In this paper,we develop the stabilization-free virtual element method for the Helmholtz transmission eigenvalue problem on anisotropic media.The eigenvalue problem is a variable-coefficient,non-elliptic,non-selfadjoi...In this paper,we develop the stabilization-free virtual element method for the Helmholtz transmission eigenvalue problem on anisotropic media.The eigenvalue problem is a variable-coefficient,non-elliptic,non-selfadjoint and nonlinear model.Separating the cases of the index of refraction n≠1 and n≡1,the stabilization-free virtual element schemes are proposed,respectively.Furthermore,we prove the spectral approximation property and error estimates in a unified theoretical framework.Finally,a series of numerical examples are provided to verify the theoretical results,show the benefits of the stabilization-free virtual element method applied to eigenvalue problems,and implement the extensions to high-order and high-dimensional cases.展开更多
The rigorous error analysis of a class of serendipity virtual element methods applied to numerically solve semilinear parabolic integro-differential equations on curved domains is the focus of this study.Different fro...The rigorous error analysis of a class of serendipity virtual element methods applied to numerically solve semilinear parabolic integro-differential equations on curved domains is the focus of this study.Different from the standard virtual element method,the serendipity virtual element method eliminates all the internal-moment degrees of freedom only under certain conditions of the mesh and the degree of approximation.Consequently,if the interpolation operators are utilized to approximate the nonlinear terms,the implementation of Newton’s iteration algorithm can be simplified.Nonhomogeneous Dirichlet boundary conditions are considered in this paper.The strategy of approximating curved domains with polygonal domains is taken into consideration,and to overcome the issue of suboptimal convergence caused by enforcing Dirichlet boundary conditions strongly,Nitsche-based projection method is employed to impose the boundary conditions weakly.For time discretization,Crank-Nicolson scheme incorporating trapezoidal quadrature rule is adopted.Based on the concrete formulation of Nitsche-based projection method,a Ritz-Volterra projection is introduced and its approximation properties are rigorously analyzed.Building upon these approximation properties,error estimates are derived for the fully discrete scheme.Additionally,the extension of the fully discrete scheme to 3D case is also included.Finally,we present two numerical experiments to corroborate the theoretical findings.展开更多
The elastic transmission eigenvalue problem is fundamental to the qualitative methods for inverse scattering involving penetrable obstacles.Although simply stated as a coupled pair of elastodynamic wave equations,the ...The elastic transmission eigenvalue problem is fundamental to the qualitative methods for inverse scattering involving penetrable obstacles.Although simply stated as a coupled pair of elastodynamic wave equations,the elastic transmission eigenvalue problem is neither self-adjoint nor elliptic.The aim of this work is to provide a systematic spectral approximation analysis for the VEM of the elastic transmission eigenvalue problem with equal elastic tensors.Considering standard assumptions on polygonal/polyhedral meshes,we prove the stability analysis of the associated VEM bilinear forms,which shall be applied to the well-defined property of the discrete solution operator.Then the correct approximation of spectrum for the proposed VEM scheme is proven.Necessitated by supporting the convergence analysis,a series of numerical examples are reported.In addition,some negative points of the current VEM scheme are considered,including the locking phenomenon and the influence of VEM stabilization parameters.Thanks to the flexibility of construction for the VEM space,the locking-free and stabilization-free VEM approaches are utilized to tackle with these negative aspects.展开更多
We discuss and analyze the virtual element method on general polygonal meshes for the time-dependent Poisson-Nernst-Planck(PNP)equations,which are a nonlinear coupled system widely used in semiconductors and ion chann...We discuss and analyze the virtual element method on general polygonal meshes for the time-dependent Poisson-Nernst-Planck(PNP)equations,which are a nonlinear coupled system widely used in semiconductors and ion channels.After presenting the semi-discrete scheme,the optimal H1 norm error estimates are presented for the time-dependent PNP equations,which are based on some error estimates of a virtual element energy projection.The Gummel iteration is used to decouple and linearize the PNP equations and the error analysis is also given for the iteration of fully discrete virtual element approximation.The numerical experiment on different polygonal meshes verifies the theoretical convergence results and shows the efficiency of the virtual element method.展开更多
