By using the properties of w-distances and Gerstewitz＇s functions, we first give a vectorial Takahashi＇s nonconvex minimization theorem with a w-distance. From this, we deduce a general vectorial Ekeland＇s variatio...By using the properties of w-distances and Gerstewitz＇s functions, we first give a vectorial Takahashi＇s nonconvex minimization theorem with a w-distance. From this, we deduce a general vectorial Ekeland＇s variational principle, where the objective function is from a complete metric space into a pre-ordered topological vector space and the perturbation contains a w-distance and a non-decreasing function of the objective function value. From the general vectorial variational principle, we deduce a vectorial Caristfs fixed point theorem with a w-distance. Finally we show that the above three theorems are equivalent to each other. The related known results are generalized and improved. In particular, some conditions in the theorems of [Y. Araya, Ekeland＇s variational principle and its equivalent theorems in vector optimization, J. Math. Anal. Appl. 346（2008）, 9-16] are weakened or even completely relieved.展开更多
This paper presents a discrete vaxiational principle and a method to build first-integrals for finite dimensional Lagrange-Maxwell mechanico-electrical systems with nonconservative forces and a dissipation function. T...This paper presents a discrete vaxiational principle and a method to build first-integrals for finite dimensional Lagrange-Maxwell mechanico-electrical systems with nonconservative forces and a dissipation function. The discrete variational principle and the corresponding Euler-Lagrange equations are derived from a discrete action associated to these systems. The first-integrals are obtained by introducing the infinitesimal transformation with respect to the generalized coordinates and electric quantities of the systems. This work also extends discrete Noether symmetries to mechanico-electrical dynamical systems. A practical example is presented to illustrate the results.展开更多
This paper summarized the recent development on Herglotz’s generalized variational principle and its symmetries and conserved quantities for nonconservative dynamical systems.Taking Lagrangian mechanics,Hamiltonian m...This paper summarized the recent development on Herglotz’s generalized variational principle and its symmetries and conserved quantities for nonconservative dynamical systems.Taking Lagrangian mechanics,Hamiltonian mechanics and Birkhoffian mechanics as three research frames,we introduce Herglotz’s generalized variational principle,dynamical equations of Herglotz type,Noether symmetry and conserved quantities,and their generalization to time-delay dynamics,fractional dynamics and time-scale dynamics,and put forward some problems as suggestions for future research.展开更多
By analogue with the methods and processes in continuous mechanics, a Lagrangian formulation and a Hamiltonian formulation of discrete mechanics are obtained. The dynamical equations including Euler Lagrange equations...By analogue with the methods and processes in continuous mechanics, a Lagrangian formulation and a Hamiltonian formulation of discrete mechanics are obtained. The dynamical equations including Euler Lagrange equations and Hamilton＇s canonical equations of the discrete nonconservative holonomic systems are derived on a discrete variational principle. Some illustrative examples are also given.展开更多
Variational principles are constructed using the semi-inverse method for two kinds of extended Korteweg-de Vries （KdV） equations, which can be regarded as simple models of the nonlinear oceanic internal waves and at...Variational principles are constructed using the semi-inverse method for two kinds of extended Korteweg-de Vries （KdV） equations, which can be regarded as simple models of the nonlinear oceanic internal waves and atmospheric long waves, respectively. The obtained variational principles have also been proved to be correct.展开更多
Let(X,φ) be a nonautonomous dynamical system.In this paper,we introduce the notions of packing topological entropy and measure-theoretical upper entropy for nonautonomous dynamical systems.Moreover,we establish the v...Let(X,φ) be a nonautonomous dynamical system.In this paper,we introduce the notions of packing topological entropy and measure-theoretical upper entropy for nonautonomous dynamical systems.Moreover,we establish the variational principle between the packing topological entropy and the measure-theoretical upper entropy.展开更多
The variational integrators of autonomous Birkhoff systems are obtained by the discrete variational principle. The geometric structure of the discrete autonomous Birkhoff system is formulated. The discretization of ma...The variational integrators of autonomous Birkhoff systems are obtained by the discrete variational principle. The geometric structure of the discrete autonomous Birkhoff system is formulated. The discretization of mathematical pendulum shows that the discrete variational method is as effective as symplectic scheme for the autonomous Birkhoff systems.展开更多
A variational principle code which can calculate self-consistently currents on the conductors is used to assess the coupling characteristic of the EAST 4-strap ion cyclotron range of frequency（ICRF） antenna. Taking ...A variational principle code which can calculate self-consistently currents on the conductors is used to assess the coupling characteristic of the EAST 4-strap ion cyclotron range of frequency（ICRF） antenna. Taking into account two layers of antenna conductors without lateral frame but with slab geometry, the antenna impedances as a function of frequency and the structure of RF field excited inside the plasma in various phasing cases are discussed in this paper.展开更多
