Some studies on dual reciprocity BEM for elastodynamic analysis

Agnantiaris, J. P.; Polyzos, D.; Beskos, D.E.

doi:10.1007/s004660050110

Cited by 26 publications

(52 citation statements)

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“…For a model problem, the Dirichlet problem for Poisson's equation in the plane, we show that the error has two components: (1) that due to approximating the particular solution (which generally depends on the error in approximating the source term) and (2) error due to the choice of boundary element method. Using this we argue that traditional BEMs using low order piecewise polynomial approximations that the dominant part of the computational error may be due to BEM error ± not the interpolation error ± so that all commonly used rbfs tend to have similar overall behavior ± in accord with the observed numerical results in Agnantiaris, Polyzos and Beskos (1996); while if higher order solvers are used the dominant error may now be the interpolation error and the effects of the choice of rbfs is evident. To verify this computationally, we present the results of a 3 Â 3 experiment solving the boundary value problem in Section 4 using three different algorithms to solve the boundary integral equations: a standard piecewise quadratic BEM (Partridge, Brebbia and Wrobel (1992); a sophisticated adaptive method due to Jeon and Atkinson (1993) based on the Nystro Èm method and the Method of Fundamental Solutions (MFS) (Golberg, Chen and Karur (1996)) and three basis functions, 1 r, TPS and MQs.…”

Section: Introductionsupporting

confidence: 66%

“…In Section 2 we discuss the appropriate implementation of TPS indicating that the failure to include the linear terms may have biased the results in Agnantiaris, Polyzos and Beskos (1996). Similar observations have been made by Karur and Ramachandran (1995).…”

Section: Introductionsupporting

confidence: 55%

“…Golberg, Chen and Karur (1996) have shown that with the proper choice of the shape parameter c, that multiquadrics (MQs) r 2 c 2 1a2 produced a signi®cant increase in accuracy over both 1 r and TPS ± up to three orders of magnitude. However, in a number of recent articles, particularly that by Agnantiaris, Polyzos and Beskos (1996), several authors have indicated that TPS showed little or no improvement over the traditional 1 r.…”

Section: Introductionmentioning

confidence: 91%

“…It is the purpose of this note to provide some clari®-cation of the results in Agnantiaris, Polyzos and Beskos (1996) and to comment on what we feel is a misinterpretation of the optimally property of TPS stated on page 275 of Agnantiaris, Polyzos and Beskos (1996). In addition, we present some new theoretical and computational results suggesting that it is important to account for the error in the choice of boundary element method used in the DRM as that error can confound the results of experimental studies which only consider the error due to the choice of interpolating functions for the inhomogeneity.…”

Section: Introductionmentioning

confidence: 98%

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mentioning

confidence: 97%

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Some comments on the use of radial basis functions in the dual reciprocity method

Golberg

Chen

Bowman

et al. 1998

Computational Mechanics

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We comment on the use of radial basis functions in the dual reciprocity method (DRM), particularly thin plate splines as used in Agnantiaris, Polyzos and Beskos (1996). We note that the omission of the linear terms could have biased the numerical results as has occurred in several previous studies. Furthermore, we show that a full understanding of the convergence behavior of the DRM requires one to consider both interpolation and BEM errors, since the latter can offset the effect of improved data approximation. For a model Poisson problem this is demonstrated theoretically and the results con®rmed by a numerical experiment.

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Section: Introductionsupporting

confidence: 66%

Section: Introductionsupporting

confidence: 55%

Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 98%

“…

…”

mentioning

confidence: 97%

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Some comments on the use of radial basis functions in the dual reciprocity method

Golberg

Chen

Bowman

et al. 1998

Computational Mechanics

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The improvement of numerical modeling in the solution of incompressible viscous flow problems using finite element method based on spherical Hankel shape functions

Farmani

Ghaeini-Hessaroeyeh

Hamzehei‐Javaran

2018

Numerical Methods in Fluids

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Summary In this paper, the finite element method with new spherical Hankel shape functions is developed for simulating 2‐dimensional incompressible viscous fluid problems. In order to approximate the hydrodynamic variables, the finite element method based on new shape functions is reformulated. The governing equations are the Navier‐Stokes equations solved by the finite element method with the classic Lagrange and spherical Hankel shape functions. The new shape functions are derived using the first and second kinds of Bessel functions. In addition, these functions have properties such as piecewise continuity. For the enrichment of Hankel radial basis functions, polynomial terms are added to the functional expansion that only employs spherical Hankel radial basis functions in the approximation. In addition, the participation of spherical Bessel function fields has enhanced the robustness and efficiency of the interpolation. To demonstrate the efficiency and accuracy of these shape functions, 4 benchmark tests in fluid mechanics are considered. Then, the present model results are compared with the classic finite element results and available analytical and numerical solutions. The results show that the proposed method, even with less number of elements, is more accurate than the classic finite element method.

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Boundary Element based Discontinuous Deformation Analysis

2016

Num Anal Meth Geomechanics

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Summary A Boundary Element based Discontinuous Deformation Analysis (BE‐DDA) method is developed by implementing the improved dual reciprocity boundary element method into the open close iterations based DDA. This newly developed BE‐DDA is capable of simulating both the deformation and movement of blocks in a blocky system. Based on geometry updating, it adopts an incremental dynamic formulation taking into consideration initial stresses and dealing with external concentrated and contact forces conveniently. The boundaries of each block in the discrete blocky system are discretized with boundary elements while the domain of each block is divided into internal cells only for the integration of the domain integral of the initial stress term. The contact forces among blocks are treated as concentrated forces and the open–close iterations are applied to ensure the computational accuracy of block interactions. In the current method, an implicit time integration scheme is adopted for numerical stability. Three examples are used to show the effectiveness of the algorithm in simulating block movement, sliding, deformation and interaction of blocks. At last, block toppling and tunnel stability examples are conducted to demonstrate that the BE‐DDA is applicable for simulation of blocky systems. Copyright © 2016 John Wiley & Sons, Ltd.

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Some studies on dual reciprocity BEM for elastodynamic analysis

Cited by 26 publications

References 0 publications

Some comments on the use of radial basis functions in the dual reciprocity method

Some comments on the use of radial basis functions in the dual reciprocity method

The improvement of numerical modeling in the solution of incompressible viscous flow problems using finite element method based on spherical Hankel shape functions

Boundary Element based Discontinuous Deformation Analysis

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