Stabilized dual-mixed method for the problem of linear elasticity with mixed boundary conditions

We extend the applicability of the augmented dual-mixed method introduced recently in Gatica (2007), Gatica et al. (2009) to the problem of linear elasticity with mixed boundary conditions. The method is based on the Hellinger–Reissner principle and the symmetry of the stress tensor is imposed in a weak sense. The Neumann boundary condition is prescribed in the finite element space. Then, suitable Galerkin least-squares type terms are added in order to obtain an augmented variational formulation which is coercive in the whole space. This allows to use any finite element subspaces to approximate the displacement, the Cauchy stress tensor and the rotation.     

A priori error estimation for the dual mixed finite element method of the elastodynamic problem in a polygonal domain, I

In this paper we analyze a new dual mixed formulation of the elastodynamic system in polygonal domains. In this formulation the symmetry of the strain tensor is relaxed by the rotation of the displacement. For the time discretization of this new dual mixed formulation, we use an explicit scheme. After the analysis of stability of the fully discrete scheme, L^∞ in time, L² in space a priori error estimates are derived for the approximation of the displacement, the strain, the pressure and the rotation. Numerical experiments confirm our theoretical predictions.     

Refined mixed finite element method for the elasticity problem in a polygonal domain

The purpose of this article is to study a mixed formulation of the elasticity problem in plane polygonal domains and its numerical approximation. In this mixed formulation the strain tensor is introduced as a new unknown and its symmetry is relaxed by a Lagrange multiplier, which is nothing else than the rotation. Because of the corner points, the displacement field is not regular in general in the vicinity of the vertices but belongs to some weighted Sobolev space. Using this information, appropriate refinement rules are imposed on the family of triangulations in order to recapture optimal error estimates. Moreover, uniform error estimates in the Lamé coefficient λ are obtained for λ large. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 323–339, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10009     

Analysis of expanded mixed methods for fourth-order elliptic problems

The recently proposed expanded mixed formulation for numerical solution of second-order elliptic problems is here extended to fourth-order elliptic problems. This expanded formulation for the differential problems under consideration differs from the classical formulation in that three variables are treated, i.e., the displacement, the stress, and the moment tensors. It works for the case where the coefficient of the differential equations is small and does not need to be inverted, or for the case in which the stress tensor of the equations does not need to be symmetric. Based on this new formulation, various mixed finite elements for fourth-order problems are considered; error estimates of quasi-optimal or optimal order depending upon the mixed elements are derived. Implementation techniques for solving the linear system arising from these expanded mixed methods are discussed, and numerical results are presented. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 483–503, 1997     

Stabilized Nonconforming Mixed Finite Element Method for Linear Elasticity on Rectangular or Cubic Meshes

Based on the primal mixed variational formulation, a stabilized nonconforming mixed finite element method is proposed for the linear elasticity on rectangular and cubic meshes. Two kinds of penalty terms are introduced in the stabilized mixed formulation, which are the jump penalty term for the displacement and the divergence penalty term for the stress. We use the classical nonconforming rectangular and cubic elements for the displacement and the discontinuous piecewise polynomial space for the stress, where the discrete space for stress are carefully chosen to guarantee the well-posedness of discrete formulation. The stabilized mixed method is locking-free. The optimal convergence order is derived in the $L^2$-norm for stress and in the broken $H^1$-norm and $L^2$-norm for displacement. A numerical test is carried out to verify the optimal convergence of the stabilized method.     

Dual hybrid methods for the elasticity and the Stokes problems: a unified approach

Summary. A unified approach to construct finite elements based on a dual-hybrid formulation of the linear elasticity problem is given. In this formulation the stress tensor is considered but its symmetry is relaxed by a Lagrange multiplier which is nothing else than the rotation. This construction is linked to the approximations of the Stokes problem in the primitive variables and it leads to a new interpretation of known elements and to new finite elements. Moreover all estimates are valid uniformly with respect to compressibility and apply in the incompressible case which is close to the Stokes problem. Received June 20, 1994 / Revised version received February 16, 1996     

On the construction of optimal mixed finite element methods for the linear elasticity problem

Summary The mixed finite element method for the linear elasticity problem is considered. We propose a systematic way of designing methods with optimal convergence rates for both the stress tensor and the displacement. The ideas are applied in some examples.     