Economic development has caused a lot of environmental problems,in turn,environmental pollution restricts economic development.Considering the influence of wind direction and speed,temperature and humidity on pollutan...Economic development has caused a lot of environmental problems,in turn,environmental pollution restricts economic development.Considering the influence of wind direction and speed,temperature and humidity on pollutants,as well as the influence of epidemic,war and exchange rate on economic development.In this paper,we develop a stochastic economic-environment model with pollution control strategies.Furthermore,sufficient and necessary conditions for the near-optimality are established.Finally,we perform some numerical simulations to demonstrate the correctness of the theoretical results,which shows that some control strategies could decrease the environmental pollution,and therefore,could alleviate economic losses caused by environmental pollution.展开更多
Conservation laws for the Birkhoffian system and the constrained Birkhoffian system of Herglotz type are studied. We propose a new differential variational principle, called the Pfaff-Birkhoff-d＇Alembert principle of...Conservation laws for the Birkhoffian system and the constrained Birkhoffian system of Herglotz type are studied. We propose a new differential variational principle, called the Pfaff-Birkhoff-d＇Alembert principle of Herglotz type. Birkhoff＇s equations for both the Birkhoffian system and the constrained Birkhoffian system of Herglotz type are obtained. According to the relationship between the isochronal variation and the nonisochronal variation, the conditions of the invariance for the Pfaff-Birkhoff-d＇Alembert principle of Herglotz type are given. Then, the conserved quantities for the Birkhoffian system and the constrained Birkhoffian system of Herglotz type are deduced. Furthermore, the inverse theorems of the conservation theorems are also established.展开更多
This paper investigates the Lie symmetries and Noether conserved quantities of discrete non-conservative mechanical systems. The variational principle of discrete mechanics, from which discrete motion equations of sys...This paper investigates the Lie symmetries and Noether conserved quantities of discrete non-conservative mechanical systems. The variational principle of discrete mechanics, from which discrete motion equations of systems are deduced, is generalized to the case of including the time variational. The requirement for an invariant group transformation is defined to be the Lie symmetry and the criterion when the Noether conserved quantities may be obtained from Lie symmetries is also presented. An example is discussed for applications of the results.展开更多
In this article, we study the following nonhomogeneous Schrodinger-Poissone quations{-△u＋λV（x）u＋K（x）Фu=f（x,u）＋g（x）,x∈R^3,-△Ф=k（x）u^2, x∈R^3}where λ 〉 0 is a parameter. Under some suitable assumpt...In this article, we study the following nonhomogeneous Schrodinger-Poissone quations{-△u＋λV（x）u＋K（x）Фu=f（x,u）＋g（x）,x∈R^3,-△Ф=k（x）u^2, x∈R^3}where λ 〉 0 is a parameter. Under some suitable assumptions on 11, K, f and g, the existence of multiple solutions is proved by using the Ekeland＇s variational principle and the Mountain Pass Theorem in critical point theory. In particular, the potential V is allowed to be signchanging.展开更多
We have developed a computer code for {／em ab initio} the variational configuration interaction calculation of the electronic structure of atoms via variationally optimized Lagurre type orbitals, treating the orbital...We have developed a computer code for {／em ab initio} the variational configuration interaction calculation of the electronic structure of atoms via variationally optimized Lagurre type orbitals, treating the orbital effective charges as variational parameters. Excited states of the same symmetry, in order to avoid the inherent restrictions of the standard method of Hylleraas--Unheim and MacDonald, are computed variationally by minimizing the recently developed minimization functionals for excited states. By computing, at the minimum, the one-electron density and the probability distribution of the two-electron angle, and the most probable two-electron angle, we investigate the atomic states of the carbon atom. We show that, without resorting to the （admittedly unproven） concept of hybridization, as an intrinsic property of the atomic wave function, the most probable value of the two-electron angle is around the known angles of carbon bonding, i.e. either 109^／circ or 120^／circ or 180^／circ, depending on each low-lying state of the bare carbon atom.展开更多
The existence and multiplicity of positive solutions are studied for a class of quasi- linear elliptic equations involving Sobolev critical exponents with mixed Dirichlet-Neumann boundary conditions by the variational...The existence and multiplicity of positive solutions are studied for a class of quasi- linear elliptic equations involving Sobolev critical exponents with mixed Dirichlet-Neumann boundary conditions by the variational methods and some analytical techniques.展开更多
A new model in nonholonomic mechanics, the Rosen-Edelstein model, has been studied. We prove that the new model is a Lagrange problem in which the action integral ∫t0^t1 Ldt can be made stationary. The theoretical ba...A new model in nonholonomic mechanics, the Rosen-Edelstein model, has been studied. We prove that the new model is a Lagrange problem in which the action integral ∫t0^t1 Ldt can be made stationary. The theoretical basis of nonholonomic mechanics is investigated and discussed. Finally, we give the range of practical applications of the Rosen-Edelstein model.展开更多
