We continue the investigation on the spectrum of operators arising from the discretization of partial differential equations. In this paper we consider a three field formulation recently introduced for the finite element least-squares approximation of linear elasticity. We discuss in particular the distribution of the discrete eigenvalues in the complex plane and how they approximate the positive real eigenvalues of the continuous problem. The dependence of the spectrum on the Lamé parameters is considered as well and its behavior when approaching the incompressible limit.
In this paper we analyze a mixed displacement-pseudostress formulation for the elasticity eigenvalue problem. We propose a finite element method to approximate the pseudostress tensor with Raviart-Thomas elements and the displacement with piecewise polynomials. With the aid of the classic theory for compact operators, we prove that our method is convergent and does not introduce spurious modes. Also, we obtain error estimates for the proposed method. Finally, we report some numerical tests supporting the theoretical results.
We introduce a pure--stress formulation of the elasticity eigenvalue problem with mixed boundary conditions. We propose an H(div)-based discontinuous Galerkin method that imposes strongly the symmetry of the stress for the discretization of the eigenproblem. Under appropriate assumptions on the mesh and the degree of polynomial approximation, we demonstrate the spectral correctness of the discrete scheme and derive optimal rates of convergence for eigenvalues and eigenfunctions. Finally, we provide numerical examples in two and three dimensions.
The dual formulation for linear elasticity, in contrast to the primal formulation, is not affected by locking, as it is based on the stresses as main unknowns. Thus it is quite attractive for nearly incompressible and incompressible materials. Discretization with mixed finite elements will lead to -- possibly large -- linear saddle point systems with a particular structure. Whereas efficient multigrid methods exist for solving problems in mixed plane elasticity, to the knowledge of the authors, no multigrid methods are readily available for the general dual formulation. Two are the main challe…
We develop a mixed finite element method for the coupled problem arising in the interaction between a free fluid governed by the Stokes equations and flow in deformable porous medium modeled by the Biot system of poroelasticity. Mass conservation, balance of stress, and the Beavers--Joseph--Saffman condition are imposed on the interface. We consider a fully mixed Biot formulation based on a weakly symmetric stress-displacement-rotation elasticity system and Darcy velocity-pressure flow formulation. A velocity-pressure formulation is used for the Stokes equations. The interface conditions are i…
We study the approximation of the spectrum of least-squares operators arising from linear elasticity. We consider a two-field (stress/displacement) and a three-field (stress/displacement/vorticity) formulation; other formulations might be analyzed with similar techniques. We prove a priori estimates and we confirm the theoretical results with simple two-dimensional numerical experiments.
We develop the \textit{a posteriori} error analysis of three mixed finite element formulations for rotation-based equations in elasticity, poroelasticity, and interfacial elasticity-poroelasticity. The discretisations use $H^1$-conforming finite elements of degree $k+1$ for displacement and fluid pressure, and discontinuous piecewise polynomials of degree $k$ for rotation vector, total pressure, and elastic pressure. Residual-based estimators are constructed, and upper and lower bounds (up to data oscillations) for all global estimators are rigorously derived. The methods are all robust with r…
We present the construction and application of a first order stabilization-free virtual element method to problems in plane elasticity. Well-posedness and error estimates of the discrete problem are established. The method is assessed on a series of well-known benchmark problems from linear elasticity and numerical results are presented that affirm the optimal convergence rate of the virtual element method in the $L^2$ norm and the energy seminorm.
A Locking-Free Weak Galerkin Finite Element Method for Elasticity Problems in the Primal Formulation
Georgia Institute of Technology（佐治亚理工学院）；Taizhou College, Nanjing Normal University（南京师范大学泰州学院）；Division of Mathematical Sciences, National Science Foundation（美国国家科学基金会数学科学处）；Jilin University（吉林大学）
This paper constructs two conforming finite element grad grad and elasticity complexes on the cuboid meshes. For the finite element grad grad complex, an $H^2$ conforming finite element space, an $\boldsymbol{H}(\operatorname{curl}; \mathbb{S})$ conforming finite element space, an $\boldsymbol{H}(\operatorname{div}; \mathbb{T})$ conforming finite element space and an $\boldsymbol{L}^2$ finite element space are constructed. Further, a finite element complex with reduced regularity is also constructed, whose degrees of freedom for the three diagonal components are coupled. For the finite element…
We present a Virtual Element Method for the 3D linear elasticity problems, based on Hellinger-Reissner variational principle. In the framework of the small strain theory, we propose a low-order scheme with a-priori symmetric stresses and continuous tractions across element interfaces. A convergence and stability analysis is developed and we confirm the theoretical predictions via some numerical tests.
