This paper presents some recent advances on the numerical solution of the classical Germain-Lagrange equation for plate bending of thin elastic plates. A meshless strategy using the Generalized Finite Difference Method (GFDM) is proposed 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 are some of the explored aspects. Simulations using very dense clouds and parallel processing scheme for efficient neighbor selection are also presented. Numerical experiments are performed for arbitrary plates and compared with analytical and Finite Element Method solutions. Finally, an overview of the procedure is presented, including a discussion of some future development.
The generalized finite difference method is a meshless method for solving partial differential equations that allows arbitrary discretizations of points. Typically, the discretizations have the same density of points in the domain. We propose a technique to get adapted discretizations for the solution of partial differential equations. This strategy allows using a smaller number of points and, therefore, a lower computational cost, to achieve the same accuracy that would be obtained with a regular discretization.
KEYWORDSadapted discretization, fourth-order approximations, generalized finite difference method
MSC CLASSIFICATION
65M06, 65M50Miguel 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.
The generalized finite difference method is a meshless method for
solving partial differential equations that allows arbitrary
discretizations of points. Typically, the discretizations have the same
density of points in the domain. We propose a technique to get adapted
discretizations for the solution of partial differential equations. This
strategy allows using a smaller number of points and a lower
computational cost to achieve the same accuracy that would be obtained
with a regular discretization.
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