The Finite Difference （FD） method is an important method for seismic numerical simulations. It helps us understand regular patterns in seismic wave propagation, analyze seismic attributes, and interpret seismic data...The Finite Difference （FD） method is an important method for seismic numerical simulations. It helps us understand regular patterns in seismic wave propagation, analyze seismic attributes, and interpret seismic data. However, because of its discretization, the FD method is only stable under certain conditions. The Arbitrary Difference Precise Integration （ADPI） method is based on the FD method and adopts an integration scheme in the time domain and an arbitrary difference scheme in the space domain. Therefore, the ADPI method is a semi-analytical method. In this paper, we deduce the formula for the ADPI method based on the 3D elastic equation and improve its stability. In forward modeling cases, the ADPI method was implemented in 2D and 3D elastic wave equation forward modeling. Results show that the travel time of the reflected seismic wave is accurate. Compared with the acoustic wave field, the elastic wave field contains more wave types, including PS- and PP- reflected waves, transmitted waves, and diffracted waves, which is important to interpretation of seismic data. The method can be easily applied to elastic wave equation numerical simulations for eoloical models.展开更多
A fast precise integration method is developed for the time integral of the hyperbolic heat conduction problem. The wave nature of heat transfer is used to analyze the structure of the matrix exponential, leading to t...A fast precise integration method is developed for the time integral of the hyperbolic heat conduction problem. The wave nature of heat transfer is used to analyze the structure of the matrix exponential, leading to the fact that the matrix exponential is sparse. The presented method employs the sparsity of the matrix exponential to improve the original precise integration method. The merits are that the proposed method is suitable for large hyperbolic heat equations and inherits the accuracy of the original version and the good computational efficiency, which are verified by two numerical examples.展开更多
This paper presents an improved precise integration algorithm fortransient analysis of heat transfer and some other problems. Theoriginal precise integration method is improved by means of the inve-rse accuracy analys...This paper presents an improved precise integration algorithm fortransient analysis of heat transfer and some other problems. Theoriginal precise integration method is improved by means of the inve-rse accuracy analysis so that the parameter N, which has been takenas a constant and an independent pa- rameter without consideration ofthe problems in the original method, can be generated automaticallyby the algorithm itself.展开更多
Nonlinear dynamic equations can be solved accurately using a precise integration method. Some algorithms exist, but the inversion of a matrix must be calculated for these al- gorithms. If the inversion of the matrix d...Nonlinear dynamic equations can be solved accurately using a precise integration method. Some algorithms exist, but the inversion of a matrix must be calculated for these al- gorithms. If the inversion of the matrix doesn’t exist or isn’t stable, the precision and stability of the algorithms will be afected. An explicit series solution of the state equation has been pre- sented. The solution avoids calculating the inversion of a matrix and its precision can be easily controlled. In this paper, an implicit series solution of nonlinear dynamic equations is presented. The algorithm is more precise and stable than the explicit series solution and isn’t sensitive to the time-step. Finally, a numerical example is presented to demonstrate the efectiveness of the algorithm.展开更多
This paper presents a finite element procedure for solving transient, multidimensional convection-diffusion equations. The procedure is based on the characteristic Galerkin method with an implicit algorithm using prec...This paper presents a finite element procedure for solving transient, multidimensional convection-diffusion equations. The procedure is based on the characteristic Galerkin method with an implicit algorithm using precise integration method. With the operator splitting procedure, the precise integration method is introduced to determine the material derivative in the convection-diffusion equation, consequently, the physical quantities of material points. An implicit algorithm with a combination of both the precise and the traditional numerical integration procedures in time domain in the Lagrange coordinates for the characteristic Galerkin method is formulated. The stability analysis of the algorithm shows that the unconditional stability of present implicit algorithm is enhanced as compared with that of the traditional implicit numerical integration procedure. The numerical results validate the presented method in solving convection-diffusion equations. As compared with SUPG method and explicit characteristic Galerkin method, the present method gives the results with higher accuracy and better stability.展开更多