In this paper we provide some more details on the numerical analysis and we present some enlightening numerical results related to the spectrum of a finite element least-squares approximation of the linear elasticity formulation introduced recently. We show that, although the formulation is robust in the incompressible limit for the source problem, its spectrum is strongly dependent on the Lamé parameters and on the underlying mesh.
We consider mixed finite element methods for linear elasticity where the symmetry of the stress tensor is weakly enforced. Both an a priori and a posteriori error analysis are given for several known families of methods that are uniformly valid in the incompressible limit. A posteriori estimates are derived for both the compressible and incompressible cases. The results are verified by numerical examples.
We develop a theory of Finite Element Systems, for the purpose of discretizing sections of vector bundles, in particular those arizing in the theory of elasticity. In the presence of curvature we prove a discrete Bianchi identity. In the flat case we prove a de Rham theorem on cohomology groups. We check that some known mixed finite elements for the stress-displacement formulation of elasticity fit our framework. We also define, in dimension two, the first conforming finite element spaces of metrics with good linearized curvature, corresponding to strain tensors with Saint-Venant compatibility…
A finite element elasticity complex on tetrahedral meshes is devised. The $H^1$ conforming finite element is the smooth finite element developed by Neilan for the velocity field in a discrete Stokes complex. The symmetric div-conforming finite element is the Hu-Zhang element for stress tensors. The construction of an $H(\textrm{inc})$-conforming finite element for symmetric tensors is the main focus of this paper. The key tools of the construction are the decomposition of polynomial tensor spaces and the characterization of the trace of the $\textrm{inc}$ operator. The polynomial elasticity co…
We present a higher order stabilization-free virtual element method applied to plane elasticity problems. We utilize a serendipity approach to reduce the total number of degrees of freedom from the corresponding high-order approximations. The well-posedness of the problem is numerically studied via an eigenanalysis. The method is then applied to several benchmark problems from linear elasticity and we show that the method delivers optimal convergence rates in $L^2$ norm and energy seminorm that match theoretical estimates as well as the convergence rates from higher order virtual element metho…
Two robust nonconforming H$^2-$elements for linear strain gradient elasticity
China Academy of Engineering Physics（中国工程物理研究院）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
In this paper we study the approximation of eigenvalues arising from the mixed Hellinger--Reissner elasticity problem by using the simple finite element using partial relaxation of $C^0$ vertex continuity of stresses introduced recently by Jun Hu and Rui Ma. We prove that the method converge when a residual type error estimator is considered and that the estimator decays optimally with respect to the number of degrees of freedom.
An important requirement in the standard finite element method (FEM) is that all elements in the underlying mesh must be tangle-free i.e., the Jacobian must be positive throughout each element. To relax this requirement, an isoparametric tangled finite element method (i-TFEM) was recently proposed for linear elasticity problems. It was demonstrated that i-TFEM leads to optimal convergence even for severely tangled meshes. In this paper, i-TFEM is generalized to nonlinear elasticity. Specifically, a variational formulation is proposed that leads to local modification in the tangent stiffness ma…
In this paper, a residual-type a posteriori error estimator is proposed and analyzed for a modified weak Galerkin finite element method solving linear elasticity problems. The estimator is proven to be both reliable and efficient because it provides upper and lower bounds on the actual error in a discrete energy norm. Numerical experiments are given to illustrate the effectiveness of the this error estimator.
In this paper we analyze a virtual element method for the two dimensional elasticity problem allowing small edges. With this approach, the classic assumptions on the geometrical features of the polygonal meshes can be relaxed. In particular, we consider only star-shaped polygons for the meshes. Suitable error estimates are presented, where a rigorous analysis on the influence of the Lamé constants in each estimate is presented. We report numerical tests to assess the performance of the method.
In this paper, an enhanced Virtual Element Method (VEM) formulation is proposed for plane elasticity. It is based on the improvement of the strain representation within the element, without altering the degree of the displacement interpolating functions on the element boundary. The idea is to fully exploit polygonal elements with a high number of sides, a peculiar VEM feature, characterized by many displacement degrees of freedom on the element boundary, even if a low interpolation order is assumed over each side. The proposed approach is framed within a generalization of the classic VEM formu…
We consider mixed finite element methods with exact symmetric stress tensors. We derive a new quasi-optimal a priori error estimate uniformly valid with respect to the compressibility. For the a posteriori error analysis we consider the Prager-Synge hypercircle principle and introduce a new estimate uniformly valid in the incompressible limit. All estimates are validated by numerical examples.
