Path following techniques for geometrically nonlinear structures based on Multi-point methods

Maghami, Ali; Shahabian, Farzad; Hosseini, Seyed Mahmoud

doi:10.1016/j.compstruc.2018.07.005

Cited by 20 publications

(5 citation statements)

References 40 publications

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“…Notably, the basic idea of IAMS is similar to path following methods (PFM) or numerical continuation, which are well-developed in solving nonlinear ODE equations, nonlinear eigenvalue problems [44] and bifurcation problems [45]. In particular, PFMs are also widely used to solve engineering problems such as quasi-static nonlinear structural [46] and fracture problems [47,48] through finite element methods (FEM). It is known that FEM is usually solved by Newton-Raphson method with available Hessian matrix, while MS simulations need to search local energy minima with million or even billion degree of freedoms [38].…”

Section: Comparison With Path Following Methods and The Significance ...mentioning

confidence: 99%

Accelerated Molecular Statics Based on Atomic Inertia Effect

Shuang¹

2020

CICP

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Molecular statics (MS) based on energy minimization serves as a useful simulation technique to study mechanical behaviors and structures at atomic level. The efficiency of MS, however, still remains a challenge due to the complexity of mathematical optimization in large dimensions. In this paper, the Inertia Accelerated Molecular Statics (IAMS) method is proposed to improve computational efficiency in MS simulations. The core idea of IAMS is to let atoms move to meta positions very close to their final equilibrium positions before minimization starts at a specific loading step. It is done by self-learning from historical movements (atomic inertia effect) without knowledge of external loadings. Examples with various configurations and loading conditions indicate that IAMS can effectively improve efficiency without loss of fidelity. In the simulation of three-point bending of nanopillar, IAMS shows efficiency improvement of up to 23 times in comparison with original MS. Particularly, the sizeindependent efficiency improvement makes IAMS more attractive for large-scale simulations. As a simple yet efficient method, IAMS also sheds light on improving the efficiency of other energy minimization-based methods.

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Section: Comparison With Path Following Methods and The Significance ...mentioning

confidence: 99%

Accelerated Molecular Statics Based on Atomic Inertia Effect

Shuang¹

2020

CICP

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“…The second example structure consists of a shallow cylindrical shell, which has been used as a benchmark problem in a number of similar studies. [45][46][47][48] Its geometric parameters, as specified in Figure 7, are the length L = 508 mm, the radius R = 2540 mm, the shell thickness t = 6.35 mm, and the section angle = 0.1 rad. The material parameters are Young's modulus E = 3.10275 kN/mm 2 and Poisson's ratio = 0.3.…”

Section: Shallow Cylindrical Shellmentioning

confidence: 99%

“…In this case, comparability of the Abaqus result with the result of our adaptive framework is based on the comparability of the total number of points on the path produced by both path following methods, that is, 27 in our adaptive framework VS 29 in Abaqus. The right hand column of Table 6 reports the corresponding length ratio r i between consecutive prediction phases defined in (47). We observe that within the first six steps, the automatic incrementation method in Abaqus increases the step length to the maximum length L max = 4L 1 .…”

Section: Our Adaptive Framework Abaqusmentioning

confidence: 99%

A stiffness parameter and truncation error criterion for adaptive path following in structural mechanics

Maghami

Schillinger

2019

Numerical Meth Engineering

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Summary We explore a truncation error criterion to steer adaptive step length refinement and coarsening in incremental‐iterative path following procedures, applied to problems in large‐deformation structural mechanics. Elaborating on ideas proposed by Bergan and collaborators in the 1970s, we first describe an easily computable scalar stiffness parameter whose sign and rate of change provide reliable information on the local behavior and complexity of the equilibrium path. We then derive a simple scaling law that adaptively adjusts the length of the next step based on the rate of change of the stiffness parameter at previous points on the path. We show that this scaling is equivalent to keeping a local truncation error constant in each step. We demonstrate with numerical examples that our adaptive method follows a path with a significantly reduced number of points compared to an analysis with uniform step length of the same fidelity level. A comparison with Abaqus illustrates that the truncation error criterion effectively concentrates points around the smallest‐scale features of the path, which is generally not possible with automatic incrementation solely based on local convergence properties.

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“…Especially, MIP Newton method can be well applied to high slenderness structures. Because of the importance of nonlinear solvers with respect to computational mechanics and their wide application, many solvers have been proposed to reduce number of iterations per load step and computation as the following: optimization-based iterative technique 6 and residual areas-based iterative technique, 7 dynamic relaxation techniques, [8][9][10] multipoint methods-based path following techniques, 11 a novel method to transform the discretized governing equations, 12 a data-driven nonlinear solver (DDNS), 13 an improved predictor-corrector method, 14 Koiter-Newton method with a superior performance for nonlinear analyses of structures, 15,16 etc. It is observed that employing time series prediction to reduce number of iterations of nonlinear solvers is very rare except the DDNS.…”

Section: Introductionmentioning

confidence: 99%

Deep learned one‐iteration nonlinear solver for solid mechanics

Nguyen

Lee

Dinh‐Tien

et al. 2022

Numerical Meth Engineering

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Nowadays, there are a lot of iterative algorithms which have been proposed for nonlinear problems of solid mechanics. The existing biggest drawback of iterative algorithms is the requirement of numerous iterations and computation to solve these problems. This can be found clearly when the large or complex problems with thousands or millions of degrees of freedom are solved. To overcome completely this difficulty, the novel one‐iteration nonlinear solver (OINS) using time series prediction and the modified Riks method (M‐R) is proposed in this paper. OINS is established upon the core idea as follows: (1) Firstly, we predict the load factor increment and the displacement vector increment and the convergent solution of the considering load step via the predictive networks which are trained by using the load factor and the displacement vector increments of the previous convergence steps and group method of data handling (GMDH); (2) Thanks to the predicted convergence solution of the load step is very close to or identical with the real one, the prediction phase used in any existing nonlinear solvers is eliminated completely in OINS. Next, the correction phase of the M‐R is adopted and the OINS iteration is started at the predicted convergence point to reach the convergent solution. The training process and the applying process of GMDH are continuously conducted and repeated during the nonlinear analysis in order to predict the convergence point at the beginning of each load step. Through numerical investigations, we prove that OINS is powerful, highly accurate and only needs about one iteration per load step. Thus, OINS significantly saves number of iterations and a huge amount of computation compared with the conventional methods. Especially, OINS not only can detect limit, inflection, and other special points but also can predict exactly various types of instabilities of structures.

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Path following techniques for geometrically nonlinear structures based on Multi-point methods

Cited by 20 publications

References 40 publications

Accelerated Molecular Statics Based on Atomic Inertia Effect

Accelerated Molecular Statics Based on Atomic Inertia Effect

A stiffness parameter and truncation error criterion for adaptive path following in structural mechanics

Deep learned one‐iteration nonlinear solver for solid mechanics

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