The objective of the paper is to develop a new algorithm for numerical solution of dynamic elastic-plastic strain hardening/softening problems. The gradient dependent model is adopted in the numerical model to overcom...The objective of the paper is to develop a new algorithm for numerical solution of dynamic elastic-plastic strain hardening/softening problems. The gradient dependent model is adopted in the numerical model to overcome the result mesh-sensitivity problem in the dynamic strain softening or strain localization analysis. The equations for the dynamic elastic-plastic problems are derived in terms of the parametric variational principle, which is valid for associated, non-associated and strain softening plastic constitutive models in the finite element analysis. The precise integration method, which has been widely used for discretization in time domain of the linear problems, is introduced for the solution of dynamic nonlinear equations. The new algorithm proposed is based on the combination of the parametric quadratic programming method and the precise integration method and has all the advantages in both of the algorithms. Results of numerical examples demonstrate not only the validity, but also the advantages of the algorithm proposed for the numerical solution of nonlinear dynamic problems.展开更多
The difficulty in solving stochastic dynamics problems lies in the need for a large number of repeated computations of deterministic dynamic equations,which has been a challenge in stochastic dynamics analysis and was...The difficulty in solving stochastic dynamics problems lies in the need for a large number of repeated computations of deterministic dynamic equations,which has been a challenge in stochastic dynamics analysis and was discussed in this study.To efficiently and accurately compute the exponential of the dynamics state matrix and the matrix functions due to external loads,an adaptively filtered precise integration method was proposed,which inherits the high precision of the precise integrationmethod,improves the computational efficiency and saves the memory required.Moreover,the perturbation method was introduced to avoid repeated computations of matrix exponential and terms due to external loads.Based on the filtering and perturbation techniques,an adaptively filtered precise integration method considering perturbation for stochastic dynamics problems was developed.Two numerical experiments,including a model of phononic crystal and a bridge model considering random parameters,were performed to test the performance of the proposed method in terms of accuracy and efficiency.Numerical results show that the accuracy and efficiency of the proposed method are better than those of the existing precise integration method,the Newmark-βmethod and the Wilson-θmethod.展开更多
An improved precise integration method （IPIM） for solving the differential Riccati equation （DRE） is presented. The solution to the DRE is connected with the exponential of a Hamiltonian matrix, and the precise in...An improved precise integration method （IPIM） for solving the differential Riccati equation （DRE） is presented. The solution to the DRE is connected with the exponential of a Hamiltonian matrix, and the precise integration method （PIM） for solving the DRE is connected with the scaling and squaring method for computing the exponential of a matrix. The error analysis of the scaling and squaring method for the exponential of a matrix is applied to the PIM of the DRE. Based ,on the error analysis, the criterion for choosing two parameters of the PIM is given. Three kinds of IPIMs for solving the DRE are proposed. The numerical examples machine accuracy solutions. show that the IPIM is stable and gives the展开更多
The quasi-Shannon interval wavelet is constructed based on the interpolation wavelet theory, and an adaptive precise integration method, which is based on extrapolation method is presented for nonlinear ordinary diffe...The quasi-Shannon interval wavelet is constructed based on the interpolation wavelet theory, and an adaptive precise integration method, which is based on extrapolation method is presented for nonlinear ordinary differential equations ( ODEs). And then, an adaptive interval wavelet precise integration method (AIWPIM) for nonlinear partial differential equations(PDEs) is proposed. The numerical results show that the computational precision of AIWPIM is higher than that of the method constructed by combining the wavelet and the 4th Runge-Kutta method, and the computational amounts of these two methods are almost equal. For convenience, the Burgers equation is taken as an example in introducing this method, which is also valid for more general cases.展开更多
In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the met...In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the method of variable coefficient dimensional expanding,the non-homogeneous ordinary differential equations（ODEs） are transformed into homogeneous ODEs.Then the interval is divided evenly,and the transfer matrix in every subinterval is worked out using the high order multiple perturbation method,and a set of algebraic equations is given in the form of matrix by the precise integration relation for each segment,which is worked out by the reduction method.Finally numerical examples are elaboratedd to validate the present method.展开更多
General purpose graphic processing unit （GPU） calculation technology is gradually widely used in various fields. Its mode of single instruction, multiple threads is capable of seismic numerical simulation which has ...General purpose graphic processing unit （GPU） calculation technology is gradually widely used in various fields. Its mode of single instruction, multiple threads is capable of seismic numerical simulation which has a huge quantity of data and calculation steps. In this study, we introduce a GPU-based parallel calculation method of a precise integration method （PIM） for seismic forward modeling. Compared with CPU single-core calculation, GPU parallel calculating perfectly keeps the features of PIM, which has small bandwidth, high accuracy and capability of modeling complex substructures, and GPU calculation brings high computational efficiency, which means that high-performing GPU parallel calculation can make seismic forward modeling closer to real seismic records.展开更多