We extend the hybridization procedure proposed in [Arnold, Brezzi, 1985, ESAIM: M2AN] to the Virtual Element Method for linear elasticity problems based on the Hellinger-Reissner principle. To illustrate such a technique, we focus on the 2D case, but other methods and 3D problems can be considered as well. We also show how to design a better approximation of the displacement field using a straightforward post-processing procedure. The numerical experiments confirm the theory for both two and three-dimensional problems.
In this article we consider the linear elasticity problem in an axisymmetric three dimensional domain, with data which are axisymmetric and have zero angular component. The weak formulation of the the three dimensional problem reduces to a two dimensional problem on the meridian domain, involving weighted integrals. The problem is formulated in a mixed method framework with both the stress and displacement treated as unknowns. The symmetry condition for the stress tensor is weakly imposed. Well posedness of the continuous weak formulation and its discretization are shown. Two approximation spa…
A Coupling Method of Mixed and Lagrange Finite Elements for Linear Elasticity Problem
LMAM and School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Chongqing Research Institute of Big Data, Peking University（北京大学重庆大数据研究院）；School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；National Center for Applied Mathematics in Hubei（湖北省应用数学中心）
We develop a Nitsche finite element method for a model of Euler--Bernoulli beams with axial stiffness embedded in a two--dimensional elastic bulk domain. The beams have their own displacement fields, and the elastic subdomains created by the beam network are triangulated independently and are coupled to the beams weakly by use of Nitsche's method in the framework of hybridization.
We present two kinds of lowest-order virtual element methods for planar linear elasticity problems. For the first one we use the nonconforming virtual element method with a stabilizing term. It can be interpreted as a modification of the nonconforming Crouzeix-Raviart finite element method as suggested in [22] to the virtual element method. For the second one we use the conforming virtual element for one component of the displacement vector and the nonconforming virtual element for the other. This approach can be seen as an extension of the idea of Kouhia and Stenberg suggested in [23] to the …
A mixed finite element method for nearly incompressible elasticity and Stokes equations using primal and dual meshes with quadrilateral and hexahedral grids
We propose a combined nodal integration and virtual element method for compressible and nearly incompressible elasticity, wherein the strain is averaged at the nodes from the strain of surrounding virtual elements. For the strain averaging procedure, a nodal averaging operator is constructed using a generalization to virtual elements of the node-based uniform strain approach for finite elements. We refer to the proposed technique as the node-based uniform strain virtual element method (NVEM). No additional degrees of freedom are introduced in this approach, thus resulting in a displacement-bas…
The purpose of this work is to study mortar methods for linear elasticity using standard low order finite element spaces. Based on residual stabilization, we introduce a stabilized mortar method for linear elasticity and compare it to the unstabilized mixed mortar method. For simplicity, both methods use a Lagrange multiplier defined on a trace mesh inherited from one side of the interface only. We derive a quasi-optimality estimate for the stabilized method and present the stability criteria of the mixed $P_1-P_1$ approximation. Our numerical results demonstrate the stability and the converge…
We introduce conformal mixed finite element methods for $2$D and $3$D incompressible nonlinear elasticity in terms of displacement, displacement gradient, the first Piola-Kirchhoff stress tensor, and pressure, where finite elements for the $\mathrm{curl}$ and the $\mathrm{div}$ operators are used to discretize strain and stress, respectively. These choices of elements follow from the strain compatibility and the momentum balance law. Some inf-sup conditions are derived to study the stability of methods. By considering $96$ choices of simplicial finite elements of degree less than or equal to $…
The immersed boundary method is a mathematical framework for modeling fluid-structure interaction. This formulation describes the momentum, viscosity, and incompressibility of the fluid-structure system in Eulerian form, and it uses Lagrangian coordinates to describe the structural deformations, stresses, and resultant forces. Integral transforms with Dirac delta function kernels connect the Eulerian and Lagrangian frames. The fluid and the structure are both typically treated as incompressible materials. Upon discretization, however, the incompressibility of the structure is only maintained a…
In this paper we analyze a virtual element method for the two dimensional elasticity spectral problem allowing small edges. Under this approach, and with the aid of the theory of compact operators, we prove convergence of the proposed VEM and error estimates, where the influence of the Lamé constants is presented. We present a series of numerical tests to assess the performance of the method where we analyze the effects of the Poisson ratio on the computation of the order of convergence, together with the effects of the stabilization term on the arising of spurious eigenvalues.
