Level set methods are widely used for predicting evolutions of complex free surface topologies,such as the crystal and crack growth,bubbles and droplets deformation,spilling and breaking waves,and two-phase flow pheno...Level set methods are widely used for predicting evolutions of complex free surface topologies,such as the crystal and crack growth,bubbles and droplets deformation,spilling and breaking waves,and two-phase flow phenomena.This paper presents a characteristic level set equation which is derived from the two-dimensional level set equation by using the characteristic-based scheme.An explicit finite volume element method is developed to discretize the equation on triangular grids.Several examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time.The proposed level set method is also coupled with the Navier-Stokes equations for two-phase immiscible incompressible flow analysis with surface tension.The Rayleigh-Taylor instability problem is used to test and evaluate the effectiveness of the proposed scheme.展开更多
We study numerical methods for level set like equations arising in image processing and curve evolution problems. Semi-implicit finite volume-element type schemes are constructed for the general level set like equati...We study numerical methods for level set like equations arising in image processing and curve evolution problems. Semi-implicit finite volume-element type schemes are constructed for the general level set like equation （image selective smoothing model） given by Alvarez et al. （Alvarez L, Lions P L, Morel J M. Image selective smoothing and edge detection by nonlinear diffusion II. SIAM J. Numer. Anal., 1992, 29： 845-866）. Through the reasonable semi-implicit discretization in time and co-volume method for space approximation, we give finite volume schemes, unconditionally stable in L∞ and W1＇2 （W1＇1） sense in isotropic （anisotropic） diffu- sion domain.展开更多
On condition that the basic equations set of atmospheric motion possesses the best stability in the smooth function classes, the structure of solution space for local analytical solution is discussed, by which the thi...On condition that the basic equations set of atmospheric motion possesses the best stability in the smooth function classes, the structure of solution space for local analytical solution is discussed, by which the third-class initial value problem with typ- icality and application is analyzed. The calculational method and concrete expressions of analytical solution about the well-posed initial value problem of the third-kind are given in the analytic function classes. Near an appointed point, the relevant theoretical and computational problems about analytical solution of initial value problem are solved completely in the meaning of local solution. Moreover, for other type ofproblems for determining solution, the computational method and process of their stable analytical solution can be obtained in a similar way given in this paper.展开更多
Because of the extensive applications of nonlinear ordinary differential equation in physics,mechanics and cybernetics,there have been many papers on the exact solution to differential equation in some major publicati...Because of the extensive applications of nonlinear ordinary differential equation in physics,mechanics and cybernetics,there have been many papers on the exact solution to differential equation in some major publications both at home and abroad in recent years Based on these papers and in virtue of Leibniz formula,and transformation set technique,this paper puts forth the solution to nonlinear ordinary differential equation set of higher-orders, moveover,its integrability is proven.The results obtained are the generalization of those in the references.展开更多
The structural damage identification through modal data often leads to solving a set of linear equations. Special numerical treatment is sometimes required for an accurate and stable solution owing to the ill conditio...The structural damage identification through modal data often leads to solving a set of linear equations. Special numerical treatment is sometimes required for an accurate and stable solution owing to the ill conditioning of the equations. Based on the singular value decomposition (SVD) of the coefficient matrix, an error based truncation algorithm is proposed in this paper. By rejection of selected small singular values, the influence of noise can be reduced. A simply-supported beam is used as a simulation example to compare the results to other methods. Illustrative numerical examples demonstrate the good efficiency and stability of the algorithm in the nondestructive identification of structural damage through modal data.展开更多
