A finite element method is introduced to solve the general elastic multi-structure problem, in which the displacements on bodies, the longitudinal displacements on plates and the longitudinal displacements on beams ar...A finite element method is introduced to solve the general elastic multi-structure problem, in which the displacements on bodies, the longitudinal displacements on plates and the longitudinal displacements on beams are discretized using conforming linear elements, the rotational angles on beams are discretized using conforming elements of second order, the transverse displacements on plates and beams are discretized by the Morley elements and the Hermite elements of third order, respectively. The generalized Korn's inequality is established on related nonconforming element spaces, which implies the unique solvability of the finite element method. Finally, the optimal error estimate in the energy norm is derived for the method.展开更多
A recent joint paper with Doina Cioranescu and Julia Orlik was concerned with the homogenization of a linearized elasticity problem with inclusions and cracks(see[Cioranescu, D., Damlamian, A. and Orlik, J., Homogeniz...A recent joint paper with Doina Cioranescu and Julia Orlik was concerned with the homogenization of a linearized elasticity problem with inclusions and cracks(see[Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis, 82, 2013, 201–232]). It required uniform estimates with respect to the homogenization parameter. A Korn inequality was used which involves unilateral terms on the boundaries where a nopenetration condition is imposed. In this paper, the author presents a general method to obtain many diverse Korn inequalities including the unilateral inequalities used in [Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis, 82, 2013, 201–232]. A preliminary version was presented in [Damlamian, A., Some unilateral Korn inequalities with application to a contact problem with inclusions, C. R. Acad. Sci. Paris, Ser. I,350, 2012, 861–865].展开更多
In this paper, we use the weak Galerkin （WG） finite element method to solve the mixed form linear elasticity problem. In the mixed form, we get the discrete of proximation of the stress tensor and the displacement f...In this paper, we use the weak Galerkin （WG） finite element method to solve the mixed form linear elasticity problem. In the mixed form, we get the discrete of proximation of the stress tensor and the displacement field. For the WG methods, we define the weak function and the weak differential operator in an optimal polynomial approximation spaces. The optimal error estimates are given and numerical results are presented to demonstrate the efficiency and the accuracy of the weak Galerkin finite element method.展开更多
The aim of this paper is to study the static problem about a general elastic multi-structure composed of an arbitrary number of elastic bodies, plates and rods. The mathematical model is derived by the variational pri...The aim of this paper is to study the static problem about a general elastic multi-structure composed of an arbitrary number of elastic bodies, plates and rods. The mathematical model is derived by the variational principle and the principle of virtual work in a vector way. The unique solvability of the resulting problem is proved by the Lax-Milgram lemma after the presentation of a generalized Korn's inequality on general elastic multi-structures. The equilibrium equations are obtained rigorously by only assuming some reasonable regularity of the solution. An important identity is also given which is essential in the finite element analysis for the problem.展开更多
The possibility of using Neumann＇s method to solve the boundary problems for thin elastic shells is studied. The variational statement of the static problems for the shells allows for a problem examination within the...The possibility of using Neumann＇s method to solve the boundary problems for thin elastic shells is studied. The variational statement of the static problems for the shells allows for a problem examination within the distribution space. The convergence of Neumann＇s method is proven for the shells with holes when the boundary of the domain is not completely fixed. The numerical implementation of Neumann＇s method normally requires significant time before any reliable results can be achieved. This paper suggests a way to improve the convergence of the process, and allows for parallel computing and evaluation during the calculations.展开更多
The authors discuss the existence and uniqueness up to isometries of Enof immersions φ : Ω R^n→ E^n with prescribed metric tensor field(g ij) : Ω→ S^n>, and discuss the continuity of the mapping(gij) →φ d...The authors discuss the existence and uniqueness up to isometries of Enof immersions φ : Ω R^n→ E^n with prescribed metric tensor field(g ij) : Ω→ S^n>, and discuss the continuity of the mapping(gij) →φ defined in this fashion with respect to various topologies. In particular, the case where the function spaces have little regularity is considered. How, in some cases, the continuity of the mapping(gij) →φ can be obtained by means of nonlinear Korn inequalities is shown.展开更多
In this note, we show that, for domains satisfying the separation property, certain weighted Korn inequality is equivalent to the John condition. Our result generalizes previous result from Jiang Kauranen [Calc. Vat. ...In this note, we show that, for domains satisfying the separation property, certain weighted Korn inequality is equivalent to the John condition. Our result generalizes previous result from Jiang Kauranen [Calc. Vat. Partial Differential Equations, 56, Art. 109, （2017）] to weighted settings.展开更多