In this paper,a robust residual-based a posteriori estimate is discussed for the Streamline Upwind/Petrov Galerkin(SUPG)virtual element method(VEM)discretization of convection dominated diffusion equation.A global upp...In this paper,a robust residual-based a posteriori estimate is discussed for the Streamline Upwind/Petrov Galerkin(SUPG)virtual element method(VEM)discretization of convection dominated diffusion equation.A global upper bound and a local lower bound for the a posteriori error estimates are derived in the natural SUPG norm,where the global upper estimate relies on some hypotheses about the interpolation errors and SUPG virtual element discretization errors.Based on the Dörfler’s marking strategy,adaptive VEM algorithm drived by the error estimators is used to solve the problem on general polygonal meshes.Numerical experiments show the robustness of the a posteriori error estimates.展开更多
Bulk-surface partial differential equations(BS-PDEs)are prevalent in manyapplications such as cellular,developmental and plant biology as well as in engineeringand material sciences.Novel numerical methods for BS-PDEs...Bulk-surface partial differential equations(BS-PDEs)are prevalent in manyapplications such as cellular,developmental and plant biology as well as in engineeringand material sciences.Novel numerical methods for BS-PDEs in three space dimensions(3D)are sparse.In this work,we present a bulk-surface virtual elementmethod(BS-VEM)for bulk-surface reaction-diffusion systems,a form of semilinearparabolic BS-PDEs in 3D.Unlike previous studies in two space dimensions(2D),the3D bulk is approximated with general polyhedra,whose outer faces constitute a flatpolygonal approximation of the surface.For this reason,the method is restricted tothe lowest order case where the geometric error is not dominant.The BS-VEM guaranteesall the advantages of polyhedral methods such as easy mesh generation andfast matrix assembly on general geometries.Such advantages are much more relevantthan in 2D.Despite allowing for general polyhedra,general nonlinear reaction kineticsand general surface curvature,the method only relies on nodal values without needingadditional evaluations usually associated with the quadrature of general reactionkinetics.This latter is particularly costly in 3D.The BS-VEM as implemented in thisstudy retains optimal convergence of second order in space.展开更多
In this work we introduce and analyze a mixed virtual element method(mixed-VEM)for the two-dimensional stationary Boussinesq problem.The continuous formulation is based on the introduction of a pseudostress tensor dep...In this work we introduce and analyze a mixed virtual element method(mixed-VEM)for the two-dimensional stationary Boussinesq problem.The continuous formulation is based on the introduction of a pseudostress tensor depending nonlinearly on the velocity,which allows to obtain an equivalent model in which the main unknowns are given by the aforementioned pseudostress tensor,the velocity and the temperature,whereas the pressure is computed via a postprocessing formula.In addition,an augmented approach together with a fixed point strategy is used to analyze the well-posedness of the resulting continuous formulation.Regarding the discrete problem,we follow the approach employed in a previous work dealing with the Navier-Stokes equations,and couple it with a VEM for the convection-diffusion equation modelling the temperature.More precisely,we use a mixed-VEM for the scheme associated with the fluid equations in such a way that the pseudostress and the velocity are approximated on virtual element subspaces of H(div)and H^(1),respectively,whereas a VEM is proposed to approximate the temperature on a virtual element subspace of H^(1).In this way,we make use of the L^(2)-orthogonal projectors onto suitable polynomial spaces,which allows the explicit integration of the terms that appear in the bilinear and trilinear forms involved in the scheme for the fluid equations.On the other hand,in order to manipulate the bilinear form associated to the heat equations,we define a suitable projector onto a space of polynomials to deal with the fact that the diffusion tensor,which represents the thermal conductivity,is variable.Next,the corresponding solvability analysis is performed using again appropriate fixed-point arguments.Further,Strang-type estimates are applied to derive the a priori error estimates for the components of the virtual element solution as well as for the fully computable projections of them and the postprocessed pressure.The corresponding rates of convergence are also established.Finally,several numerical examples illustrating the performance of the mixed-VEM scheme and confirming these theoretical rates are presented.展开更多
