Multi-domain dual reciprocity BEM approach for the Navier-Stokes system of equations

Florez, W. F.; Power, H.; Chejne, Farid

doi:10.1002/1099-0887(200010)16:10<671::aid-cnm366>3.0.co;2-v

Cited by 28 publications

(17 citation statements)

References 10 publications

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“…Successful implementations of the MD-DRM approach for nonlinear Newtonian viscous¯uid problems have been previously reported in [16,17], which integral equation formulation is based upon the primitive variables, i.e. velocity and pressure, in terms of the Stokes fundamental solution.…”

Section: Introductionmentioning

confidence: 91%

“…In this way, we obtain a boundary only integral representation formula for the velocity ®eld at each subdomain c kj xu j x À C n t kj x; yu j ydC y À The particular solution of the above non-homogeneous auxiliary Stokes system of equations and its corresponding surface traction were previously obtained by Florez et al [16]. For economy in space in this work we refer to the expressions of such auxiliary ®elds to our previous works [16,17].…”

Section: Drm Multiple Domain Formulationmentioning

confidence: 99%

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Multi-domain dual reciprocity for the solution of inelastic non-Newtonian flow problems at low Reynolds number

Florez

Power²,

Janna³

2001

Computational Mechanics

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An ef®cient boundary element solution of the motion of inelastic non-Newtonian¯uids at low Reynolds number is presented in this paper. For the numerical solution all the domain integrals of the boundary element formulation have been transformed into equivalent boundary integrals by means of the dual reciprocity method (DRM). To achieve an accurate approximation of the non-linear and non-Newtonian terms two major improvements have been made to the DRM, namely the use of augmented thin plate splines as interpolation functions, and the partition of the entire domain into smaller subregions or domain decomposition. In each subregion or domain element the DRM was applied together with some additional equations that ensure continuity on the interfaces between adjacent subdomains. After applying the boundary conditions the ®nal systems of equations will be sparse and the approximation of the nonlinear terms will be more localised than in the traditional DRM. This new method known as multidomain dual reciprocity (MD-DRM) has been used to solve several non-Newtonian problems including the pressure driven¯ow of a power law¯uid, the Couette¯ow and two simulations of industrial polymer mixers. IntroductionThe term non-Newtonian denotes a¯uid whose constitutive equation is not the ordinary Newtonian form that leads to the Navier±Stokes equations. The behaviour of the non-Newtonian¯uids is strongly dependent upon the viscosity variations within the domain. Most non-Newtonian¯uids such as polymers have a viscosity which is a decreasing function of the shear rate, this characteristic is known as shear thinning [1]. Among all shear-thinning¯u ids, polymers are one of the most importants due to their rapid expansion and numerous industrial applications. The viscosity of an inelastic non-Newtonian¯uid can be calculated through several mathematical models such as the power law model; the Carreau model and the hyperbolic tangent model (see [1,2]). All these models are inelastic for they do not include any of the hysteresis or memory effects described by Bird et al. [3], Larson [4], and Tanner [5]. Since the viscosity of inelastic non-Newtonian¯uids is shear rate dependent, it follows that the mathematical models governing the¯ow motion of such uids are nonlinear even in the case when inertia effects can be neglected, i.e. at low Reynolds number. This type of ow process requires sophisticated numerical solution techniques.The application of the boundary element method (BEM) to a non-Newtonian problem requires ®nding a fundamental solution of the system of governing equations including all the nonlinear terms from the stress tensor. However, since such fundamental solution for a general model is not possible to be known, it is necessary to lump the nonlinear terms into a pseudo-body force leading to domain integrals that can be evaluated by using the cell integration approach (Cell-BEM) [6]. Although this method is effective and general, it makes the BEM lose its boundary only nature resulting in a numerical scheme several orders of ma...

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Section: Introductionmentioning

confidence: 91%

Section: Drm Multiple Domain Formulationmentioning

confidence: 99%

Multi-domain dual reciprocity for the solution of inelastic non-Newtonian flow problems at low Reynolds number

Florez

Power²,

Janna³

2001

Computational Mechanics

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“…Apart from Poisson-type equation, DRBEM is also employed for integral equation involving biharmonic operator r 4 , especially structural analysis of plates like bending problem [118], buckling problem [65], large deflection problem [255] and vibration problem [54]. Moreover, the application of DRBEM to date has successfully been implemented in areas including heat conduction problems [12,22,234,[244][245][246][265][266][267], heat convection problem [41], convective-diffusive heat problem [228,229], fluid flow problems governed by the Navier-Stokes equation [42,70] and crack problem [7,252]. As DRBEM proved to have practical applicability, it was later extended to three dimensional problems [1,73,250].…”

Section: Dual Reciprocity Bemmentioning

confidence: 99%

Development and implementation of some BEM variants—A critical review

Kadarman

Djojodihardjo

2010

Engineering Analysis with Boundary Elements

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a b s t r a c tDue to rapid development of boundary element method (BEM), this article explores the evolution of BEM over the past half century. We here summarize the overall development and implementation of several well-known BEM variants that includes collocation BEM, galerkin BEM, dual reciprocity BEM, complex variable BEM and analog equation method. Their theoretical and mathematical backgrounds are carefully described and a generalized Laplace's equation (and Poisson's equation) is utilized in demonstrating the different approaches involved. An up-to-date review on characteristics and implementation for each of the five variants is presented and also highlighted their significant contributions in boundary element research. In addition, this article tries to cover whole aspect of interests including efficiency, applicability and accuracy in order to give better understanding of BEM evolution. Comparisons and techniques of improvement for these variants are also discussed.

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“…Florez et al [4] improved the performance of the DRM in the BEM numerical solution of the NavierStokes equations through a multidomain decomposition technique. Following those results, the performance the DRM-MD was investigated [5], implementing quadratic shape functions of the boundary elements for both the approximation of the geometry and the surface variables of the governing equations.…”

Section: Introductionmentioning

confidence: 99%

Local regular dual reciprocity method for 2D convection-diffusion equation

Caruso¹,

Portapila²,

Power³

2012

WIT Transactions on Modelling and Simulation

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In this paper a new technique considering the dual reciprocity method (DRM) with only internal collocation points is considered along with local radial basis function interpolation. This approach gives rise to regular integral equations. Numerical results for the convection-diffusion equation are presented for different Peclet numbers. Comparisons with other numerical techniques are shown in order to illustrate the good solutions obtained by this method.

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Multi-domain dual reciprocity BEM approach for the Navier-Stokes system of equations

Cited by 28 publications

References 10 publications

Multi-domain dual reciprocity for the solution of inelastic non-Newtonian flow problems at low Reynolds number

Multi-domain dual reciprocity for the solution of inelastic non-Newtonian flow problems at low Reynolds number

Development and implementation of some BEM variants—A critical review

Local regular dual reciprocity method for 2D convection-diffusion equation

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