How to improve efficiency and robustness of the Newton method in geometrically non-linear structural problem discretized via displacement-based finite elements

Magisano, Domenico; Leonetti, Leonardo; Garcea, Giovanni

doi:10.1016/j.cma.2016.10.023

Cited by 66 publications

(71 citation statements)

References 33 publications

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“…The key component is that the failure of the component will lead the disproportionate damage to the structure, which is determined by the impact of the failure on the structure as a whole, and the importance coefficient of the component is the quantitative evaluation index of the key component. [36] The importance coefficient of component is used to reflect the extent of influence of individual component failure on the overall structural performance under extreme conditions. Assessing component importance is one of the foundations for studying structural robustness.…”

Section: Importance Coefficient Of Componentsmentioning

confidence: 99%

Robustness analysis of vertical resistance to progressive collapse of diagrid structures in tall buildings

Liu

Fang

2020

Structural Design Tall Build

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In order to investigate the vertical resistance to progressive collapse and the evolution of its robustness in the process of loading, a quantitative method of the robustness index of diagrid structure is established based on the mechanical properties of the components and the vertical bearing capacity of the structure. A scale experiment model of diagrid tube structure is established in laboratory which verified the availability of the proposed robustness quantification method. Results demonstrated that, the axial force of inclined column is pressure, and the axial force of ring beam is tension under vertical load. The ring beam is the main restraint to the side bulge of the diagrid structure. The welded strength of the inclined column have great influence on the vertical bearing capacity, so it is necessary to consider the reduction of the strength of welded joint. The influence of the welding line of the joint can lead to the plastic stage of the local components of the structure, which makes the structural robustness worse. In addition, the experimental study and numerical simulation results verify that the proposed method of robustness coefficient quantification for diagrid tube structure can accurately and reasonably evaluate the resistance of vertical progressive collapse performance.

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Section: Importance Coefficient Of Componentsmentioning

confidence: 99%

Robustness analysis of vertical resistance to progressive collapse of diagrid structures in tall buildings

Liu

Fang

2020

Structural Design Tall Build

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“…The equilibrium path is obtained from the following condition:

false(normal Φ {false[u false]}^{'} - {normal Φ}^{'} false[ũ false] - λ p false) δ u = 0 \forall δ u

which in FE format becomes

bolds false[boldu false] - \overset{p}{true} - λ \overset{p}{true} = bold0 .

In particular, the internal force vector s [ u ], the load vector

\overset{p}{true}

and the imperfection vector

\overset{p}{true}

are defined by the energy equivalences

s^{T} δ boldu \equiv {normal Φ}^{'} false[u false] δ u, {\overset{p}{true}}^{T} δ boldu \equiv \overset{p}{true} δ u, {\overset{p}{true}}^{T} δ boldu \equiv {normal Φ}^{'} false[ũ false] δ u, \forall δ boldu .

Equation can be solved using standard path‐following techniques for an assigned imperfection

ũ

. Note that in the hybrid solid‐shell FE model, the internal force vector of the imperfect structure is obtained by simply subtracting a constant vector

\overset{p}{true}

, evaluated once and for all at the beginning of the analysis, to the internal forces vector s [ u ] of the perfect structure.…”

Section: An Accurate a Posteriori Account Of Geometrical Imperfectionsmentioning

confidence: 99%

Accurate and efficient a posteriori account of geometrical imperfections in Koiter finite element analysis

