An eigenanalysis-based bifurcation indicator proposed in the framework of a reduced-order modeling technique for non-linear structural analysis

Aerospace Science and Technology

Ruess

2016

Self Cite

Liang, K., and Nonlinear buckling analysis of the conical and cylindrical shells using the SGL strain based reduced order model and the PHC method. Aerospace Science and Technology, 55, There may be differences between this version and the published version. AbstractThin-walled conical and cylindrical shells subjected to axial compression often show a snap-back response in the presence of buckling. Newton iterations based path-following methods cannot trace reliably the snap-back response due to the extremely sharp turning angle near the limit point, and the original Koiter-Newton method also meets difficulties to achieve a complete post-buckling response beyond the limit point. In this paper, the improved KoiterNewton method is proposed to trace the post-buckling path of cylinders and cones, in the framework of the reducedorder modeling technique. The polynomial homotopy continuation (PHC) method is used to solve the lower-order nonlinear reduced order model reliably and efficiently. The simplified Green-Lagrange (SGL) kinematics which consider the stress redistribution after buckling are implemented into the construction of the reduced order model to produce accurate results for curved shells. The numerical results presented reveal that the improved method is a robust and efficient technology to achieve the entire nonlinear response for the snap-back case.

Section: Introductionmentioning

confidence: 91%

“…Snap-through and snap-back responses are two main phenomenons usually associated with the buckling of shell structures [6]. Some variants of the classical Newton method, i.e.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear buckling analysis of the conical and cylindrical shells using the SGL strain based reduced order model and the PHC method

Aerospace Science and Technology

Ruess

2016

Self Cite

“…We will demonstrate the numerical performance in terms of convergence measures compared to the pure displacement‐based formulation as introduced in Liang et al The focus of these tests is primarily on the reduction of the number of Newton iteration steps in the corrector phase in comparison to the original approach, which requires the consideration of the full‐order model thus dominating the numerical complexity of the analysis. A comparison of the Koiter‐Newton method with traditional path‐following methods, revealing the method's ability to perform a full nonlinear analysis with considerably fewer load steps, has been carefully documented in previous studies() and therefore is not considered here though this strength of the method is also an immanent property of the modified version proposed in this contribution.…”

Section: Numerical Testsmentioning

confidence: 99%

“…The method allows to trace the entire equilibrium path and to handle reliably snap‐back and snap‐through phenomena . In an extended version, the method provides a bifurcation indicator based on the constructed reduced‐order model, which enables to trace the corresponding bifurcation branches . The method has proven to be a robust and computationally efficient solution approach though the corrector phase cannot fully profit from the reduced‐order model and requires a Newton iteration in each load step, involving the repeated solution of the linear system of equations of the full‐order model.…”

Section: Introductionmentioning

confidence: 99%

“…15 In an extended version, the method provides a bifurcation indicator based on the constructed reduced-order model, which enables to trace the corresponding bifurcation branches. 16 The method has proven to be a robust and computationally efficient solution approach though the corrector phase cannot fully profit from the reduced-order model and requires a Newton iteration in each load step, involving the repeated solution of the linear system of equations of the full-order model. In the past, it has been observed in several studies that mixed (stress and displacement) finite elements show superior properties compared to pure displacement-based elements in the context of a standard Newton approach for nonlinear analyses.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Magisano

Numerical Meth Engineering

Garcea

et al. 2017

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 modified Newton‐type Koiter‐Newton method for tracing the geometrically nonlinear response of structures

Numerical Meth Engineering

Abdalla

Qin

2017

Self Cite

Summary The Koiter‐Newton (KN) method is a combination of local multimode polynomial approximations inspired by Koiter's initial postbuckling theory and global corrections using the standard Newton method. In the original formulation, the local polynomial approximation, called a reduced‐order model, is used to make significantly more accurate predictions compared to the standard linear prediction used in conjunction with Newton method. The correction to the exact equilibrium path relied exclusively on Newton‐Raphson method using the full model. In this paper, we proposed a modified Newton‐type KN method to trace the geometrically nonlinear response of structures. The developed predictor‐corrector strategy is applied to each predicted solution of the reduced‐order model. The reduced‐order model can be used also in the correction phase, and the exact full nonlinear model is applied only to calculate force residuals. Remainder terms in both the displacement expansion and the reduced‐order model are well considered and constantly updated during correction. The same augmented finite element model system is used for both the construction of the reduced‐order model and the iterations for correction. Hence, the developed method can be seen as a particular modified Newton method with a constant iteration matrix over the single KN step. This significantly reduces the computational cost of the method. As a side product, the method has better error control, leading to more robust step size adaptation strategies. Numerical results demonstrate the effectiveness of the method in treating nonlinear buckling problems.