Several algorithms were proposed relating to the development of a framework of the perturbation-based stochastic finite element method (PSFEM) for large variation nonlinear dynamic problems. For this purpose, algorith...Several algorithms were proposed relating to the development of a framework of the perturbation-based stochastic finite element method (PSFEM) for large variation nonlinear dynamic problems. For this purpose, algorithms and a framework related to SFEM based on the stochastic virtual work principle were studied. To prove the validity and practicality of the algorithms and framework, numerical examples for nonlinear dynamic problems with large variations were calculated and compared with the Monte-Carlo Simulation method. This comparison shows that the proposed approaches are accurate and effective for the nonlinear dynamic analysis of structures with random parameters.展开更多
In this article, we have two parts. In the first part, we are concerned with the locally Hlder continuity of quasi-minima of the following integral functional ∫Ωf（x, u, Du）dx, （1） where Ω is an open subset of E...In this article, we have two parts. In the first part, we are concerned with the locally Hlder continuity of quasi-minima of the following integral functional ∫Ωf（x, u, Du）dx, （1） where Ω is an open subset of Euclidean N-space （N ≥ 3）, u：Ω → R,the Carath′eodory function f satisfies the critical Sobolev exponent growth condition ｜Du｜^p＊ ｜u｜^p＊-a（x） ≤ f（x,u,Du） ≤ L（｜Du｜^p＋｜u｜^p＊ ＋ a（x））, （2） where L≥1, 1pN,p^＊ = Np/N-p , and a（x） is a nonnegative function that lies in a suitable Lp space. In the second part, we study the locally Hlder continuity of ω-minima of （1）. Our method is to compare the ω-minima of （1） with the minima of corresponding function determined by its critical Sobolev exponent growth condition. Finally, we obtain the regularity by Ekeland’s variational principal.展开更多
This paper studies analytically and numerically the dynamics of two-dimensional elliptical Gaussian solitons in a ＂double-self-focusing＂ synthetic nonlocal media featuring elliptical and circular Gaussian response w...This paper studies analytically and numerically the dynamics of two-dimensional elliptical Gaussian solitons in a ＂double-self-focusing＂ synthetic nonlocal media featuring elliptical and circular Gaussian response with different degrees of nonlocality. Based on the variational approach, it obtains the approximately analytical solution of such Gaussian elliptical solitons. It also computes the stability of the solitons by numerical simulations.展开更多
A hybrid natural element method（HNEM） for two-dimensional viscoelasticity problems under the creep condition is proposed. The natural neighbor interpolation is used as the test function, and the discrete equation sy...A hybrid natural element method（HNEM） for two-dimensional viscoelasticity problems under the creep condition is proposed. The natural neighbor interpolation is used as the test function, and the discrete equation system of the HNEM for viscoelasticity problems is obtained using the Hellinger–Reissner variational principle. In contrast to the natural element method（NEM）, the HNEM can directly obtain the nodal stresses, which have higher precisions than those obtained using the moving least-square（MLS） approximation. Some numerical examples are given to demonstrate the validity and superiority of this HNEM.展开更多
The drying of liquid droplets is a common phenomenon in daily life,and has long attracted special interest in scientific research.We propose a simple model to quantify the shape evolution of drying droplets.The model ...The drying of liquid droplets is a common phenomenon in daily life,and has long attracted special interest in scientific research.We propose a simple model to quantify the shape evolution of drying droplets.The model takes into account the friction constant between the contact line(CL)and the substrate,the capillary forces,and the evaporation rate.Two typical evaporation processes observed in experiments,i.e.,the constant contact radius(CCR)and the constant contact angle(CCA),are demonstrated by the model.Moreover,the simple model shows complicated evaporation dynamics,for example,the CL first spreads and then recedes during evaporation.Analytical models of no evaporation,CCR,and CCA cases are given,respectively.The scaling law of the CL or the contact angle as a function of time obtained by analytical model is consistent with the full numerical model,and they are all subjected to experimental tests.The general model facilitates a quantitative understanding of the physical mechanism underlying the drying of liquid droplets.展开更多
基金Supported by the National Natural Science Foundation of China(10871141)
文摘By using the properties of w-distances and Gerstewitz＇s functions, we first give a vectorial Takahashi＇s nonconvex minimization theorem with a w-distance. From this, we deduce a general vectorial Ekeland＇s variational principle, where the objective function is from a complete metric space into a pre-ordered topological vector space and the perturbation contains a w-distance and a non-decreasing function of the objective function value. From the general vectorial variational principle, we deduce a vectorial Caristfs fixed point theorem with a w-distance. Finally we show that the above three theorems are equivalent to each other. The related known results are generalized and improved. In particular, some conditions in the theorems of [Y. Araya, Ekeland＇s variational principle and its equivalent theorems in vector optimization, J. Math. Anal. Appl. 346（2008）, 9-16] are weakened or even completely relieved.