This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matr...This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method.展开更多
By using the precise integration method, the numerical solution of linear quadratic Gaussian (LQG) optimal control problem was discussed. Based on the separation principle, the LQG central problem decomposes, or separ...By using the precise integration method, the numerical solution of linear quadratic Gaussian (LQG) optimal control problem was discussed. Based on the separation principle, the LQG central problem decomposes, or separates, into an optimal state-feedback control problem and an optimal state estimation problem. That is the off-line solution of two sets of Riccati differential equations and the on-line integration solution of the state vector from a set of time-variant differential equations. The present algorithms are not only appropriate to solve the two-point boundary-value problem and the corresponding Riccati differential equation, but also can be used to solve the estimated state from the time-variant differential equations. The high precision of precise integration is of advantage for the control and estimation. Numerical examples demonstrate the high precision and effectiveness of the algorithm.展开更多
The Non-uniform rational B-spline （NURBS） enhanced scaled boundary finite element method in combination with the modified precise integration method is proposed for the transient heat conduction problems in this pap...The Non-uniform rational B-spline （NURBS） enhanced scaled boundary finite element method in combination with the modified precise integration method is proposed for the transient heat conduction problems in this paper. The scaled boundary finite element method is a semi-analytical technique, which weakens the governing differential equations along the circumferential direction and solves those analytically in the radial direction. In this method, only the boundary is discretized in the finite element sense leading to a re- duction of the spatial dimension by one with no fundamental solution required. Neverthe- less, in case of the complex geometry, a huge number of elements are generally required to properly approximate the exact shape of the domain and distorted meshes are often un- avoidable in the conventional finite element approach, which leads to huge computational efforts and loss of accuracy. NURBS are the most popular mathematical tool in CAD industry due to its flexibility to fit any free-form shape. In the proposed methodology, the arbitrary curved boundary of problem domain is exactly represented with NURBS basis functions, while the straight part of the boundary is discretized by the conventional Lagrange shape functions. Both the concepts of isogeometric analysis and scaled boundary finite element method are combined to form the governing equations of transient heat conduction analy- sis and the solution is obtained using the modified precise integration method. The stiffness matrix is obtained from a standard quadratic eigenvalue problem and the mass matrix is determined from the low-frequency expansion. Finally the governing equations become a system of first-order ordinary differential equations and the time domain response is solved numerically by the modified precise integration method. The accuracy and stability of the proposed method to deal with the transient heat conduction problems are demonstrated by numerical examples.展开更多
Precise integration methods to solve structural dynamic responses and the corresponding time integration formula are composed of two parts： the multiplication of an exponential matrix with a vector and the integratio...Precise integration methods to solve structural dynamic responses and the corresponding time integration formula are composed of two parts： the multiplication of an exponential matrix with a vector and the integration term. The second term can be solved by the series solution. Two hybrid granularity parallel algorithms are designed, that is, the exponential matrix and the first term are computed by the fine-grained parallel algorithra and the second term is computed by the coarse-grained parallel algorithm. Numerical examples show that these two hybrid granularity parallel algorithms obtain higher speedup and parallel efficiency than two existing parallel algorithms.展开更多
To solve the Riccati equation of LQ control problem, the computation of interval mixed variable energy matrices is the first step. Taylor expansion can be used to compute the matrices. According to the analogy between...To solve the Riccati equation of LQ control problem, the computation of interval mixed variable energy matrices is the first step. Taylor expansion can be used to compute the matrices. According to the analogy between structural mechanics and optimal control and the mechanical implication of the matrices, a computational method using state transition matrix of differential equation was presented. Numerical examples are provided to show the effectiveness of the present approach.展开更多
For the constrained nonlinear optimal control problem, by taking the first term of Taylor series, the dynamic equation is linearized. Thus by, introducing into the dual variable (Lagrange multiplier vector), the dynam...For the constrained nonlinear optimal control problem, by taking the first term of Taylor series, the dynamic equation is linearized. Thus by, introducing into the dual variable (Lagrange multiplier vector), the dynamic equation can be transformed into Hamilton system from Lagrange system on the basis of the original variable. Under the whole state, the problem discussed can be described from a new view, and the equation can be precisely solved by, the time precise integration method established in linear dynamic system. A numerical example shows the effectiveness of the method.展开更多
