Positive results are proved here about the ability of balanced methods to reproduce the mean square stability of the impulsive stochastic differential equations. It is shown that the balanced methods with strong conve...Positive results are proved here about the ability of balanced methods to reproduce the mean square stability of the impulsive stochastic differential equations. It is shown that the balanced methods with strong convergence can preserve the mean square stability with the sufficiently small stepsize. Weak variants and their mean square stability are also considered. Several numerical experiments are given for illustration and show that the fully implicit methods are superior to those of the explicit methods in terms of mean-square stabilities for relatively large stepsizes especially.展开更多
A class of general modified split-step balanced methods proposed in the paper can be applied to solve stiff stochastic differential systems with m-dimensional multiplicative noise. Compared to some other already repor...A class of general modified split-step balanced methods proposed in the paper can be applied to solve stiff stochastic differential systems with m-dimensional multiplicative noise. Compared to some other already reported split-step balanced methods, the drift increment function of the methods can be taken from any chosen ane-step ordinary differential equations （ODEs） solver. The schemes is proved to be strong convergent with order one. For the mean-square stability analysis, the investigation is confined to two cases. Some numerical experiments are reported to testify the performance and the effectiveness of the methods.展开更多
This paper deals with numerical solutions of nonlinear stiff stochastic differential equations with jump-diffusion and piecewise continuous arguments.By combining compensated split-step methods and balanced methods,a ...This paper deals with numerical solutions of nonlinear stiff stochastic differential equations with jump-diffusion and piecewise continuous arguments.By combining compensated split-step methods and balanced methods,a class of compensated split-step balanced(CSSB)methods are suggested for solving the equations.Based on the one-sided Lipschitz condition and local Lipschitz condition,a strong convergence criterion of CSSB methods is derived.It is proved under some suitable conditions that the numerical solutions produced by CSSB methods can preserve the mean-square exponential stability of the corresponding analytical solutions.Several numerical examples are presented to illustrate the obtained theoretical results and the effectiveness of CSSB methods.Moreover,in order to show the computational advantage of CSSB methods,we also give a numerical comparison with the adapted split-step backward Euler methods with or without compensation and tamed explicit methods.展开更多
A vibro-impact system is a hot topic in the study on nonlinear dynamics due to its generality and importance in engineering.In general,the alternating frequency-time harmonic balance(AFT-HB)method can be used to solve...A vibro-impact system is a hot topic in the study on nonlinear dynamics due to its generality and importance in engineering.In general,the alternating frequency-time harmonic balance(AFT-HB)method can be used to solve elastic collision.However,since the system is non-smooth,the required Fourier/harmonic truncation order is high in order to achieve the theoretical convergence rate,resulting in expensive computational cost.Furthermore,for rigid body collision,the periodic response of the system cannot be solved with the AFT-HB method due to the discontinuous velocity of the system.In order to accelerate the convergence and solve highly non-smooth systems,an enriched harmonic balance(HB)method is proposed,which is derived from the AFT-HB method in the framework of event-driven Gauss quadrature.The basic idea is to augment the Fourier bases by introducing a non-smooth Bernoulli base such that the non-smooth Bernoulli base compensates for the non-smooth part of the solution and the smooth part of the solution is approximated by the Fourier bases,thus achieving accelerated convergence.Based on the enriched HB method,gear pair systems with gear backlash and oscillator systems with rigid impact are solved,and the dynamic response characteristics are analyzed in this work.Then,based on the Floquet theory,the event-driven monodromy matrix method for non-smooth systems is used to analyze the stability and bifurcation of the periodic solutions.The numerical example shows that the results obtained from the enriched HB method are consistent with those from the Runge-Kutta method,which proves that the presented method is an effective method for analyzing the dynamic response characteristic of the vibro-impact system.展开更多
Cantilever casting concrete arch bridge using form traveller has a broad application prospect.However,it is difficult to obtain reasonable initial cable force in construction stage.In this study,stress balance and inf...Cantilever casting concrete arch bridge using form traveller has a broad application prospect.However,it is difficult to obtain reasonable initial cable force in construction stage.In this study,stress balance and influence matrix methods were developed to determine the initial cable force of cantilever casting concrete arch bridge.The stress balance equation and influence matrix of arch rib critical section were established,and the buckle cable force range was determined by the allowable stress of arch rib critical section.Then a group of buckle cable forces were selected and substituted into the stress balance equation,and the reasonable initial buckle cable force was determined through iteration.Based on the principle of force balance,the initial anchor cable force was determined.In an engineering application example,it is shown that the stress balance and influence matrix methods for the determination of initial cable force are feasible and reliable.The initial cable forces of arch rib segments only need to be adjusted once in the corresponding construction process,which improves the working efficiency and reduces the construction risk.It is found that the methods have great advantages for determining initial cable force in cantilever casting construction process of concrete arch bridge.展开更多
