Complementary error bounds for foolproof finite element mesh generation

The use of complementary variational principles in finite element analysis is examined. It is shown that complementary finite element solutions provide an element by element measure of the accuracy of the solution. By solving a problem repeatedly, beginning with a coarse mesh and refining those elements having the largest errors, an automatic, foolproof finite element mesh generation procedure is developed. Finite element solutions obtained by the new procedure have the property that the finest elements are concentrated in regions of greatest need while large elements are found in less important regions. A computer program which implements the new algorithm is described and examples of finite element solutions generated by the program are presented.     

Inherited error in finite element analyses of structures

Mathematicians are quick to point out that round-off and truncation errors induced by the digital computer are only half of the manipulation errors in numerical analyses. The other half are the errors in quantizing the mathematical problem for computer solution: errors inherited for the equation solving process.This paper examines the relevance of inherited errors in structural analyses using the finite element concept and the digital computer. It illustrates error magnitudes using numerical experiments of simple structures. It constructs a theory explaining the errors for these systems. Having identified the most significant controllable computer error source, it describes a process for minimizing its contribution to the inherited error.The paper concludes that orders of magnitude between errors reported by various investigators can be explained by differences in inherited error. The most significant effect of these errors can be identified with inconsistency in problem formulation. These inconsistencies can be eliminated by exploiting the existence of rigid body states in the finite element models. Thereby, solution errors introduced by inherited errors can be reduced to intrinsic errors in parameters defining the geometry, material characteristics, and boundary conditions.     

Nonconforming mixed finite element methods for linear elasticity using rectangular elements in two and three dimensions

Abstract We present nonconforming, rectangular mixed finite element methods based on the Hellinger-Reissner variational principle in both two and three dimensions and show stability and convergence. An optimal error estimate of is obtained for the displacement, along with a suboptimal, , error estimate for the stress, in both dimensions.     

Accurate recovery-based upper error bounds for the extended finite element framework

This paper introduces a recovery-type error estimator yielding upper bounds of the error in energy norm for linear elastic fracture mechanics problems solved using the extended finite element method (XFEM). The paper can be considered as an extension and enhancement of a previous work in which the upper bounds of the error were developed in a FEM framework. The upper bound property requires the recovered solution to be equilibrated and continuous. The proposed technique consists of using a recovery technique, especially adapted to the XFEM framework that yields equilibrium at a local level (patch by patch). Then a postprocess based on the partition of unity concept is used to obtain continuity. The result is a very accurate but only nearly-statically admissible recovered stress field, with small equilibrium defaults introduced by the postprocess. Sharp upper bounds are obtained using a new methodology accounting for the equilibrium defaults, as demonstrated by the numerical tests.     

On certain computable tests and componentwise error bounds

Improved forms of the existence and uniqueness tests due to Pandian are given and are related to results due to Moore and Kioustelidis and to Shen and Neumaier.     

An analysis of some mixed-enhanced finite element for plane linear elasticity

The paper investigates Mixed-Enhanced Strain finite elements developed within the context of the u/p formulation for nearly incompressible linear elasticity problems. A rigorous convergence and stability analysis is detailed, providing also L²-error estimates for the displacement field. Extensive numerical tests are developed, showing in particular the accordance of the computational results with the theoretical predictions.     

Worst case bounds on the point-wise discretization error in boundary element method for the elasticity problem

In this work, point-wise discretization error is bounded via interval approach for the elasticity problem using interval boundary element formulation. The formulation allows for computation of the worst case bounds on the boundary values for the elasticity problem. From these bounds the worst case bounds on the true solution at any point in the domain of the system can be computed. Examples are presented to demonstrate the effectiveness of the treatment of local discretization error in elasticity problem via interval methods.     

On a posteriori error estimates for the linear triangular finite element

Based on equilibration of side fluxes, an a posteriori error estimator is obtained for the linear triangular element for the Poisson equation, which can be computed locally. We present a procedure for constructing the estimator in which we use the Lagrange multiplier similar to the usual equilibrated residual method introduced by Ainsworth and Oden. The estimator is shown to provide guaranteed upper bound, and local lower bounds on the error up to a multiplicative constant depending only on the geometry. Based on this, we give another error estimator which can be directly constructed without solving local Neumann problems and also provide the two-sided bounds on the error. Finally, numerical tests show our error estimators are very efficient.     

Frequency-dependent error bounds for uncertain linear models

We study frequency-dependent error bounds for approximation and truncation of linear dynamic models with uncertainty. A possible application is in model simplification relevant for control design. The uncertainty is described by quadratic constraints, and the error bounds are calculated based on solutions to linear matrix inequalities     

Some results of finite element applications in finite elasticity

This paper examines a number of problems connected with the finite element analysis of finite elastic deformations. A brief review of formulation of equations governing finite deformations of highly elastic elements is given. The convergence of finite element approximations for static problems in elasticity is studied. Incremental stiffness equations are derived in general form and various types of incremental loading techniques are examined. A number of representative solved problems in finite elasticity are given.     

Determining invertible two and three dimensional quadratic isoparametric finite element transformations

This paper presents an algorithm determining the invertibility of any planar, triangular quadratic isoparametric finite element transformation. Extensions of the algorithm to three-dimensional isoparametric finite element transformations yield conditions which guarantee invertibility of 10-node tetrahedra and 8-node bricks.     