This paper presented a new convenient method for viscoelastic problems which are generally solved through complex Laplace transform. State space equation is derived from the differential form of constitutive equation ...This paper presented a new convenient method for viscoelastic problems which are generally solved through complex Laplace transform. State space equation is derived from the differential form of constitutive equation of linear viscoelastic solid that can be solved by a precise integration method which is used in many fields with the advantages of high precision and convenience. For linear viscoelastic solids and crack, the finite elements program of the precise integration method is developed, which appears to be efficient and precise.展开更多
Wave equation migration is often applied to solve seismic imaging problems. Usually, the finite difference method is used to obtain the numerical solution of the wave equation. In this paper, the arbitrary difference ...Wave equation migration is often applied to solve seismic imaging problems. Usually, the finite difference method is used to obtain the numerical solution of the wave equation. In this paper, the arbitrary difference precise integration (ADPI) method is discussed and applied in seismic migration. The ADPI method has its own distinctive idea. When dispersing coordinates in the space domain, it employs a relatively unrestrained form instead of the one used by the conventional finite difference method. Moreover, in the time domain it adopts the sub domain precise integration method. As a result, it not only takes the merits of high precision and narrow bandwidth, but also can process various boundary conditions and describe the feature of an inhomogeneous medium better. Numerical results show the benefit of the presented algorithm using the ADPI method.展开更多
The vehicle-road coupling dynamics problem is a prominent issue in transportation,drawing significant attention in recent years.These dynamic equations are characterized by high-dimensionality,coupling,and time-varyin...The vehicle-road coupling dynamics problem is a prominent issue in transportation,drawing significant attention in recent years.These dynamic equations are characterized by high-dimensionality,coupling,and time-varying dynamics,making the exact solutions challenging to obtain.As a result,numerical integration methods are typically employed.However,conventional methods often suffer from low computational efficiency.To address this,this paper explores the application of the parameter freezing precise exponential integrator to vehicle-road coupling models.The model accounts for road roughness irregularities,incorporating all terms unrelated to the linear part into the algorithm's inhomogeneous vector.The general construction process of the algorithm is detailed.The validity of numerical results is verified through approximate analytical solutions(AASs),and the advantages of this method over traditional numerical integration methods are demonstrated.Multiple parameter freezing precise exponential integrator schemes are constructed based on the Runge-Kutta framework,with the fourth-order four-stage scheme identified as the optimal one.The study indicates that this method can quickly and accurately capture the dynamic system's vibration response,offering a new,efficient approach for numerical studies of high-dimensional vehicle-road coupling systems.展开更多
基金supported by the National Science and Technology Major Project of China(Grant No. 2011ZX05004-003,2011ZX05014-006-006)the National Key Basic Research Program of China(Grant No. 2013CB228602)the Natural Science Foundation of China(Grant No. 40974066)
文摘The Finite Difference （FD） method is an important method for seismic numerical simulations. It helps us understand regular patterns in seismic wave propagation, analyze seismic attributes, and interpret seismic data. However, because of its discretization, the FD method is only stable under certain conditions. The Arbitrary Difference Precise Integration （ADPI） method is based on the FD method and adopts an integration scheme in the time domain and an arbitrary difference scheme in the space domain. Therefore, the ADPI method is a semi-analytical method. In this paper, we deduce the formula for the ADPI method based on the 3D elastic equation and improve its stability. In forward modeling cases, the ADPI method was implemented in 2D and 3D elastic wave equation forward modeling. Results show that the travel time of the reflected seismic wave is accurate. Compared with the acoustic wave field, the elastic wave field contains more wave types, including PS- and PP- reflected waves, transmitted waves, and diffracted waves, which is important to interpretation of seismic data. The method can be easily applied to elastic wave equation numerical simulations for eoloical models.
基金supported by the National Natural Science Foundation of China (Nos. 10902020 and 10721062)
文摘A fast precise integration method is developed for the time integral of the hyperbolic heat conduction problem. The wave nature of heat transfer is used to analyze the structure of the matrix exponential, leading to the fact that the matrix exponential is sparse. The presented method employs the sparsity of the matrix exponential to improve the original precise integration method. The merits are that the proposed method is suitable for large hyperbolic heat equations and inherits the accuracy of the original version and the good computational efficiency, which are verified by two numerical examples.