Comparisons of the common methods for obtaining the periodic responses show that the harmonic balance method with alternating frequency/time （HB-AFT） do- main technique has some advantages in dealing with nonlinear ...Comparisons of the common methods for obtaining the periodic responses show that the harmonic balance method with alternating frequency/time （HB-AFT） do- main technique has some advantages in dealing with nonlinear problems of fractional exponential models. By the HB-AFT method, a rigid rotor supported by ball bearings with nonlinearity of Hertz contact and ball passage vibrations is considered. With the aid of the Floquet theory, the movement characteristics of interval stability are deeply studied. Besides, a simple strategy to determine the monodromy matrix is proposed for the stability analysis.展开更多
We have deduced incremental harmonic balance an iteration scheme in the （IHB） method using the harmonic balance plus the Newton-Raphson method. Since the convergence of the iteration is dependent upon the initial va...We have deduced incremental harmonic balance an iteration scheme in the （IHB） method using the harmonic balance plus the Newton-Raphson method. Since the convergence of the iteration is dependent upon the initial values in the iteration, the convergent region is greatly restricted for some cases. In this contribution, in order to enlarge the convergent region of the IHB method, we constructed the zeroth-order deformation equation using the homotopy analysis method, in which the IHB method is employed to solve the deformation equation with an embedding parameter as the active increment. Taking the Duffing and the van der Pol equations as examples, we obtained the highly accurate solutions. Importantly, the presented approach renders a convenient way to control and adjust the convergence.展开更多
In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the （1＋1）-dimensional KdV equation. Some exact travelling wave solu- tions with parameters have been obtained, which...In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the （1＋1）-dimensional KdV equation. Some exact travelling wave solu- tions with parameters have been obtained, which cover the existing solutions. Compared to other methods, the presented method is more direct, more concise, more effective, and easier for calculations. In addition, it can be used to solve other nonlinear evolution equations in mathematical physics.展开更多
A modified homogeneous balance method is proposed by improving some key steps in the homogeneousbalance method.Bilinear equations of some nonlinear evolution equations are derived by using the modified homogeneousbala...A modified homogeneous balance method is proposed by improving some key steps in the homogeneousbalance method.Bilinear equations of some nonlinear evolution equations are derived by using the modified homogeneousbalance method.Generalized Boussinesq equation,KP equation,and mKdV equation are chosen as examples to illustrateour method.This approach is also applicable to a large variety of nonlinear evolution equations.展开更多
We applied the material balance principle of the denudation volume and sedimentary flux to study the denudation-accumulation system between the Longmen Mountains （Mts.） and the foreland basin. The amount of sediment...We applied the material balance principle of the denudation volume and sedimentary flux to study the denudation-accumulation system between the Longmen Mountains （Mts.） and the foreland basin. The amount of sediment in each sedimentation stage of the basin was estimated to obtain the denudation volume, erosion thickness and deposit thickness since the Late Triassic Epoch, to enable us to recover the paleoelevation of the provenance and the sedimentary area. The results show the following： （1） Since the Late Triassic Epoch, the elevation of the surface of the Longmen Mts. has uplifted from 0 m to 2751 m, and the crust of the Longmen Mts. has uplifted by 9.8 km. Approximately 72% of the materials introduced have been denuded from the mountains. （2） It is difficult to recover the paleoelevation of each stage of the Longmen Mts. foreland basin quantitatively by the present-day techniques and data. （3） The formation of the Longmen Mts. foreland basin consisted of three stages of thrust belt tectonic load and three stages of thrust belt erosional unload. During tectonic loading stages （Late Triassic Epoch, Late Jurassic-Early Cretaceous, Late Cretaceous-Miocene）, the average elevation of Longmen Mts. was lower （approximately 700-1700 m）. During erosional unloading stages （Early and Middle Jurassic, Middle Cretaceous and Jiaguan, Late Cenozoic）, the average elevation of Longmen Mts. was high at approximately 2000-2800m.展开更多
This paper revisits the Space-Time Gradient(STG) method which was developed for efficient analysis of unsteady flows due to rotor–stator interaction and presents the method from an alternative time-clocking perspecti...This paper revisits the Space-Time Gradient(STG) method which was developed for efficient analysis of unsteady flows due to rotor–stator interaction and presents the method from an alternative time-clocking perspective. The STG method requires reordering of blade passages according to their relative clocking positions with respect to blades of an adjacent blade row. As the space-clocking is linked to an equivalent time-clocking, the passage reordering can be performed according to the alternative time-clocking. With the time-clocking perspective, unsteady flow solutions from different passages of the same blade row are mapped to flow solutions of the same passage at different time instants or phase angles. Accordingly, the time derivative of the unsteady flow equation is discretized in time directly, which is more natural than transforming the time derivative to a spatial one as with the original STG method. To improve the solution accuracy, a ninth order difference scheme has been investigated for discretizing the time derivative. To achieve a stable solution for the high order scheme, the implicit solution method of Lower-Upper Symmetric GaussSeidel/Gauss-Seidel(LU-SGS/GS) has been employed. The NASA Stage 35 and its blade-countreduced variant are used to demonstrate the validity of the time-clocking based passage reordering and the advantages of the high order difference scheme for the STG method. Results from an existing harmonic balance flow solver are also provided to contrast the two methods in terms of solution stability and computational cost.展开更多
