This article surveys a general approach to error control and adaptive mesh design in Galerkin finite element methods that is based on duality principles as used in optimal control. Most of the existing work on a posteriori error analysis deals with error estimation in global norms like the ‘energy norm’ or the L2 norm, involving usually unknown ‘stability constants’. However, in most applications, the error in a global norm does not provide useful bounds for the errors in the quantities of real physical interest. Further, their sensitivity to local error sources is not properly represented by global stability constants. These deficiencies are overcome by employing duality techniques, as is common in a priori error analysis of finite element methods, and replacing the global stability constants by computationally obtained local sensitivity factors. Combining this with Galerkin orthogonality, a posteriori estimates can be derived directly for the error in the target quantity. In these estimates local residuals of the computed solution are multiplied by weights which measure the dependence of the error on the local residuals. Those, in turn, can be controlled by locally refining or coarsening the computational mesh. The weights are obtained by approximately solving a linear adjoint problem. The resulting a posteriori error estimates provide the basis of a feedback process for successively constructing economical meshes and corresponding error bounds tailored to the particular goal of the computation. This approach, called the ‘dual-weighted-residual method’, is introduced initially within an abstract functional analytic setting, and is then developed in detail for several model situations featuring the characteristic properties of elliptic, parabolic and hyperbolic problems. After having discussed the basic properties of duality-based adaptivity, we demonstrate the potential of this approach by presenting a selection of results obtained for practical test cases. These include problems from viscous fluid flow, chemically reactive flow, elasto-plasticity, radiative transfer, and optimal control. Throughout the paper, open theoretical and practical problems are stated together with references to the relevant literature.

An optimal control approach to a posteriori error estimation in finite element methods

Bulk forming of sheet metal

Since the mid-1960s when the forms of curved shell finite elements were originated, including those pioneered by Professor Gallagher, the published literature on the subject has grown extensively. The first two present authors and Liaw presented a survey of such literature in 1990 in this journal. Professor Gallagher maintained an active interest in this subject during his entire academic career, publishing milestone research works and providing periodic reviews of the literature. In this paper, we endeavor to summarize the important literature on shell finite elements over the past 15 years. It is hoped that this will be a befitting tribute to the pioneering achievements and sustained legacy of our beloved Professor Gallagher in the area of shell finite elements. This survey includes: the degenerated shell approach; stress-resultant-based formulations and Cosserat surface approach; reduced integration with stabilization; incompatible modes approach; enhanced strain formulations; 3-D elasticity elements; drilling d.o.f. elements; co-rotational approach; and higher-order theories for composites. Copyright © 2000 John Wiley & Sons, Ltd.

A survey of recent shell finite elements

A 4-node finite shell element for the implementation of general hyperelastic 3D-elasticity at finite strains

We consider variationally consistent discretization schemes for mechanical contact problems. Most of the results can also be applied to other variational inequalities, such as those for phase transition problems in porous media, for plasticity or for option pricing applications from finance. The starting point is to weakly incorporate the constraint into the setting and to reformulate the inequality in the displacement in terms of a saddle-point problem. Here, the Lagrange multiplier represents the surface forces, and the constraints are restricted to the boundary of the simulation domain. Having a uniform inf-sup bound, one can then establish optimal low-order a priori convergence rates for the discretization error in the primal and dual variables. In addition to the abstract framework of linear saddle-point theory, complementarity terms have to be taken into account. The resulting inequality system is solved by rewriting it equivalently by means of the non-linear complementarity function as a system of equations. Although it is not differentiable in the classical sense, semi-smooth Newton methods, yielding super-linear convergence rates, can be applied and easily implemented in terms of a primal–dual active set strategy. Quite often the solution of contact problems has a low regularity, and the efficiency of the approach can be improved by using adaptive refinement techniques. Different standard types, such as residual- and equilibrated-based a posteriori error estimators, can be designed based on the interpretation of the dual variable as Neumann boundary condition. For the fully dynamic setting it is of interest to apply energy-preserving time-integration schemes. However, the differential algebraic character of the system can result in high oscillations if standard methods are applied. A possible remedy is to modify the fully discretized system by a local redistribution of the mass. Numerical results in two and three dimensions illustrate the wide range of possible applications and show the performance of the space discretization scheme, non-linear solver, adaptive refinement process and time integration.

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Variationally consistent discretization schemes and numerical algorithms for contact problems

In this paper we present two types of local error estimators for the primal finite-element-method (FEM) by duality arguments. They are first derived from the (explicit) residual error estimation method (REM) and then—as a new contribution—from the (implicit) posterior equilibrium method (PEM) using improved boundary tractions, gained by local post-processing with local Neumann problems, with applications in elastic problems. For the displacements a local error estimator with an upper bound is derived and also a local estimator for stresses. Furthermore—for better numerical efficiency—the residua are projected energy-invariant onto reference elements, where the local Neumann problems have to be solved. Comparative examples between REM- and PEM-type local estimators show superior effectivity indices for the latter one. Copyright © 2001 John Wiley & Sons, Ltd.

Local error estimates of FEM for displacements and stresses in linear elasticity by solving local Neumann problems

Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems

Coupled model- and solution-adaptivity in the finite-element method

Error-controlled adaptive goal-oriented modeling and finite element approximations in elasticity

The main purpose of this paper is two-fold: first, to present a critical review of available error-controlled adaptive finite element methods—with absolute global and goal-oriented error estimates—for approximate solutions of a given mathematical model and the model error of the considered mathematical model and its dimensions, enhanced with our own recent results in fracture mechanics. The second related important matter is which physical/mathematical models—including various boundary layers and other disturbances—and which accuracy of the finite element solutions in which norm are prescribed and necessary for the safety and reliability of a certain engineering problem, e.g. in structural engineering, of course depending on materials, systems and the type as well as the magnitude of prescribed loadings (i.e. actions in a factorized concept) acting on a specific structure. It is hard to understand why the current regulations for structural engineering design in Europe—especially the European Standards (Eurocodes) and the relevant German DIN-Codes (DIN: Deutsches Institut fur Normung e. V.)—do not lay out any rules or bounds for the prescribed accuracy of the material and structural resistance, i.e. the validity of mechanical modelling and the accuracy of associated numerical methods, usually today the finite element method. It can be observed that behind the existing codes there still lies the old thinking of, e.g., classical elastic beam theory of so-called second order for stability problems and the corresponding elastic potential for plates and shells in buckling. Modern flexible finite element modelling by deriving theories in variational form for thin-walled beams, plates and shells from rather general 3D theories, for both linear and non-linear problems, capturing inelastic deformations and layer effects by material and dimensional expansions are not addressed in the Eurocodes for structural engineering. The necessary consequence is that joint work of computational mechanics specialists and structural engineers, working in national and international code-committees, is absolutely necessary, in order to avoid a lot of costly troubles and dangers of today by improving and updating the codes. Copyright © 2004 John Wiley & Sons, Ltd.

Stephan Ohnimus

Papers

Local error estimates of FEM for displacements and stresses in linear elasticity by solving local Neumann problems

Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems

Coupled model- and solution-adaptivity in the finite-element method

Error-controlled adaptive goal-oriented modeling and finite element approximations in elasticity

Adaptive finite element analysis and modelling of solids and structures. Findings, problems and trends†