基金the National Natural Science Foundation of China (No.19872016,19872017)the National Key Basic Research Special Foundation (G1999032805)the Foundation for University Key Teachers by the Ministry of Education of China
文摘This paper presents an improved precise integration algorithm fortransient analysis of heat transfer and some other problems. Theoriginal precise integration method is improved by means of the inve-rse accuracy analysis so that the parameter N, which has been takenas a constant and an independent pa- rameter without consideration ofthe problems in the original method, can be generated automaticallyby the algorithm itself.
基金Project supported by the National Natural Science Foundation of China(Nos.60273048and60174023).
文摘Nonlinear dynamic equations can be solved accurately using a precise integration method. Some algorithms exist, but the inversion of a matrix must be calculated for these al- gorithms. If the inversion of the matrix doesn’t exist or isn’t stable, the precision and stability of the algorithms will be afected. An explicit series solution of the state equation has been pre- sented. The solution avoids calculating the inversion of a matrix and its precision can be easily controlled. In this paper, an implicit series solution of nonlinear dynamic equations is presented. The algorithm is more precise and stable than the explicit series solution and isn’t sensitive to the time-step. Finally, a numerical example is presented to demonstrate the efectiveness of the algorithm.
文摘This paper presents a finite element procedure for solving transient, multidimensional convection-diffusion equations. The procedure is based on the characteristic Galerkin method with an implicit algorithm using precise integration method. With the operator splitting procedure, the precise integration method is introduced to determine the material derivative in the convection-diffusion equation, consequently, the physical quantities of material points. An implicit algorithm with a combination of both the precise and the traditional numerical integration procedures in time domain in the Lagrange coordinates for the characteristic Galerkin method is formulated. The stability analysis of the algorithm shows that the unconditional stability of present implicit algorithm is enhanced as compared with that of the traditional implicit numerical integration procedure. The numerical results validate the presented method in solving convection-diffusion equations. As compared with SUPG method and explicit characteristic Galerkin method, the present method gives the results with higher accuracy and better stability.
文摘The objective of the paper is to develop a new algorithm for numerical solution of dynamic elastic-plastic strain hardening/softening problems. The gradient dependent model is adopted in the numerical model to overcome the result mesh-sensitivity problem in the dynamic strain softening or strain localization analysis. The equations for the dynamic elastic-plastic problems are derived in terms of the parametric variational principle, which is valid for associated, non-associated and strain softening plastic constitutive models in the finite element analysis. The precise integration method, which has been widely used for discretization in time domain of the linear problems, is introduced for the solution of dynamic nonlinear equations. The new algorithm proposed is based on the combination of the parametric quadratic programming method and the precise integration method and has all the advantages in both of the algorithms. Results of numerical examples demonstrate not only the validity, but also the advantages of the algorithm proposed for the numerical solution of nonlinear dynamic problems.
基金the support of the National Natural Science Foundation of China(Grant Nos.11472067 and 51609034)the Science Foundation of Liaoning Province of China(No.2021-MS-119)+1 种基金the Dalian Youth Science and Technology Star Project(No.2018RQ06)the Fundamental Research Funds for the Central Universities(Grant No.DUT20GJ216).
文摘The difficulty in solving stochastic dynamics problems lies in the need for a large number of repeated computations of deterministic dynamic equations,which has been a challenge in stochastic dynamics analysis and was discussed in this study.To efficiently and accurately compute the exponential of the dynamics state matrix and the matrix functions due to external loads,an adaptively filtered precise integration method was proposed,which inherits the high precision of the precise integrationmethod,improves the computational efficiency and saves the memory required.Moreover,the perturbation method was introduced to avoid repeated computations of matrix exponential and terms due to external loads.Based on the filtering and perturbation techniques,an adaptively filtered precise integration method considering perturbation for stochastic dynamics problems was developed.Two numerical experiments,including a model of phononic crystal and a bridge model considering random parameters,were performed to test the performance of the proposed method in terms of accuracy and efficiency.Numerical results show that the accuracy and efficiency of the proposed method are better than those of the existing precise integration method,the Newmark-βmethod and the Wilson-θmethod.
