## Adaptive Finite Elements in Linear and Nonlinear Solid and Structural MechanicsThis course with 6 lecturers intends to present a systematic survey of recent re search results of well-known scientists on error-controlled adaptive finite element methods in solid and structural mechanics with emphasis to problem-dependent concepts for adaptivity, error analysis as well as h- and p-adaptive refinement techniques including meshing and remeshing. Challenging applications are of equal importance, including elastic and elastoplastic deformations of solids, con tact problems and thin-walled structures. Some major topics should be pointed out, namely: (i) The growing importance of goal-oriented and local error estimates for quan tities of interest—in comparison with global error estimates—based on dual finite element solutions; (a) The importance of the p-version of the finite element method in conjunction with parameter-dependent hierarchical approximations of the mathematical model, for example in boundary layers of elastic plates; (Hi) The choice of problem-oriented error measures in suitable norms, consider ing residual, averaging and hierarchical error estimates in conjunction with the efficiency of the associated adaptive computations; (iv) The importance of implicit local postprocessing with enhanced test spaces in order to get constant-free, i. e. absolute-not only relative-discretizati- error estimates; (v) The coupling of error-controlled adaptive discretizations and the mathemat ical modeling in related subdomains, such as boundary layers. The main goals of adaptivity are reliability and efficiency, combined with in sight and access to controls which are independent of the applied discretization methods. By these efforts, new paradigms in Computational Mechanics should be realized, namely verifications and even validations of engineering models. |

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### Contents

Duality Techniques for Error Estimation and Mesh Adaptation in Finite | 1 |

by R Rannacher | 14 |

Hierarchical Model and Solution Adaptivity of Thinwalled Structures | 59 |

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### Common terms and phrases

adaptive finite element adaptive meshes algorithm anisotropic ansatz approach approximation approximation error Babuska bilinear boundary layers boundary tractions computed consider constitutive equations constitutive relation error constraints contact problems convergence CRE estimator Curnier defined deformations denotes derived differential equations dimensional discretization error displacement domain dual problem dual solution effectivity index elastoplasticity energy norm Engineering equilibrated equilibrium error analysis error control error indicators example Figure finite element method formulation Galerkin method geometric given global error grids Hansbo hierarchic integration interpolation isotropic iteration kinematics L2-norm Ladeveze Lagrangian multiplier linear elasticity load Mechanics mesh adaptation mesh refinement model error nodal nonlinear number of elements numerical obtained Oden optimal p-extensions parameter patch polynomial posteriori error estimate Prandtl-Reuss Rannacher residual error estimator shape functions shapefunctions shell models solved space space-time spatial Stein step strategy stress superconvergence tangential technique tensor test functions variational vector Wriggers yields Zienkiewicz