The author first reviews the classical Korn inequality and its proof.Following recent works of S.Kesavan,P.Ciarlet,Jr.,and the author,it is shown how the Korn inequality can be recovered by an entirely different proof...The author first reviews the classical Korn inequality and its proof.Following recent works of S.Kesavan,P.Ciarlet,Jr.,and the author,it is shown how the Korn inequality can be recovered by an entirely different proof.This new proof hinges on appropriate weak versions of the classical Poincar'e and Saint-Venant lemma.In fine,both proofs essentially depend on a crucial lemma of J.L.Lions,recalled at the beginning of this paper.展开更多
Inequalities of Korn's type on a surface with boundary have been proved in many papersusing different techniques (see e.g. [4, 5, 11]). The author proves here an inequality of Korn'stype on a compact surface w...Inequalities of Korn's type on a surface with boundary have been proved in many papersusing different techniques (see e.g. [4, 5, 11]). The author proves here an inequality of Korn'stype on a compact surface without boundary. The idea is to use a finite number of maps fordefining the surface and the inequality of Korn's type without boundary conditions for everymap and to recast these in a general functional analysis setting about quotient spaces.展开更多
By applying the inequality of Korn＇s type without boundary conditions on a general surface, we prove that the scaled displacement of the two-dimensional linearly viscoelastic Koiter＇s shell converges to the solution...By applying the inequality of Korn＇s type without boundary conditions on a general surface, we prove that the scaled displacement of the two-dimensional linearly viscoelastic Koiter＇s shell converges to the solution of two-dimensional model system of linearly viscoelastic ＂membrane＂ shell.展开更多
In the use of finite element methods to the planar elasticity problems,one diffculty is to overcome locking when elasticity constant λ→∞.In the case of traction boundary condition,another diffculty is to make the d...In the use of finite element methods to the planar elasticity problems,one diffculty is to overcome locking when elasticity constant λ→∞.In the case of traction boundary condition,another diffculty is to make the discrete Korn＇s second inequality valid.In this paper,a triangular element is presented.We prove that this element is locking-free,the discrete Korn＇s second inequality holds and the convergence order is two.展开更多
Let φ be a generalized Orlicz function satisfying(A0),(A1),(A2),(aInc)and(aDec). We prove that the mapping f■f^#:=supB|1/B|∫B|f(x)-fp|dx is continuous on L^φ(·)(R^n) by extrapolation. Based on this result we ...Let φ be a generalized Orlicz function satisfying(A0),(A1),(A2),(aInc)and(aDec). We prove that the mapping f■f^#:=supB|1/B|∫B|f(x)-fp|dx is continuous on L^φ(·)(R^n) by extrapolation. Based on this result we generalize Korn's inequality to the setting of generalized Orlicz spaces, i.e., ‖■f‖Lφ(·)(Ω)■‖Df‖Lφ(·)(Ω). Using the Calderón–Zygmund theory on generalized Orlicz spaces, we obtain that the divergence equation divu = f has a solution u ∈(W^1φ(·)(Ω)0)^n such that ‖■f‖Lφ(·)(Ω)■‖f‖Lφ(Ω).展开更多
Consider an elastic thin three-dimensional body made of a periodic distribution of elastic inclusions.When both the thickness of the beam and the size of the heterogeneities tend simultaneously to zero the authors obt...Consider an elastic thin three-dimensional body made of a periodic distribution of elastic inclusions.When both the thickness of the beam and the size of the heterogeneities tend simultaneously to zero the authors obtain three different one-dimensional models of beam depending upon the limit of the ratio of these two small parameters.展开更多
“伙记,我33岁,但我今天觉得自己像已是53岁了。”Munky摊躺在软皮沙发上抱怨道。这位Korn的吉他手昨夜和朋友一起从洛杉矶来到英格兰,只睡了三个小时的觉。而此时已是清晨,他要在West London录音室度过漫长的一天。Munky坐下来用很大...“伙记,我33岁,但我今天觉得自己像已是53岁了。”Munky摊躺在软皮沙发上抱怨道。这位Korn的吉他手昨夜和朋友一起从洛杉矶来到英格兰,只睡了三个小时的觉。而此时已是清晨,他要在West London录音室度过漫长的一天。Munky坐下来用很大的声音反复播放《Take a look in the mirror(照照镜子)》展开更多
He was born to a family of statesmen, with Chinese blood running in his veins.He witnessed the establishment of diplomatic relations between China and Thailand, with the seeds of friendship sprouting in his heart.He h...He was born to a family of statesmen, with Chinese blood running in his veins.He witnessed the establishment of diplomatic relations between China and Thailand, with the seeds of friendship sprouting in his heart.He has remained true to his original aspirations.展开更多
In this paper we consider the Lamé system on a polygonal convex domain with mixed boundary conditions of Dirichlet-Neumann type.An explicit L2 norm estimate for the gradient of the solution of this problem is est...In this paper we consider the Lamé system on a polygonal convex domain with mixed boundary conditions of Dirichlet-Neumann type.An explicit L2 norm estimate for the gradient of the solution of this problem is established.This leads to an explicit bound of the H1 norm of this solution.Note that the obtained upper-bound is not optimal.展开更多
The use of the finite finite element method leads to replace the initial domain by an approach- ing domain.In part I,the uniform validity of the friedrichs’and Korn’s inequalities in these ap- proaching domains will...The use of the finite finite element method leads to replace the initial domain by an approach- ing domain.In part I,the uniform validity of the friedrichs’and Korn’s inequalities in these ap- proaching domains will be proved under some appropriate assumptions.展开更多
基金supported by the National Natural Science Foundation(Grant No.10371076)E-Institutes of Shanghai Municipal Education Commission(Grant No.E03004)The Science Foundation of Shanghai(Grant No.04JC14062).