基金Project supported by State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences and the National Natural Science Foundation of China （Grant Nos 10672143 and 10471145） and the Natural Science Foundation of Henan Province Government, China （Grant Nos 0311011400 and 0511022200）.
文摘This paper presents a discrete vaxiational principle and a method to build first-integrals for finite dimensional Lagrange-Maxwell mechanico-electrical systems with nonconservative forces and a dissipation function. The discrete variational principle and the corresponding Euler-Lagrange equations are derived from a discrete action associated to these systems. The first-integrals are obtained by introducing the infinitesimal transformation with respect to the generalized coordinates and electric quantities of the systems. This work also extends discrete Noether symmetries to mechanico-electrical dynamical systems. A practical example is presented to illustrate the results.
基金supported by the National Natural Science Foundations of China (Nos. 11972241,11572212,11272227)the Natural Science Foundation of Jiangsu Province(No. BK20191454).
文摘This paper summarized the recent development on Herglotz’s generalized variational principle and its symmetries and conserved quantities for nonconservative dynamical systems.Taking Lagrangian mechanics,Hamiltonian mechanics and Birkhoffian mechanics as three research frames,we introduce Herglotz’s generalized variational principle,dynamical equations of Herglotz type,Noether symmetry and conserved quantities,and their generalization to time-delay dynamics,fractional dynamics and time-scale dynamics,and put forward some problems as suggestions for future research.
基金Project supported by the National Natural Science Foundation of China （Grant No 19572018）.
文摘By analogue with the methods and processes in continuous mechanics, a Lagrangian formulation and a Hamiltonian formulation of discrete mechanics are obtained. The dynamical equations including Euler Lagrange equations and Hamilton＇s canonical equations of the discrete nonconservative holonomic systems are derived on a discrete variational principle. Some illustrative examples are also given.
基金Project supported by the National Natural Science Foundation of China (Grant Nos.61070041 and 40775064)
文摘Variational principles are constructed using the semi-inverse method for two kinds of extended Korteweg-de Vries （KdV） equations, which can be regarded as simple models of the nonlinear oceanic internal waves and atmospheric long waves, respectively. The obtained variational principles have also been proved to be correct.
基金the National Natural Science Foundation of China (11871188, 12031019)。
文摘Let(X,φ) be a nonautonomous dynamical system.In this paper,we introduce the notions of packing topological entropy and measure-theoretical upper entropy for nonautonomous dynamical systems.Moreover,we establish the variational principle between the packing topological entropy and the measure-theoretical upper entropy.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10872084 and 10932002)the Research Program of Higher Education of Liaoning Province,China (Grant No. 2008S098)+3 种基金the Program of Supporting Elitists of Higher Education of Liaoning Province,China (Grant No. 2008RC20)the Program of Constructing Liaoning Provincial Key Laboratory,China (Grant No. 2008403009)the Foundation Research Plan of Liaoning educational Bureau,China (Grant No. L2010147)the Youth fund of Liaoning University,China (Grant No. 2008LDQN04)
文摘The variational integrators of autonomous Birkhoff systems are obtained by the discrete variational principle. The geometric structure of the discrete autonomous Birkhoff system is formulated. The discretization of mathematical pendulum shows that the discrete variational method is as effective as symplectic scheme for the autonomous Birkhoff systems.
