Finite elements for symmetric tensors in three dimensions

We construct finite element subspaces of the space of symmetric tensors with square-integrable divergence on a three-dimensional domain. These spaces can be used to approximate the stress field in the classical Hellinger-Reissner mixed formulation of the elasticty equations, when standard discontinuous finite element spaces are used to approximate the displacement field. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and there is one for each positive value of the polynomial degree used for the displacements. For each degree, these provide a stable finite element discretization. The construction of the spaces is closely tied to discretizations of the elasticity complex and can be viewed as the three-dimensional analogue of the triangular element family for plane elasticity previously proposed by Arnold and Winther.

     

New mixed finite elements for plane elasticity and Stokes equations

We consider mixed finite elements for the plane elasticity system and the Stokes equation. For the unmodified Hellinger-Reissner formulation of elasticity in which the stress and displacement fields are the primary unknowns, we derive two new nonconforming mixed finite elements of triangle type. Both elements use piecewise rigid motions to approximate the displacement and piecewise polynomial functions to approximate the stress, where no vertex degrees of freedom are involved. The two stress finite element ...     

Some remarks on the constant in the strengthened CBS inequality: Estimate for hierarchical finite element discretizations of elasticity problems

For a class of two‐dimensional boundary value problems including diffusion and elasticity problems, it is proved that the constants in the corresponding strengthened Cauchy‐Buniakowski‐Schwarz (CBS) inequality in the cases of two‐level hierarchical piecewise‐linear/piecewise‐linear and piecewise‐linear/piecewise‐quadratic finite element discretizations with triangular meshes differ by the factor 0.75. For plane linear elasticity problems and triangulations with right isosceles triangles, formulas are presented that show the dependence of the constant in the CBS inequality on the Poisson's ratio. Furthermore, numerically determined bounds of the constant in the CBS inequality are given for plane linear elasticity problems discretized by means of arbitrary triangles and for three‐dimensional elasticity problems discretized by means of tetrahedral elements. Finally, the robustness of iterative solvers for elasticity problems is discussed briefly. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 469–487, 1999     

An auxiliary space preconditioner for linear elasticity based on the generalized finite element method

We construct and analyze a preconditioner of the linear elasticity system discretized by conforming linear finite elements in the framework of the auxiliary space method. The auxiliary space preconditioner is based on two auxiliary spaces corresponding to discretizations of the scalar Poisson equation by linear finite elements and the generalized finite element method. Copyright © 2011 John Wiley & Sons, Ltd.     

弱有限元方法在线弹性问题中的应用

本文考虑弱有限元（简称WG）方法在线弹性问题中的应用.WG方法是传统有限元方法的推广,用于偏微分方程的数值求解.和传统有限元一样,它的基本思想源于变分原理.WG方法的特点是使用在剖分单元内部和剖分单元边界上分别有定义的分片多项式函数（即弱函数）作为近似函数来逼近真解,并针对弱函数定义相应的弱微分算子代入数值格式进行计算.除此之外,WG方法允许在数值格式中引进稳定子以实现近似函数的弱连续性.WG方法具有允许使用任意多边形或多面体剖分,数值格式与逼近函数构造简单,易于满足相应的稳定性条件等优点.本文考虑WG方法在求解线弹性问题中的应用.围绕线弹性问题数值求解中常见的三个问题,即：数值格式的强制性,闭锁性,应力张量的对称性介绍WG方法在线弹性问题求解中的应用.     

Symmetric and conforming mixed finite elements for plane elasticity using rational bubble functions

We construct stable, conforming and symmetric finite elements for the mixed formulation of the linear elasticity problem in two dimensions. In our approach we add three divergence-free rational functions to piecewise polynomials to form the stress finite element space. The relation with the elasticity elements and a class of generalized $C^1$ Zienkiewicz elements is also discussed.     

平面线弹性纯位移边值问题低阶非协调矩形元的Robust多重网格方法

1引 言 对于各向同性,均匀介质的平面线弹性问题,当Lamé常数λ→∞(泊松率v→0.5)时,即对于几乎不可压介质,通常的协调有限元格式的解往往不再收敛到原问题的解,或者达不到最优收敛阶,这就是所谓的闭锁现象(见[3],[7],[8]及[10]).究其原因,在通常的有限元分析中,其误差估计的系数与λ有关,当λ→∞时,该系数将趋于无穷大.因此为克服闭锁现象就需要构造特殊的有限元格式,使得当λ→∞时,有限元逼近解仍然收敛到原问题的解.     