The second-order serendipity virtual element method is studied for the semilinear pseudo-parabolic equations on curved domains in this paper.Nonhomogeneous Dirichlet boundary conditions are taken into account,the exis...The second-order serendipity virtual element method is studied for the semilinear pseudo-parabolic equations on curved domains in this paper.Nonhomogeneous Dirichlet boundary conditions are taken into account,the existence and uniqueness are investigated for the weak solution of the nonhomogeneous initial-boundary value problem.The Nitschebased projection method is adopted to impose the boundary conditions in a weak way.The interpolation operator is used to deal with the nonlinear term.The Crank-Nicolson scheme is employed to discretize the temporal variable.There are two main features of the proposed scheme:(i)the internal degrees of freedom are avoided no matter what type of mesh is utilized,and(ii)the Jacobian is simple to calculate when Newton’s iteration method is applied to solve the fully discrete scheme.The error estimates are established for the discrete schemes and the theoretical results are illustrated through some numerical examples.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
A novel polygonal finite element method （PFEM） based on partition of unity is proposed, termed the virtual node method （VNM）. To test the performance of the present method, numerical examples are given for solid m...A novel polygonal finite element method （PFEM） based on partition of unity is proposed, termed the virtual node method （VNM）. To test the performance of the present method, numerical examples are given for solid mechanics problems. With a polynomial form, the VNM achieves better results than those of traditional PFEMs, including the Wachspress method and the mean value method in standard patch tests. Compared with the standard triangular FEM, the VNM can achieve better accuracy. With the ability to construct shape functions on polygonal elements, the VNM provides greater flexibility in mesh generation. Therefore, several fracture problems are studied to demonstrate the potential implementation. With the advantage of the VNM, the convenient refinement and remeshing strategy are applied.展开更多
The virtual element method(VEM)can be seen as an extension of the classical finite element method(FEM)based on Galerkin projection.It allows meshes with highly irregular shaped elements,including concave shapes.So far...The virtual element method(VEM)can be seen as an extension of the classical finite element method(FEM)based on Galerkin projection.It allows meshes with highly irregular shaped elements,including concave shapes.So far the virtual element method has been applied to various engineering problems such as elasto-plasticity,multiphysics,damage and fracture mechanics.This work focuses on the extension of the virtual element method to efficient modeling of nonlinear elasto-dynamics undergoing large deformations.Within this framework,we employ low-order ansatz functions in two and three dimensions for elements that can have arbitrary polygonal shape.The formulations considered in this contribution are based on minimization of potential function for both the static and the dynamic behavior.Generally the construction of a virtual element is based on a projection part and a stabilization part.While the stiffness matrix needs a suitable stabilization,the mass matrix can be calculated using only the projection part.For the implicit time integration scheme,Newmark-Method is used.To show the performance of the method,various two-and three-dimensional numerical examples in are presented.展开更多
In this paper, we discuss virtual element method (VEM) approximation of optimal control problem governed by Brinkman equations with control constraints. Based on the polynomial projections and variational discretizati...In this paper, we discuss virtual element method (VEM) approximation of optimal control problem governed by Brinkman equations with control constraints. Based on the polynomial projections and variational discretization of the control variable, we build up the virtual element discrete scheme of the optimal control problem and derive the discrete first order optimality system. A priori error estimates for the state, adjoint state and control variables in L<sup>2</sup> and H<sup>1</sup> norm are derived. The theoretical findings are illustrated by the numerical experiments.展开更多
