The finite segment modelling for the flexible beam-formed structural elements is presented, in which the discretization views of the finite segment method and the difference from the finite element method are introduc...The finite segment modelling for the flexible beam-formed structural elements is presented, in which the discretization views of the finite segment method and the difference from the finite element method are introduced. In terms of the nodal model, the joint properties are described easily by the model of the finite segment method, and according to the element properties, the assumption of the small strain is only met in the finite segment method, i. e., the geometric nonlinear deformation of the flexible bodies is allowable. Consequently,the finite segment method is very suited to the flexible multibody structure. The finite segment model is used and the are differentiation is adopted for the differential beam segments. The stiffness equation is derived by the use of the principle of virtual work. The new modelling method shows its normalization, clear physical and geometric meanings and simple computational process.展开更多
An arc_length method is presented to solve the ordinary differential equations (ODEs) with certain types of singularity such as stiff property or discontinuity on continuum problem. By introducing one or two arc_lengt...An arc_length method is presented to solve the ordinary differential equations (ODEs) with certain types of singularity such as stiff property or discontinuity on continuum problem. By introducing one or two arc_length parameters as variables, the differential equations with singularity are transformed into non_singularity equations, which can be solved by usual methods. The method is also applicable for partial differential equations (PDEs), because they may be changed into systems of ODEs by discretization. Two examples are given to show the accuracy, efficiency and application.展开更多
A nonlinear numerical integration method, based on forward and backward Euler integration methods, is proposed for solving the stiff dynamic equations of a flexible multibody system, which are transformed from the sec...A nonlinear numerical integration method, based on forward and backward Euler integration methods, is proposed for solving the stiff dynamic equations of a flexible multibody system, which are transformed from the second order to the first order by adopting state variables. This method is of A0 stability and infinity stability. The numerical solutions violating the constraint equations are corrected by Blajer's modification approach. Simulation results of a slider-crank mechanism by the proposed method are in good agreement with ones from other literature.展开更多
Numerical algorithms for stiff stochastic differential equations are developed using lin-ear approximations of the fast diffusion processes,under the assumption of decoupling between fast and slow processes.Three nume...Numerical algorithms for stiff stochastic differential equations are developed using lin-ear approximations of the fast diffusion processes,under the assumption of decoupling between fast and slow processes.Three numerical schemes are proposed,all of which are based on the linearized formulation albeit with different degrees of approximation.The schemes are of comparable complexity to the classical explicit Euler-Maruyama scheme but can achieve better accuracy at larger time steps in stiff systems.Convergence analysis is conducted for one of the schemes,that shows it to have a strong convergence order of 1/2 and a weak convergence order of 1.Approximations arriving at the other two schemes are discussed.Numerical experiments are carried out to examine the convergence of the schemes proposed on model problems.展开更多
To improve the adaptability of TBMs in diverse geological environments,this paper proposes a reconfigurable Type-V thrust mechanism(V-TM)with rearrangeable working states,in which structural stiffness can be automatic...To improve the adaptability of TBMs in diverse geological environments,this paper proposes a reconfigurable Type-V thrust mechanism(V-TM)with rearrangeable working states,in which structural stiffness can be automatically altered during operation.Therefore,millions of configurations can be obtained,and thousands of instances of working status per configuration can be set respectively.Nonetheless,the complexity of configurations and diversity of working states contributes to further complications for the structural stiffness algorithm.This results in challenges such as difficulty calculating the payload compliance index and the environment adaptability index.To solve this problem,we use the configuration matrix to describe the relationship between propelling jacks under reconfiguration and adopt pattern vectors to describe the working state of each hydraulic cylinder.Then,both the dynamic compatible equation between propeller forces of the hydraulic cylinders and driving forces,and the kinematic harmonizing equation between the hydraulic cylinder displacements and their deformations are established.Next,we derive the stiffness analytical equation using Hooke’s law and the Jacobian Matrix.The proposed approach provides an effective algorithm to support structural rigidity analysis,and lays a solid theoretical foundation for calculating the performance indexes of the V-TM.We then analyze the rigidity characteristics of typical configurations under different working states,and obtain the main factors affecting structural stiffness of the V-TM.The results show the deviation degree of structural parameters in hydraulic cylinders within the same group,and the working status of propelling jacks.Finally,our constructive conclusions contribute valuable information for matching and optimization by drawing on the factors that affect the structural rigidity of the V-TM.展开更多