The incremental harmonic balance method was extended to analyze the flutter of systems with multiple structural strong nonlinearities. The strongly nonlinear cubic plunging and pitching stiffness terms were considered...The incremental harmonic balance method was extended to analyze the flutter of systems with multiple structural strong nonlinearities. The strongly nonlinear cubic plunging and pitching stiffness terms were considered in the flutter equations of two-dimensional airfoil. First, the equations were transferred into matrix form, then the vibration process was divided into the persistent incremental processes of vibration moments. And the expression of their solutions could be obtained by using a certain amplitude as control parameter in the harmonic balance process, and then the bifurcation, limit cycle flutter phenomena and the number of harmonic terms were analyzed. Finally, numerical results calculated by the Runge-Kutta method were given to verify the results obtained by the proposed procedure. It has been shown that the incremental harmonic method is effective and precise in the analysis of strongly nonlinear flutter with multiple structural nonlinearities.展开更多
The dynamical properties of fractional-order Duffing–van der Pol oscillator are studied, and the amplitude–frequency response equation of primary resonance is obtained by the harmonic balance method. The stability c...The dynamical properties of fractional-order Duffing–van der Pol oscillator are studied, and the amplitude–frequency response equation of primary resonance is obtained by the harmonic balance method. The stability condition for steady-state solution is obtained based on Lyapunov theory. The comparison of the approximate analytical results with the numerical results is fulfilled, and the approximations obtained are in good agreement with the numerical solutions. The bifurcations of primary resonance for system parameters are analyzed. The results show that the harmonic balance method is effective and convenient for solving this problem, and it provides a reference for the dynamical analysis of similar nonlinear systems.展开更多
Based on the homogenous balance method and with the help of mathematica, the Backlund transformation and the transfer heat equation are derived. Analyzing the heat-transfer equation, the multiple soliton solutions and...Based on the homogenous balance method and with the help of mathematica, the Backlund transformation and the transfer heat equation are derived. Analyzing the heat-transfer equation, the multiple soliton solutions and other exact analytical solution for Whitham-Broer-Kaup equations(WBK) are derived. These solutions contain Fan's, Xie's and Yan's results and other new types of analytical solutions, such as rational function solutions and periodic solutions. The method can also be applied to solve more nonlinear differential equations.展开更多
A new modulation approach was presented for the control of neutral-point （NP） voltage variation in the three-level NP-clamped voltage source inverter, and the average NP current model was established based on vector...A new modulation approach was presented for the control of neutral-point （NP） voltage variation in the three-level NP-clamped voltage source inverter, and the average NP current model was established based on vector diagram partition. Thus, theory base was built for balancing control of NP potential. Theoretical analysis and experimental results indicate that the proposed method for NP balancing control vector synthe- sizing concept based can make the average NP current zero, and do not influence NP potential within every sample period. The effectiveness of proposed research approach was verified by simulative and experimental results.展开更多
The combined finiteediscrete element method （FDEM） belongs to a family of methods of computationalmechanics of discontinua. The method is suitable for problems of discontinua, where particles aredeformable and can f...The combined finiteediscrete element method （FDEM） belongs to a family of methods of computationalmechanics of discontinua. The method is suitable for problems of discontinua, where particles aredeformable and can fracture or fragment. The applications of FDEM have spread over a number of disciplinesincluding rock mechanics, where problems like mining, mineral processing or rock blasting canbe solved by employing FDEM. In this work, a novel approach for the parallelization of two-dimensional（2D） FDEM aiming at clusters and desktop computers is developed. Dynamic domain decompositionbased parallelization solvers covering all aspects of FDEM have been developed. These have beenimplemented into the open source Y2D software package and have been tested on a PC cluster. Theoverall performance and scalability of the parallel code have been studied using numerical examples. Theresults obtained confirm the suitability of the parallel implementation for solving large scale problems. 2014 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting byElsevier B.V. All rights reserved.展开更多
The （G＇/G, 1/G）-expansion method for finding exact travelling wave solutions of nonlinear evolution equations, which can be thought of as an extension of the （G＇/G）-expansion method proposed recently, is present...The （G＇/G, 1/G）-expansion method for finding exact travelling wave solutions of nonlinear evolution equations, which can be thought of as an extension of the （G＇/G）-expansion method proposed recently, is presented. By using this method abundant travelling wave so- lutions with arbitrary parameters of the Zakharov equations are successfully obtained. When the parameters are replaced by special values, the well-known solitary wave solutions of the equations are rediscovered from the travelling waves.展开更多
