Boundary element methods for transient convective diffusion. Part I: General formulation and 1D implementation

Grigoriev, M. M.; Dargush, Gary F.

doi:10.1016/s0045-7825(03)00388-8

Cited by 18 publications

(14 citation statements)

References 35 publications

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“…Hence there is always a need to modify BEM application for those problems that unambivalently incorporate the problem domain into problem formulation such as is found with body-force terms, nonlinearity, transience, heterogeneity, source and sink terms etc. Current efforts to prevail over this challenge have resulted in a variety of hybrid BEM techniques Grigoriev [23], Taigbenu [35], Onyejekwe [36], Onyejekwe [37], Taigbenu and Onyejekwe [38], Hibersek and Skerget [39], Onyejekwe [40], Grigoriev and Dargush [41], Perata and Popov [42], Portapilla and Power [43], Sladeck et al [44], Toutip [25].…”

Section: Introductionmentioning

confidence: 99%

An Element-Driven Boundary Integral Treatment for Nonlinearity: A Coupled Nonlinear Two-Dimensional Burger’s Equation Test Case

Onyejekwe¹

2017

IJFMTS

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Abstract:To further improve on the competitiveness of the boundary element method (BEM), a hybrid version of it is used for a numerical solution of two dimensional nonlinear coupled viscous Burger's equation. Adopting this approach to a discretized 2D spatial domain, the resulting integral equations arising from the singular integral theory are applied locally to each of the elements. The resulting nonlinear discrete equations are finally solved by the Picard iteration algorithm. The simulation results obtained, not only concur with analytical solutions, but also display high accuracy and are in agreement with those available in literature.

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Section: Introductionmentioning

confidence: 99%

An Element-Driven Boundary Integral Treatment for Nonlinearity: A Coupled Nonlinear Two-Dimensional Burger’s Equation Test Case

Onyejekwe¹

2017

IJFMTS

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“…Recently, Grigoriev and Dargush [1][2][3] presented higher-order boundary element methods for unsteady convective di usion problems using the time-dependent convective-di usion kernels originally obtained by Carslaw and Jaeger [13]. Similar to their earlier work [14], linear, quadratic and quartic time interpolation functions were utilized for a temporal discretization of the boundary integral equation.…”

Section: Introductionmentioning

confidence: 99%

“…Recent studies [1][2][3][4][5][6][7][8][9][10][11][12] have revealed that the use of convective kernels within the boundary element framework provides an automatic upwinding in the most natural way for the entire range of Reynolds (or, Peclet) number, from zero to inÿnity. Despite the attractiveness of the convective boundary element methods (BEM) a proper numerical implementation of the convective fundamental solutions appears extremely di cult.…”

Section: Introductionmentioning

confidence: 99%

“…Similar to their earlier work [14], linear, quadratic and quartic time interpolation functions were utilized for a temporal discretization of the boundary integral equation. In References [1][2][3], the authors considered both one-and two-dimensional BEM formulations. Although the implicit time marching formulation utilized in References [2,3] did not impose any restrictions on the time step size t, the applicability of the numerical algorithm was restricted to the following time step sizes:…”

Section: Introductionmentioning

confidence: 99%

“…Consequently, the BEM became less e cient and even in some cases failed to converge if the time steps were too large. In this paper, the formulation presented in References [1][2][3] is extended to alleviate constraint (1). The use of time steps up to three orders of magnitude larger than t c is facilitated by splitting the time interval t into singular and non-singular intervals.…”

Section: Introductionmentioning

confidence: 99%

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Efficiency of boundary element methods for time‐dependent convective heat diffusion at high Peclet numbers

Grigoriev

Dargush

2004

Commun. Numer. Meth. Engng.

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SUMMARYA higher-order boundary element method (BEM) recently developed by the current authors (Comput Methods Appl Mech Eng 2003; 192:4281-4298; 4299-4312; 4313-4335) for time-dependent convective heat di usion in two-dimensions appears to be a very attractive tool for e cient simulations of transient linear ows. However, the previous BEM formulation is restricted to relatively small time step sizes (i.e. t 6 4Ä=V2 ) owing to the convergence issues of the time series for the kernel representation within a time interval. This paper extends the boundary element formulation in a way to allow time step sizes several orders of magnitude larger than in the previous approach. We consider an example problem of thermal propagation, and investigate the accuracy and e ciency of BEM formulations for Peclet numbers in the range from 10 3 to 10 5 .

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A method for numerical modelling of convection–reaction–diffusion using electrical analogues

Shafaq

Kennedy

Delauré

2015

Int J Numerical Modelling

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In this paper, a new method, called the lumped-component circuit method (LCM), is developed for one-dimensioal and two-dimensional convection-reaction-diffusion with low to moderate Peclet numbers, tested for modelling both steady-state and transient problems, and compared with standard finite volume method (FVM) schemes. The method has been developed principally for solving equations with piecewise-constant coefficients using nodes that are not positioned to correspond to the coefficient discontinuities. In such situations, the FVM solutions do not converge consistently as the node spacing is decreased, but LCM solutions do. In general, the LCM method is more accurate than the FVM schemes tested, and, while the computational cost of LCM is higher, results suggest that it can be more efficient. Like the transmission line method (TLM), it is an indirect scheme in which the problem to be solved is first represented by an analogous transmission line (TL). Unlike with TLM, however, the TL is then modelled using a lumped-component circuit, the voltages at nodes within that circuit being calculated.

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Boundary element methods for transient convective diffusion. Part I: General formulation and 1D implementation

Cited by 18 publications

References 35 publications

An Element-Driven Boundary Integral Treatment for Nonlinearity: A Coupled Nonlinear Two-Dimensional Burger’s Equation Test Case

An Element-Driven Boundary Integral Treatment for Nonlinearity: A Coupled Nonlinear Two-Dimensional Burger’s Equation Test Case

Efficiency of boundary element methods for time‐dependent convective heat diffusion at high Peclet numbers

A method for numerical modelling of convection–reaction–diffusion using electrical analogues

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