The virtual laminated element method (VLEM) can resolve structural shap e optimization problems with a new method. According to the characteristics of V LEM , only some characterized layer thickness values need be def...The virtual laminated element method (VLEM) can resolve structural shap e optimization problems with a new method. According to the characteristics of V LEM , only some characterized layer thickness values need be defined as design v ariables instead of boundary node coordinates or some other parameters determini ng the system boundary. One of the important features of this method is that it is not necessary to regenerate the FE(finite element) grid during the optimizati on process so as to avoid optimization failures resulting from some distortion grid elements. Th e thickness distribution in thin plate optimization problems in other studies be fore is of stepped shape. However, in this paper, a continuous thickness distrib ution can be obtained after optimization using VLEM, and is more reasonable. Fur thermore, an approximate reanalysis method named ″behavior model technique″ ca n be used to reduce the amount of structural reanalysis. Some typical examples are offered to prove the effectiveness and practicality of the proposed method.展开更多
In this paper,we develop a fully discrete virtual element scheme based on the local pressure projection stabilization for a three-field poroelasticity problem with a storage coefficient c00.We not only provide the wel...In this paper,we develop a fully discrete virtual element scheme based on the local pressure projection stabilization for a three-field poroelasticity problem with a storage coefficient c00.We not only provide the well-posedness of the proposed scheme by proving a weaker form of the discrete inf-sup condition,but also show optimal error estimates for all unknowns,whose generic constants are independent of the Lam´e coefficient.Moreover,our proposed scheme avoids pressure oscillation and applies to general polygonal elements,including hanging-node elements.Finally,we numerically validate the good performance of our virtual element scheme.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.12301532,12271523,12071481,12371198,12371374)by the Defense Science Foundation of China(Grant Nos.2021-JCJQ-JJ-0538,2022-JCJQ-JJ-0879)+3 种基金by the National Key Research and Development Program of China(Grant No.2020YFA0709803)by the Science and Technology Innovation Program of Hunan Province(Grant Nos.2022RC1192,2021RC3082)by the Natural Science Foundation of Hunan(Grant No.2021JJ20053)by the National University of Defense Technology(Grant Nos.2023-lxy-fhjj-002,202401-YJRC-XX-001).
文摘In this paper,we develop the stabilization-free virtual element method for the Helmholtz transmission eigenvalue problem on anisotropic media.The eigenvalue problem is a variable-coefficient,non-elliptic,non-selfadjoint and nonlinear model.Separating the cases of the index of refraction n≠1 and n≡1,the stabilization-free virtual element schemes are proposed,respectively.Furthermore,we prove the spectral approximation property and error estimates in a unified theoretical framework.Finally,a series of numerical examples are provided to verify the theoretical results,show the benefits of the stabilization-free virtual element method applied to eigenvalue problems,and implement the extensions to high-order and high-dimensional cases.
基金supported by the National Natural Science Foundation of China(Project Nos.12471373,12071100).
文摘The rigorous error analysis of a class of serendipity virtual element methods applied to numerically solve semilinear parabolic integro-differential equations on curved domains is the focus of this study.Different from the standard virtual element method,the serendipity virtual element method eliminates all the internal-moment degrees of freedom only under certain conditions of the mesh and the degree of approximation.Consequently,if the interpolation operators are utilized to approximate the nonlinear terms,the implementation of Newton’s iteration algorithm can be simplified.Nonhomogeneous Dirichlet boundary conditions are considered in this paper.The strategy of approximating curved domains with polygonal domains is taken into consideration,and to overcome the issue of suboptimal convergence caused by enforcing Dirichlet boundary conditions strongly,Nitsche-based projection method is employed to impose the boundary conditions weakly.For time discretization,Crank-Nicolson scheme incorporating trapezoidal quadrature rule is adopted.Based on the concrete formulation of Nitsche-based projection method,a Ritz-Volterra projection is introduced and its approximation properties are rigorously analyzed.Building upon these approximation properties,error estimates are derived for the fully discrete scheme.Additionally,the extension of the fully discrete scheme to 3D case is also included.Finally,we present two numerical experiments to corroborate the theoretical findings.
