Reduced-order strategy for meshless solution of plate bending problems with the generalized finite difference method

Ferreira, Augusto César Albuquerque; Ribeiro, Paulo Marcelo Vieira

doi:10.1590/1679-78255191

Cited by 9 publications

(8 citation statements)

References 25 publications

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“…To account for the elastic Winkler foundation, k W h 4 /D is added to the term associated to the node of interest in the stencil describing the governing equations (Equations (8e), (9g), (10), and ( 11)). Equation (8e) i.e.…”

Section: Elastic Winkler Foundation Over An Area Of the Platementioning

confidence: 99%

“…the governing equations and constrained boundary conditions were derived based on the first-order shear deformation theory and Hamilton's principle. Ferreira et al [10] proposed a meshless strategy using the Generalized Finite Difference Method upon substitution of the original fourth-order differential equation by a system composed of two second-order partial differential equations; mixed boundary conditions, variable nodal density and curved contours were explored.…”

Section: Introductionmentioning

confidence: 99%

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Kirchhoff-Love Plate Theory: First-Order Analysis, Second-Order Analysis, Plate Buckling Analysis, and Vibration Analysis Using the Finite Difference Method

Fogang¹

2021

Preprint

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This paper presents an approach to the Kirchhoff-Love plate theory (KLPT) using the finite difference method (FDM). The KLPT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations. The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. Generally in the case of KLPT, the finite difference approximations are derived based on the fourth-order polynomial hypothesis (FOPH) and second-order polynomial hypothesis (SOPH) for the deflection surface. The FOPH is made for the fourth and third derivative of the deflection surface while the SOPH is made for its second and first derivative; this leads to a 13-point stencil for the governing equation. In addition, the boundary conditions and not the governing equations are applied at the plate edges. In this paper, the FOPH was made for all of the derivatives of the deflection surface; this led to a 25-point stencil for the governing equation. Furthermore, additional nodes were introduced at plate edges and at positions of discontinuity (continuous supports/hinges, incorporated beams, stiffeners, brutal change of stiffness, etc.), the number of additional nodes corresponding to the number of boundary conditions at the node of interest. The introduction of additional nodes allowed us to apply the governing equations at the plate edges and to satisfy the boundary and continuity conditions. First-order analysis, second-order analysis, buckling analysis, and vibration analysis of plates were conducted with this model. Moreover, plates of varying thickness and plates with stiffeners were analyzed. Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, with damping taken into account. In first-order, second-order, buckling, and vibration analyses of rectangular plates, the results obtained in this paper were in good agreement with those of well-established methods, and the accuracy was increased through a grid refinement.

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Section: Elastic Winkler Foundation Over An Area Of the Platementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Kirchhoff-Love Plate Theory: First-Order Analysis, Second-Order Analysis, Plate Buckling Analysis, and Vibration Analysis Using the Finite Difference Method

Fogang¹

2021

Preprint

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“…We emphasize that for each point of the discretization, the approximate values of the derivatives depend only on three factors: the number of points of the star, the arrangement of the points around the central point and the weighting function, as can be seen in Equation (12). It is in Equation (13) where the ODE comes into play. The strategy that we propose uses techniques that are already known and used to improve the approximation of the derivatives, but instead of applying them to the whole discretization, we do it selectively in the conflictive points.…”

Section: F I G U R E Formation Of Global Vector Gmentioning

confidence: 99%

“…The use of second-order approximations with the GFDM has received attention in the last two decades and has been successfully consolidated. Its applications vary among many types of problems, such us: seismic wave propagation, 7 hyperbolic nonlinear equations, 8 transient heat flow in anisotropic composites, 9 mathematical models of tumor growth, 10 inverse heat conduction problem, 11 discontinuous crack-faces, 12 plate bending problems, 13 etc.…”

Section: Introductionmentioning

confidence: 99%

A strategy to avoid ill‐conditioned stars in the generalized finite difference method for solving one‐dimensional problems

Ferreira

Ureña

Ramos

2021

Comp and Math Methods

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In this paper, we solve linear boundary value problems of second-order in ordinary differential equations with the generalized finite difference method and compare the numerical accuracy for different orders of approximations. We develop a strategy for dealing with ill-conditioned stars based on the condition number of the matrix of derivatives. In addition, we consider a scheme implemented with parallel processing for the formation of the stars and the calculation of the derivatives. We present some examples with high gradients in irregular discretizations exaggerated on purpose, to highlight the efficiency of the proposed strategy. K E Y W O R D Sfourth-order approximations, generalized finite difference method, ill-conditioned stars, parallel processing 1 Miguel Ureña, PhD. contributed to this article in his personal capacity. The views expressed are his own and do not necessarily represent the views of Statistics Spain.

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“…Thai, H. T et al [39] developed an analytical solution for refined plate. Ferreira et al [40] presented a meshless strategy using the Generalized Finite Difference Method (GFDM) for plate bending problems. They showed that by using this method, the original fourth-order differential equation could be substituted by a system composed of two second-order partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

Analyzing of thick plates with cutouts using the meshless (EFG) method based on higher order shear deformation theories for solving shear-locking issue

Vakili

Shahabian

Rad

2021

NMCE

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Although the finite element method (FEM) is a well-established method for modelling the thick plates, in some cases FEM encounters some difficulties such as shear locking and decrease in the accuracy of results caused by stress concentration around the openings. In this paper for the first time, the EFG method based on the higher-order shear deformation theories is developed for analysis of thick plates with cutout to overcome these drawbacks. It should be mentioned that the EFG method does not need any mesh generation in problem domain and its boundaries. The Radial Point Interpolation method (RPIM) is used to discrete the problem domain. Several numerical examples are analyzed using proposed method and effects of aspect ratios, boundary conditions and location of cutout are discussed in details. Results show that by choosing the appropriate shape functions for the deflection and rotations, the presented EFG method has successfully overcome the shear-locking problem. Based on numerical results, the best position of circular cutout, which minimizes the maximum deflection is determined. The approximate equations for determination of maximum deflection are presented using the cubic polynomial method. Numerical implementations show that the presented method has high efficiency, good accuracy and easy implementation.

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Reduced-order strategy for meshless solution of plate bending problems with the generalized finite difference method

Cited by 9 publications

References 25 publications

Kirchhoff-Love Plate Theory: First-Order Analysis, Second-Order Analysis, Plate Buckling Analysis, and Vibration Analysis Using the Finite Difference Method

Kirchhoff-Love Plate Theory: First-Order Analysis, Second-Order Analysis, Plate Buckling Analysis, and Vibration Analysis Using the Finite Difference Method

A strategy to avoid ill‐conditioned stars in the generalized finite difference method for solving one‐dimensional problems

Analyzing of thick plates with cutouts using the meshless (EFG) method based on higher order shear deformation theories for solving shear-locking issue

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