A mixed finite element method for the unilateral contact problem in elasticity

In this paper, we provide a new mixed finite element approximation of the varia-tional inequality resulting from the unilateral contact problem in elasticity. We use the continuous piecewise P2-P1 finite element to approximate the displacement field and the normal stress component on the contact region. Optimal convergence rates are obtained under the reasonable regularity hypotheses. Numerical example verifies our results.     

Mixed finite element methods for unilateral problems: convergence analysis and numerical studies

In this paper, we propose and study different mixed variational methods in order to approximate with finite elements the unilateral problems arising in contact mechanics. The discretized unilateral conditions at the candidate contact interface are expressed by using either continuous piecewise linear or piecewise constant Lagrange multipliers in the saddle-point formulation. A priori error estimates are established and several numerical studies corresponding to the different choices of the discretized unilateral conditions are achieved.

     

Piecewise linear finite element methods are not localized

Recent results of Schatz show that standard Galerkin finite element methods employing piecewise polynomial elements of degree two and higher to approximate solutions to elliptic boundary value problems are localized in the sense that the global dependence of pointwise errors is of higher order than the overall order of the error. These results do not indicate that such localization occurs when piecewise linear elements are used. We show via simple one-dimensional examples that Schatz's estimates are sharp in that localization indeed does not occur when piecewise linear elements are used.

     

An augmented mixed finite element method for 3D linear elasticity problems

In this paper we introduce and analyze a new augmented mixed finite element method for linear elasticity problems in 3D. Our approach is an extension of a technique developed recently for plane elasticity, which is based on the introduction of consistent terms of Galerkin least-squares type. We consider non-homogeneous and homogeneous Dirichlet boundary conditions and prove that the resulting augmented variational formulations lead to strongly coercive bilinear forms. In this way, the associated Galerkin schemes become well posed for arbitrary choices of the corresponding finite element subspaces. In particular, Raviart-Thomas spaces of order 0 for the stress tensor, continuous piecewise linear elements for the displacement, and piecewise constants for the rotation can be utilized. Moreover, we show that in this case the number of unknowns behaves approximately as 9.5 times the number of elements (tetrahedrons) of the triangulation, which is cheaper, by a factor of 3, than the classical PEERS in 3D. Several numerical results illustrating the good performance of the augmented schemes are provided.     

Rectangular mixed elements for elasticity with weakly imposed symmetry condition

We present new rectangular mixed finite elements for linear elasticity. The approach is based on a modification of the Hellinger–Reissner functional in which the symmetry of the stress field is enforced weakly through the introduction of a Lagrange multiplier. The elements are analogues of the lowest order elements described in Arnold et al. (Math Comput 76:1699–1723, 2007). Piecewise constants are used to approximate the displacement and the rotation. The first order BDM elements are used to approximate each row of the stress field.     

Some equilibrium finite element methods for two-dimensional elasticity problems

Summary We consider some equilibrium finite element methods for two-dimensional elasticity problems. The stresses and the displacements are approximated by using piecewise linear functions. We establishL ₂-estimates of orderO(h ²) for both stresses and displacements.     

Domain decomposition preconditioners for multiscale problems in linear elasticity

We analyze two‐level overlapping Schwarz domain decomposition methods for vector‐valued piecewise linear finite element discretizations of the PDE system of linear elasticity. The focus of our study lies in the application to compressible, particle‐reinforced composites in 3D with large jumps in their material coefficients. We present coefficient‐explicit bounds for the condition number of the two‐level additive Schwarz preconditioned linear system. Thereby, we do not require that the coefficients are resolved by the coarse mesh. The bounds show a dependence of the condition number on the energy of the coarse basis functions, the coarse mesh, and the overlap parameters, as well as the coefficient variation. Similar estimates have been developed for scalar elliptic PDEs by Graham et al. 1 The coarse spaces to which they apply here are assumed to contain the rigid body modes and can be considered as generalizations of the space of piecewise linear vector‐valued functions on a coarse triangulation. The developed estimates provide a concept for the construction of coarse spaces, which can lead to preconditioners that are robust with respect to high contrasts in Young's modulus and the Poisson ratio of the underlying composite. To confirm the sharpness of the theoretical findings, we present numerical results in 3D using vector‐valued linear, multiscale finite element and energy‐minimizing coarse spaces. The theory is not restricted to the isotropic (Lamé) case, extends to the full‐tensor case, and allows applications to multiphase materials with anisotropic constituents in two and three spatial dimensions. However, the bounds will depend on the ratio of largest to smallest eigenvalue of the elasticity tensor.     

