ERROR ANALYSIS OF THE SECOND-ORDER SERENDIPITY VIRTUAL ELEMENT METHOD FOR SEMILINEAR PSEUDO-PARABOLIC EQUATIONS ON CURVED DOMAINS

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.

The Bulk-Surface Virtual Element Method for Reaction-Diffusion PDEs:Analysis and Applications

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.

A New Locking-Free Virtual Element Method for Linear Elasticity Problems

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.

Virtual Element Discretization of Optimal Control Problem Governed by Brinkman Equations

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 L2 and H1 norm are derived. The theoretical findings are illustrated by the numerical experiments.

A MIXED VIRTUAL ELEMENT METHOD FOR THE BOUSSINESQ PROBLEM ON POLYGONAL MESHES

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.

A novel virtual node method for polygonal elements

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.

Virtual Element Formulation for Finite Strain Elastodynamics Dedicated to Professor Karl Stark Pister for his 95th birthday

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.

Shape optimization of plate with static and dynamic constraints via virtual laminated element

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.

基于S-R和分解定理的二维几何非线性问题的虚单元法求解

应变-旋转(Strain-Rotation,S-R)和分解定理为分析几何非线性问题提供了合理可靠的理论基础,但用有限元求解时会遇到大变形发生后的网格畸变问题。近年提出的虚单元法(Virtual element method,VEM)适用于一般的多边形网格,因此,该文尝试使用一阶虚单元求解基于S-R和分解定理的二维几何非线性问题,以克服网格畸变的影响。基于重新定义的多项式位移空间基函数,推演获得一阶虚单元分析线弹性力学问题时允许位移空间向多项式位移空间的投影表达式;按照虚单元法双线性格式的计算规则,分析处理基于更新拖带坐标法和势能率原理的增量变分方程;进而建立离散系统方程及其矩阵表达形式,并编制MATLAB求解程序;采用常规多边形网格和畸变网格,应用该文算法分析均布荷载下的悬臂梁和均匀内压下的厚壁圆筒变形。结果与已有文献和ANSYS软件的对比表明:该文算法在两种网格中均可有效执行且具备足够数值精度。总体该文算法为基于S-R和分解定理的二维几何非线性问题求解提供了一种鲁棒方法。

砌体轴心和局部受压离散元仿真在教学中的应用

砌体是块体和砂浆的黏结组合材料,为了让学生更好地理解砖块体和砂浆对外力的分担作用,运用离散元方法建立了砖块体团簇模型、砂浆团簇模型,进而“砌筑”了普通砖砌体数值试件,对砖砌体分别开展轴心受压和局部受压仿真。仿真复现了砌体受压时的应力-应变关系和破坏模式,应力峰值处伴随大量砖颗粒-砖颗粒胶结接触破坏。砌体应力主要由胶结接触承担,其中砖-砖胶结接触>砖-砂浆胶结接触>砂浆-砂浆胶结接触,砂浆主要起到连接块体的作用;法向接触力承力大于切向接触力。通过仿真教学,有助于学生建立砌体宏观力学表现受控于微观结构和微观力学的认知,加深了学生对砌体受力机理和强度的理解。

A Priori Error Analysis for NCVEM Discretization of Elliptic Optimal Control Problem

In this paper, we propose the nonconforming virtual element method (NCVEM) discretization for the pointwise control constraint optimal control problem governed by elliptic equations. Based on the NCVEM approximation of state equation and the variational discretization of control variables, we construct a virtual element discrete scheme. For the state, adjoint state and control variable, we obtain the corresponding prior estimate in H1 and L2 norms. Finally, some numerical experiments are carried out to support the theoretical results.

一种三维Poisson-Nernst-Planck方程的虚单元计算

利用虚单元方法在多面体网格上求解一种三维稳态Poisson-Nernst-Planck(PNP)方程,并给出PNP方程的虚单元离散形式,推导电势方程及离子浓度方程的刚度矩阵与荷载向量的矩阵表达式.数值实验结果表明,在3种不同的多面体网格下实现了PNP方程的虚单元计算,数值解在L^(2)和H^(1)范数下均达到最优阶.

一种求解线弹性问题的无闭锁低阶虚拟元方法

研究了二维区域上线弹性问题的低阶虚拟元方法.用不连续的分段线性向量值函数增扩低阶协调虚拟元空间来构造离散空间,设计了一种离散方法,证明了能量范数下的误差是最优收敛的,和Lamé常数λ无关.最后给出数值算例验证了理论结果.

线性Poisson-Boltzmann方程的虚单元L^(2)误差估计

针对一类线性Poisson-Boltzmann方程的虚单元L^(2)误差估计进行分析.首先引入正则化的线性Poisson-Boltzmann方程,将原问题转化为非奇性Poisson-Boltzmann方程.然后给出L^(2)范数的误差估计.最后在四边形和五边形混合多边形网格上进行数值实验,数值结果验证了理论分析的正确性.