Analysis of the coupling of BEM, FEM and mixed-FEM for a two-dimensional fluid–solid interaction problem

In this paper we consider a fluid–solid interaction problem posed in the plane. We employ a mixed variational formulation in the obstacle, in which the Cauchy stress tensor and the rotation are the only unknowns. This new mixed formulation is coupled, through suitable transmission conditions on the wet interface, with a Helmholtz equation satisfied by the pressure of the fluid in the unbounded domain. We use a traditional primal variational formulation in this part of the domain and incorporate the far field information through boundary integral equations. We approximate the resulting weak formulation by a Galerkin scheme based on PEERS in the solid and on a FEM-BEM approach in the fluid part. We show that our scheme is uniquely solvable and convergent, and then provide optimal error estimates. Finally, we illustrate our analysis with some computational experiments.     

A stabilized mixed formulation for finite strain deformation for low-order tetrahedral solid elements

A stabilized mixed finite element formulation for four-noded tetrahedral elements is introduced for robustly solving small and large deformation problems. The uniqueness of the formulation lies within the fact that it is general in that it can be applied to any type of material model without requiring specialized geometric or material parameters. To overcome the problem of volumetric locking, a mixed element formulation that utilizes linear displacement and pressure fields was implemented. The stabilization is provided by enhancing the rate of deformation tensor with a term derived using a bubble function approach. The element was implemented through a user-programmable element of the commercial finite element software ANSYS. Using the ANSYS platform, the performance of the element was evaluated by comparing the predicted results with those obtained using mixed quadratic tetrahedral elements and hexahedral elements with a B-bar formulation. Based on the quality of the results, the new element formulation shows significant potential for use in simulating complex engineering processes.     

Dual mixed finite element methods for the elasticity problem with Lagrange multipliers

We study a dual mixed formulation of the elasticity system in a polygonal domain of the plane with mixed boundary conditions and its numerical approximation. The (essential) Neumann boundary conditions (or traction boundary condition) are imposed using a discontinuous Lagrange multiplier corresponding to the trace of the displacement field. Moreover, a strain tensor is introduced as a new unknown and its symmetry is relaxed, also by the use of a Lagrange multiplier (the rotation). The singular behaviour of the solution requires us to use refined meshes to restore optimal rates of convergence. Uniform error estimates in the Lamé coefficient λ$\lambda$ are obtained for large λ$\lambda$. The hybridization of the problem is performed and numerical tests are presented confirming our theoretical results.     

Coupling of mixed finite element and stabilized boundary element methods for a fluid–solid interaction problem in 3D

We introduce and analyze the coupling of a mixed finite element and a boundary element for a three‐dimensional time‐harmonic fluid–solid interaction problem. We consider a formulation in which the Cauchy stress tensor and the rotation are the main variables in the elastic structure and use the usual pressure formulation in the acoustic fluid. The mixed variational formulation in the solid is completed with boundary integral equations relating the Cauchy data of the acoustic problem on the coupling interface. A crucial point in our formulation is the stabilization technique introduced by Hiptmair and coworkers to avoid the well‐known instability issue appearing in the boundary element method treatment of the exterior Helmholtz problem. The main novelty of this formulation, with respect to a previous approach, consists in reducing the computational domain to the solid media and providing a more accurate treatment of the far field effect. We show that the continuous problem is well‐posed and propose a conforming Galerkin method based on the lowest‐order Arnold–Falk–Winther mixed finite element. Finally, we prove that the numerical scheme is convergent with optimal order.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1211–1233, 2014     

A mixed finite element method for a quasi‐Newtonian fluid flow

We propose a mixed formulation for quasi‐Newtonian fluid flow obeying the power law where the stress tensor is introduced as a new variable. Based on such a formulation, a mixed finite element is constructed and analyzed. This finite element method possesses local (i.e., at element level) conservation properties (conservation of the momentum and the mass) as in the finite volume methods. We give existence and uniqueness results for the continuous problem and its approximation and we prove error bounds. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