基金Project supported by the National Natural Science Foundation of China(Nos.10902020 and 10721062)
文摘An improved precise integration method （IPIM） for solving the differential Riccati equation （DRE） is presented. The solution to the DRE is connected with the exponential of a Hamiltonian matrix, and the precise integration method （PIM） for solving the DRE is connected with the scaling and squaring method for computing the exponential of a matrix. The error analysis of the scaling and squaring method for the exponential of a matrix is applied to the PIM of the DRE. Based ,on the error analysis, the criterion for choosing two parameters of the PIM is given. Three kinds of IPIMs for solving the DRE are proposed. The numerical examples machine accuracy solutions. show that the IPIM is stable and gives the
文摘The quasi-Shannon interval wavelet is constructed based on the interpolation wavelet theory, and an adaptive precise integration method, which is based on extrapolation method is presented for nonlinear ordinary differential equations ( ODEs). And then, an adaptive interval wavelet precise integration method (AIWPIM) for nonlinear partial differential equations(PDEs) is proposed. The numerical results show that the computational precision of AIWPIM is higher than that of the method constructed by combining the wavelet and the 4th Runge-Kutta method, and the computational amounts of these two methods are almost equal. For convenience, the Burgers equation is taken as an example in introducing this method, which is also valid for more general cases.
基金supported by the National Natural Science Foundation of China (11132004 and 51078145)the Natural Science Foundation of Guangdong Province (9251064101000016)
文摘In this paper we present a precise integration method based on high order multiple perturbation method and reduction method for solving a class of singular twopoint boundary value problems.Firstly,by employing the method of variable coefficient dimensional expanding,the non-homogeneous ordinary differential equations（ODEs） are transformed into homogeneous ODEs.Then the interval is divided evenly,and the transfer matrix in every subinterval is worked out using the high order multiple perturbation method,and a set of algebraic equations is given in the form of matrix by the precise integration relation for each segment,which is worked out by the reduction method.Finally numerical examples are elaboratedd to validate the present method.
基金supported by the National Natural Science Foundation of China (Nos 40974066 and 40821062)National Basic Research Program of China (No 2007CB209602)
文摘General purpose graphic processing unit （GPU） calculation technology is gradually widely used in various fields. Its mode of single instruction, multiple threads is capable of seismic numerical simulation which has a huge quantity of data and calculation steps. In this study, we introduce a GPU-based parallel calculation method of a precise integration method （PIM） for seismic forward modeling. Compared with CPU single-core calculation, GPU parallel calculating perfectly keeps the features of PIM, which has small bandwidth, high accuracy and capability of modeling complex substructures, and GPU calculation brings high computational efficiency, which means that high-performing GPU parallel calculation can make seismic forward modeling closer to real seismic records.
基金Project supported by the National Natural Science Foundation of China(No.10672194)the China-Russia Cooperative Project(the National Natural Science Foundation of China and the Russian Foundation for Basic Research)(No.10811120012)
文摘This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method.
文摘By using the precise integration method, the numerical solution of linear quadratic Gaussian (LQG) optimal control problem was discussed. Based on the separation principle, the LQG central problem decomposes, or separates, into an optimal state-feedback control problem and an optimal state estimation problem. That is the off-line solution of two sets of Riccati differential equations and the on-line integration solution of the state vector from a set of time-variant differential equations. The present algorithms are not only appropriate to solve the two-point boundary-value problem and the corresponding Riccati differential equation, but also can be used to solve the estimated state from the time-variant differential equations. The high precision of precise integration is of advantage for the control and estimation. Numerical examples demonstrate the high precision and effectiveness of the algorithm.
基金support by the National Natural Science Foundation of China(grant No.51779033,51409038)the National Key Research and Development Plan(grant No.2016YFB0201001)the National Natural Science Foundation of China(grant No.51421064)
文摘The Non-uniform rational B-spline （NURBS） enhanced scaled boundary finite element method in combination with the modified precise integration method is proposed for the transient heat conduction problems in this paper. The scaled boundary finite element method is a semi-analytical technique, which weakens the governing differential equations along the circumferential direction and solves those analytically in the radial direction. In this method, only the boundary is discretized in the finite element sense leading to a re- duction of the spatial dimension by one with no fundamental solution required. Neverthe- less, in case of the complex geometry, a huge number of elements are generally required to properly approximate the exact shape of the domain and distorted meshes are often un- avoidable in the conventional finite element approach, which leads to huge computational efforts and loss of accuracy. NURBS are the most popular mathematical tool in CAD industry due to its flexibility to fit any free-form shape. In the proposed methodology, the arbitrary curved boundary of problem domain is exactly represented with NURBS basis functions, while the straight part of the boundary is discretized by the conventional Lagrange shape functions. Both the concepts of isogeometric analysis and scaled boundary finite element method are combined to form the governing equations of transient heat conduction analy- sis and the solution is obtained using the modified precise integration method. The stiffness matrix is obtained from a standard quadratic eigenvalue problem and the mass matrix is determined from the low-frequency expansion. Finally the governing equations become a system of first-order ordinary differential equations and the time domain response is solved numerically by the modified precise integration method. The accuracy and stability of the proposed method to deal with the transient heat conduction problems are demonstrated by numerical examples.