基金Supported by the National Natural Science Foundation of China(Grant Nos.12301532,12271523,12071481,12371198,12371374)Defense Science Foundation of China(Grant Nos.2021-JCJQ-JJ-0538,2022-JCJQ-JJ-0879)+3 种基金National Key Research and Development Program of China(Grant No.2020YFA0709803)Science and Technology Innovation Program of Hunan Province(Grant Nos.2022RC1192,2021RC3082)Natural Science Foundation of Hunan(Grant No.2021JJ20053)National University of Defense Technology(Grant Nos.2023-lxy-fhjj-002,202401-YJRC-XX-001)。
文摘The elastic transmission eigenvalue problem is fundamental to the qualitative methods for inverse scattering involving penetrable obstacles.Although simply stated as a coupled pair of elastodynamic wave equations,the elastic transmission eigenvalue problem is neither self-adjoint nor elliptic.The aim of this work is to provide a systematic spectral approximation analysis for the VEM of the elastic transmission eigenvalue problem with equal elastic tensors.Considering standard assumptions on polygonal/polyhedral meshes,we prove the stability analysis of the associated VEM bilinear forms,which shall be applied to the well-defined property of the discrete solution operator.Then the correct approximation of spectrum for the proposed VEM scheme is proven.Necessitated by supporting the convergence analysis,a series of numerical examples are reported.In addition,some negative points of the current VEM scheme are considered,including the locking phenomenon and the influence of VEM stabilization parameters.Thanks to the flexibility of construction for the VEM space,the locking-free and stabilization-free VEM approaches are utilized to tackle with these negative aspects.
基金supported by the China NSF(NSFC 12161026)by the Special Fund for Scientific and Technological Bases and Talents of Guangxi(Grant No.Guike AD23026048)+2 种基金by the Guangxi Natural Science Foundation,China(Grant No.2020GXNSFAA159098)supported by the China NSF(NSFC 12371373)supported by the GUET Excellent Graduate Thesis Program(Grant No.2020YJSPYA02).
文摘We discuss and analyze the virtual element method on general polygonal meshes for the time-dependent Poisson-Nernst-Planck(PNP)equations,which are a nonlinear coupled system widely used in semiconductors and ion channels.After presenting the semi-discrete scheme,the optimal H1 norm error estimates are presented for the time-dependent PNP equations,which are based on some error estimates of a virtual element energy projection.The Gummel iteration is used to decouple and linearize the PNP equations and the error analysis is also given for the iteration of fully discrete virtual element approximation.The numerical experiment on different polygonal meshes verifies the theoretical convergence results and shows the efficiency of the virtual element method.
基金supported by the National Natural Science Foundation of China(Grant Nos.11971276,11301311)by the Natural Science Foundation of Shandong Province(Grant No.ZR2016JL004).
文摘In this paper,a robust residual-based a posteriori estimate is discussed for the Streamline Upwind/Petrov Galerkin(SUPG)virtual element method(VEM)discretization of convection dominated diffusion equation.A global upper bound and a local lower bound for the a posteriori error estimates are derived in the natural SUPG norm,where the global upper estimate relies on some hypotheses about the interpolation errors and SUPG virtual element discretization errors.Based on the Dörfler’s marking strategy,adaptive VEM algorithm drived by the error estimators is used to solve the problem on general polygonal meshes.Numerical experiments show the robustness of the a posteriori error estimates.
基金Regione Puglia(Italy)through the research programme REFIN-Research for Innovation(protocol code 901D2CAA,project No.UNISAL026)MF acknowledges support from the Italian National Institute of High Mathematics(INdAM)through the INdAM-GNCS Project no.CUP E55F22000270001+3 种基金the Global Challenges Research Fund through the Engineering and Physical Sciences Research Council grant number EP/T00410X/1:UK-Africa Postgraduate Advanced Study Institute in Mathematical Sciences,the Health Foundation(1902431)the NIHR(NIHR133761)and by the Discovery Grant awarded by Canadian Natural Sciences and Engineering Research Council(2023-2028)AM acknowledges support from the Royal Society Wolfson Research Merit Award funded generously by the Wolfson Foundation(2016-2021)AM is a Distinguished Visiting Scholar to the Department of Mathematics,University of Johannesburg,South Africa,and the University of Pretoria in South Africa.IS and MF are members of the INdAM-GNCS activity group.The work of IS is supported by the PRIN 2020 research project(no.2020F3NCPX)”Mathematics for Industry 4.0”,and from the”National Centre for High Performance Computing,Big Data and Quantum Computing”funded by European Union-NextGenerationEU,PNRR project code CN00000013,CUP F83C22000740001.
