The oscillation property （OP） is a fundamental and important qualitative property for the vibrations of single span one-dimensional continuums such as strings, bars, torsion bars, and Euler beams. Any properly discr...The oscillation property （OP） is a fundamental and important qualitative property for the vibrations of single span one-dimensional continuums such as strings, bars, torsion bars, and Euler beams. Any properly discretized continuum model should keep the OP. In literatures, the OP of discrete beam models is discussed essentially by means of matrix factorization. The discussion is model-specific and boundary-condition- specific. Besides, matrix factorization is difficult in handling finite element （FE） models of beams. In this paper, according to a sufficient condition for the OP, a new approach to discuss the property is proposed. The local criteria on discrete displacements rather than global matrix factorizations are given to verify the OP. Based on the proposed approach, known results such as the OP for the 2-node FE beams via the Heilinger- Reissener principle （HR-FE beams） as well as the 5-point finite difference （FD） beams are verified. New results on the OP for the 2-node PE-FE beams and the FE Timoshenko beams with small slenderness are given. Through a simple manipulation, the qualitative property of discrete multibearing beams can also be discussed by the proposed approach.展开更多
Here we investigate the kinematic transports of the defects in the nematic liquid crystal system by numerical experiments.The model is a shear flow case of the viscoelastic continuummodel simplified fromthe Ericksen-L...Here we investigate the kinematic transports of the defects in the nematic liquid crystal system by numerical experiments.The model is a shear flow case of the viscoelastic continuummodel simplified fromthe Ericksen-Leslie system.The numerical experiments are carried out by using a differencemethod.Based on these numerical experiments we find some interesting and important relationships between the kinematic transports and the characteristics of the flow.We present the development and interaction of the defects.These results are partly consistent with the observation from the experiments.Thus this scheme illustrates,to some extent,the kinematic effects of the defects.展开更多
Finite difference computations that involve spatial adaptation commonly employ an equidistribution principle.In these cases,a new mesh is constructed such that a given monitor function is equidistributed in some sense...Finite difference computations that involve spatial adaptation commonly employ an equidistribution principle.In these cases,a new mesh is constructed such that a given monitor function is equidistributed in some sense.Typical choices of the monitor function involve the solution or one of its many derivatives.This straightforward concept has proven to be extremely effective and practical.However,selections of core monitoring functions are often challenging and crucial to the computational success.This paper concerns six different designs of the monitoring function that targets a highly nonlinear partial differential equation that exhibits both quenching-type and degeneracy singularities.While the first four monitoring strategies are within the so-called primitive regime,the rest belong to a later category of the modified type,which requires the priori knowledge of certain important quenching solution characteristics.Simulated examples are given to illustrate our study and conclusions.展开更多
In this paper,we study numerically quantized vortex dynamics and their interaction in the two-dimensional(2D)Ginzburg-Landau equation(GLE)with a dimensionless parameter#>0 on bounded domains under either Dirichlet ...In this paper,we study numerically quantized vortex dynamics and their interaction in the two-dimensional(2D)Ginzburg-Landau equation(GLE)with a dimensionless parameter#>0 on bounded domains under either Dirichlet or homogeneous Neumann boundary condition.We begin with a reviewof the reduced dynamical laws for time evolution of quantized vortex centers in GLE and show how to solve these nonlinear ordinary differential equations numerically.Then we present efficient and accurate numerical methods for discretizing the GLE on either a rectangular or a disk domain under either Dirichlet or homogeneous Neumann boundary condition.Based on these efficient and accurate numerical methods for GLE and the reduced dynamical laws,we simulate quantized vortex interaction of GLE with different#and under different initial setups including single vortex,vortex pair,vortex dipole and vortex lattice,compare them with those obtained from the corresponding reduced dynamical laws,and identify the cases where the reduced dynamical laws agree qualitatively and/or quantitatively as well as fail to agree with those from GLE on vortex interaction.Finally,we also obtain numerically different patterns of the steady states for quantized vortex lattices under the GLE dynamics on bounded domains.展开更多
Great progress has been made in study on dynamic behavior of the damaged structures subject to deterministic excitation.The stochastic response analysis of the damaged structures,however,has not yet attracted people&#...Great progress has been made in study on dynamic behavior of the damaged structures subject to deterministic excitation.The stochastic response analysis of the damaged structures,however,has not yet attracted people's attention.Taking the damaged elastic beams for example,the analysis procedure for stochastic response of the damaged structures subject to stochastic excitations is investigated in this paper.First,the damage constitutive relations and the corresponding damage evolution equation of one-dimensional elastic structures are briefly discussed.Second,the stochastic dynamic equation with respect to transverse displacement of the damaged elastic beams is deduced.The finite difference method and Newmark method are adopted to solve the stochastic partially-differential equation and corresponding boundary conditions.The stochastic response characteristic,damage evolution law,the effect of noise intensity on damage evolution and the first-passage time of damage are discussed in detail.The present work extends the research field of damaged structures,and the proposed procedure can be generalized to analyze the dynamic behavior of more complex structures,such as damaged plates and shells.展开更多
The numerical solution of blow-up problems for nonlinear wave equations on unbounded spatial domains is considered.Applying the unified approach,which is based on the operator splitting method,we construct the efficie...The numerical solution of blow-up problems for nonlinear wave equations on unbounded spatial domains is considered.Applying the unified approach,which is based on the operator splitting method,we construct the efficient nonlinear local absorbing boundary conditions for the nonlinear wave equation,and reduce the nonlinear problem on the unbounded spatial domain to an initial-boundary-value problem on a bounded domain.Then the finite difference method is used to solve the reduced problem on the bounded computational domain.Finally,a broad range of numerical examples are given to demonstrate the effectiveness and accuracy of our method,and some interesting propagation and behaviors of the blow-up problems for nonlinear wave equations are observed.展开更多
基金Project supported by the National Natural Science Foundation of China(Nos.10972005 and 11272011)
文摘The oscillation property （OP） is a fundamental and important qualitative property for the vibrations of single span one-dimensional continuums such as strings, bars, torsion bars, and Euler beams. Any properly discretized continuum model should keep the OP. In literatures, the OP of discrete beam models is discussed essentially by means of matrix factorization. The discussion is model-specific and boundary-condition- specific. Besides, matrix factorization is difficult in handling finite element （FE） models of beams. In this paper, according to a sufficient condition for the OP, a new approach to discuss the property is proposed. The local criteria on discrete displacements rather than global matrix factorizations are given to verify the OP. Based on the proposed approach, known results such as the OP for the 2-node FE beams via the Heilinger- Reissener principle （HR-FE beams） as well as the 5-point finite difference （FD） beams are verified. New results on the OP for the 2-node PE-FE beams and the FE Timoshenko beams with small slenderness are given. Through a simple manipulation, the qualitative property of discrete multibearing beams can also be discussed by the proposed approach.
