The main aim of this article is to study the approximation of a locking-free anisotropic nonconforming finite element for the pure displacement boundary value problem of planar linear elasticity. The optimal error est...The main aim of this article is to study the approximation of a locking-free anisotropic nonconforming finite element for the pure displacement boundary value problem of planar linear elasticity. The optimal error estimates are obtained by using some novel approaches and techniques. The method proposed in this article is robust in the sense that the convergence estimates in the energy and L^2-norms are independent-of the Lame parameter λ.展开更多
This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this elem...This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L^2-norm are obtained. The restrictions of regularity assumption and quasi-uniform assumption or the inverse assumption on the meshes required in the conventional finite element methods analysis are to be got rid of and the applicable scope of the nonconforming finite elements is extended.展开更多
A family of conforming finite elements are designed on rectangular grids for solving the Reissner-Mindlin plate equation.The rotation is approximated by C^(1)-Q_(k+1)in one direction and C^(0)-Q_(k)in the other direct...A family of conforming finite elements are designed on rectangular grids for solving the Reissner-Mindlin plate equation.The rotation is approximated by C^(1)-Q_(k+1)in one direction and C^(0)-Q_(k)in the other direction finite elements.The displacement is approximated by C^(1)-Q_(k+1,k+1).The method is locking-free without using any projection/reduction operator.Theoretical proof and numerical confirmation are presented.展开更多
In this paper, we discusse the locking phenomenon of the finite element method for the pure displacement boundary value problem in the planar elasticity as Lame constant λ- ∞. The locking-free scheme of Crouziex-Rav...In this paper, we discusse the locking phenomenon of the finite element method for the pure displacement boundary value problem in the planar elasticity as Lame constant λ- ∞. The locking-free scheme of Crouziex-Raviart element was pro- posed and anaIyzed by Brenner et al.[2] and [3]. We firstly present the derivation of Brenner’s scheme, then propose and analyse a locking-free scheme of noncon- forming rectangle finite element.展开更多
The main aim of this paper is to study the nonconforming linear triangular Crouzeix- Raviart type finite element approximation of planar linear elasticity problem with the pure displacement boundary value on anisotrop...The main aim of this paper is to study the nonconforming linear triangular Crouzeix- Raviart type finite element approximation of planar linear elasticity problem with the pure displacement boundary value on anisotropic general triangular meshes satisfying the maximal angle condition and coordinate system condition. The optimal order error estimates of energy norm and L2-norm are obtained, which are independent of lame parameter λ. Numerical results are given to demonstrate the validity of our theoretical analysis.Mathematics subject classification： 65N30, 65N15.展开更多
In this paper, a locking-free nonconforming rectangular finite element scheme is presented for the planar elasticity problem with pure displacement boundary condition. Meanwhile, we prove that this element is also con...In this paper, a locking-free nonconforming rectangular finite element scheme is presented for the planar elasticity problem with pure displacement boundary condition. Meanwhile, we prove that this element is also convergent for stationary Stokes problem.展开更多
Two new locking-free nonconforming finite elements for the pure displacement planar elasticity problem are presented. Convergence rates of the elements are uniformly optimal with respect to A. The energy norm and L2 n...Two new locking-free nonconforming finite elements for the pure displacement planar elasticity problem are presented. Convergence rates of the elements are uniformly optimal with respect to A. The energy norm and L2 norm errors are proved to be O（h2） and O（h3）, respectively. Numerical tests confirm the theoretical analysis.展开更多
This paper devises a new lowest-order conforming virtual element method(VEM)for planar linear elasticity with the pure displacement/traction boundary condition.The main trick is to view a generic polygon K as a new on...This paper devises a new lowest-order conforming virtual element method(VEM)for planar linear elasticity with the pure displacement/traction boundary condition.The main trick is to view a generic polygon K as a new one K with additional vertices consisting of interior points on edges of K,so that the discrete admissible space is taken as the V1 type virtual element space related to the partition{K}instead of{K}.The method is proved to converge with optimal convergence order both in H^(1)and L^(2)norms and uniformly with respect to the Lam´e constantλ.Numerical tests are presented to illustrate the good performance of the proposed VEM and confirm the theoretical results.展开更多
In this paper, we extend two rectangular elements for Reissner-Mindlin plate [9] to the quadrilateral case. Optimal H and L error bounds independent of the plate hickness are derived under a mild assumption on the mes...In this paper, we extend two rectangular elements for Reissner-Mindlin plate [9] to the quadrilateral case. Optimal H and L error bounds independent of the plate hickness are derived under a mild assumption on the mesh partition.展开更多