文摘Bulk-surface partial differential equations(BS-PDEs)are prevalent in manyapplications such as cellular,developmental and plant biology as well as in engineeringand material sciences.Novel numerical methods for BS-PDEs in three space dimensions(3D)are sparse.In this work,we present a bulk-surface virtual elementmethod(BS-VEM)for bulk-surface reaction-diffusion systems,a form of semilinearparabolic BS-PDEs in 3D.Unlike previous studies in two space dimensions(2D),the3D bulk is approximated with general polyhedra,whose outer faces constitute a flatpolygonal approximation of the surface.For this reason,the method is restricted tothe lowest order case where the geometric error is not dominant.The BS-VEM guaranteesall the advantages of polyhedral methods such as easy mesh generation andfast matrix assembly on general geometries.Such advantages are much more relevantthan in 2D.Despite allowing for general polyhedra,general nonlinear reaction kineticsand general surface curvature,the method only relies on nodal values without needingadditional evaluations usually associated with the quadrature of general reactionkinetics.This latter is particularly costly in 3D.The BS-VEM as implemented in thisstudy retains optimal convergence of second order in space.
基金supported by CONICYT-Chile through the project AFB170001 of the PIA Program:Concurso Apoyo a Centros Cientificos y Tecnologicos de Excelencia con Financiamiento Basal,and the Becas-CONICYT Programme for foreign studentsby Centro de Investigacion en Ingenieria Matematica(CI^(2)MA),Universidad de Con-cepcionby Uniyersidad Nacional,Costa Ricea,through the prejeet 0103-18.
文摘In this work we introduce and analyze a mixed virtual element method(mixed-VEM)for the two-dimensional stationary Boussinesq problem.The continuous formulation is based on the introduction of a pseudostress tensor depending nonlinearly on the velocity,which allows to obtain an equivalent model in which the main unknowns are given by the aforementioned pseudostress tensor,the velocity and the temperature,whereas the pressure is computed via a postprocessing formula.In addition,an augmented approach together with a fixed point strategy is used to analyze the well-posedness of the resulting continuous formulation.Regarding the discrete problem,we follow the approach employed in a previous work dealing with the Navier-Stokes equations,and couple it with a VEM for the convection-diffusion equation modelling the temperature.More precisely,we use a mixed-VEM for the scheme associated with the fluid equations in such a way that the pseudostress and the velocity are approximated on virtual element subspaces of H(div)and H^(1),respectively,whereas a VEM is proposed to approximate the temperature on a virtual element subspace of H^(1).In this way,we make use of the L^(2)-orthogonal projectors onto suitable polynomial spaces,which allows the explicit integration of the terms that appear in the bilinear and trilinear forms involved in the scheme for the fluid equations.On the other hand,in order to manipulate the bilinear form associated to the heat equations,we define a suitable projector onto a space of polynomials to deal with the fact that the diffusion tensor,which represents the thermal conductivity,is variable.Next,the corresponding solvability analysis is performed using again appropriate fixed-point arguments.Further,Strang-type estimates are applied to derive the a priori error estimates for the components of the virtual element solution as well as for the fully computable projections of them and the postprocessed pressure.The corresponding rates of convergence are also established.Finally,several numerical examples illustrating the performance of the mixed-VEM scheme and confirming these theoretical rates are presented.
基金supported by the National Natural Science Foundation of China(Grant No.12071100)by the Fundamental Research Funds for the Central Universities(Grant No.2022FRFK060019).