§1.Problem and AssumptionsThis paper deals with the solutions of the following differential inclusion problem:fAuEf(x,u),xEn;lu=0.xEa0,(1)whereAu(x)=-EDi[a;(x,Du(x))],1mQ C R'N is a bounded domain with piecew...§1.Problem and AssumptionsThis paper deals with the solutions of the following differential inclusion problem:fAuEf(x,u),xEn;lu=0.xEa0,(1)whereAu(x)=-EDi[a;(x,Du(x))],1mQ C R'N is a bounded domain with piecewise Lipschitz boundary a2,Du=(D1u,D2u,…,auDNu),Du=ax,i=1,2,…,N,and f:2×R→2R is a set-valued function.Let p≥2 be an integer and q its conjugate exponent,that is,141=1.Let W1."(2)denotepqthe usual Sobolev space and(W1.P(2))*its dual space,<·,·>the duality pairing between theelements of(W1·P(2))*and W1.P(2),||·||1.and||·|,the norms in W1·P(2)and inL'(2)respectively.A partial ordering is defined in LP(2)and in W1.P(2)by u»if and onlyif v–u belongs to the set L'.(Q)of all nonnegative elements of LP(2).展开更多
In the present paper, geometry of the Boolean space B<sup>n</sup> in terms of Hausdorff distances between subsets and subset sums is investigated. The main results are the algebraic and analytical expressi...In the present paper, geometry of the Boolean space B<sup>n</sup> in terms of Hausdorff distances between subsets and subset sums is investigated. The main results are the algebraic and analytical expressions for representing of classical figures in B<sup>n</sup> and the functions of distances between them. In particular, equations in sets are considered and their interpretations in combinatory terms are given.展开更多
In the paper ,we study longtime dynamic behavior of dissipative soliton equation existence of attractor ,geometrical structure of attractor dynamic behavior under the parametric perturbation of dissipative soliton equ...In the paper ,we study longtime dynamic behavior of dissipative soliton equation existence of attractor ,geometrical structure of attractor dynamic behavior under the parametric perturbation of dissipative soliton equation.estimate of fractal dimension of attractor.展开更多
In this paper,we develop strict stability concepts of ODE to impulsive hybrid set valued differential equations.By Lyapunov’s original method,we get some basic strict stability criteria of impulsive hybrid set valued...In this paper,we develop strict stability concepts of ODE to impulsive hybrid set valued differential equations.By Lyapunov’s original method,we get some basic strict stability criteria of impulsive hybrid set valued equations.展开更多
Stability related to theoretical model for catastrophic weather prediction, which includes non-hydrostatic perfect elastic model and anelastic model, is discussed and analyzed in detail. It is proved that non-hydrosta...Stability related to theoretical model for catastrophic weather prediction, which includes non-hydrostatic perfect elastic model and anelastic model, is discussed and analyzed in detail. It is proved that non-hydrostatic perfect elastic equations set is stable in the class of infinitely differentiable function. However, for the anelastic equations set, its continuity equation is changed in form because of the particular hypothesis for fluid, so ＂the matching consisting of both viscosity coefficient and incompressible assumption＂ appears, thereby the most important equations set of this class in practical prediction shows the same instability in topological property as Navier-Stokes equation, which should be avoided first in practical numerical prediction. In light of this, the referenced suggestions to amend the applied model are finally presented.展开更多
The transient simulation technology of natural gas pipeline networks plays an increasingly prominent role in the scheduling management of natural gas pipeline network system.The increasingly large and complex natural ...The transient simulation technology of natural gas pipeline networks plays an increasingly prominent role in the scheduling management of natural gas pipeline network system.The increasingly large and complex natural gas pipeline network requires more strictly on the calculation efficiency of transient simulation.To this end,this paper proposes a new method for the transient simulation of natural gas pipeline networks based on fracture-dimension-reduction algorithm.Firstly,a pipeline network model is abstracted into a station model,inter-station pipeline network model and connection node model.Secondlly,the pressure at the connection node connecting the station and the inter-station pipeline network is used as the basic variable to solve the general solution of the flow rate at the connection node,reconstruct the simulation model of the inter-station pipeline network,and reduce the equation set dimension of the inter-station pipeline network model.Thirdly,the transient simulation model of the natural gas pipeline network system is constructed based on the reconstructed simulation model of the inter-station pipeline network.Fnally,the calculation accuracy and efficiency of the proposed algorithm are compared and analyzed for the two working conditions of slow change of compressor speed and rapid shutdown of the compressor.And the following research results are obtained.First,the fracture-dimension-reduction algorithm has a high calculation accuracy,and the relative error of compressor outlet pressure and user pressure is less than 0.1%.Second,the calculation efficiency of the new fracture-dimension-reduction