This paper generalizes two nonconforming rectangular elements of the Reissner-Mindlin plate to the quadrilateral mesh. The first quadrilateral element uses the usual conforming bilinear element to approximate both com...This paper generalizes two nonconforming rectangular elements of the Reissner-Mindlin plate to the quadrilateral mesh. The first quadrilateral element uses the usual conforming bilinear element to approximate both components of the rotation, and the modified nonconforming rotated Q1 element enriched with the intersected term on each element to approximate the displacement, whereas the second one uses the enriched modified nonconforming rotated Q1 element to approximate both the rotation and the displacement. Both elements employ a more complicated shear force space to overcome the shear force locking, which will be described in detail in the introduction. We prove that both methods converge at optimal rates uniformly in the plate thickness t and the mesh distortion parameter in both the H1-and the L2-norms, and consequently they are locking free.展开更多
Based on the primal mixed variational formulation,a stabilized nonconforming mixed finite element method is proposed for the linear elasticity problem by adding the jump penalty term for the displacement.Here we use t...Based on the primal mixed variational formulation,a stabilized nonconforming mixed finite element method is proposed for the linear elasticity problem by adding the jump penalty term for the displacement.Here we use the piecewise constant space for stress and the Crouzeix-Raviart element space for displacement.The mixed method is locking-free,i.e.,the convergence does not deteriorate in the nearly incompressible or incompressible case.The optimal convergence order is shown in the L^(2)-norm for stress and in the broken H1-norm and L2-norm for displacement,respectively.Finally,some numerical results are given to demonstrate the optimal convergence and stability of the mixed method.展开更多
A mixed finite element method combining an iso-parametric Q 2-P 1 element and an isoparametric P^-Pi element is developed for the computation of multiple cavities in incompressible nonlinear elasticity. The method is ...A mixed finite element method combining an iso-parametric Q 2-P 1 element and an isoparametric P^-Pi element is developed for the computation of multiple cavities in incompressible nonlinear elasticity. The method is analytically proved to be locking-free and convergent, and it is also shown to be numerically accurate and efficient by numerical experiments. Furthermore, the newly developed accurate method enables us to find an interesting new bifurcation phenomenon in multi-cavity growth.展开更多
In this paper,we extend the work of Brenner and Sung[Math.Comp.59,321–338(1992)]and present a regularity estimate for the elastic equations in concave domains.Based on the regularity estimate we prove that the consta...In this paper,we extend the work of Brenner and Sung[Math.Comp.59,321–338(1992)]and present a regularity estimate for the elastic equations in concave domains.Based on the regularity estimate we prove that the constants in the error estimates of the nonconforming Crouzeix-Raviart element approximations for the elastic equations/eigenvalue problem are independent of Laméconstant,which means the nonconforming Crouzeix-Raviart element approximations are locking-free.We also establish two kinds of two-grid discretization schemes for the elastic eigenvalue problem,and analyze that when the mesh sizes of coarse grid and fine grid satisfy some relationship,the resulting solutions can achieve the optimal accuracy.Numerical examples are provided to show the efficiency of two-grid schemes for the elastic eigenvalue problem.展开更多
基金The research is supported by NSF of China (10371113 10471133)
文摘The main aim of this article is to study the approximation of a locking-free anisotropic nonconforming finite element for the pure displacement boundary value problem of planar linear elasticity. The optimal error estimates are obtained by using some novel approaches and techniques. The method proposed in this article is robust in the sense that the convergence estimates in the energy and L^2-norms are independent-of the Lame parameter λ.
基金Supported by the National Natural Science Foundation of China(10371113,10671184)
文摘This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L^2-norm are obtained. The restrictions of regularity assumption and quasi-uniform assumption or the inverse assumption on the meshes required in the conventional finite element methods analysis are to be got rid of and the applicable scope of the nonconforming finite elements is extended.
文摘A family of conforming finite elements are designed on rectangular grids for solving the Reissner-Mindlin plate equation.The rotation is approximated by C^(1)-Q_(k+1)in one direction and C^(0)-Q_(k)in the other direction finite elements.The displacement is approximated by C^(1)-Q_(k+1,k+1).The method is locking-free without using any projection/reduction operator.Theoretical proof and numerical confirmation are presented.
文摘In this paper, we discusse the locking phenomenon of the finite element method for the pure displacement boundary value problem in the planar elasticity as Lame constant λ- ∞. The locking-free scheme of Crouziex-Raviart element was pro- posed and anaIyzed by Brenner et al.[2] and [3]. We firstly present the derivation of Brenner’s scheme, then propose and analyse a locking-free scheme of noncon- forming rectangle finite element.
基金Acknowledgments. This work was supported by National Natural Science Foundation of China （No. 10971203）, Specialized Research Fund for the Doctoral Program of Higher Education （No. 20094101110006）, the Educational Department Foundation of Henan Province of China （No.2009B110013）.