It is difficult to determine the effective reservoir rock compressibility,aquifer volume and water influx of high pressure and ultra-high pressure gas reservoirs,so when the traditional apparent reservoir pressure and...It is difficult to determine the effective reservoir rock compressibility,aquifer volume and water influx of high pressure and ultra-high pressure gas reservoirs,so when the traditional apparent reservoir pressure and cumulative gas production curve extrapolation method and its modified version are used to calculate the initial gas in-place of such gas reservoirs,the calculation accuracy is lower.The material balance equation in the form of power function for such gas reservoirs was established based on Gonzalez method to improve the accuracy and reliability of reserve evaluation.Then,based on 20 high pressure and ultra-high pressure reservoirs that had been developed abroad,the empirical value of the power exponent was defined,and the influences of the depletion degree of apparent reservoir pressure and the degree of reserve recovery on the reliability of reserve calculation were analyzed.Besides,the critical value of the key parameter affecting the reliability of reserve evaluation(depletion degree of apparent reservoir pressure)was determined and compared with the critical value of the two-linear trends.Finally,an example calculation was carried out.And the following research results were obtained.First,the empirical value of the power exponent calculated using the material balance method in the form of power function is 1.02847,with an upper limit of 1.11567.Second,the depletion degree of apparent reservoir pressure corresponding to the inflection point of the classical two-linear trends ranges from 0.14 to 0.38 with an average of 0.23,while that corresponding to the extrapolation point of the second straight line is between 0.23 and 0.50 with an average of 0.33,and the corresponding degree of reserve recovery is in the range of 33%-65%with an average of 45%.Third,the reserves of such high pressure gas reservoirs calculated with this new method has an error rate less than 10%if the depletion degree of apparent reservoir pressure is greater than 0.33.In conclusion,for such high pressure gas reservoirs and the stress-sensitive fractured gas reservoirs,the material balance method in the form of power function proposed in this paper can avoid the uncertainty parameters(e.g.effective reservoir rock compressibility,aquifer volume and water influx)and it is advantageous with simple calculation process,better practicability and small error.展开更多
The harmonic balance method(HBM)is one of the most widely used methods in solving nonlinear vibration problems,and its accuracy and computational efficiency largely depend on the number of the harmonics selected.The a...The harmonic balance method(HBM)is one of the most widely used methods in solving nonlinear vibration problems,and its accuracy and computational efficiency largely depend on the number of the harmonics selected.The adaptive harmonic balance(AHB)method is an improved HBM method.This paper presents a modified AHB method with the asymptotic harmonic selection(AHS)procedure.This new harmonic selection procedure selects harmonics from the frequency spectra of nonlinear terms instead of estimating the contribution of each harmonic to the whole nonlinear response,by which the additional calculation is avoided.A modified continuation method is proposed to deal with the variable size of nonlinear algebraic equations at different values of path parameters,and then all solution branches of the amplitude-frequency response are obtained.Numerical experiments are carried out to verify the performance of the AHB-AHS method.Five typical nonlinear dynamic equations with different types of nonlinearities and excitations are chosen as the illustrative examples.Compared with the classical HBM and Runge-Kutta methods,the proposed AHB-AHS method is of higher accuracy and better convergence.The AHB-AHS method proposed in this paper has the potential to investigate the nonlinear vibrations of complex high-dimensional nonlinear systems.展开更多
Dry friction damping structures are widely-used in aero-engines to mitigate vibration.The nonlinear nature of friction and the two-dimensional in-plane motion on the contact interface bring challenges to accurately an...Dry friction damping structures are widely-used in aero-engines to mitigate vibration.The nonlinear nature of friction and the two-dimensional in-plane motion on the contact interface bring challenges to accurately and efficiently predict the forced response of frictionally damped structures.The state-of-the-art Multi-Harmonic Balance Method(MHBM)on quasi-3D contact model in engineering cannot precisely capture the kinematics on the friction interface although the efficiency is high.The full-3D contact model can describe the constitutive relationship of the interface in a more accurate manner;however,the efficiency and convergence are not guaranteed for large-scale models.In this paper,a semi-analytical MHBM on full-3D contact model is proposed.The original Trajectory Tracking Method(TTM)for evaluating the contact force is reformulated to make the calculation more concise and the derivation of the Analytical Jacobian Matrix(AJM)feasible.Based on the chain rule of derivation,the AJM which is the core to upgrade the performance is deduced.Through a shrouded blade finite element model,the accuracy and efficiency of the proposed method are compared with both the MHBM on full-3D contact model with numerical Jacobian matrix and the MHBM on quasi-3D contact model with AJM.The results show that the AJM improves significantly the efficiency of the MHBM on full-3D contact model.The time cost of the proposed method is in the same order of magnitude as that of the MHBM on quasi-3D contact model.We also confirm that the full-3D contact model is necessary for the dynamic analyses of shrouded blades.If one uses the quasi-3D model,the estimation relative error of damping can even reach 31.8%in some cases.In addition,the AJM also brings benefits for stability analysis.It is highly recommended that engineers use the MHBM on full-3D contact model for the dynamic analysis and design of shrouded blades.展开更多
基金National Natural Science Foundations of China(Nos.11561028,11101101,11461032,11401267)Natural Science Foundations of Jiangxi Province,China(Nos.20151BAB201013,20151BAB201010,20151BAB201015)
文摘Positive results are proved here about the ability of balanced methods to reproduce the mean square stability of the impulsive stochastic differential equations. It is shown that the balanced methods with strong convergence can preserve the mean square stability with the sufficiently small stepsize. Weak variants and their mean square stability are also considered. Several numerical experiments are given for illustration and show that the fully implicit methods are superior to those of the explicit methods in terms of mean-square stabilities for relatively large stepsizes especially.