algorithm is high,and compared with the nonlinear equations solv ing method,the speed-up ratios under the two conditions are high up to 17.3 and 12.2 respectively.Third,the speed-up ratio of the fracture-dimension-reduction algorithm is linearly related to the equation set dimension of the transient simulation model of the pipeline network system.The larger the equation set dimension,the higher the speed-up ratio,which means the more complex the pipeline network model,the more remarkable the calculation speed-up effect.In conclusion,this new method improves the calculation speed while keeping the calculation accuracy,which is of great theoretical value and reference significance for improving the calculation efficiency of the transient simulation of complex natural gas pipeline network systems.展开更多
The principle aim of this essay is to illustrate how different phenomena is captured by different discretizations of the Hopf equation and general hyperbolic conservation laws. This includes dispersive schemes, shock ...The principle aim of this essay is to illustrate how different phenomena is captured by different discretizations of the Hopf equation and general hyperbolic conservation laws. This includes dispersive schemes, shock capturing schemes as well as schemes for computing multi-valued solutions of the underlying equation. We introduce some model equations which describe the behavior of the discrete equation more accurate than the original equation. These model equations can either be conveniently discretized for producing novel numerical schemes or further analyzed to enrich the theory of nonlinear partial differential equations.展开更多
The generalized finite difference method (GFDM) used for irregular grids is first introduced into the numerical study of thelevel set equation, which is coupled with the theory of detonation shock dynamics (DSD) descr...The generalized finite difference method (GFDM) used for irregular grids is first introduced into the numerical study of thelevel set equation, which is coupled with the theory of detonation shock dynamics (DSD) describing the propagation of thedetonation shock front. The numerical results of a rate-stick problem, a converging channel problem and an arc channel prob-lem for specified boundaries show that GFDM is effective on solving the level set equation in the irregular geometrical domain.The arrival time and the normal velocity distribution of the detonation shock front of these problems can then be obtainedconveniently with this method. The numerical results also confirm that when there is a curvature effect, the theory of DSDmust be considered for the propagation of detonation shock surface, while classic Huygens construction is not suitable anymore.展开更多
Let X,Y be Banach spaces, A,D ： X - Y and B,C ： Y - X be the bounded linear operators satisfying operator equation set ACD=DBD DBA=ACAIn this paper, we show that AC and BD share some basic operator properties such a...Let X,Y be Banach spaces, A,D ： X - Y and B,C ： Y - X be the bounded linear operators satisfying operator equation set ACD=DBD DBA=ACAIn this paper, we show that AC and BD share some basic operator properties such as the injectivity and the invertibility. Moreover, we show that AG＇ and BD share many common local spectral properties including SVEP, Bishop property （β） and Dunford property （C）.展开更多
基金King Mongkut’s University of Technology North Bangkok (KMUTNB)the Office of the Higher Education Commission (OHEC)the National Metal and Materials Technology Center (MTEC) for supporting this research work
文摘Level set methods are widely used for predicting evolutions of complex free surface topologies,such as the crystal and crack growth,bubbles and droplets deformation,spilling and breaking waves,and two-phase flow phenomena.This paper presents a characteristic level set equation which is derived from the two-dimensional level set equation by using the characteristic-based scheme.An explicit finite volume element method is developed to discretize the equation on triangular grids.Several examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time.The proposed level set method is also coupled with the Navier-Stokes equations for two-phase immiscible incompressible flow analysis with surface tension.The Rayleigh-Taylor instability problem is used to test and evaluate the effectiveness of the proposed scheme.
文摘We study numerical methods for level set like equations arising in image processing and curve evolution problems. Semi-implicit finite volume-element type schemes are constructed for the general level set like equation （image selective smoothing model） given by Alvarez et al. （Alvarez L, Lions P L, Morel J M. Image selective smoothing and edge detection by nonlinear diffusion II. SIAM J. Numer. Anal., 1992, 29： 845-866）. Through the reasonable semi-implicit discretization in time and co-volume method for space approximation, we give finite volume schemes, unconditionally stable in L∞ and W1＇2 （W1＇1） sense in isotropic （anisotropic） diffu- sion domain.
