An efficient mixed variational reduced order model formulation for non-linear analyses of elastic shells

The Koiter-Newton method had recently demonstrated a superior performance for non-linear analyses of structures, compared to traditional path-following strategies. The method follows a predictor-corrector scheme to trace the entire equilibrium path. During a predictor step a reduced order model is constructed based on Koiter's asymptotic post-buckling theory which is followed by a Newton iteration in the corrector phase to regain the equilibrium of forces. In this manuscript, we introduce a robust mixed solid-shell formulation to further enhance the efficiency of stability analyses in various aspects. We show that a Hellinger-Reissner variational formulation facilitates the reduced order model construction omitting an expensive evaluation of the inherent fourth order derivatives of the strain energy. We demonstrate that extremely large step sizes with a reasonable out-of-balance residual can be obtained with substantial impact on the total number of steps needed to trace the complete equilibrium path. More importantly, the numerical effort of the corrector phase involving a Newton iteration of the full order model is drastically reduced thus revealing the true strength of the proposed formulation. We study a number of problems from engineering and compare the results to the conventional approach in order to highlight the gain in numerical efficiency for stability problems

A quasi-static nonlinear analysis for assessing the fire resistance of 3d frames exploiting time-dependent yield surface

In this work an automatic procedure for evaluating the axial force-biaxial bending yield surface of reinforced concrete sections in fire is proposed. It provides an accurate time-dependent expression of the yield condition by a section analysis carried out once and for all, accounting for the strength reduction of the materials, which is a function of the fire duration. The equilibrium state of 3D frames with such yield conditions, once discretized using beam finite elements, is formulated as a nonlinear vectorial equation defining a curve in the hyperspace of the discrete variables and the fire duration. A generalized path-following strategy is proposed for tracing this curve and evaluating, if it exists, the limit fire duration, that is the time of exposure which leads to structural collapse. Compared to the previous proposals on the topic, which are limited to local sectional checks, this work is the first to present a global analysis for assessing the fire resistance of 3D frames, providing a time history of the fire event and taking account of the stress redistribution. Numerical examples are given to illustrate and validate the proposal

Data for the article "Sensitivity analysis to geometrical imperfections in shell buckling via a mixed generalized path-following method"

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Optimal design of cnt-nanocomposite nonlinear shells

Carbon nanotube/polymer nanocomposite plate-and shell-like structures will be the next generation lightweight structures in advanced applications due to the superior multifunctional properties combined with lightness. Here material optimization of carbon nanotube/polymer nanocomposite beams and shells is tackled via ad hoc nonlinear finite element schemes so as to control the loss of stability and overall nonlinear response. Three types of optimizations are considered: variable through-the-thickness volume fraction of random carbon nanotubes (CNTs) distributions, variable volume fraction of randomly oriented CNTs within the mid-surface, aligned CNTs with variable orientation with respect to the mid-surface. The collapse load, which includes both limit points and deformation thresholds, is chosen as the objective/cost function. An efficient computation of the cost function is carried out using the Koiter reduced order model obtained starting from an isogeometric solid-shell model to accurately describe the point-wise material distribution. The sensitivity to geometrical imperfections is also investigated. The optimization is carried out making use of the Global Convergent Method of Moving Asymptotes. The extensive numerical analyses show that varying the volume fraction distribution as well as the CNTs orientation can lead to significantly enhanced performances towards the loss of elastic stability making these lightweight structures more stable. The most striking result is that for curved shells, the unstable postbuckling response of the baseline material can be turned into a globally stable response maintaining the same amount of nanostructural reinforcement but simply tailoring strategically its distribution

A Numerical Strategy for Multistable Nanocomposite Shells

This work presents an efficient continuation strategy based on the Riks method to describe the stable and of unstable branches of the response of carbon nanotubes (CNT)/polymer nanocomposite shells. Exploiting the superior properties of this class of nanostructured materials in the context of elastic instabilities and multistability has the potential to pave the way towards a variety of novel smart engineering applications. The equilibrium paths and the static bifurcations of CNT nanocomposites are numerically investigated highlighting the effects of material parameters such as the orientation and weight fraction of high aspect ratio CNTs integrated in a thermoplastic polymer