Two experimental methods were adopted to verify the correctness and practicability of the shape meter method: one is to roll aluminum plate, calculate the shape stiffness of mill and rolled piece, and then measure alu...Two experimental methods were adopted to verify the correctness and practicability of the shape meter method: one is to roll aluminum plate, calculate the shape stiffness of mill and rolled piece, and then measure aluminum plate crown to verify shape stiffness equation; the other is to calculate the measured off line data of hot continuous roll and verify the shape mathematical model for measuring and controlling by self adaptation method.展开更多
The comparison of five atmospheric chemical modeling schemes was presented from accuracy,efficiency and progamming in term of Carbon bond Mechanism IV of atmospheric chemical reactions. The major results were as fo...The comparison of five atmospheric chemical modeling schemes was presented from accuracy,efficiency and progamming in term of Carbon bond Mechanism IV of atmospheric chemical reactions. The major results were as follows. The classical Gear scheme can provide an accurate solution and it is easy for programming by a computer automatically. The sparse matrix Gear type scheme can also provide an accurate solution but much faster than classical Gear scheme. QSSA,EBI and hybrid schemes can run with much longer time step without sacrificing of accuracy, therefore, much efficiently. If analytical solution is obtained by EBI scheme the accuracy and efficiency are much better.展开更多
The improvements of high-throughput experimental devices such as microarray and mass spectrometry have allowed an effective acquisition of biological comprehensive data which include genome, transcriptome, proteome, a...The improvements of high-throughput experimental devices such as microarray and mass spectrometry have allowed an effective acquisition of biological comprehensive data which include genome, transcriptome, proteome, and metabolome (multi-layered omics data). In Systems Biology, we try to elucidate various dynamical characteristics of biological functions with applying the omics data to detailed mathematical model based on the central dogma. However, such mathematical models possess multi-time-scale properties which are often accompanied by time-scale differences seen among biological layers. The differences cause time stiff problem, and have a grave influence on numerical calculation stability. In the present conventional method, the time stiff problem remained because the calculation of all layers was implemented by adaptive time step sizes of the smallest time-scale layer to ensure stability and maintain calculation accuracy. In this paper, we designed and developed an effective numerical calculation method to improve the time stiff problem. This method consisted of ahead, backward, and cumulative algorithms. Both ahead and cumulative algorithms enhanced calculation efficiency of numerical calculations via adjustments of step sizes of each layer, and reduced the number of numerical calculations required for multi-time-scale models with the time stiff problem. Backward algorithm ensured calculation accuracy in the multi-time-scale models. In case studies which were focused on three layers system with 60 times difference in time-scale order in between layers, a proposed method had almost the same calculation accuracy compared with the conventional method in spite of a reduction of the total amount of the number of numerical calculations. Accordingly, the proposed method is useful in a numerical analysis of multi-time-scale models with time stiff problem.展开更多
Many problems in engineering sciences can be described by linear,inhomogeneous,m-th order ordinary differential equations(ODEs)with variable coefficients.For this wide class of problems,we here present a new,simple,fl...Many problems in engineering sciences can be described by linear,inhomogeneous,m-th order ordinary differential equations(ODEs)with variable coefficients.For this wide class of problems,we here present a new,simple,flexible,and robust solution method,based on piecewise exact integration of local approximation polynomials as well as on averaging local integrals.The method is designed for modern mathematical software providing efficient environments for numerical matrix-vector operation-based calculus.Based on cubic approximation polynomials,the presented method can be expected to perform(i)similar to the Runge-Kutta method,when applied to stiff initial value problems,and(ii)significantly better than the finite difference method,when applied to boundary value problems.Therefore,we use the presented method for the analysis of engineering problems including the oscillation of a modulated torsional spring pendulum,steady-state heat transfer through a cooling web,and the structural analysis of a slender tower based on second-order beam theory.Related convergence studies provide insight into the satisfying characteristics of the proposed solution scheme.展开更多