基金National Natural Science Foundation of China(No.11171352)
文摘A class of general modified split-step balanced methods proposed in the paper can be applied to solve stiff stochastic differential systems with m-dimensional multiplicative noise. Compared to some other already reported split-step balanced methods, the drift increment function of the methods can be taken from any chosen ane-step ordinary differential equations （ODEs） solver. The schemes is proved to be strong convergent with order one. For the mean-square stability analysis, the investigation is confined to two cases. Some numerical experiments are reported to testify the performance and the effectiveness of the methods.
基金supported by National Natural Science Foundation of China(Grant No.11971010)Scientific Research Project of Education Department of Hubei Province(Grant No.B2019184)。
文摘This paper deals with numerical solutions of nonlinear stiff stochastic differential equations with jump-diffusion and piecewise continuous arguments.By combining compensated split-step methods and balanced methods,a class of compensated split-step balanced(CSSB)methods are suggested for solving the equations.Based on the one-sided Lipschitz condition and local Lipschitz condition,a strong convergence criterion of CSSB methods is derived.It is proved under some suitable conditions that the numerical solutions produced by CSSB methods can preserve the mean-square exponential stability of the corresponding analytical solutions.Several numerical examples are presented to illustrate the obtained theoretical results and the effectiveness of CSSB methods.Moreover,in order to show the computational advantage of CSSB methods,we also give a numerical comparison with the adapted split-step backward Euler methods with or without compensation and tamed explicit methods.
基金Project supported by the National Natural Science Foundation of China(No.12372028)the Guangdong Basic and Applied Basic Research Foundation(No.2022A1515011809)。
文摘A vibro-impact system is a hot topic in the study on nonlinear dynamics due to its generality and importance in engineering.In general,the alternating frequency-time harmonic balance(AFT-HB)method can be used to solve elastic collision.However,since the system is non-smooth,the required Fourier/harmonic truncation order is high in order to achieve the theoretical convergence rate,resulting in expensive computational cost.Furthermore,for rigid body collision,the periodic response of the system cannot be solved with the AFT-HB method due to the discontinuous velocity of the system.In order to accelerate the convergence and solve highly non-smooth systems,an enriched harmonic balance(HB)method is proposed,which is derived from the AFT-HB method in the framework of event-driven Gauss quadrature.The basic idea is to augment the Fourier bases by introducing a non-smooth Bernoulli base such that the non-smooth Bernoulli base compensates for the non-smooth part of the solution and the smooth part of the solution is approximated by the Fourier bases,thus achieving accelerated convergence.Based on the enriched HB method,gear pair systems with gear backlash and oscillator systems with rigid impact are solved,and the dynamic response characteristics are analyzed in this work.Then,based on the Floquet theory,the event-driven monodromy matrix method for non-smooth systems is used to analyze the stability and bifurcation of the periodic solutions.The numerical example shows that the results obtained from the enriched HB method are consistent with those from the Runge-Kutta method,which proves that the presented method is an effective method for analyzing the dynamic response characteristic of the vibro-impact system.
基金Projects(51478049,51778068)supported by the National Natural Science Foundation of ChinaProject(14JJ2075,2019JJ40301)supported by the Hunan Natural Science Foundation of China+1 种基金Project(17A010)supported by the Scientific Research Fund of Hunan Provincial Education Department of ChinaProject(2017GK4034)supported by the Major Technological Achievements Transformation Program of Hunan Strategic Emerging Industries of China
文摘Cantilever casting concrete arch bridge using form traveller has a broad application prospect.However,it is difficult to obtain reasonable initial cable force in construction stage.In this study,stress balance and influence matrix methods were developed to determine the initial cable force of cantilever casting concrete arch bridge.The stress balance equation and influence matrix of arch rib critical section were established,and the buckle cable force range was determined by the allowable stress of arch rib critical section.Then a group of buckle cable forces were selected and substituted into the stress balance equation,and the reasonable initial buckle cable force was determined through iteration.Based on the principle of force balance,the initial anchor cable force was determined.In an engineering application example,it is shown that the stress balance and influence matrix methods for the determination of initial cable force are feasible and reliable.The initial cable forces of arch rib segments only need to be adjusted once in the corresponding construction process,which improves the working efficiency and reduces the construction risk.It is found that the methods have great advantages for determining initial cable force in cantilever casting construction process of concrete arch bridge.