基金Project supported by the National Natural Science Foundation of China (Major Program of the Tenth Five-Year Plan) (No.90411006).
文摘On condition that the basic equations set of atmospheric motion possesses the best stability in the smooth function classes, the structure of solution space for local analytical solution is discussed, by which the third-class initial value problem with typ- icality and application is analyzed. The calculational method and concrete expressions of analytical solution about the well-posed initial value problem of the third-kind are given in the analytic function classes. Near an appointed point, the relevant theoretical and computational problems about analytical solution of initial value problem are solved completely in the meaning of local solution. Moreover, for other type ofproblems for determining solution, the computational method and process of their stable analytical solution can be obtained in a similar way given in this paper.
文摘Because of the extensive applications of nonlinear ordinary differential equation in physics,mechanics and cybernetics,there have been many papers on the exact solution to differential equation in some major publications both at home and abroad in recent years Based on these papers and in virtue of Leibniz formula,and transformation set technique,this paper puts forth the solution to nonlinear ordinary differential equation set of higher-orders, moveover,its integrability is proven.The results obtained are the generalization of those in the references.
文摘The structural damage identification through modal data often leads to solving a set of linear equations. Special numerical treatment is sometimes required for an accurate and stable solution owing to the ill conditioning of the equations. Based on the singular value decomposition (SVD) of the coefficient matrix, an error based truncation algorithm is proposed in this paper. By rejection of selected small singular values, the influence of noise can be reduced. A simply-supported beam is used as a simulation example to compare the results to other methods. Illustrative numerical examples demonstrate the good efficiency and stability of the algorithm in the nondestructive identification of structural damage through modal data.
文摘§1.Problem and AssumptionsThis paper deals with the solutions of the following differential inclusion problem:fAuEf(x,u),xEn;lu=0.xEa0,(1)whereAu(x)=-EDi[a;(x,Du(x))],1mQ C R'N is a bounded domain with piecewise Lipschitz boundary a2,Du=(D1u,D2u,…,auDNu),Du=ax,i=1,2,…,N,and f:2×R→2R is a set-valued function.Let p≥2 be an integer and q its conjugate exponent,that is,141=1.Let W1."(2)denotepqthe usual Sobolev space and(W1.P(2))*its dual space,<·,·>the duality pairing between theelements of(W1·P(2))*and W1.P(2),||·||1.and||·|,the norms in W1·P(2)and inL'(2)respectively.A partial ordering is defined in LP(2)and in W1.P(2)by u»if and onlyif v–u belongs to the set L'.(Q)of all nonnegative elements of LP(2).
文摘In the present paper, geometry of the Boolean space B<sup>n</sup> in terms of Hausdorff distances between subsets and subset sums is investigated. The main results are the algebraic and analytical expressions for representing of classical figures in B<sup>n</sup> and the functions of distances between them. In particular, equations in sets are considered and their interpretations in combinatory terms are given.
文摘In the paper ,we study longtime dynamic behavior of dissipative soliton equation existence of attractor ,geometrical structure of attractor dynamic behavior under the parametric perturbation of dissipative soliton equation.estimate of fractal dimension of attractor.
基金supported by the National Natural Science Foundation of China(10971045)the Natural Science Foundation of Hebei Province(A2009000151)
文摘In this paper,we develop strict stability concepts of ODE to impulsive hybrid set valued differential equations.By Lyapunov’s original method,we get some basic strict stability criteria of impulsive hybrid set valued equations.