In this paper,we present the backward stochastic Taylor expansions for a Ito process,including backward Ito-Taylor expansions and backward Stratonovich-Taylor expansions.We construct the general full implicit strong T...In this paper,we present the backward stochastic Taylor expansions for a Ito process,including backward Ito-Taylor expansions and backward Stratonovich-Taylor expansions.We construct the general full implicit strong Taylor approximations(including Ito-Taylor and Stratonovich-Taylor schemes)with implicitness in both the deterministic and the stochastic terms for the stiff stochastic differential equations(SSDE)by employing truncations of backward stochastic Taylor expansions.We demonstrate that these schemes will converge strongly with corresponding order 1,2,3,....Mean-square stability has been investigated for full implicit strong Stratonovich-Taylor scheme with order 2,and it has larger meansquare stability region than the explicit and the semi-implicit strong Stratonovich-Taylor schemes with order 2.We can improve the stability of simulations considerably without too much additional computational effort by using our full implicit schemes.The full implicit strong Taylor schemes allow a larger range of time step sizes than other schemes and are suitable for SSDE with stiffness on both the drift and the diffusion terms.Our numerical experiment show these points.展开更多
A novel canonical Euler splitting method is proposed for nonlinear compositestiff functional differential-algebraic equations, the stability and convergence of themethod is evidenced, theoretical results are further c...A novel canonical Euler splitting method is proposed for nonlinear compositestiff functional differential-algebraic equations, the stability and convergence of themethod is evidenced, theoretical results are further confirmed by some numerical experiments.Especially, the numerical method and its theories can be applied to specialcases, such as delay differential-algebraic equations and integral differential-algebraicequations.展开更多
Presents a class of modified parallel Rosenbrock methods (MPROW) which possesses more free parameters to improve further the various properties of the methods and will be similarly written as MPROW. Information on par...Presents a class of modified parallel Rosenbrock methods (MPROW) which possesses more free parameters to improve further the various properties of the methods and will be similarly written as MPROW. Information on parallel Rosenbrock methods; Convergence and stability analysis; Discussion on two-stage third-order 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
Nonlinear methods are combined with Runge Kutta methods to develop A stable explicit nonlinear Runge Kutta methods for solving stiff differential equations and a class of the third order formulae are constructed.I...Nonlinear methods are combined with Runge Kutta methods to develop A stable explicit nonlinear Runge Kutta methods for solving stiff differential equations and a class of the third order formulae are constructed.It avoids solving the nonlinear equations which implicit methods must solve. Implementation is very simple and the computation cost for each step is small.This paper uses a shift transformation to avoid the order reduction of nonlinear methods at y = 0 . Thus the method is very practicable. Numerical tests show that the method is more efficient than explicit methods or implicit methods of the same order.展开更多
Stabilized or Chebyshev explicit methods have been widely used in the past to solve stiff ordinary differential equations. Making use of special properties of Chebyshev-like polynomials, these methods have favorable s...Stabilized or Chebyshev explicit methods have been widely used in the past to solve stiff ordinary differential equations. Making use of special properties of Chebyshev-like polynomials, these methods have favorable stability properties compared to standard explicit methods while remaining explicit. A new class of such methods, called ROCK, introduced in [Numer. Math., 90, 1-18, 2001] has recently been extended to stiff stochastic differential equations under the name S-ROCK [C. R. Acad. Sci. Paris, 345(10), 2007 and Commun. Math. Sci, 6(4), 2008]. In this paper we discuss the extension of the S-ROCK methods to systems with discrete noise and propose a new class of methods for such problems, the τ-ROCK methods. One motivation for such methods is the simulation of multi-scale or stiff chemical kinetic systems and such systems are the focus of this paper, but our new methods could potentially be interesting for other stiff systems with discrete noise. Two versions of the τ-ROCK methods are discussed and their stability behavior is analyzed on a test problem. Compared to the τ-leaping method, a significant speed-up can be achieved for some stiff kinetic systems. The behavior of the proposed methods are tested on several numerical experiments.展开更多