基金We are grateful to Prof.Qiang Du of Pennsylvania State University and Dr.Yanzhi Zhang of Missouri University of Science and Technology for many helpful discussions.This work was partially donewhile Hui Zhang was visiting National University of Singapore.HereHui Zhang is very grateful to Prof.Weizhu Bao for his hospitable friend.Hui Zhang is partially supported by NSFC grant No.11471046 and NSFC-RGC No.11261160486 and the Ministry of Education Program for New Century Excellent Talents Project NCET-12-0053.
文摘Here we investigate the kinematic transports of the defects in the nematic liquid crystal system by numerical experiments.The model is a shear flow case of the viscoelastic continuummodel simplified fromthe Ericksen-Leslie system.The numerical experiments are carried out by using a differencemethod.Based on these numerical experiments we find some interesting and important relationships between the kinematic transports and the characteristics of the flow.We present the development and interaction of the defects.These results are partly consistent with the observation from the experiments.Thus this scheme illustrates,to some extent,the kinematic effects of the defects.
文摘Finite difference computations that involve spatial adaptation commonly employ an equidistribution principle.In these cases,a new mesh is constructed such that a given monitor function is equidistributed in some sense.Typical choices of the monitor function involve the solution or one of its many derivatives.This straightforward concept has proven to be extremely effective and practical.However,selections of core monitoring functions are often challenging and crucial to the computational success.This paper concerns six different designs of the monitoring function that targets a highly nonlinear partial differential equation that exhibits both quenching-type and degeneracy singularities.While the first four monitoring strategies are within the so-called primitive regime,the rest belong to a later category of the modified type,which requires the priori knowledge of certain important quenching solution characteristics.Simulated examples are given to illustrate our study and conclusions.
基金supported by the Singapore A*STAR SERC“Complex Systems”Research Programme grant 1224504056the Academic Research Fund of Ministry of Education of Singapore grant R-146-000-120-112。
文摘In this paper,we study numerically quantized vortex dynamics and their interaction in the two-dimensional(2D)Ginzburg-Landau equation(GLE)with a dimensionless parameter#>0 on bounded domains under either Dirichlet or homogeneous Neumann boundary condition.We begin with a reviewof the reduced dynamical laws for time evolution of quantized vortex centers in GLE and show how to solve these nonlinear ordinary differential equations numerically.Then we present efficient and accurate numerical methods for discretizing the GLE on either a rectangular or a disk domain under either Dirichlet or homogeneous Neumann boundary condition.Based on these efficient and accurate numerical methods for GLE and the reduced dynamical laws,we simulate quantized vortex interaction of GLE with different#and under different initial setups including single vortex,vortex pair,vortex dipole and vortex lattice,compare them with those obtained from the corresponding reduced dynamical laws,and identify the cases where the reduced dynamical laws agree qualitatively and/or quantitatively as well as fail to agree with those from GLE on vortex interaction.Finally,we also obtain numerically different patterns of the steady states for quantized vortex lattices under the GLE dynamics on bounded domains.
基金supported by the National Natural Science Foundation of China (Grant No. 11072076)
文摘Great progress has been made in study on dynamic behavior of the damaged structures subject to deterministic excitation.The stochastic response analysis of the damaged structures,however,has not yet attracted people's attention.Taking the damaged elastic beams for example,the analysis procedure for stochastic response of the damaged structures subject to stochastic excitations is investigated in this paper.First,the damage constitutive relations and the corresponding damage evolution equation of one-dimensional elastic structures are briefly discussed.Second,the stochastic dynamic equation with respect to transverse displacement of the damaged elastic beams is deduced.The finite difference method and Newmark method are adopted to solve the stochastic partially-differential equation and corresponding boundary conditions.The stochastic response characteristic,damage evolution law,the effect of noise intensity on damage evolution and the first-passage time of damage are discussed in detail.The present work extends the research field of damaged structures,and the proposed procedure can be generalized to analyze the dynamic behavior of more complex structures,such as damaged plates and shells.
基金supported by FRG of Hong Kong Baptist University,and RGC of Hong Kong.
文摘The numerical solution of blow-up problems for nonlinear wave equations on unbounded spatial domains is considered.Applying the unified approach,which is based on the operator splitting method,we construct the efficient nonlinear local absorbing boundary conditions for the nonlinear wave equation,and reduce the nonlinear problem on the unbounded spatial domain to an initial-boundary-value problem on a bounded domain.Then the finite difference method is used to solve the reduced problem on the bounded computational domain.Finally,a broad range of numerical examples are given to demonstrate the effectiveness and accuracy of our method,and some interesting propagation and behaviors of the blow-up problems for nonlinear wave equations are observed.