基金supported by the National Natural Science Foundation of China(No.10632040)
文摘Comparisons of the common methods for obtaining the periodic responses show that the harmonic balance method with alternating frequency/time （HB-AFT） do- main technique has some advantages in dealing with nonlinear problems of fractional exponential models. By the HB-AFT method, a rigid rotor supported by ball bearings with nonlinearity of Hertz contact and ball passage vibrations is considered. With the aid of the Floquet theory, the movement characteristics of interval stability are deeply studied. Besides, a simple strategy to determine the monodromy matrix is proposed for the stability analysis.
基金supported by the National Natural Science Foundation of China (10772202)Doctoral Program Foundation of Ministry of Education of China (20050558032)Guangdong Province Natural Science Foundation (07003680, 05003295)
文摘We have deduced incremental harmonic balance an iteration scheme in the （IHB） method using the harmonic balance plus the Newton-Raphson method. Since the convergence of the iteration is dependent upon the initial values in the iteration, the convergent region is greatly restricted for some cases. In this contribution, in order to enlarge the convergent region of the IHB method, we constructed the zeroth-order deformation equation using the homotopy analysis method, in which the IHB method is employed to solve the deformation equation with an embedding parameter as the active increment. Taking the Duffing and the van der Pol equations as examples, we obtained the highly accurate solutions. Importantly, the presented approach renders a convenient way to control and adjust the convergence.
基金supported by the National Natural Science Foundation of China (No.10461005)the Ph.D.Programs Foundation of Ministry of Education of China (No.20070128001)the High Education Science Research Program of Inner Mongolia (No.NJZY08057)
文摘In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the （1＋1）-dimensional KdV equation. Some exact travelling wave solu- tions with parameters have been obtained, which cover the existing solutions. Compared to other methods, the presented method is more direct, more concise, more effective, and easier for calculations. In addition, it can be used to solve other nonlinear evolution equations in mathematical physics.
基金Supported by the National Natural Science Foundation of China under Grant No. 11071209the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province under Grant No. 10KJBll0011
文摘A modified homogeneous balance method is proposed by improving some key steps in the homogeneousbalance method.Bilinear equations of some nonlinear evolution equations are derived by using the modified homogeneousbalance method.Generalized Boussinesq equation,KP equation,and mKdV equation are chosen as examples to illustrateour method.This approach is also applicable to a large variety of nonlinear evolution equations.
基金the Project of the National Natural Science Foudation of China (Grant No.41372114,41340005,41172162,40972083)
文摘We applied the material balance principle of the denudation volume and sedimentary flux to study the denudation-accumulation system between the Longmen Mountains （Mts.） and the foreland basin. The amount of sediment in each sedimentation stage of the basin was estimated to obtain the denudation volume, erosion thickness and deposit thickness since the Late Triassic Epoch, to enable us to recover the paleoelevation of the provenance and the sedimentary area. The results show the following： （1） Since the Late Triassic Epoch, the elevation of the surface of the Longmen Mts. has uplifted from 0 m to 2751 m, and the crust of the Longmen Mts. has uplifted by 9.8 km. Approximately 72% of the materials introduced have been denuded from the mountains. （2） It is difficult to recover the paleoelevation of each stage of the Longmen Mts. foreland basin quantitatively by the present-day techniques and data. （3） The formation of the Longmen Mts. foreland basin consisted of three stages of thrust belt tectonic load and three stages of thrust belt erosional unload. During tectonic loading stages （Late Triassic Epoch, Late Jurassic-Early Cretaceous, Late Cretaceous-Miocene）, the average elevation of Longmen Mts. was lower （approximately 700-1700 m）. During erosional unloading stages （Early and Middle Jurassic, Middle Cretaceous and Jiaguan, Late Cenozoic）, the average elevation of Longmen Mts. was high at approximately 2000-2800m.
基金co-supported by the National Natural Science Foundation of China(No.51976172)the National Science and Technology Major Project of China(No.2017-Ⅱ-0009-0023)。
文摘This paper revisits the Space-Time Gradient(STG) method which was developed for efficient analysis of unsteady flows due to rotor–stator interaction and presents the method from an alternative time-clocking perspective. The STG method requires reordering of blade passages according to their relative clocking positions with respect to blades of an adjacent blade row. As the space-clocking is linked to an equivalent time-clocking, the passage reordering can be performed according to the alternative time-clocking. With the time-clocking perspective, unsteady flow solutions from different passages of the same blade row are mapped to flow solutions of the same passage at different time instants or phase angles. Accordingly, the time derivative of the unsteady flow equation is discretized in time directly, which is more natural than transforming the time derivative to a spatial one as with the original STG method. To improve the solution accuracy, a ninth order difference scheme has been investigated for discretizing the time derivative. To achieve a stable solution for the high order scheme, the implicit solution method of Lower-Upper Symmetric GaussSeidel/Gauss-Seidel(LU-SGS/GS) has been employed. The NASA Stage 35 and its blade-countreduced variant are used to demonstrate the validity of the time-clocking based passage reordering and the advantages of the high order difference scheme for the STG method. Results from an existing harmonic balance flow solver are also provided to contrast the two methods in terms of solution stability and computational cost.
