An efficient mixed variational reduced‐order model formulation for nonlinear analyses of elastic shells

Abstract

Summary The Koiter‐Newton method had recently demonstrated a superior performance for nonlinear 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 postbuckling theory that 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 novel reduced‐order modeling method for nonlinear buckling analysis and optimization of geometrically imperfect cylinders

“…To achieve an entire path‐following calculation in nonlinear structural analysis, the Koiter reduction method has been improved by approximating the full‐order FE model to be its second‐order asymptotic expansion using the initial path tangent, buckling modes, and the corresponding second‐order modes 18,19 . In addition, the Koiter–Newton (KN) reduced‐order modeling method 20,21 inspired by Koiter's initial postbuckling theory and Newton arc‐length techniques has been proposed and further developed 22,23 . The unique feature of the method is to use Koiter's asymptotic expansion to construct the reduced‐order model at any equilibrium configuration from the beginning of the equilibrium path, rather than to use it only at the bifurcation point.…”

“…18,19 In addition, the Koiter-Newton (KN) reduced-order modeling method 20,21 inspired by Koiter's initial postbuckling theory and Newton arc-length techniques has been proposed and further developed. 22,23 The unique feature of the method is to use Koiter's asymptotic expansion to construct the reduced-order model at any equilibrium configuration from the beginning of the equilibrium path, rather than to use it only at the bifurcation point. The proposed method can trace the entire equilibrium path in a stepwise manner using an efficient predictor-corrector phase in each step, and the method has been completely implemented into a standard finite element procedure.…”

Material Design for Optimal Postbuckling Behaviour of Composite Shells

“…The strategy also provides an inexpensive sensitivity analysis with a statistical estimation of the worst-case imperfection, assumed to be a combination of the linearized buckling modes of the perfect structure. A hybrid solution strategy, referred to as the Koiter–Newton approach, was further investigated in [ 29 , 30 ].…”

Robust and Efficient Large Deformation Analysis of Kirchhoff–love Shells: Locking, Patch Coupling and Iterative Solution

“…Moreover, a significantly lower number of integration points is employed compared to standard Gauss quadrature, improving also the computational efficiency. In path-following methods for geometrically nonlinear analyses of beams and shells, many authors observed a more robust and efficient iterative solution for mixed formulations [8]. The performance of Newton's method drastically deteriorates in displacement formulations when the membrane/flexural stiffness ratios get higher [9,10].…”