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Abstract: A computational methodology for goal-oriented combined discretization and expansive (refined) model adaptivity by overall implicit error control of quantities of interest is presented, requiring estimators of primal and dual discretization and model errors. In the case of dimensional within model adaptivity, prolongations of coarse model solutions into the solution space of a fine model for defining a consistent model error are necessary, which can be achieved at the element level by two strategies. The first one is an orthogonalized kinematic prolongation of nodal displacements, whereas the second one uses prolongations of the external loads which are then used to solve additional local variational problems thus yielding prolongated solutions which a priori fulfill the required orthogonality relations at the element level. Finally, a numerical example of an elastic continuous T-beam is presented with comparative results where goal-oriented error estimation is applied to linear elasticity with a discrete Reissner–Mindlin plate model as the coarse model and the 3D theory as the fine model. [Copyright &y& Elsevier]

Abstract: In finite element simulation of sheet metal forming, shell elements are widely used. The limits of applicability of the shell elements are sometimes disregarded, which leads to an error in predictions of important values such as springback geometry. The underlying kinematic assumptions of the shell elements do not hold where the thickness of the metal sheet approaches the value of the radius of curvature. Complex three-dimensional material behavior effects cannot be represented precisely as the result of the simplified kinematics. Here we present a model adaptivity scheme based on a model error indicator. The model-adaptive technique presented in this paper aides to resolve only the critical areas of the structure with a three-dimensional discretization while keeping reasonable computational cost by utilizing shell elements for the rest of the structure. The model error indicator serves as a guide for subsequent automatic adaptive re-meshing of the work-piece followed by a model-adaptive finite element analysis. The accuracy of the approximation obtained by the model-adaptive technique coincides well with that of a more expensive solution obtained with solid elements only. [Copyright &y& Elsevier]

A modification of the hp–d method is presented, which allows to locally enhance a dimensionally reduced finite element approximation with an Ansatz based on a more accurate, i.e. dimensionally higher, model. The procedure is based on a hierarchical domain decomposition which is applied to couple discretizations of different types of mechanical models, like beams, plates and solids. Thereby, the error of the underlying mathematical model can be reduced in critical regions by introducing locally a physically more sophisticated model. Two examples are given to demonstrate the basic properties of the proposed method. In the first example the solution of a one-dimensional beam model is improved locally by a two-dimensional finite element computation. In the second example the local enhancement of a Reissner–Mindlin plate by a fully three-dimensional continuum approach is considered, yielding a significant reduction of the model error. [Copyright &y& Elsevier]