Numerical manifold method for vibration analysis of Kirchhoff's plates of arbitrary geometry

Guo, Hongwei; Zheng, Hong; Zhuang, Xiaoying

doi:10.1016/j.apm.2018.10.006

Cited by 64 publications

(13 citation statements)

References 73 publications

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“…The relevant integro-differential elastostatic formulation has been converted into a differential problem by means of the Helmholtz's averaging kernel. Solution strategies recently adopted for local elastic Kirchhoff plates of arbitrary geometry (see, e.g., [17]) and based on machine learning methods (see, e.g., [19,38]) can be further extended to small-scale two-dimensional continua. It is worth noting that limiting elastostatic solutions of the examined case studies for vanishing mixture parameter are coincident with those obtained in [7] exploiting the purely nonlocal stress-driven methodology.…”

Section: Plate With Simply Supported Edges Under Distributed Loadingmentioning

confidence: 99%

Two-phase elastic axisymmetric nanoplates

Vaccaro

Sedighi

2022

Engineering with Computers

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In the present work, the two-phase integral theory of elasticity developed in Barretta et al. (Phys E 97:13–30, 2018) for nano-beams is generalized to model two-dimensional nano-continua. Notably, a well-posed mixture local/stress-driven nonlocal elasticity is proposed to accurately predict size effects in Kirchhoff axisymmetric nanoplates. The key idea is to express the elastic radial curvature as a convex combination of local and nonlocal integral responses, that is a coherent choice motivated by virtue of the plate axisymmetry. The relevant structural problem is shown to be governed by a set of integro-differential equations, whose solution is computationally onerous. Thus, Helmholtz’s averaging kernel is advantageously adopted, since it enables explicit inversion of the integral constitutive law by virtue of an equivalence property. Specifically, the elastostatic problem of axisymmetry nanoplates is equivalently formulated in a differential form whose solution in terms of transverse displacement field is governed by nonlocal and mixture parameters. A parametric study is performed for case studies of applicative interest, and numerical solutions are finally provided and discussed. The presented methodology can be adopted to design and optimization of plate-based nano-electro-mechanical-systems (NEMS).

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Section: Plate With Simply Supported Edges Under Distributed Loadingmentioning

confidence: 99%

Two-phase elastic axisymmetric nanoplates

Vaccaro

Sedighi

2022

Engineering with Computers

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“…e phase field modelling was also used to deal with the crack propagation [26][27][28][29]. A numerical manifold method was developed to deal with the Kirchhoff plate [30][31][32].…”

Section: Introductionmentioning

confidence: 99%

Numerical Investigation on Improving the Computational Efficiency of the Material Point Method

Shi

et al. 2021

Mathematical Problems in Engineering

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Based on the basic theory of the material point method (MPM), the factors affecting the computational efficiency are analyzed and discussed, and the problem of improving calculation efficiency is studied. This paper introduces a mirror reflection boundary condition to the MPM to solve axisymmetric problems; to improve the computational efficiency of solving large deformation problems, the concept of "dynamic background domain (DBD)" is also proposed in this paper. Taking the explosion and/or shock problems as an example, the numerical simulation are calculated, and the typical characteristic parameters and the CPU time are compared. The results show that the processing method introducing mirror reflection boundary condition and MPM with DBD can improve the calculation efficiency of the corresponding problems, which, under the premise of ensuring its calculation accuracy, provide useful reference for further promoting the engineering application of this method.

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“…Thin plates are widely employed as basic structural components in engineering fields [Ventsel and Krauthammer (2001)], which combines light weight, efficient load-carrying capacity, economy with technological effectiveness. Their mechanical behaviours have long been studied by various methods such as finite element method [Bathe (2006); Hughes (2012); Zhuang, Huang, Zhu et al (2013)], boundary element method [Katsikadelis (2016); Brebbia and Walker (2016)], meshfree method [Nguyen, Rabczuk, Bordas et al (2008)], isogeometric analysis [Nguyen, Anitescu, Bordas et al (2015)], and numerical manifold method [Zheng, Liu and Ge (2013); Guo and Zheng (2018); Guo, Zheng and Zhuang (2019)]. The Kirchhoff bending problem is a classical fourth-order problem, its mechanical behaviour is described by fourth-order partial differential equation which poses difficulties to construct a shape function to be globally C 1 continuous but piecewise C 2 continuous, namely, H 2 regular, for those mesh-based numerical method.…”

Section: Introductionmentioning

confidence: 99%

A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate

Guo¹,

Zhuang²,

Rabczuk³

2019

Computers, Materials &Amp; Continua

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331

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In this paper, a deep collocation method (DCM) for thin plate bending problems is proposed. This method takes advantage of computational graphs and backpropagation algorithms involved in deep learning. Besides, the proposed DCM is based on a feedforward deep neural network (DNN) and differs from most previous applications of deep learning for mechanical problems. First, batches of randomly distributed collocation points are initially generated inside the domain and along the boundaries. A loss function is built with the aim that the governing partial differential equations (PDEs) of Kirchhoff plate bending problems, and the boundary/initial conditions are minimised at those collocation points. A combination of optimizers is adopted in the backpropagation process to minimize the loss function so as to obtain the optimal hyperparameters. In Kirchhoff plate bending problems, the C 1 continuity requirement poses significant difficulties in traditional mesh-based methods. This can be solved by the proposed DCM, which uses a deep neural network to approximate the continuous transversal deflection, and is proved to be suitable to the bending analysis of Kirchhoff plate of various geometries.

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Numerical manifold method for vibration analysis of Kirchhoff's plates of arbitrary geometry

Cited by 64 publications

References 73 publications

Two-phase elastic axisymmetric nanoplates

Two-phase elastic axisymmetric nanoplates

Numerical Investigation on Improving the Computational Efficiency of the Material Point Method

A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate

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