Mixed finite element methods for the Signorini problem with friction

In this article, we propose and study different mixed variational methods in order to approximate the Signorini problem with friction using finite elements. The discretized normal and tangential constraints at the contact interface are expressed by using either continuous piecewise linear or piecewise constant Lagrange multipliers in the saddle?point formulation. A priori error estimates are established and several numerical examples corresponding to the different choices of the discretized normal and tangential constraints are carried out. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Computing material collapse displacement fields on a cray X-MP/48 by the LP primal affine scaling algorithm

A pair of dual, highly degenerate linear programs modeling the static and kinematic principles of limit analysis are studied. They arise from mixed finite element discretizations of the continuous saddle point problem and piecewise linear approximations of the convex unbounded Mises set of admissible stresses. We use piecewise constant stresses and piecewise bilinear displacements, and present computational results with the Affine Scaling Algorithm a la Dikin (1967) and Karmarker (1984). Graphical displays are presented for material collapse fields of a rectangular solid with thin cuts under tension, in problem sizes up to 9000 variables and 7700 equations. To our knowledge, these stress fields have not been computed before.The research underlying this report was supported by NSF Grants ECS-8500940 at Carnegie-Mellon University and the University of Iowa and by ECS-8696087 at the University of Iowa. A preliminary version of this paper was presented at the Supercomputers and Large-Scale Optimization Workshop at the University of Minnesota, Minneapolis, May 16–18, 1988.     

An algebraic multigrid method for elasticity based on an auxiliary topology with edge matrices

This article introduces a novel approach to algebraic multigrid methods for large systems of linear equations coming from finite element discretizations of certain elliptic second-order partial differential equations. Based on a discrete energy made up of edge and vertex contributions, we are able to develop coarsening criteria that guarantee two-level convergence even for systems of equations such as linear elasticity . This energy also allows us to construct prolongations with prescribed sparsity pattern that still preserve kernel vectors exactly. These allow for a straightforward optimization that simplifies parallelization and reduces communication on coarse levels. Numerical experiments demonstrate efficiency and robustness of the method and scalability of the implementation.     

The hierarchical basis multigrid method

Summary We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric Gauß-Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigridV-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.     

Least-squares finite element methods for the elasticity problem

A new first-order system formulation for the linear elasticity problem in displacement-stress form is proposed. The formulation is derived by introducing additional variables of derivatives of the displacements, whose combinations represent the usual stresses. Standard and weighted least-squares finite element methods are then applied to this extended system. These methods offer certain advantages such as that they need not satisfy the inf-sup condition which is required in the mixed finite element formulation, that a single continuous piecewise polynomial space can be used for the approximation of all the unknowns, that the resulting algebraic systems are symmetric and positive definite, and that accurate approximations of the displacements and the stresses can be obtained simultaneously. With displacement boundary conditions, it is shown that both methods achieve optimal rates of convergence in the H¹-norm and in the L²-norm for all the unknowns. Numerical experiments with various Poisson ratios are given to demonstrate the theoretical error estimates.     

CONVERGENCE AND OPTIMALITY OF ADAPTIVE MIXED METHODS FOR POISSON'S EQUATION IN THE FEEC FRAMEWORK

Finite Element Exterior Calculus (FEEC) was developed by Arnold, Falk, Winther and others over the last decade to exploit the observation that mixed variational problems can be posed on a Hilbert complex, and Galerkin-type mixed methods can then be obtained by solving finite-dimensional subcomplex problems. Chen, Holst, and Xu (Math. Comp. 78 (2009) 35-53) established convergence and optimality of an adaptive mixed finite element method using Raviart-Thomas or Brezzi-Douglas-Marini elements for Poisson's equation on contractible domains in $\mathbb{R}^2$, which can be viewed as a boundary problem on the de Rham complex. Recently Demlow and Hirani (Found. Math. Comput. 14 (2014) 1337-1371) developed fundamental tools for a posteriori analysis on the de Rham complex. In this paper, we use tools in FEEC to construct convergence and complexity results on domains with general topology and spatial dimension. In particular, we construct a reliable and efficient error estimator and a sharper quasi-orthogonality result using a novel technique. Without marking for data oscillation, our adaptive method is a contraction with respect to a total error incorporating the error estimator and data oscillation.