A mixed nonconforming finite element for linear elasticity

This article considers a mixed finite element method for linear elasticity. It is based on a modified mixed formulation that enforces the continuity of the stress weakly by adding a jump term of the approximated stress on interior edges. The symmetric stress are approximated by nonconforming linear elements and the displacement by piecewise constants. We establish ??(h) error bound in the (broken) L² norm for the divergence of the stress and ??(h) error bound in the L² norm for both the displacement and the stress tensor. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

A new mixed finite element method for the Stokes problem

We propose a new mixed formulation of the Stokes problem where the extra stress tensor is considered. Based on such a formulation, a mixed finite element is constructed and analyzed. This new finite element has properties analogous to the finite volume methods, namely, the local conservation of the momentum and the mass. Optimal error estimates are derived. For the numerical implementation of this finite element, a hybrid form is presented. This work is a first step towards the treatment of viscoelastic fluid flows by mixed finite element methods.     

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 ...     

板的大变形分析的混合变分泛函*

本文导出了板在大变形分析的混合变分表达式,在本式中,平衡方程和协调方程是分别用应力函数和位移分量等同满足的,而应力应变关系是在最小二乘方的意义上满足的.解了一个例,并和文献中已知的结果进行了比较.此外,我们写出了特别适用于板的屈曲失稳分析的泛函,并举例题解证明理论的有效.     

THE PARTIAL PROJECTION METHOD IN THE FINITE ELEMENT DISCRETIZATION OF THE REISSNER-MINDLIN PLATE MODEL

1.IntroductionItiswellknownthatthestandaxdfiniteelementdiscretizatiollsoftheReissner-Mindlinplateproblemproducepoorapproximationswhenthethicknessistoosmall...incomparisonwiththediameteroftheregionoccupiedbythemidsectionoftheplate.TherootisthesthcaJled"locking"phenomenonwhichisbynowwellunderstood.Amongseveralapproachestoavoidinglockingisamodificationofthestandardfiniteelementschemesbyinterpolatingorprojectingthediscretetrallsverseshearforceintoalower-orderfiniteelemelltspaJce.Thiskindofmethod…     

A family of higher order mixed finite element methods for plane elasticity

Summary The Dirichler problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the Raviart-Thomas mixed finite elements for a scalar, second-order elliptic equation. Estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor inL ²(), and optimal order negative norm estimates are obtained inH ^s () for a range ofs depending on the index of the finite element space. An optimal order estimate inL () for the displacement error is given. Also, a quasioptimal estimate is derived in an appropriate space. All estimates are valid uniformly with respect to the compressibility and apply in the incompressible case. The formulation of the elements is presented in detail.This work was performed while Professor Arnold was a NATO Postdoctoral Fellow     

Mixed dynamic problem for a half plane and its application to rock mechanics

The mixed dynamic problem of the theory of elasticity is solved for an isotropic half plane. The dynamic equations are reduced to integration of fourth-degree equations in partial derivatives with constant coefficients, after whose solution, the components of the stress tensor and displacement vector are written in a form similar to that introduced by Lekhnitskii for an anisotropic body. The stress state of a rock mass subjected to rapid face advance in a seam is investigated using the solution obtained. The stress distribution is analyzed numerically. The existence of a critical rate at which the stress increases without restriction is demonstrated.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 56–61, 1990.     

Continuous finite element methods for Reissner-Mindlin plate problem

On triangle or quadrilateral meshes, two finite element methods are proposed for solving the Reissner-Mindlin plate problem either by augmenting the Galerkin formulation or modifying the plate-thickness. In these methods, the transverse displacement is approximated by conforming (bi)linear macroelements or (bi)quadratic elements, and the rotation by conforming (bi)linear elements. The shear stress can be locally computed from transverse displacement and rotation. Uniform in plate thickness, optimal error bounds are obtained for the transverse displacement, rotation, and shear stress in their natural norms. Numerical results are presented to illustrate the theoretical results.