A locking-free stabilized mixed finite element method for linear elasticity: the high order case

In this paper, we propose a locking-free stabilized mixed finite element method for the linear elasticity problem, which employs a jump penalty term for the displacement approximation. The continuous piecewise k-order polynomial space is used for the stress and the discontinuous piecewise \((k-1)\)-order polynomial space for the displacement, where we require that \(k\ge 3\) in the two dimensions and \(k\ge 4\) in the three dimensions. The method is proved to be stable and k-order convergent for the stress in \(H(\mathrm {div})\)-norm and for the displacement in \(L^2\)-norm. Further, the convergence does not deteriorate in the nearly incompressible or incompressible case. Finally, the numerical results are presented to illustrate the optimal convergence of the stabilized mixed method.     

Adaptive error control for multigrid finite element

We consider the problem of adaptive error control in the finite element method including the error resulting from, inexact solution of the discrete equations. We prove a posteriori error estimates for a prototype elliptic model problem discretized by the finite element with a canomical multigrid algorithm. The proofs are based on a combination of so-called strong stability and, the orthogonality inherent in both the finite element method can the multigrid algorithm.     

Robust FETI-DP methods for heterogeneous three dimensional elasticity problems

Iterative substructuring methods with Lagrange multipliers are considered for heterogeneous linear elasticity problems with large discontinuities in the material stiffnesses. In particular, results for algorithms belonging to the family of dual-primal FETI methods are presented. The core issue of these algorithms is the construction of an appropriate global problem, in order to obtain a robust method which converges independently of the material discontinuities. In this article, several necessary and sufficient conditions arising from the theory are numerically tested and confirmed. Furthermore, results of numerical experiments are presented for situations which are not covered by the theory, such as curved edges and material discontinuities not aligned with the interface, and an attempt is made to develop rules for these cases.     

A posteriori error estimates for space-time finite element approximation of quasistatic hereditary linear viscoelasticity problems

We give a space-time Galerkin finite element discretisation of the quasistatic compressible linear viscoelasticity problem as described by an elliptic partial differential equation with a fading memory Volterra integral. The numerical scheme consists of a continuous Galerkin approximation in space based on piecewise polynomials of degree p>0 (cG(p)), with a discontinuous Galerkin piecewise constant (dG(0)) or linear (dG(1)) approximation in time. A posteriori Galerkin-error estimates are derived by exploiting the Galerkin framework and optimal stability estimates for a related dual backward problem. The a posteriori error estimates are quite flexible: strong L_p-energy norms of the errors are estimated using time derivatives of the residual terms when the data are smooth, while weak-energy norms are used when the data are non-smooth (in time).We also give upper bounds on the dG(0)cG(1) a posteriori error estimates which indicate optimality. However, a complete analysis is not given.     

Optimal error bounds for the MITC4 plate bending element

Abstract One of the most popular finite element method for Reissner-Mindlin plates is the so-called MITC4 method. Unfortunately, until recently, the best error estimate available in the literature for this element needs a non-optimal and generally unrealistic regularity for the solution. Here we derive optimal error estimates for this well-known low-order finite element method, which are valid for a general family of meshes. A similar result has recently been obtained elsewhere, with the same mesh restrictions. The exposition presented here is totally different.     

Adaptive finite element procedures in elasticity and plasticity

The paper discusses error estimation and h-adaptive finite element procedures for elasticity and plasticity problems. For the spatial discretization error, an enhanced Superconvergent Patch Recovery (SPR) technique which improves the error estimation by including fulfillment of equilibrium and boundary conditions in the smoothing procedure is discussed. It is known that an accurate error estimation on an early stage of analysis results in a more rapid and optimal adaptive process. It is shown that node patches and element patches give similar quality of the postprocessed solution. For dynamic problems, a postprocessed type of error estimate and an adaptive procedure for the semidiscrete finite element method are discussed. It is shown that the procedure is able to update the spatial mesh and the time step size so that both spatial and time discretization errors are controlled within specified tolerances. A time-discontinuous Galerkin method for solving the second-order ordinary differential equations in structural dynamics is also presented. Many advantages of the new approach such as high order accuracy, possibility to filter effects of spurious modes and convenience to apply adaptive analysis are observed. For plasticity problems, some recent work that improved plastic strains and plastic localization is discussed.     

Upper bounds for error rates of linear combinations of classifiers

A useful notion of weak dependence between many classifiers constructed with the same training data is introduced. It is shown that if both this weak dependence is low and the expected margins are large, then decision rules based on linear combinations of these classifiers can achieve error rates that decrease exponentially fast. Empirical results with randomized trees and trees constructed via boosting and bagging show that weak dependence is present in these type of trees. Furthermore, these results also suggest that there is a trade-off between weak dependence and expected margins, in the sense that to compensate for low expected margins, there should be low mutual dependence between the classifiers involved in the linear combination     

Sharp error bounds for piecewise linear interpolation of planar curves

Curves are commonly drawn by piecewise linear interpolation, but to worry about the error is rather seldom. In the present paper we give a strong mathematical error analysis for curve segments with bounded curvature and length. Though the result seems very clear, the proof turned out to be unexpectedly hard, comparable to that of the famous four vertex theorem.     

Adaptive multiple-levelh-refinement in automated finite element analyses

This paper discusses an automatic, adaptive finite element modeling system consisting of mesh generation, finite element analysis, and error estimation. The individual components interact with one another and efficiently reduce the finite element error to within an acceptable value and perform only a minimum number of finite element analyses.One of the necessary components in the automated system is a multiple-level local remeshing algorithm. Givenh-refinement information provided by an a posteriori error estimator, and adjacency information available in the mesh data structures, the local remeshing algorithm grades the refinement toward areas requesting refinement. It is shown that the optimal asymptotic convergence rate is achieved, demonstrating the effectiveness of the intelligent multiple-level localh-refinement.