基金supported by the National Magnetic Confinement Fusion Science Program,China(Grant No.2015GB101001)the National Natural Science Foundation of China(Grant Nos.11375236 and 11375235)
文摘A variational principle code which can calculate self-consistently currents on the conductors is used to assess the coupling characteristic of the EAST 4-strap ion cyclotron range of frequency（ICRF） antenna. Taking into account two layers of antenna conductors without lateral frame but with slab geometry, the antenna impedances as a function of frequency and the structure of RF field excited inside the plasma in various phasing cases are discussed in this paper.
基金supported by the National Natural Science Foundation of China(12071407,12171193)the Natural Science Foundation of Hubei Province(2024AFB170)。
文摘Economic development has caused a lot of environmental problems,in turn,environmental pollution restricts economic development.Considering the influence of wind direction and speed,temperature and humidity on pollutants,as well as the influence of epidemic,war and exchange rate on economic development.In this paper,we develop a stochastic economic-environment model with pollution control strategies.Furthermore,sufficient and necessary conditions for the near-optimality are established.Finally,we perform some numerical simulations to demonstrate the correctness of the theoretical results,which shows that some control strategies could decrease the environmental pollution,and therefore,could alleviate economic losses caused by environmental pollution.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11572212 and 11272227)the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province,China(Grant No.KYZZ16-0479)the Innovation Program for Postgraduate of Suzhou University of Science and Technology,China(Grant No.SKCX16-058)
文摘Conservation laws for the Birkhoffian system and the constrained Birkhoffian system of Herglotz type are studied. We propose a new differential variational principle, called the Pfaff-Birkhoff-d＇Alembert principle of Herglotz type. Birkhoff＇s equations for both the Birkhoffian system and the constrained Birkhoffian system of Herglotz type are obtained. According to the relationship between the isochronal variation and the nonisochronal variation, the conditions of the invariance for the Pfaff-Birkhoff-d＇Alembert principle of Herglotz type are given. Then, the conserved quantities for the Birkhoffian system and the constrained Birkhoffian system of Herglotz type are deduced. Furthermore, the inverse theorems of the conservation theorems are also established.
基金Project supported by the National Natural Science Foundation of China (Grant No 10672143)the Natural Science Foundation of Henan Province,China (Grant No 0511022200)
文摘This paper investigates the Lie symmetries and Noether conserved quantities of discrete non-conservative mechanical systems. The variational principle of discrete mechanics, from which discrete motion equations of systems are deduced, is generalized to the case of including the time variational. The requirement for an invariant group transformation is defined to be the Lie symmetry and the criterion when the Noether conserved quantities may be obtained from Lie symmetries is also presented. An example is discussed for applications of the results.
基金supported by the Tianyuan Special Foundation(11526148)the second author is supported by the National Natural Science Foundation of China(11571187)
文摘In this article, we study the following nonhomogeneous Schrodinger-Poissone quations{-△u＋λV（x）u＋K（x）Фu=f（x,u）＋g（x）,x∈R^3,-△Ф=k（x）u^2, x∈R^3}where λ 〉 0 is a parameter. Under some suitable assumptions on 11, K, f and g, the existence of multiple solutions is proved by using the Ekeland＇s variational principle and the Mountain Pass Theorem in critical point theory. In particular, the potential V is allowed to be signchanging.
基金supported by the National High Technology Research and Development Program of China (Grant No. 2004AA306H10)the operational program "Competitiveness" of the Greek General Secretariat of Research and Technology(Grant No. 04EP111/ENTEP-2004)
文摘We have developed a computer code for {／em ab initio} the variational configuration interaction calculation of the electronic structure of atoms via variationally optimized Lagurre type orbitals, treating the orbital effective charges as variational parameters. Excited states of the same symmetry, in order to avoid the inherent restrictions of the standard method of Hylleraas--Unheim and MacDonald, are computed variationally by minimizing the recently developed minimization functionals for excited states. By computing, at the minimum, the one-electron density and the probability distribution of the two-electron angle, and the most probable two-electron angle, we investigate the atomic states of the carbon atom. We show that, without resorting to the （admittedly unproven） concept of hybridization, as an intrinsic property of the atomic wave function, the most probable value of the two-electron angle is around the known angles of carbon bonding, i.e. either 109^／circ or 120^／circ or 180^／circ, depending on each low-lying state of the bare carbon atom.
基金Supported by National Natural Science Foundation of China (11071198 11101347)+2 种基金Postdoctor Foundation of China (2012M510363)the Key Project in Science and Technology Research Plan of the Education Department of Hubei Province (D20112605 D20122501)
文摘The existence and multiplicity of positive solutions are studied for a class of quasi- linear elliptic equations involving Sobolev critical exponents with mixed Dirichlet-Neumann boundary conditions by the variational methods and some analytical techniques.
