Search

Summary: This paper is concerned with the analysis and implementation of robust finite element approximation methods for mixed formulations of linear elasticity problems where the elastic body is almost incompressible. Several alternative a posteriori error estimators for the energy norm of the finite element error are introduced and analysed. We establish upper and lower bounds for the energy error in terms of these error estimators and show that the constants in the bounds are independent of the Lamé coefficients: The proposed estimators are robust in the incompressible limit. We also consider the requirement for pressure stabilisation when using lowest‐order conforming approximation. Computational results are presented that validate the theoretical estimates. The software used to generate these results is available online. [ABSTRACT FROM AUTHOR]

Modern software's allow computers to perform symbolic calculations in addition to simple numerical computations for which they were originally designed. This fact opens new possibilities in numerical simulation. In this paper a finite element implementation using the commercial software Wolfram Mathematica is used to solve a plane stress elasticity problem. The code adaptability from the provided examples to more challenging problems can easily be performed using the present code. Numerical examples with post‐processing illustrates the new tool and its flexibility. It has been proven in this study the feasibility of Wolfram Mathematica in learning and programming finite element methods for the solution of two‐dimensional problems in elasticity. [ABSTRACT FROM AUTHOR]

The constant γ in the strengthened Cauchy-Buniakowski-Schwarc (CBS) inequality plays a key role in the convergence analysis of the multilevel iterative methods. We consider in this paper the approximation of the two-dimensional elasticity problem by bilinear rectangle finite elements. Two semi-coarsening refinement procedures are studied. We prove for both cases new estimates of the constant γ, uniformly on the Poisson ratio. As a result of the presented analysis we obtain an optimal order algebraic multiLevel iteration (AMLI) method for the case of balanced semi-coarsening mesh refinement. The total computational complexity of the algorithm is proportional to the size of the discrete problem with a proportionality constant independent of the Poisson ratio, that is, the algorithm is of optimal order for almost incompressible elasticity problems. Copyright © 1999 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY Arterial walls in the healthy physiological regime are characterized by quasi-incompressible, anisotropic, hyperelastic material behavior. Polyconvex material functions representing such materials typically incorporate a penalty function to account for the incompressibility. Unfortunately, the penalty will affect the conditioning of the stiffness matrices. For high penalty parameters, the performance of iterative solvers will degrade, and when direct solvers are used, the quality of the solutions will deteriorate. In this paper, an augmented Lagrange approach is used to cope with the quasi-incompressibility condition. Here, the penalty parameter can be chosen much smaller, and as a consequence, the arising linear systems of equations have better properties. An improved convergence is then observed for the finite element tearing and interconnecting-dual primal domain decomposition method, which is used as an iterative solver. Numerical results for an arterial geometry obtained from ultrasound imaging are presented. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper reports the use of polynomial wavelets as approximation functions in hybrid-mixed stress finite element models applied to the solution of plane elasticity problems. The stress and displacement fields in the domain and the displacements on the static boundary are independently approximated. The kinematic boundary conditions are locally satisfied. All remaining equations are enforced in a weighted residual form so designed as to ensure that the discrete model embodies the relevant properties of continuum systems, namely the static-kinematic duality and elastic reciprocity. A set of numerical applications is presented to illustrate the use of the hybrid-mixed model and to assess its efficiency and accuracy. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The paper deals with the symbolic integration of a 4-noded isoparametric finite element for plane elasticity. An efficient approach to generate explicit formulas for computing the elementary stiffness matrix is discussed. The procedure is based on the use of the Derive symbolic manipulation code as well as in a posteriori manipulation of the expressions obtained. The accuracy of the results is tested in extremely distorted and geometrically ill-conditioned elements. Three practical engineering models are presented and the accuracy of the results is discussed. A computer time comparison between both numerical and symbolic integration approaches is also included, showing that relevant CPU savings are obtained when applying symbolic integration. [ABSTRACT FROM AUTHOR]

Multigrid methods for a stencil-based implementation of a finite element method for planar elasticity, using semi-structured triangular grids, are presented. Local Fourier analysis (LFA) is applied to identify the correct multigrid components. To this end, LFA for multigrid methods on regular triangular grids is extended to the case of the problem of planar elasticity, although its application to other systems is straightforward. For the discrete elasticity operator obtained with linear finite elements, different collective smoothers, such as three-color smoother and some zebra-type smoothers, are analyzed, and LFA results for these smoothers are shown. The multigrid method is constructed in a block-wise form. In particular, different smoothers and different numbers of pre- and post-smoothing steps are considered in each triangle of the coarsest triangulation of the domain. Some numerical experiments are presented to illustrate the efficiency of this multigrid algorithm. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Highly scalable parallel domain decomposition methods for elliptic partial differential equations are considered with a special emphasis on problems arising in elasticity. The focus of this survey article is on Finite Element Tearing and Interconnecting (FETI) methods, a family of nonoverlapping domain decomposition methods where the continuity between the subdomains, in principle, is enforced by the use of Lagrange multipliers. Exact onelevel and dual-primal FETI methods as well as related inexact dual-primal variants are described and theoretical convergence estimates are presented together with numerical results confirming the parallel scalability properties of these methods. New aspects such as a hybrid onelevel FETI/FETI-DP approach and the behavior of FETI-DP for anisotropic elasticity problems are presented. Parallel and numerical scalability of the methods for more than 65 000 processor cores of the JUGENE supercomputer is shown. An application of a dual-primal FETI method to a nontrivial biomechanical problem from nonlinear elasticity, modeling arterial wall stress, is given, showing the robustness of our domain decomposition methods for such problems. [ABSTRACT FROM AUTHOR]