基金the National Natural Science Foundation of China(No.60273048).
文摘Precise integration methods to solve structural dynamic responses and the corresponding time integration formula are composed of two parts： the multiplication of an exponential matrix with a vector and the integration term. The second term can be solved by the series solution. Two hybrid granularity parallel algorithms are designed, that is, the exponential matrix and the first term are computed by the fine-grained parallel algorithra and the second term is computed by the coarse-grained parallel algorithm. Numerical examples show that these two hybrid granularity parallel algorithms obtain higher speedup and parallel efficiency than two existing parallel algorithms.
文摘To solve the Riccati equation of LQ control problem, the computation of interval mixed variable energy matrices is the first step. Taylor expansion can be used to compute the matrices. According to the analogy between structural mechanics and optimal control and the mechanical implication of the matrices, a computational method using state transition matrix of differential equation was presented. Numerical examples are provided to show the effectiveness of the present approach.
文摘For the constrained nonlinear optimal control problem, by taking the first term of Taylor series, the dynamic equation is linearized. Thus by, introducing into the dual variable (Lagrange multiplier vector), the dynamic equation can be transformed into Hamilton system from Lagrange system on the basis of the original variable. Under the whole state, the problem discussed can be described from a new view, and the equation can be precisely solved by, the time precise integration method established in linear dynamic system. A numerical example shows the effectiveness of the method.
基金National Natural Science Foundation ofChina(No.10 172 0 78)
文摘This paper presented a new convenient method for viscoelastic problems which are generally solved through complex Laplace transform. State space equation is derived from the differential form of constitutive equation of linear viscoelastic solid that can be solved by a precise integration method which is used in many fields with the advantages of high precision and convenience. For linear viscoelastic solids and crack, the finite elements program of the precise integration method is developed, which appears to be efficient and precise.
文摘Wave equation migration is often applied to solve seismic imaging problems. Usually, the finite difference method is used to obtain the numerical solution of the wave equation. In this paper, the arbitrary difference precise integration (ADPI) method is discussed and applied in seismic migration. The ADPI method has its own distinctive idea. When dispersing coordinates in the space domain, it employs a relatively unrestrained form instead of the one used by the conventional finite difference method. Moreover, in the time domain it adopts the sub domain precise integration method. As a result, it not only takes the merits of high precision and narrow bandwidth, but also can process various boundary conditions and describe the feature of an inhomogeneous medium better. Numerical results show the benefit of the presented algorithm using the ADPI method.
基金Supported by the National Natural Science Foundation of China(No.U22A20246)the Key Project of Natural Science Foundation of Hebei Province of China(Basic Research Base Project)(No.A2023210064)the Science and Technology Program of Hebei Province of China(Nos.246Z1904G and 225676162GH)。
文摘The vehicle-road coupling dynamics problem is a prominent issue in transportation,drawing significant attention in recent years.These dynamic equations are characterized by high-dimensionality,coupling,and time-varying dynamics,making the exact solutions challenging to obtain.As a result,numerical integration methods are typically employed.However,conventional methods often suffer from low computational efficiency.To address this,this paper explores the application of the parameter freezing precise exponential integrator to vehicle-road coupling models.The model accounts for road roughness irregularities,incorporating all terms unrelated to the linear part into the algorithm's inhomogeneous vector.The general construction process of the algorithm is detailed.The validity of numerical results is verified through approximate analytical solutions(AASs),and the advantages of this method over traditional numerical integration methods are demonstrated.Multiple parameter freezing precise exponential integrator schemes are constructed based on the Runge-Kutta framework,with the fourth-order four-stage scheme identified as the optimal one.The study indicates that this method can quickly and accurately capture the dynamic system's vibration response,offering a new,efficient approach for numerical studies of high-dimensional vehicle-road coupling systems.