文摘A finite element method is introduced to solve the general elastic multi-structure problem, in which the displacements on bodies, the longitudinal displacements on plates and the longitudinal displacements on beams are discretized using conforming linear elements, the rotational angles on beams are discretized using conforming elements of second order, the transverse displacements on plates and beams are discretized by the Morley elements and the Hermite elements of third order, respectively. The generalized Korn's inequality is established on related nonconforming element spaces, which implies the unique solvability of the finite element method. Finally, the optimal error estimate in the energy norm is derived for the method.
文摘A recent joint paper with Doina Cioranescu and Julia Orlik was concerned with the homogenization of a linearized elasticity problem with inclusions and cracks(see[Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis, 82, 2013, 201–232]). It required uniform estimates with respect to the homogenization parameter. A Korn inequality was used which involves unilateral terms on the boundaries where a nopenetration condition is imposed. In this paper, the author presents a general method to obtain many diverse Korn inequalities including the unilateral inequalities used in [Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis, 82, 2013, 201–232]. A preliminary version was presented in [Damlamian, A., Some unilateral Korn inequalities with application to a contact problem with inclusions, C. R. Acad. Sci. Paris, Ser. I,350, 2012, 861–865].
基金The authors would like to thank China National Natural Science Foundation （91630201, U1530116, 11726102, 11771179）, and the Program for Cheung Kong Scholars of Ministry of Education of China, Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, 3ilin University, Changchun, 130012, P.R. China.
文摘In this paper, we use the weak Galerkin （WG） finite element method to solve the mixed form linear elasticity problem. In the mixed form, we get the discrete of proximation of the stress tensor and the displacement field. For the WG methods, we define the weak function and the weak differential operator in an optimal polynomial approximation spaces. The optimal error estimates are given and numerical results are presented to demonstrate the efficiency and the accuracy of the weak Galerkin finite element method.
基金This work was partly supported by the 973 projectthe National Natural Science Foundation of China(Grant No.10371076)+1 种基金E-Institutes of Shanghai Municipal Education Commission(Grant No.E03004)The Science Foundation of Shanghai(Grant No.04JC14062).
文摘The aim of this paper is to study the static problem about a general elastic multi-structure composed of an arbitrary number of elastic bodies, plates and rods. The mathematical model is derived by the variational principle and the principle of virtual work in a vector way. The unique solvability of the resulting problem is proved by the Lax-Milgram lemma after the presentation of a generalized Korn's inequality on general elastic multi-structures. The equilibrium equations are obtained rigorously by only assuming some reasonable regularity of the solution. An important identity is also given which is essential in the finite element analysis for the problem.
文摘The possibility of using Neumann＇s method to solve the boundary problems for thin elastic shells is studied. The variational statement of the static problems for the shells allows for a problem examination within the distribution space. The convergence of Neumann＇s method is proven for the shells with holes when the boundary of the domain is not completely fixed. The numerical implementation of Neumann＇s method normally requires significant time before any reliable results can be achieved. This paper suggests a way to improve the convergence of the process, and allows for parallel computing and evaluation during the calculations.