A quasi-static nonlinear analysis for assessing the fire resistance of 3d frames exploiting time-dependent yield surface

In this work an automatic procedure for evaluating the axial force-biaxial bending yield surface of reinforced concrete sections in fire is proposed. It provides an accurate time-dependent expression of the yield condition by a section analysis carried out once and for all, accounting for the strength reduction of the materials, which is a function of the fire duration. The equilibrium state of 3D frames with such yield conditions, once discretized using beam finite elements, is formulated as a nonlinear vectorial equation defining a curve in the hyperspace of the discrete variables and the fire duration. A generalized path-following strategy is proposed for tracing this curve and evaluating, if it exists, the limit fire duration, that is the time of exposure which leads to structural collapse. Compared to the previous proposals on the topic, which are limited to local sectional checks, this work is the first to present a global analysis for assessing the fire resistance of 3D frames, providing a time history of the fire event and taking account of the stress redistribution. Numerical examples are given to illustrate and validate the proposal

A simplified Kirchhoff–Love large deformation model for elastic shells and its effective isogeometric formulation

Isogeometric Kirchhoff–Love elements have received an increasing attention in geometrically nonlinear analysis of elastic shells. Nevertheless, some difficulties still remain. Among the others, the highly nonlinear expression of the strain measure, which leads to a complicated and costly computation of the discrete operators, and the existence of locking, which prevents the use of coarse meshes for slender shells and low order NURBS, are key issues that need to be addressed. In this work, exploiting the hypothesis of small membrane strains, we propose a simplified strain measure with a third order polynomial dependence on the displacement variables which allows an efficient evaluation of the discrete quantities. Numerical results show practically no difference to the original model, even for very large displacements and composite structures. Patch-wise reduced integrations are then investigated to deal with membrane locking in large deformation problems. An optimal integration scheme for third order C2 NURBS, in terms of accuracy and efficiency, is identified. Finally, the recently proposed Newton method with mixed integration points is used for the solution of the discrete nonlinear equations with a great reduction of the iterative burden with respect to the standard Newton scheme

A robust penalty coupling of non-matching isogeometric Kirchhoff–Love shell patches in large deformations

Isogeometric Kirchhoff–Love elements have been receiving increasing attention in geometrically nonlinear analysis of thin shells because they make it possible to meet the C1 requirement in the interior of surface patches and to avoid the use of finite rotations. However, engineering structures of appreciable complexity are typically modeled using multiple patches and, often, neither rotational continuity nor conforming discretization can be practically obtained at patch interfaces. Simple penalty approaches for coupling adjacent patches, applicable to either smooth or non-smooth interfaces and either matching or non-matching discretizations, have been proposed. Although the problem dependence of the penalty coefficient can be reduced by scaling factors which take into account geometrical and material parameters, only high values of the penalty coefficient can guarantee a negligible coupling error in all possible cases. However, this can lead to an ill conditioned problem and to an increasing iterative effort for solving the nonlinear discrete equations. In this work, we show how to avoid this drawback by rewriting the penalty terms in an Hellinger–Reissner form, introducing independent fields work-conjugated to the coupling equations. This technique avoids convergence problems, making the analysis robust also for very high values of the penalty coefficient, which can be then employed to avert coupling errors. Moreover, a proper choice of the basis functions for the new fields provides an accurate coupling also for general non-matching cases, preventing overconstrained solutions. The additional variables are condensed out and then not involved in the global system of equations to be solved. A highly efficient approach based on a mixed integration point strategy and an interface-wise reduced integration rule makes the condensation inexpensive preserving the sparsity of the condensed stiffness matrix and the coupling accuracy