基金Project supported by the National Natural Science Foundation of China (Major Program of the Tenth Five-Year Plan) (No.90411006)the Post-Doctoral Science Foundation of Jiangsu Province of China(No.0602024C)
文摘Stability related to theoretical model for catastrophic weather prediction, which includes non-hydrostatic perfect elastic model and anelastic model, is discussed and analyzed in detail. It is proved that non-hydrostatic perfect elastic equations set is stable in the class of infinitely differentiable function. However, for the anelastic equations set, its continuity equation is changed in form because of the particular hypothesis for fluid, so ＂the matching consisting of both viscosity coefficient and incompressible assumption＂ appears, thereby the most important equations set of this class in practical prediction shows the same instability in topological property as Navier-Stokes equation, which should be avoided first in practical numerical prediction. In light of this, the referenced suggestions to amend the applied model are finally presented.
文摘The transient simulation technology of natural gas pipeline networks plays an increasingly prominent role in the scheduling management of natural gas pipeline network system.The increasingly large and complex natural gas pipeline network requires more strictly on the calculation efficiency of transient simulation.To this end,this paper proposes a new method for the transient simulation of natural gas pipeline networks based on fracture-dimension-reduction algorithm.Firstly,a pipeline network model is abstracted into a station model,inter-station pipeline network model and connection node model.Secondlly,the pressure at the connection node connecting the station and the inter-station pipeline network is used as the basic variable to solve the general solution of the flow rate at the connection node,reconstruct the simulation model of the inter-station pipeline network,and reduce the equation set dimension of the inter-station pipeline network model.Thirdly,the transient simulation model of the natural gas pipeline network system is constructed based on the reconstructed simulation model of the inter-station pipeline network.Fnally,the calculation accuracy and efficiency of the proposed algorithm are compared and analyzed for the two working conditions of slow change of compressor speed and rapid shutdown of the compressor.And the following research results are obtained.First,the fracture-dimension-reduction algorithm has a high calculation accuracy,and the relative error of compressor outlet pressure and user pressure is less than 0.1%.Second,the calculation efficiency of the new fracture-dimension-reduction algorithm is high,and compared with the nonlinear equations solv ing method,the speed-up ratios under the two conditions are high up to 17.3 and 12.2 respectively.Third,the speed-up ratio of the fracture-dimension-reduction algorithm is linearly related to the equation set dimension of the transient simulation model of the pipeline network system.The larger the equation set dimension,the higher the speed-up ratio,which means the more complex the pipeline network model,the more remarkable the calculation speed-up effect.In conclusion,this new method improves the calculation speed while keeping the calculation accuracy,which is of great theoretical value and reference significance for improving the calculation efficiency of the transient simulation of complex natural gas pipeline network systems.
基金the National Science Foundation under Grant DMS05-05975.
文摘The principle aim of this essay is to illustrate how different phenomena is captured by different discretizations of the Hopf equation and general hyperbolic conservation laws. This includes dispersive schemes, shock capturing schemes as well as schemes for computing multi-valued solutions of the underlying equation. We introduce some model equations which describe the behavior of the discrete equation more accurate than the original equation. These model equations can either be conveniently discretized for producing novel numerical schemes or further analyzed to enrich the theory of nonlinear partial differential equations.
基金supported by the National Natural Science Foundation of China (Grant No. 11002029)
文摘The generalized finite difference method (GFDM) used for irregular grids is first introduced into the numerical study of thelevel set equation, which is coupled with the theory of detonation shock dynamics (DSD) describing the propagation of thedetonation shock front. The numerical results of a rate-stick problem, a converging channel problem and an arc channel prob-lem for specified boundaries show that GFDM is effective on solving the level set equation in the irregular geometrical domain.The arrival time and the normal velocity distribution of the detonation shock front of these problems can then be obtainedconveniently with this method. The numerical results also confirm that when there is a curvature effect, the theory of DSDmust be considered for the propagation of detonation shock surface, while classic Huygens construction is not suitable anymore.
基金Supported by National Natural Science Foundation of China(Grant No.11371279)
文摘Let X,Y be Banach spaces, A,D ： X - Y and B,C ： Y - X be the bounded linear operators satisfying operator equation set ACD=DBD DBA=ACAIn this paper, we show that AC and BD share some basic operator properties such as the injectivity and the invertibility. Moreover, we show that AG＇ and BD share many common local spectral properties including SVEP, Bishop property （β） and Dunford property （C）.