基金supported by a grant from the Research Grants Council of the Hong Kong Special Administration Region,China(Nos.9041637,CiyuU100711)
文摘The authors discuss the existence and uniqueness up to isometries of Enof immersions φ : Ω R^n→ E^n with prescribed metric tensor field(g ij) : Ω→ S^n>, and discuss the continuity of the mapping(gij) →φ defined in this fashion with respect to various topologies. In particular, the case where the function spaces have little regularity is considered. How, in some cases, the continuity of the mapping(gij) →φ can be obtained by means of nonlinear Korn inequalities is shown.
基金Partially supported by NNSF of China(Grant No.11671039)
文摘In this note, we show that, for domains satisfying the separation property, certain weighted Korn inequality is equivalent to the John condition. Our result generalizes previous result from Jiang Kauranen [Calc. Vat. Partial Differential Equations, 56, Art. 109, （2017）] to weighted settings.
文摘The author first reviews the classical Korn inequality and its proof.Following recent works of S.Kesavan,P.Ciarlet,Jr.,and the author,it is shown how the Korn inequality can be recovered by an entirely different proof.This new proof hinges on appropriate weak versions of the classical Poincar'e and Saint-Venant lemma.In fine,both proofs essentially depend on a crucial lemma of J.L.Lions,recalled at the beginning of this paper.
文摘Inequalities of Korn's type on a surface with boundary have been proved in many papersusing different techniques (see e.g. [4, 5, 11]). The author proves here an inequality of Korn'stype on a compact surface without boundary. The idea is to use a finite number of maps fordefining the surface and the inequality of Korn's type without boundary conditions for everymap and to recast these in a general functional analysis setting about quotient spaces.
基金National Natural Science Foundation of China(No.10271030)Foundation of Qufu Normal University for Ph.D
文摘By applying the inequality of Korn＇s type without boundary conditions on a general surface, we prove that the scaled displacement of the two-dimensional linearly viscoelastic Koiter＇s shell converges to the solution of two-dimensional model system of linearly viscoelastic ＂membrane＂ shell.
文摘In the use of finite element methods to the planar elasticity problems,one diffculty is to overcome locking when elasticity constant λ→∞.In the case of traction boundary condition,another diffculty is to make the discrete Korn＇s second inequality valid.In this paper,a triangular element is presented.We prove that this element is locking-free,the discrete Korn＇s second inequality holds and the convergence order is two.
基金Supported by the National Natural Science Foundation of China (Grant No.11726622)Scientific Research Fund of Young Teachers in Longqiao College (Grant No. LQKJ2020-01)。
文摘Let φ be a generalized Orlicz function satisfying(A0),(A1),(A2),(aInc)and(aDec). We prove that the mapping f■f^#:=supB|1/B|∫B|f(x)-fp|dx is continuous on L^φ(·)(R^n) by extrapolation. Based on this result we generalize Korn's inequality to the setting of generalized Orlicz spaces, i.e., ‖■f‖Lφ(·)(Ω)■‖Df‖Lφ(·)(Ω). Using the Calderón–Zygmund theory on generalized Orlicz spaces, we obtain that the divergence equation divu = f has a solution u ∈(W^1φ(·)(Ω)0)^n such that ‖■f‖Lφ(·)(Ω)■‖f‖Lφ(Ω).
文摘Consider an elastic thin three-dimensional body made of a periodic distribution of elastic inclusions.When both the thickness of the beam and the size of the heterogeneities tend simultaneously to zero the authors obtain three different one-dimensional models of beam depending upon the limit of the ratio of these two small parameters.
文摘“伙记,我33岁,但我今天觉得自己像已是53岁了。”Munky摊躺在软皮沙发上抱怨道。这位Korn的吉他手昨夜和朋友一起从洛杉矶来到英格兰,只睡了三个小时的觉。而此时已是清晨,他要在West London录音室度过漫长的一天。Munky坐下来用很大的声音反复播放《Take a look in the mirror(照照镜子)》
文摘He was born to a family of statesmen, with Chinese blood running in his veins.He witnessed the establishment of diplomatic relations between China and Thailand, with the seeds of friendship sprouting in his heart.He has remained true to his original aspirations.
文摘In this paper we consider the Lamé system on a polygonal convex domain with mixed boundary conditions of Dirichlet-Neumann type.An explicit L2 norm estimate for the gradient of the solution of this problem is established.This leads to an explicit bound of the H1 norm of this solution.Note that the obtained upper-bound is not optimal.
基金Programme Sino-Francais de Recherche Avanc es(PRA)and the Natural ScienceFoundations of China
文摘The use of the finite finite element method leads to replace the initial domain by an approach- ing domain.In part I,the uniform validity of the friedrichs’and Korn’s inequalities in these ap- proaching domains will be proved under some appropriate assumptions.
