Simulation of two-dimensional sloshing phenomenon by generalized finite difference method

Benito

Math Methods in App Sciences

et al. 2017

The generalized finite differences method allows the use of irregular clouds of nodes. The optimal values of the key parameters of the method vary depending on how the nodes in the cloud are distributed, and this can be complicated especially in 3D. Therefore, we establish 2 criteria to allow the automation of the selection process of the key parameters. These criteria depend on 2 discrete functions, one of them penalizes distances and the other one penalizes imbalances. In addition, we show how to generate irregular clouds of nodes more efficient than finer regular clouds of nodes. We propose an improved and more versatile h‐adaptive method that allows adding, moving, and removing nodes. To decide which nodes to act on, we use an indicator of the error a posteriori. This h‐adaptive method gives results more accurate than those ones presented for the generalized finite differences method so far and, in addition, with fewer nodes. In addition, this method can be used in time‐dependant problems to increase the temporal step or to avoid instabilities. As an example, we apply it in problems of seismic waves propagation.

“…Let a be the coefficient vector in (1) and so a T D u = f (x). On including the value of D u given in (13) in Equation 1, the following expression is obtained…”

Section: Theoretical Backgroundmentioning

confidence: 99%

Section: Error Indicatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Adaptive strategies to improve the application of the generalized finite differences method in 2D and 3D

Benito

Math Methods in App Sciences

et al. 2017

Numerical Meth Engineering

“…Up to now, the GFDM combined with low‐order differentiation formula for the time derivative has been widely used to solve time‐dependent problems . These algorithms have observably advanced our knowledge in science and engineering, which, in turn, also revealed the weaknesses of existing approaches.…”

Section: Introductionmentioning

confidence: 99%

A combined scheme of generalized finite difference method and Krylov deferred correction technique for highly accurate solution of transient heat conduction problems

Zhang

et al. 2018

Self Cite

This paper presents a numerical framework for the highly accurate solutions of transient heat conduction problems. The numerical framework discretizes the temporal direction of the problems by introducing the Krylov deferred correction (KDC) approach, which is arbitrarily high order of accuracy while remaining the computational complexity same as in the time-marching of first-order methods. The discretization by employing the KDC method yields a boundary value problem of the inhomogeneous modified Helmholtz equation at each time step. The meshless generalized finite difference method (GFDM) or meshless finite difference method (MFDM), a meshless method, is then applied to the solution of resulting boundary value problems at each time step. Six numerical experiments in one-, two-, and three-dimensional cases show that the proposed hybrid KDC-GFDM scheme allows big time step size for a long-time dynamic simulation and has a great potential for the problems with complex boundaries.In addition, some comparisons are also presented between the present method, the COMSOL software, and the GFDM with implicit Euler method. KEYWORDS generalized finite difference method, Krylov deferred correction method, long-time simulation, meshless method, transient heat conduction 1 INTRODUCTION Numerical analysis and modeling of transient heat conduction problems with inhomogeneous heat sources have very important applications in engineering fields, such as ignition of reactive solids and microwave heating. Various Int J Numer Methods Eng. 2019;117:63-83.wileyonlinelibrary.com/journal/nme

“…The GFDM has been also used for solving 3D partial differential equations by Izadian et al [13], Gavete et al [14], and Hua et al [15]. The GFDM was also applied for different practical tasks as in Chan et al [16] for solving two-dimensional nonlinear obstacle problems, in Zhang et al [17,18] for simulating two-dimensional sloshing phenomena and for propagation of nonlinear water waves, in Mochnacki and Majchrzak [19] for modelling of casting solidification, and in Li and Fan [20] for solving two-dimensional shallow water equations.…”

Section: Introductionmentioning

confidence: 99%

Solving Elliptical Equations in 3D by Means of an Adaptive Refinement in Generalized Finite Differences

Gavete

Mathematical Problems in Engineering

Benito

et al. 2018

We apply a 3D adaptive refinement procedure using meshless generalized finite difference method for solving elliptic partial differential equations. This adaptive refinement, based on an octree structure, allows adding nodes in a regular way in order to obtain smooth transitions with different nodal densities in the model. For this purpose, we define an error indicator as stop condition of the refinement, a criterion for choosing nodes with the highest errors, and a limit for the number of nodes to be added in each adaptive stage. This kind of equations often appears in engineering problems such as simulation of heat conduction, electrical potential, seepage through porous media, or irrotational flow of fluids. The numerical results show the high accuracy obtained.