Solid-shell finite element and isogeometric models for nonlinear analysis and design of elastic shells using Newton, Koiter and Koiter-Newton solution strategies

Abstract

This thesis aims at developing a reliable and efficient numerical framework for the analysis and the design of slender elastic shells, in particular when composite materials are adopted, taking account of the geometrically nonlinear behaviour. Different aspects of this challenging topic are tackled: discretisation techniques, numerical solution strategies and optimal design. The first chapter, after a short summary of the Riks and Koiter methods, discusses the important advantages of using a mixed (stress-displacement) solid model for analysing shell structures over traditional shell models and the implications of this on the performances of the solution strategies. The second chapter introduces a mixed solid-shell model and reformulates the Koiter method to obtain an effective tool for analysing imperfection sensitive structures. This approach is the starting point of the third chapter, which proposes a stochastic optimisation strategy for the layup of composite shells, able to take account of the worst geometrical imperfection. The fourth chapter extends the benefits of the mixed formulation in the Newton iterative scheme to any displacement-based finite element model by means of a novel strategy, called Mixed Integration Point. The fifth chapter illustrates an efficient implementation of the novel Koiter-Newton method, able to recover the equilibrium path of a structure accurately with a few Newton iterations, combining an accurate Koiter predictor with the reduced iterative effort due to a mixed formulation. The solid-shell discrete model is reformulated in the sixth chapter, following the isogeometric concept, by using NURBS functions to interpolate geometry and displacement field on the middle surface of the shell in order to take advantage of their high continuity and of the exact geometry description. The approach is made accurate and efficient in large deformation problems by combining the Mixed Integration Point strategy with a suitable patch-wise reduced integration. The resulting discrete model proves to be much more convenient than low order finite elements, especially in the analysis of curved shells undergoing buckling. This is shown in the seventh chapter, which proposes an efficient isogeometric Koiter analysis.