Structural and sensitivity analysis for the primal and dual problems in the physical and material spaces

The present work is concerned with a complete and consistent representation of structural and sensitivity analysis for the primal and dual problems in structural mechanics. A complete description means that besides classical changes in the physical space (displacement space) also changes in the material space (geometry or design space) are allowed. This point of view yield a complexity of eight problems in structural mechanics. Based on a variational approach, these eight problems are completely derived in a continuous and finite element formulation for the model problem of nonlinear elasticity. The above mentioned formulations are applied to mesh optimization (r-adaptivity) and shape optimization problems. First of all, in the framework of a complete primal problem, classical global r-adaptive mesh optimization strategies are considered and different error measures are introduced. Thereafter, a novel goal-oriented r-adaptive mesh optimization algorithm is proposed, in which the finite element mesh is optimized in such a way, that a chosen quantity of interest can be computed with high accuracy. Furthermore, shape optimization problems are investigated and the coherence to configurational mechanics is demonstrated. Moreover, error estimators and improvement algorithms for first-order sensitivity relations are derived. Novel theorems for exact sensitivity relations for the state and a chosen quantity of interest are presented. These results are the basis for error estimators and for any improvement algorithms of design sensitivity relations. All of the required higher-order variations of the weak form of equilibrium are explicitly derived for shape design sensitivities. The efficiency and reliability of the error estimators and improvement algorithms are verified by means of numerical examples.