文摘The second-order serendipity virtual element method is studied for the semilinear pseudo-parabolic equations on curved domains in this paper.Nonhomogeneous Dirichlet boundary conditions are taken into account,the existence and uniqueness are investigated for the weak solution of the nonhomogeneous initial-boundary value problem.The Nitschebased projection method is adopted to impose the boundary conditions in a weak way.The interpolation operator is used to deal with the nonlinear term.The Crank-Nicolson scheme is employed to discretize the temporal variable.There are two main features of the proposed scheme:(i)the internal degrees of freedom are avoided no matter what type of mesh is utilized,and(ii)the Jacobian is simple to calculate when Newton’s iteration method is applied to solve the fully discrete scheme.The error estimates are established for the discrete schemes and the theoretical results are illustrated through some numerical examples.
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.
文摘A novel polygonal finite element method （PFEM） based on partition of unity is proposed, termed the virtual node method （VNM）. To test the performance of the present method, numerical examples are given for solid mechanics problems. With a polynomial form, the VNM achieves better results than those of traditional PFEMs, including the Wachspress method and the mean value method in standard patch tests. Compared with the standard triangular FEM, the VNM can achieve better accuracy. With the ability to construct shape functions on polygonal elements, the VNM provides greater flexibility in mesh generation. Therefore, several fracture problems are studied to demonstrate the potential implementation. With the advantage of the VNM, the convenient refinement and remeshing strategy are applied.
基金The authors gratefully acknowledges support for this research by the“German Research Foundation”(DFG)in(i)the Collaborative Research Center CRC 1153 and(ii)the Priority Program SPP 2020.
文摘The virtual element method(VEM)can be seen as an extension of the classical finite element method(FEM)based on Galerkin projection.It allows meshes with highly irregular shaped elements,including concave shapes.So far the virtual element method has been applied to various engineering problems such as elasto-plasticity,multiphysics,damage and fracture mechanics.This work focuses on the extension of the virtual element method to efficient modeling of nonlinear elasto-dynamics undergoing large deformations.Within this framework,we employ low-order ansatz functions in two and three dimensions for elements that can have arbitrary polygonal shape.The formulations considered in this contribution are based on minimization of potential function for both the static and the dynamic behavior.Generally the construction of a virtual element is based on a projection part and a stabilization part.While the stiffness matrix needs a suitable stabilization,the mass matrix can be calculated using only the projection part.For the implicit time integration scheme,Newmark-Method is used.To show the performance of the method,various two-and three-dimensional numerical examples in are presented.
文摘In this paper, we discuss virtual element method (VEM) approximation of optimal control problem governed by Brinkman equations with control constraints. Based on the polynomial projections and variational discretization of the control variable, we build up the virtual element discrete scheme of the optimal control problem and derive the discrete first order optimality system. A priori error estimates for the state, adjoint state and control variables in L<sup>2</sup> and H<sup>1</sup> norm are derived. The theoretical findings are illustrated by the numerical experiments.
文摘The virtual laminated element method (VLEM) can resolve structural shap e optimization problems with a new method. According to the characteristics of V LEM , only some characterized layer thickness values need be defined as design v ariables instead of boundary node coordinates or some other parameters determini ng the system boundary. One of the important features of this method is that it is not necessary to regenerate the FE(finite element) grid during the optimizati on process so as to avoid optimization failures resulting from some distortion grid elements. Th e thickness distribution in thin plate optimization problems in other studies be fore is of stepped shape. However, in this paper, a continuous thickness distrib ution can be obtained after optimization using VLEM, and is more reasonable. Fur thermore, an approximate reanalysis method named ″behavior model technique″ ca n be used to reduce the amount of structural reanalysis. Some typical examples are offered to prove the effectiveness and practicality of the proposed method.
基金supported in part by the National Natural Science Foundation of China(Grant No.12371405).
文摘In this paper,we develop a fully discrete virtual element scheme based on the local pressure projection stabilization for a three-field poroelasticity problem with a storage coefficient c00.We not only provide the well-posedness of the proposed scheme by proving a weaker form of the discrete inf-sup condition,but also show optimal error estimates for all unknowns,whose generic constants are independent of the Lam´e coefficient.Moreover,our proposed scheme avoids pressure oscillation and applies to general polygonal elements,including hanging-node elements.Finally,we numerically validate the good performance of our virtual element scheme.