基金Project supported by the Ph. D. Programs Foundation of Ministry of Education of China (No.20050558032) the Natural Science Foundation of Guangdong Province of China (No.05003295) the Foundation of Sun Yat-sen University Advanced Research Center (No.06M8) the Young Teacher Scientific Research Foundation of Sun Sat-sen University (No.1131011)
文摘The incremental harmonic balance method was extended to analyze the flutter of systems with multiple structural strong nonlinearities. The strongly nonlinear cubic plunging and pitching stiffness terms were considered in the flutter equations of two-dimensional airfoil. First, the equations were transferred into matrix form, then the vibration process was divided into the persistent incremental processes of vibration moments. And the expression of their solutions could be obtained by using a certain amplitude as control parameter in the harmonic balance process, and then the bifurcation, limit cycle flutter phenomena and the number of harmonic terms were analyzed. Finally, numerical results calculated by the Runge-Kutta method were given to verify the results obtained by the proposed procedure. It has been shown that the incremental harmonic method is effective and precise in the analysis of strongly nonlinear flutter with multiple structural nonlinearities.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11872254 and 11672191)
文摘The dynamical properties of fractional-order Duffing–van der Pol oscillator are studied, and the amplitude–frequency response equation of primary resonance is obtained by the harmonic balance method. The stability condition for steady-state solution is obtained based on Lyapunov theory. The comparison of the approximate analytical results with the numerical results is fulfilled, and the approximations obtained are in good agreement with the numerical solutions. The bifurcations of primary resonance for system parameters are analyzed. The results show that the harmonic balance method is effective and convenient for solving this problem, and it provides a reference for the dynamical analysis of similar nonlinear systems.
基金Supported by the National Nature Science Foundation of China(10371070)Supported by the Nature Science Foundation of Educational Committee of Liaoning Province(2021401157)
文摘Based on the homogenous balance method and with the help of mathematica, the Backlund transformation and the transfer heat equation are derived. Analyzing the heat-transfer equation, the multiple soliton solutions and other exact analytical solution for Whitham-Broer-Kaup equations(WBK) are derived. These solutions contain Fan's, Xie's and Yan's results and other new types of analytical solutions, such as rational function solutions and periodic solutions. The method can also be applied to solve more nonlinear differential equations.
文摘A new modulation approach was presented for the control of neutral-point （NP） voltage variation in the three-level NP-clamped voltage source inverter, and the average NP current model was established based on vector diagram partition. Thus, theory base was built for balancing control of NP potential. Theoretical analysis and experimental results indicate that the proposed method for NP balancing control vector synthe- sizing concept based can make the average NP current zero, and do not influence NP potential within every sample period. The effectiveness of proposed research approach was verified by simulative and experimental results.
文摘The combined finiteediscrete element method （FDEM） belongs to a family of methods of computationalmechanics of discontinua. The method is suitable for problems of discontinua, where particles aredeformable and can fracture or fragment. The applications of FDEM have spread over a number of disciplinesincluding rock mechanics, where problems like mining, mineral processing or rock blasting canbe solved by employing FDEM. In this work, a novel approach for the parallelization of two-dimensional（2D） FDEM aiming at clusters and desktop computers is developed. Dynamic domain decompositionbased parallelization solvers covering all aspects of FDEM have been developed. These have beenimplemented into the open source Y2D software package and have been tested on a PC cluster. Theoverall performance and scalability of the parallel code have been studied using numerical examples. Theresults obtained confirm the suitability of the parallel implementation for solving large scale problems. 2014 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting byElsevier B.V. All rights reserved.
基金Supported by the International Cooperation and Exchanges Foundation of Henan Province (084300510060)the Youth Science Foundation of Henan University of Science and Technology of China (2008QN026)
文摘The （G＇/G, 1/G）-expansion method for finding exact travelling wave solutions of nonlinear evolution equations, which can be thought of as an extension of the （G＇/G）-expansion method proposed recently, is presented. By using this method abundant travelling wave so- lutions with arbitrary parameters of the Zakharov equations are successfully obtained. When the parameters are replaced by special values, the well-known solitary wave solutions of the equations are rediscovered from the travelling waves.
基金Project supported by Major Special Science and Technology Project of PetroChina Company Limited“Study and application of key technologies for developing deepeultra-deep gas in Kuqa depression”(No.:2018E-1803).