Garcea

Liguori

Leonetti

et al. 2017

Numerical Meth Engineering

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The Koiter method recovers the equilibrium path of an elastic structure using a reduced model, obtained by means of a quadratic asymptotic expansion of the finite element model. Its main feature is the possibility of efficiently performing sensitivity analysis by including a posteriori the effects of the imperfections in the reduced nonlinear equations. The state-of-art treatment of geometrical imperfections is accurate only for small imperfection amplitudes and linear pre-critical behaviour. This work enlarges the validity of the method to a wider range of practical problems through a new approach, which accurately takes into account the imperfection without losing the benefits of the a posteriori treatment. A mixed solid-shell finite element is used to build the discrete model. A large number of numerical tests, regarding nonlinear buckling problems, modal interaction, unstable post-critical and imperfection sensitive structures, validates the proposal. 1155Standard path-following approaches, aimed at recovering the equilibrium path for a single loading case and assigned imperfections, are not suitable for this purpose because of the high computational burden of the single run [13] and are unusable if no information about the worst imperfection shapes is available. For these reasons, the FE implementation of asymptotic methods [14-24] based on Koiter's theory of elastic stability [25] has recently become [26][27][28][29][30][31][32][33][34][35][36] more and more attractive. The Koiter method consists of the construction of a reduced model, in which the FE model is replaced by its second-order asymptotic expansion using the initial path tangent, m buckling modes and the corresponding second-order modes, named quadratic correctives. In this way, once the reduced model is built, the equilibrium path of the structure can be obtained by solving the nonlinear reduced system of m equations in m + 1 unknowns, which represent the modal amplitudes and the load factor. The coefficients of the reduced system are evaluated using strain energy variations up to the fourth order. Shell structures can require a very large number of FE DOFs to avoid significant discretization errors, while m is usually at most a few tens. Clearly, the convenience of the method with respect to the standard path-following strategy is evident.Since the first proposals [6,15,37,38], the method has been continuously enhanced in terms of both accuracy and computational efficiency. In particular, a mixed (stress-displacement) formulation is required to avoid an interpolation locking phenomenon in the evaluation of the coefficients of the reduced system [6,[38][39][40][41] and to make the asymptotic expansion accurate for a wider range, avoiding the extrapolation locking [14,27] common in the displacement-based approach and providing accurate results also for nonlinear pre-critical behaviours. Geometrically exact shells and beams [42,43] or corotational approaches [32,44] have been proposed to achieve structural model objectivity. Both the strategies ma...

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“…Details of the Newton iteration with the Riks constraint can be found in previous studies. 19,21 The construction of the reduced-order model is dominated by the factorization of the governing system of equations of the augmented problem (15) and (16), respectively. It is important to note that both systems of equations of dimension (N + m + 1) have an identical system matrix, hence factorization is needed only once.…”

Section: Koiter-newton Path-following Analysismentioning

confidence: 99%

“…The U‐shape cantilever beam considered here is a standard benchmark problem to verify the nonlinear properties of the method, including local buckling phenomena . Nevertheless, the problem is a severe test case for the Koiter‐Newton approach due to a distinct nonlinear precritical behavior, which must be represented in the reduced‐order model by the degree of freedom associated with the external load.…”

Section: Numerical Testsmentioning

confidence: 99%

“…() In general, the mixed formulation allows for larger step sizes and requires less iteration steps to regain equilibrium as highlighted in Garcea et al for beam elements. A more recent work observed the beneficial convergence properties of mixed‐model formulations in a general context, which can be extended to displacement‐based finite element formulations following the mixed integration points strategy proposed in Magisano et al The studies show that the evolution of the iterative displacement process is forced to satisfy the constitutive equations in each iteration step and this constraint leads to a drop of the convergence rate. In contrast, the stress components in the mixed model are introduced as independent variables, which only satisfy the discrete constitutive equations at convergence.…”

Section: Introductionmentioning

confidence: 99%

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

Magisano

Liang

Garcea

et al. 2017

Numerical Meth Engineering

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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.

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How to improve efficiency and robustness of the Newton method in geometrically non-linear structural problem discretized via displacement-based finite elements

Cited by 66 publications

References 33 publications

Robustness analysis of vertical resistance to progressive collapse of diagrid structures in tall buildings

Robustness analysis of vertical resistance to progressive collapse of diagrid structures in tall buildings

Accurate and efficient a posteriori account of geometrical imperfections in Koiter finite element analysis

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

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