鸟粪形貌特性对220 kV复合绝缘子闪络影响的仿真分析

通过COMSOL有限元仿真软件研究不同形貌特性的鸟粪对绝缘子闪络的影响。通过研究鸟粪的长度和黏度对闪络的影响以及鸟粪下落导致绝缘子闪络的过程,本文得出了不同黏度多段鸟粪构成的组合空气间隙击穿电压的计算方法。结果表明:鸟粪黏度越高,鸟粪的连续性越好,构成的空气间隙击穿电压越低,容易造成复合绝缘子闪络。因此若想提高防鸟罩对高黏度长鸟粪的防护能力,防鸟罩需具有减少鸟粪长度的能力。本研究在防鸟罩表面设计导流槽,将原本一段高黏度且长度较长的鸟粪通过导流槽分割成多段长度较短的鸟粪,通过减少鸟粪长度从而增加空气间隙长度,进而可以提高击穿电压,降低鸟粪闪络概率。

一类三维椭圆界面问题的虚单元计算

椭圆界面问题是一种常出现在力学、材料科学、生物化学等领域的数值模拟中的经典模型问题。该模型问题具有间断系数和不规则界面两大特点。采用虚单元方法求解三维椭圆界面问题,与传统有限元法相比,该方法更适用于一般的多面体网格,这使得网格的选取变得自由与多样。基于这种特点,给出虚单元法求解一类三维椭圆界面问题的计算公式,并使用3种不同的多面体网格完成数值实验。数值结果表明,虚单元方法应用于该问题是有效的。

一类二维椭圆界面问题的虚单元方法计算

针对一类椭圆界面问题,采用一种近年来新提出的虚单元方法进行求解。基于虚单元法适用于任意多边形网格的特点,将该方法应用于六种不同的多边形网格下,求解一类椭圆界面问题,并给出虚单元法的计算公式。数值结果表明,虚单元方法应用于椭圆界面问题是有效的。

基于虚拟材料法的金属膜片联轴器振动传递特性分析

在对大型船舶金属联轴器进行振动性能试验时由于试验台架体积大、成本高,导致进行试验十分困难,因此研究高效率和高精度的仿真计算方法具有重要意义。传统精确的数值仿真网格多、效率低,为此采用虚拟材料法对船舶金属联轴器进行数值仿真并进行试验验证。首先通过计算和试验得到螺栓的虚拟材料层各参数,然后对大型金属联轴器的所有螺栓连接部位进行简化建模。其次通过该方法得到金属膜片联轴器的刚度特性、模态特性、阻抗特性和振动传递特性,并与传统建模方法的计算效率进行对比。最后搭建联轴器振动性能实验台架进行验证。研究结果表明:采用虚拟材料法的联轴器振动性能仿真效率提高73.9%;四组采用虚拟材料法的联轴器膜片组件刚度与实验测得的联轴器刚度的平均误差为2.61%,与带螺栓模型测得的刚度平均误差为1.15%。金属膜片联轴器前5阶模态频率仿真与试验的平均误差为9.91%,在10~2000 Hz频带,金属膜片联轴器轴向和径向阻抗的仿真与试验的误差分别为1.9和2.8 dB,金属联轴器轴向和径向的振级落差仿真与试验的误差分别为4.69和3.41 dB。试验与仿真结果误差较小,符合工程精度要求,该方法的计算效率和准确性得到验证。

基于新型虚单元法的超弹性材料变形研究

虚单元法是一种先进的求解固体力学问题的数值方法,在过去10年间,该算法在线弹性问题中得到了较为广泛地开发和应用.文章尝试给出一种通用的高阶虚单元法格式,可用于计算超弹性问题以及更普遍的非线性问题.与传统求解力学问题的虚单元法的思想不同,其主要思想是对泊松方程求解映射算子,并将该映射算子直接用于位移场的近似,从而可求解众多非线性力学问题.由于采用了标量场的映射算子来近似矢量场,因此该算法格式简单,并且可以轻易扩展到高阶格式或者三维问题求解.将从泊松方程出发,介绍虚单元法中椭圆映射算子的计算方法,在此基础上,详细推导虚单元法在求解超弹性问题时的具体格式,并给出切线刚度矩阵的计算方法.最后,给出了几个典型的超弹性数值算例,从而证明该虚单元法格式的有效性.

一类含时Poisson-Nernst-Planck方程的虚单元计算

针对一类含时Poisson-Nernst-Planck(PNP)方程,为避免在解决实际问题时有限元法中的网格适应性问题,构造了L^(2)投影算子与Gummel迭代相结合的虚单元算法。该算法允许以更简单的方式设计和分析新的格式,可以灵活处理各种网格,对于多边形或多面体单元甚至非凸单元组成的网格剖分都可以很好地处理,使得虚单元法可以适应于任意多边形网格,大大降低了网格的生成难度。给出了虚单元算法在三角形网格、四边形网格、非凸网格下的数值算例。数值实验结果表明,在这3种多边形网格上,L^(2)和H^(1)模的收敛阶分别为二阶和一阶,均达到了最优阶。