基金National Natural Science Foundation of China!59575026
文摘The finite segment modelling for the flexible beam-formed structural elements is presented, in which the discretization views of the finite segment method and the difference from the finite element method are introduced. In terms of the nodal model, the joint properties are described easily by the model of the finite segment method, and according to the element properties, the assumption of the small strain is only met in the finite segment method, i. e., the geometric nonlinear deformation of the flexible bodies is allowable. Consequently,the finite segment method is very suited to the flexible multibody structure. The finite segment model is used and the are differentiation is adopted for the differential beam segments. The stiffness equation is derived by the use of the principle of virtual work. The new modelling method shows its normalization, clear physical and geometric meanings and simple computational process.
文摘An arc_length method is presented to solve the ordinary differential equations (ODEs) with certain types of singularity such as stiff property or discontinuity on continuum problem. By introducing one or two arc_length parameters as variables, the differential equations with singularity are transformed into non_singularity equations, which can be solved by usual methods. The method is also applicable for partial differential equations (PDEs), because they may be changed into systems of ODEs by discretization. Two examples are given to show the accuracy, efficiency and application.
基金The Science-Technology Foundation for Young Scientist of Fujian Province (No.2005J053)
文摘A nonlinear numerical integration method, based on forward and backward Euler integration methods, is proposed for solving the stiff dynamic equations of a flexible multibody system, which are transformed from the second order to the first order by adopting state variables. This method is of A0 stability and infinity stability. The numerical solutions violating the constraint equations are corrected by Blajer's modification approach. Simulation results of a slider-crank mechanism by the proposed method are in good agreement with ones from other literature.
文摘Numerical algorithms for stiff stochastic differential equations are developed using lin-ear approximations of the fast diffusion processes,under the assumption of decoupling between fast and slow processes.Three numerical schemes are proposed,all of which are based on the linearized formulation albeit with different degrees of approximation.The schemes are of comparable complexity to the classical explicit Euler-Maruyama scheme but can achieve better accuracy at larger time steps in stiff systems.Convergence analysis is conducted for one of the schemes,that shows it to have a strong convergence order of 1/2 and a weak convergence order of 1.Approximations arriving at the other two schemes are discussed.Numerical experiments are carried out to examine the convergence of the schemes proposed on model problems.
基金Supported by National Natural Science Foundation of China(Grant No.51675180)National Key Basic Research Program of China(973 Program,Grant No.2013CB037503)
文摘To improve the adaptability of TBMs in diverse geological environments,this paper proposes a reconfigurable Type-V thrust mechanism(V-TM)with rearrangeable working states,in which structural stiffness can be automatically altered during operation.Therefore,millions of configurations can be obtained,and thousands of instances of working status per configuration can be set respectively.Nonetheless,the complexity of configurations and diversity of working states contributes to further complications for the structural stiffness algorithm.This results in challenges such as difficulty calculating the payload compliance index and the environment adaptability index.To solve this problem,we use the configuration matrix to describe the relationship between propelling jacks under reconfiguration and adopt pattern vectors to describe the working state of each hydraulic cylinder.Then,both the dynamic compatible equation between propeller forces of the hydraulic cylinders and driving forces,and the kinematic harmonizing equation between the hydraulic cylinder displacements and their deformations are established.Next,we derive the stiffness analytical equation using Hooke’s law and the Jacobian Matrix.The proposed approach provides an effective algorithm to support structural rigidity analysis,and lays a solid theoretical foundation for calculating the performance indexes of the V-TM.We then analyze the rigidity characteristics of typical configurations under different working states,and obtain the main factors affecting structural stiffness of the V-TM.The results show the deviation degree of structural parameters in hydraulic cylinders within the same group,and the working status of propelling jacks.Finally,our constructive conclusions contribute valuable information for matching and optimization by drawing on the factors that affect the structural rigidity of the V-TM.
