No abstract

A variational formulation of shape design sensitivity analysis is outlined, starting from a differential geometry-based representation of continuum mechanics.A rigorous analysis using convected curvilinear coordinates yields a decomposition of all continuum mechanical functions into independent geometry and displacement mappings. Using this representation of geometry and displacements defined on a fixed parameter space, their influence on physical quantities can easily be separated. Consequently, the variations of continuum mechanical quantities with respect to either geometry or displacements can be performed similarly using the well-known linearization techniques in nonlinear mechanics.The proposed methodology for performing variational design sensitivity analysis is formulated for general nonlinear hyperelastic material behavionr using either the Lagrangian or Eulerian description. The differences and similarities of the formulation presented compared with the material derivative approach and the domain parametrization approach are highlighted and discussed.

Many interesting tasks in technology need the solution of complex boundary value problems within the mathematical theory of elastoplasticity. Error controlled adaptive strategies should be used in order to achieve a prescribed accuracy of the computed solutions at minimum cost. Local error indicators in the primal form of the ®nite-element-method for Hencky-and Prandtl±Reuû-plasticity without and with nonlinear hardening are presented, controlling global errors of equilibrium, plastic strain rates, the yield condition and the numerical integration of the¯ow rule. A 2D ®nite element code was extended for the transfer of physical parameters between meshes and for a combined space and time adaptive strategy. All error indicators are tested with the benchmark example of a rectangular plane strain problem with a hole. IntroductionIn the adaptive Finite-Element process, aside from the stepwise optimal choice of the spatial discretization we have to pay attention to the implicit time-dependency of the deformation process.Therefore, special attention has to be devoted to adaptive re®nement and coarsening algorithms for mesh and load-step sizing. The usual decomposition of FEM in space and FDM in time is used which does not permit a rigorous coupled a posteriori error analysis.Thus, an heuristic assumption based on the comparison of Prandtl±Reuû elastoplasticity with Hencky plasticity, i.e. nonlinear elasticity, is used to split the in¯uence of the error of plastic deformation into spatial and time discretization errors [5,7].In this paper three separate local spatial error indicators are derived from global error investigations, namely for equilibrium, plastic strains using the plastic dissipation and the yield condition. Each of them yields a scaled local indicator for mesh re®nement. The error within the integration of the¯ow rule by the backward-Euler scheme is estimated by the maximal change of the normal with respect to the yield surface between two time (load) steps. Additionally, the error by updating and interpolation of the material parameters to an adapted mesh has to be considered, at least by a further iteration step of the weak equilibrium conditions before the next load step.A new technique is given for a staggered combination of adaptivity in space and time and compared with the procedure by Ladeve Áze et al. [10].In the next chapter, the basic equations for elastoplastic deformations with hardening are given pointing out the differences between the Hencky-deformation theory and the Prandtl±Reuû-¯ow-theory. The choice of an adequate ®nite element is treated in section three. The fourth section regards algorithmic questions such as the combination of space and time adaptivity during the computation, the data transfer between different meshes, as mentioned above, and the differences between global and local error estimations. The ®fth section describes the benchmark example used for the numerical tests. Section six presents the different sources of errors in elastoplastic deformations as well as the error es...

The numerical solution of shape optimization problems is considered. The algorithm of successive optimization based on finite element techniques and design sensitivity analysis is applied. Mesh refinement is used to improve the quality of finite element analysis and the computed numerical solution. The norm of the variation of the Lagrange augmented functional with respect to boundary variation (residuals in necessary optimality conditions) is taken as an a posteriori error estimator for optimality conditions and the Zienkiewicz-Zhu error estimator is used to improve the quality of structural analysis. The examples presented show meaningful effects obtained by means of mesh refinement with a new error estimator.

The aim of this paper is to improve the shape of specimens for biaxial experiments with respect to optimal stress states, characterized by the stress triaxiality. Gradient-based optimization strategies are used to achieve this goal. Thus, it is crucial to know how the stress state changes if the geometric shape of the specimen is varied. The design sensitivity analysis (DSA) of the stress triaxiality is performed using a variational approach based on an enhanced kinematic concept that offers a rigorous separation of structural and physical quantities. In the present case of elastoplastic material behavior, the deformation history has to be taken into account for the structural analysis as well as for the determination of response sensitivities. The presented method is flexible in terms of the choice of design variables. In a first step, the approach is used to identify material parameters. Thus, material parameters are chosen as design variables. Subsequently, the design parameters are chosen as geometric quantities so as to optimize the specimen shape with the aim to obtain a preferably homogeneous stress triaxiality distribution in the relevant cross section of the specimen.