文摘It is difficult to determine the effective reservoir rock compressibility,aquifer volume and water influx of high pressure and ultra-high pressure gas reservoirs,so when the traditional apparent reservoir pressure and cumulative gas production curve extrapolation method and its modified version are used to calculate the initial gas in-place of such gas reservoirs,the calculation accuracy is lower.The material balance equation in the form of power function for such gas reservoirs was established based on Gonzalez method to improve the accuracy and reliability of reserve evaluation.Then,based on 20 high pressure and ultra-high pressure reservoirs that had been developed abroad,the empirical value of the power exponent was defined,and the influences of the depletion degree of apparent reservoir pressure and the degree of reserve recovery on the reliability of reserve calculation were analyzed.Besides,the critical value of the key parameter affecting the reliability of reserve evaluation(depletion degree of apparent reservoir pressure)was determined and compared with the critical value of the two-linear trends.Finally,an example calculation was carried out.And the following research results were obtained.First,the empirical value of the power exponent calculated using the material balance method in the form of power function is 1.02847,with an upper limit of 1.11567.Second,the depletion degree of apparent reservoir pressure corresponding to the inflection point of the classical two-linear trends ranges from 0.14 to 0.38 with an average of 0.23,while that corresponding to the extrapolation point of the second straight line is between 0.23 and 0.50 with an average of 0.33,and the corresponding degree of reserve recovery is in the range of 33%-65%with an average of 45%.Third,the reserves of such high pressure gas reservoirs calculated with this new method has an error rate less than 10%if the depletion degree of apparent reservoir pressure is greater than 0.33.In conclusion,for such high pressure gas reservoirs and the stress-sensitive fractured gas reservoirs,the material balance method in the form of power function proposed in this paper can avoid the uncertainty parameters(e.g.effective reservoir rock compressibility,aquifer volume and water influx)and it is advantageous with simple calculation process,better practicability and small error.
基金Project supported by the National Natural Science Foundation of China(Nos.11972129 and12372008)the National Major Science and Technology Projects of China(No.2017-IV-0008-0045)+3 种基金the Natural Science Foundation of Heilongjiang Province of China(No.YQ2022A008)the Fundamental Research Funds for the Central Universities of China(No.HIT.OCEF.2023006)the Polish National Science Centre of Poland under the OPUS 18 grant(No.2019/35/B/ST8/00980)the Tianjin University Independent Innovation Foundation of China(No.2023XJS-0038)。
文摘The harmonic balance method(HBM)is one of the most widely used methods in solving nonlinear vibration problems,and its accuracy and computational efficiency largely depend on the number of the harmonics selected.The adaptive harmonic balance(AHB)method is an improved HBM method.This paper presents a modified AHB method with the asymptotic harmonic selection(AHS)procedure.This new harmonic selection procedure selects harmonics from the frequency spectra of nonlinear terms instead of estimating the contribution of each harmonic to the whole nonlinear response,by which the additional calculation is avoided.A modified continuation method is proposed to deal with the variable size of nonlinear algebraic equations at different values of path parameters,and then all solution branches of the amplitude-frequency response are obtained.Numerical experiments are carried out to verify the performance of the AHB-AHS method.Five typical nonlinear dynamic equations with different types of nonlinearities and excitations are chosen as the illustrative examples.Compared with the classical HBM and Runge-Kutta methods,the proposed AHB-AHS method is of higher accuracy and better convergence.The AHB-AHS method proposed in this paper has the potential to investigate the nonlinear vibrations of complex high-dimensional nonlinear systems.
基金financially supported by the National Natural Science Foundation of China(Nos.52175071,91860205)the Major Projects of Aero-engines and Gas turbines(No.J2019-IV-023-0091)。
文摘Dry friction damping structures are widely-used in aero-engines to mitigate vibration.The nonlinear nature of friction and the two-dimensional in-plane motion on the contact interface bring challenges to accurately and efficiently predict the forced response of frictionally damped structures.The state-of-the-art Multi-Harmonic Balance Method(MHBM)on quasi-3D contact model in engineering cannot precisely capture the kinematics on the friction interface although the efficiency is high.The full-3D contact model can describe the constitutive relationship of the interface in a more accurate manner;however,the efficiency and convergence are not guaranteed for large-scale models.In this paper,a semi-analytical MHBM on full-3D contact model is proposed.The original Trajectory Tracking Method(TTM)for evaluating the contact force is reformulated to make the calculation more concise and the derivation of the Analytical Jacobian Matrix(AJM)feasible.Based on the chain rule of derivation,the AJM which is the core to upgrade the performance is deduced.Through a shrouded blade finite element model,the accuracy and efficiency of the proposed method are compared with both the MHBM on full-3D contact model with numerical Jacobian matrix and the MHBM on quasi-3D contact model with AJM.The results show that the AJM improves significantly the efficiency of the MHBM on full-3D contact model.The time cost of the proposed method is in the same order of magnitude as that of the MHBM on quasi-3D contact model.We also confirm that the full-3D contact model is necessary for the dynamic analyses of shrouded blades.If one uses the quasi-3D model,the estimation relative error of damping can even reach 31.8%in some cases.In addition,the AJM also brings benefits for stability analysis.It is highly recommended that engineers use the MHBM on full-3D contact model for the dynamic analysis and design of shrouded blades.