基金Item Sponsored by National Natural Science Foundation of China(19974035)Natural Science Foundation of Hebei Province(599240)
文摘Two experimental methods were adopted to verify the correctness and practicability of the shape meter method: one is to roll aluminum plate, calculate the shape stiffness of mill and rolled piece, and then measure aluminum plate crown to verify shape stiffness equation; the other is to calculate the measured off line data of hot continuous roll and verify the shape mathematical model for measuring and controlling by self adaptation method.
文摘The comparison of five atmospheric chemical modeling schemes was presented from accuracy,efficiency and progamming in term of Carbon bond Mechanism IV of atmospheric chemical reactions. The major results were as follows. The classical Gear scheme can provide an accurate solution and it is easy for programming by a computer automatically. The sparse matrix Gear type scheme can also provide an accurate solution but much faster than classical Gear scheme. QSSA,EBI and hybrid schemes can run with much longer time step without sacrificing of accuracy, therefore, much efficiently. If analytical solution is obtained by EBI scheme the accuracy and efficiency are much better.
文摘The improvements of high-throughput experimental devices such as microarray and mass spectrometry have allowed an effective acquisition of biological comprehensive data which include genome, transcriptome, proteome, and metabolome (multi-layered omics data). In Systems Biology, we try to elucidate various dynamical characteristics of biological functions with applying the omics data to detailed mathematical model based on the central dogma. However, such mathematical models possess multi-time-scale properties which are often accompanied by time-scale differences seen among biological layers. The differences cause time stiff problem, and have a grave influence on numerical calculation stability. In the present conventional method, the time stiff problem remained because the calculation of all layers was implemented by adaptive time step sizes of the smallest time-scale layer to ensure stability and maintain calculation accuracy. In this paper, we designed and developed an effective numerical calculation method to improve the time stiff problem. This method consisted of ahead, backward, and cumulative algorithms. Both ahead and cumulative algorithms enhanced calculation efficiency of numerical calculations via adjustments of step sizes of each layer, and reduced the number of numerical calculations required for multi-time-scale models with the time stiff problem. Backward algorithm ensured calculation accuracy in the multi-time-scale models. In case studies which were focused on three layers system with 60 times difference in time-scale order in between layers, a proposed method had almost the same calculation accuracy compared with the conventional method in spite of a reduction of the total amount of the number of numerical calculations. Accordingly, the proposed method is useful in a numerical analysis of multi-time-scale models with time stiff problem.
基金Fruitful discussions with Gerhard Hofinger,from Feb 2007 until Dec 2010 research assistant at the Institute for Mechanics of Materials and Structures,Vienna University of Technology,are gratefully acknowledged.
文摘Many problems in engineering sciences can be described by linear,inhomogeneous,m-th order ordinary differential equations(ODEs)with variable coefficients.For this wide class of problems,we here present a new,simple,flexible,and robust solution method,based on piecewise exact integration of local approximation polynomials as well as on averaging local integrals.The method is designed for modern mathematical software providing efficient environments for numerical matrix-vector operation-based calculus.Based on cubic approximation polynomials,the presented method can be expected to perform(i)similar to the Runge-Kutta method,when applied to stiff initial value problems,and(ii)significantly better than the finite difference method,when applied to boundary value problems.Therefore,we use the presented method for the analysis of engineering problems including the oscillation of a modulated torsional spring pendulum,steady-state heat transfer through a cooling web,and the structural analysis of a slender tower based on second-order beam theory.Related convergence studies provide insight into the satisfying characteristics of the proposed solution scheme.