文摘The main aim of this paper is to study the nonconforming linear triangular Crouzeix- Raviart type finite element approximation of planar linear elasticity problem with the pure displacement boundary value on anisotropic general triangular meshes satisfying the maximal angle condition and coordinate system condition. The optimal order error estimates of energy norm and L2-norm are obtained, which are independent of lame parameter λ. Numerical results are given to demonstrate the validity of our theoretical analysis.Mathematics subject classification： 65N30, 65N15.
文摘In this paper, a locking-free nonconforming rectangular finite element scheme is presented for the planar elasticity problem with pure displacement boundary condition. Meanwhile, we prove that this element is also convergent for stationary Stokes problem.
基金Project supported by the National Natural Science Foundation of China (Nos. 10771198 and 11071226)the Foundation of International Science and Technology Cooperation of Henan Province
文摘Two new locking-free nonconforming finite elements for the pure displacement planar elasticity problem are presented. Convergence rates of the elements are uniformly optimal with respect to A. The energy norm and L2 norm errors are proved to be O（h2） and O（h3）, respectively. Numerical tests confirm the theoretical analysis.
基金supported by NSFC(Grant No.12071289)the Fundamental Research Funds for the Central Universities.
文摘This paper devises a new lowest-order conforming virtual element method(VEM)for planar linear elasticity with the pure displacement/traction boundary condition.The main trick is to view a generic polygon K as a new one K with additional vertices consisting of interior points on edges of K,so that the discrete admissible space is taken as the V1 type virtual element space related to the partition{K}instead of{K}.The method is proved to converge with optimal convergence order both in H^(1)and L^(2)norms and uniformly with respect to the Lam´e constantλ.Numerical tests are presented to illustrate the good performance of the proposed VEM and confirm the theoretical results.
基金Subsidized by the Special Funds for Major State Basic Research Projects G1999032804.
文摘In this paper, we extend two rectangular elements for Reissner-Mindlin plate [9] to the quadrilateral case. Optimal H and L error bounds independent of the plate hickness are derived under a mild assumption on the mesh partition.
基金the National Natural Science Foundation of China (Grant No. 10601003)National Excellent Doctoral Dissertation of China (Grant No. 200718)
文摘This paper generalizes two nonconforming rectangular elements of the Reissner-Mindlin plate to the quadrilateral mesh. The first quadrilateral element uses the usual conforming bilinear element to approximate both components of the rotation, and the modified nonconforming rotated Q1 element enriched with the intersected term on each element to approximate the displacement, whereas the second one uses the enriched modified nonconforming rotated Q1 element to approximate both the rotation and the displacement. Both elements employ a more complicated shear force space to overcome the shear force locking, which will be described in detail in the introduction. We prove that both methods converge at optimal rates uniformly in the plate thickness t and the mesh distortion parameter in both the H1-and the L2-norms, and consequently they are locking free.
基金supported by National Natural Science Foundation of China(No.11701522)Key scientific research projects in colleges and universities in Henan Province(No.18A110030)Research Foundation for Advanced Talents of Henan University of Technology(No.2018BS013).
文摘Based on the primal mixed variational formulation,a stabilized nonconforming mixed finite element method is proposed for the linear elasticity problem by adding the jump penalty term for the displacement.Here we use the piecewise constant space for stress and the Crouzeix-Raviart element space for displacement.The mixed method is locking-free,i.e.,the convergence does not deteriorate in the nearly incompressible or incompressible case.The optimal convergence order is shown in the L^(2)-norm for stress and in the broken H1-norm and L2-norm for displacement,respectively.Finally,some numerical results are given to demonstrate the optimal convergence and stability of the mixed method.
文摘A mixed finite element method combining an iso-parametric Q 2-P 1 element and an isoparametric P^-Pi element is developed for the computation of multiple cavities in incompressible nonlinear elasticity. The method is analytically proved to be locking-free and convergent, and it is also shown to be numerically accurate and efficient by numerical experiments. Furthermore, the newly developed accurate method enables us to find an interesting new bifurcation phenomenon in multi-cavity growth.
基金supported by the National Natural Science Foundation of China (Grant No.11761022)。
文摘In this paper,we extend the work of Brenner and Sung[Math.Comp.59,321–338(1992)]and present a regularity estimate for the elastic equations in concave domains.Based on the regularity estimate we prove that the constants in the error estimates of the nonconforming Crouzeix-Raviart element approximations for the elastic equations/eigenvalue problem are independent of Laméconstant,which means the nonconforming Crouzeix-Raviart element approximations are locking-free.We also establish two kinds of two-grid discretization schemes for the elastic eigenvalue problem,and analyze that when the mesh sizes of coarse grid and fine grid satisfy some relationship,the resulting solutions can achieve the optimal accuracy.Numerical examples are provided to show the efficiency of two-grid schemes for the elastic eigenvalue problem.