基金Project supported by the National Natural Science Foundation of China （Grant Nos 10272021 and 10572021） and the Doctoral Programme Foundation of Institution of Higher Education of China （Grant No 20040007022）.
文摘A new model in nonholonomic mechanics, the Rosen-Edelstein model, has been studied. We prove that the new model is a Lagrange problem in which the action integral ∫t0^t1 Ldt can be made stationary. The theoretical basis of nonholonomic mechanics is investigated and discussed. Finally, we give the range of practical applications of the Rosen-Edelstein model.
文摘Several algorithms were proposed relating to the development of a framework of the perturbation-based stochastic finite element method (PSFEM) for large variation nonlinear dynamic problems. For this purpose, algorithms and a framework related to SFEM based on the stochastic virtual work principle were studied. To prove the validity and practicality of the algorithms and framework, numerical examples for nonlinear dynamic problems with large variations were calculated and compared with the Monte-Carlo Simulation method. This comparison shows that the proposed approaches are accurate and effective for the nonlinear dynamic analysis of structures with random parameters.
基金Supported by the Program of Fujian Province-HongKong
文摘In this article, we have two parts. In the first part, we are concerned with the locally Hlder continuity of quasi-minima of the following integral functional ∫Ωf（x, u, Du）dx, （1） where Ω is an open subset of Euclidean N-space （N ≥ 3）, u：Ω → R,the Carath′eodory function f satisfies the critical Sobolev exponent growth condition ｜Du｜^p＊ ｜u｜^p＊-a（x） ≤ f（x,u,Du） ≤ L（｜Du｜^p＋｜u｜^p＊ ＋ a（x））, （2） where L≥1, 1pN,p^＊ = Np/N-p , and a（x） is a nonnegative function that lies in a suitable Lp space. In the second part, we study the locally Hlder continuity of ω-minima of （1）. Our method is to compare the ω-minima of （1） with the minima of corresponding function determined by its critical Sobolev exponent growth condition. Finally, we obtain the regularity by Ekeland’s variational principal.
基金Project supported by the National Natural Science Foundation of China (Grant No. 60808002)the Shanghai Leading Academic Discipline Program,China (Grant No. S30105)
文摘This paper studies analytically and numerically the dynamics of two-dimensional elliptical Gaussian solitons in a ＂double-self-focusing＂ synthetic nonlocal media featuring elliptical and circular Gaussian response with different degrees of nonlocality. Based on the variational approach, it obtains the approximately analytical solution of such Gaussian elliptical solitons. It also computes the stability of the solitons by numerical simulations.
基金Project supported by the Natural Science Foundation of Shanghai,China(Grant No.13ZR1415900)
文摘A hybrid natural element method（HNEM） for two-dimensional viscoelasticity problems under the creep condition is proposed. The natural neighbor interpolation is used as the test function, and the discrete equation system of the HNEM for viscoelasticity problems is obtained using the Hellinger–Reissner variational principle. In contrast to the natural element method（NEM）, the HNEM can directly obtain the nodal stresses, which have higher precisions than those obtained using the moving least-square（MLS） approximation. Some numerical examples are given to demonstrate the validity and superiority of this HNEM.
基金Project supported by the National Natural Science Foundation of China(Grant No.21822302)the joint NSFC-ISF Research Program,China(Grant No.21961142020)+1 种基金the Fundamental Research Funds for the Central Universities,Chinathe National College Students'Innovative and Entrepreneurial Training Plan Program,China(Grant No.201910006142).
文摘The drying of liquid droplets is a common phenomenon in daily life,and has long attracted special interest in scientific research.We propose a simple model to quantify the shape evolution of drying droplets.The model takes into account the friction constant between the contact line(CL)and the substrate,the capillary forces,and the evaporation rate.Two typical evaporation processes observed in experiments,i.e.,the constant contact radius(CCR)and the constant contact angle(CCA),are demonstrated by the model.Moreover,the simple model shows complicated evaporation dynamics,for example,the CL first spreads and then recedes during evaporation.Analytical models of no evaporation,CCR,and CCA cases are given,respectively.The scaling law of the CL or the contact angle as a function of time obtained by analytical model is consistent with the full numerical model,and they are all subjected to experimental tests.The general model facilitates a quantitative understanding of the physical mechanism underlying the drying of liquid droplets.
