctoral Program of Higher Education of China (No. 20050487047)
文摘The dynamic response of moored crane-ship is studied. Governing equations for the dynamic response of a crane-ship coupled with the pendulum motion of the payload are derived based on Lagrange’s equations. The boom is modeled based on finite element method, while the payload is modeled as a planar pendulum of point mass. The dynamic response was studied using numerical method. The calculation results show that the large-amplitude responses occur at wave periods near the natural period of the payload. Load swing angle is smaller for crane-ship with flexible boom, in comparison with rigid boom. The ship surge mo- tions have large vibrations for crane-ship with flexible boom, which were not observed for a rigid boom. The analysis identifies the significance of key parameters and reveals how the system design can be adjusted to avoid critical conditions.
文摘The virtual laminated element method (VLEM) can resolve structural shap e optimization problems with a new method. According to the characteristics of V LEM , only some characterized layer thickness values need be defined as design v ariables instead of boundary node coordinates or some other parameters determini ng the system boundary. One of the important features of this method is that it is not necessary to regenerate the FE(finite element) grid during the optimizati on process so as to avoid optimization failures resulting from some distortion grid elements. Th e thickness distribution in thin plate optimization problems in other studies be fore is of stepped shape. However, in this paper, a continuous thickness distrib ution can be obtained after optimization using VLEM, and is more reasonable. Fur thermore, an approximate reanalysis method named ″behavior model technique″ ca n be used to reduce the amount of structural reanalysis. Some typical examples are offered to prove the effectiveness and practicality of the proposed method.
基金the National Natural Science Foundation of China(Grant No.10502016)Development Program for Outstanding Young Teachers in Harbin Institute of Technology(Grant No.HITQNJS.2006.020)
文摘A three-dimensional finite element analysis of process-induced residual stress in resin transfer molding (RTM) process is presented. The finite element method (FEM) was employed to solve the coupled equations involved in the transient heat transfer and the cure kinetics of the resin, and the distributions of internal temperature and cure degree of the composite at any instant time were obtained. The self-consistent field micro-mechanics model was used to predict the cure-dependent mechanical properties of the composites. Thermal expansion and cure shrinkage were included in the analysis. The thermo-elastic mechanical governing equations were solved using the incremental stress-strain relationship based FEM and the residual stress development was predicted. The present results were validated by the comparisons with the pertinent literature. The numerical example of a half cylinder was presented. The results show that it is necessary to carry out the three-dimensional analysis due to the complex distributions of temperatures, cure degrees and process-induced stress for thick parts, which can be predicted at any point within composite structures in the present analysis.
文摘There are various analytical, empirical and numerical methods to calculate groundwater inflow into tun- nels excavated in rocky media. Analytical methods have been widely applied in prediction of groundwa- ter inflow to tunnels due to their simplicity and practical base theory. Investigations show that the real amount of water infiltrating into jointed tunnels is much less than calculated amount using analytical methods and obtained results are very dependent on tunnel's geometry and environmental situations. In this study, using multiple regression analysis, a new empirical model for estimation of groundwater seepage into circular tunnels was introduced. Our data was acquired from field surveys and laboratory analysis of core samples. New regression variables were defined after perusing single and two variables relationship between groundwater seepage and other variables. Finally, an appropriate model for estima- tion of leakage was obtained using the stepwise algorithm. Statistics like R, R2, R2e and the histogram of residual values in the model represent a good reputation and fitness for this model to estimate the groundwater seepage into tunnels. The new experimental model was used for the test data and results were satisfactory. Therefore, multiple regression analysis is an effective and efficient way to estimate the groundwater seeoage into tunnels.
基金Supported by National Natural Science Foundation of China(No.50539010)
文摘In this paper,three different modeling ranges were selected in the structural analysis for a hydropower house.The analysis was carried out using ABAQUS 6.6.The modeling range has a remarkable effect on finite element method(FEM) calculation result at the middle position of typical cross-sections where the concrete is relatively thin,and at the region close to turbine floor.If the ventilation barrel,floor slabs and columns above turbine floor are excluded from FEM model,the maximum rise difference of pedestal structure increases by about 24% compared with that of the whole model.It is indicated that different modeling ranges indeed affect FEM calculation result,and the structure above turbine floor in the FEM model should be included.
文摘We consider a statically determinate structural truss problem where all of the physical model parameters are uncertain: not just the material values and applied loads, but also the positions of the nodes are assumed to be inexact but bounded and are represented by intervals. Such uncertainty may typically arise from imprecision during the process of manufacturing or construction, or round-off errors. In this case the application of the finite element method results in a system of linear equations with numerous interval parameters which cannot be solved conventionally. Applying a suitable variable substitution, an iteration method for the solution of a parametric system of linear equations is firstly employed to obtain initial bounds on the node displacements. Thereafter, an interval tightening (pruning) technique is applied, firstly on the element forces and secondly on the node displacements, in order to obtain tight guaranteed enclosures for the interval solutions for the forces and displacements.
基金Project(51038004) supported by the National Natural Science Foundation of China
文摘In order to verify the validity of finite element numerical simulation method for asphalt mixture, which consists of aggregates, mastic (where mastic is a kind of fine mixture composed of asphalt binder mixed with fines and fine aggregates) and air voids, based on three-dimensional (3D) heterogeneous specimen, X-ray computerized tomography (X-ray CT) was used to scan the asphalt specimens to obtain the real internal microstrnctures of asphalt mixture. CT images were reconstructed to build up 3D digital specimen, and the viscoelastic properties of mastic were described with Burgers model The uniaxial creep numerical simulations of three different levels of aggregate gradation were conducted. The simulation results agree well with the experimental results. The numerical simulation of asphalt mixture incorporated with real 3D microstructure based on finite element method is a promising application to conduct research of asphalt concrete. Additionally, this method can increase the mechanistic understanding of global viscoelastic properties of asphalt mixtures by linking the real 3D microstructure.
文摘Recently, solutions to inverse problems have been required in various engineering fields. The neural network inversion method has been studied as one of the neural network-based solutions. On the other hand, the extension of the neural network to a higher-dimensional domain, e.g., complex-value or quaternion, has been proposed, and a number of higher-dimensional neural network models have been proposed. Using the quatemion, we have the advantage of expressing 3D (three-dimensional) object attitudes easily. In the quaternion domain, we can define inverse problems where the cause and the result are expressed by the quaternion. In this paper, we extend the neural network inversion method to the quatemion domain. Further, we provide the results of the computer experiments to demonstrate the process and effectiveness of our method.