基金supported by the Fundamental Research Funds for the Central Universities of China,and the second author is supported by the National Natural Fund Projects of China(Nos.11771100,12071332).
文摘In this paper,we present the backward stochastic Taylor expansions for a Ito process,including backward Ito-Taylor expansions and backward Stratonovich-Taylor expansions.We construct the general full implicit strong Taylor approximations(including Ito-Taylor and Stratonovich-Taylor schemes)with implicitness in both the deterministic and the stochastic terms for the stiff stochastic differential equations(SSDE)by employing truncations of backward stochastic Taylor expansions.We demonstrate that these schemes will converge strongly with corresponding order 1,2,3,....Mean-square stability has been investigated for full implicit strong Stratonovich-Taylor scheme with order 2,and it has larger meansquare stability region than the explicit and the semi-implicit strong Stratonovich-Taylor schemes with order 2.We can improve the stability of simulations considerably without too much additional computational effort by using our full implicit schemes.The full implicit strong Taylor schemes allow a larger range of time step sizes than other schemes and are suitable for SSDE with stiffness on both the drift and the diffusion terms.Our numerical experiment show these points.
基金National Natural Science Foundation of China(Grant No.11971412)Key Project of Education Department of Hunan Province(Grant No.20A484)Project of Hunan National Center for Applied Mathematics(Grant No.2020ZYT003).
文摘A novel canonical Euler splitting method is proposed for nonlinear compositestiff functional differential-algebraic equations, the stability and convergence of themethod is evidenced, theoretical results are further confirmed by some numerical experiments.Especially, the numerical method and its theories can be applied to specialcases, such as delay differential-algebraic equations and integral differential-algebraicequations.
文摘Presents a class of modified parallel Rosenbrock methods (MPROW) which possesses more free parameters to improve further the various properties of the methods and will be similarly written as MPROW. Information on parallel Rosenbrock methods; Convergence and stability analysis; Discussion on two-stage third-order 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Supported by the National Natural Science Foundationof China(No.195 710 4 6 )
文摘Nonlinear methods are combined with Runge Kutta methods to develop A stable explicit nonlinear Runge Kutta methods for solving stiff differential equations and a class of the third order formulae are constructed.It avoids solving the nonlinear equations which implicit methods must solve. Implementation is very simple and the computation cost for each step is small.This paper uses a shift transformation to avoid the order reduction of nonlinear methods at y = 0 . Thus the method is very practicable. Numerical tests show that the method is more efficient than explicit methods or implicit methods of the same order.
基金supported by the National Science Foundation of China under grant 10871010the National Basic Research Program under grant 2005CB321704
文摘Stabilized or Chebyshev explicit methods have been widely used in the past to solve stiff ordinary differential equations. Making use of special properties of Chebyshev-like polynomials, these methods have favorable stability properties compared to standard explicit methods while remaining explicit. A new class of such methods, called ROCK, introduced in [Numer. Math., 90, 1-18, 2001] has recently been extended to stiff stochastic differential equations under the name S-ROCK [C. R. Acad. Sci. Paris, 345(10), 2007 and Commun. Math. Sci, 6(4), 2008]. In this paper we discuss the extension of the S-ROCK methods to systems with discrete noise and propose a new class of methods for such problems, the τ-ROCK methods. One motivation for such methods is the simulation of multi-scale or stiff chemical kinetic systems and such systems are the focus of this paper, but our new methods could potentially be interesting for other stiff systems with discrete noise. Two versions of the τ-ROCK methods are discussed and their stability behavior is analyzed on a test problem. Compared to the τ-leaping method, a significant speed-up can be achieved for some stiff kinetic systems. The behavior of the proposed methods are tested on several numerical experiments.