The solution of transport problems using dual the ... · terms expressed as domain integrals,...

The solution of transport problems using dual

reciprocity boundary element A review

H. Power, C.A. Brebbia

Wessex Institute of Technology, Ashurst Lodge, Ashurst, South-

ampton SO40 7AA

Abstract

This paper surveys the most relevant works on the solution ofthe transport equation by the use of the Dual Reciprocity BoundaryElement technique (DRM-BEM). An evaluation is carried out onthe numerical performance of three of the most popular DRM-BEMschemes A brief review of each method is presented, followed by anumerical application, for which the performance of each scheme iscompared and evaluated.

1 Introduction

The two-dimensional steady-state transport equation (convection- diffusion)including first-order reaction can be written in the form

where v* = u*(z,%/) and Vy = Vy(x,y) are the components of the velocityvector v, D is the diffusion coefficient (assuming the medium is homoge-neous and isotropic) and k represents the reaction coefficient. The variable can be interpreted as temperature for heat transfer problems, concentrationfor dispersion problems, etc, and will be herein referred to as a potential.The mathematical description of the problem is complemented by boundaryconditions of the Dirichlet, Neumann or Robin (mixed) types.

For a vector field having continuous first derivatives, the above par-tial differential equation can be reduced to an equivalent integral represen-tational formula of only boundary type, by the use of the correspondingGreen's theorem (see Mikhlin [1]) in terms of the fundamental solution ofthe adjoint equation of equation (1). Although theoretically the adjointpartial differential equation has a fundamental solution (see Courant andHilbert [2] for its general form), in most cases this solution cannot be ex-pressed in a closed form, requiring numerical integration in order to be

Transactions on Ecology and the Environment vol 9, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541

138 Environmental Problems in Coastal Regions

evaluated. The use of such fundamental solution for practical programmingpurposes is not very convenient.

In the case of constant velocity, the above differential equation can betransformed into the following boundary integral representation formula:

Environmental Problems in Coastal Regions 139

The equivalent integral representation formula can be written as [5]:

where * is the fundamental solution of Laplace's equation.Alternatively, the variable velocity components Vx(x,y) and Vy(x,y) can

be decomposed into average (constant) terms u* and Vy and perturbations

Px = Px(x,y) and Py = Py(a?,y), i.e.

This allows rewriting equation (1) as

dx dy dx dy

The above differential equation can be transformed into the following

integral representation formula [6]:

Jr dn Jr 9n Jp


2 DRM formulation with Laplaces' funda-

mental solution

Considering the Poisson equation

W(x) = 6(6 ĵ , ̂), x e n (8)

the solution (f> can be obtained using the fundamental solution * for theLaplace equation and the Green formula

I *Jn

(9)

where q = d/dn and q* = d*/dn. In order to obtain a boundary-onlyformulation of the problem, using the Dual Reciprocity Method, the termb(,d(/>/dx,d(i>/dy) is expressed in terms of set of known functions /(x,y')and unknown coefficients a,, as:

where y* (i = 1, ...,n) is a set of collocation points. In this way the volumeintegral in (9) becomes

(11)= f>, / /(x,y')̂(

^ JnTo reduce the last domain integral to equivalent boundary integrals, let

us define an auxiliary potential '(*), such that

= /(x,y') (12)

Once again we can apply the Green formula to equation (12)

? = / f? dT - ! q*$ dT + / ̂*/(x, y') dfi (13)Jr Jr Jn

Substituting equations (11) and (13) into equation (9) we have a formu-lation in term of boundary-only integrals:

* dT - ^ (14)

The coefficients a, in the above equation are obtained by evaluatingexpression (10) at all the collocation points and inverting



where #, is the matrix obtained by the evaluation of the function /(y',y')

at all points z,j = l,...,n.In the boundary element literature the Green's representation formula

is obtained by applying the reciprocity principle and the weighted residualcriterion, instead of the classical mathematical approach based upon theGauss theorem (the fundamental theorem of calculus). To obtain equation( 14) we have applied twice the Green formula for the Laplace equation,and this explains the name of Dual Reciprocity given to the method.

2.1 Choice of function / :

Previous work on dual reciprocity has shown (see Partridge et.al. [10]), thatalthough a variety of functions can in principle be used as a basic approxi-mation function, best results are normally obtained with simple expansions,the most popular of which is / = 1 + R, where R is the distance betweenpre-specified fixed collocation points, y™, and a field point x where thefunction is approximated, i.e.. R =\ x - y™ | . In the reference given, aswell as in previous DRM literature, the choice is based on experience ratherthan formal mathematical analysis. However, recent mathematical work,related to the theory of mathematical interpolation, and unrelated to theintegral equations literature, based on the so called radial functions, haspartially sustained these numerical findings. The approximation function,/ = ! + /?, defined previously, in general use in DRM is just one such radialfunction. This mathematical work provides general convergence criteria forthe interpolation series based on radial functions. The convergence is de-rived from the fact that the behaviour of these interpolation series is local,i.e.. increasing the number of fixed nodal points makes the interpolationmore localized. It is interesting to note that although the basic functions aredefined globally, the interpolation series have local behaviour. For the caseof radial functions defined by the above "/" function in one dimensionalspace, these new theories have shown that the resulting interpolation serieshappens to be a natural spline interpolation. Much of this mathematicalwork has addressed the properties of these approximations when the datapoints or collocation points y™, form an infinite regular grid, because thisstructure allows the order of accuracy to be derived when an interpolationor quasi-interpolation procedure is applied to a smooth underlying function.Unfortunately, the subject of non-regular grids has insufficient theoreticalsupport, and is a recent topic of mathematical research. In practical applica-tion of the DRM, the grid for the interpolation series is usually non-regular(for more detail on the theory of radial basis function approximation seePowell [11] or Powell [12]).

By evaluating expression (4) at all nodal points and inverting, one arrives

at



Micchelli [13] has proved that for the case when the nodal points are alldistinct, the matrix (F')~*, resulting from the basic function defined pre-viously, for all positive integers p, (the dimension of the matrix), and n,(the dimension of the space, /?",) is always non-singular, i.e.. the matrix

is invertible. Therefore, as long as the function 6(x) is regular, then, theabove vector coefficients a™, that are used in the dual reciprocity schemes,

are well defined.From (12) and the use of 1 + R as interpolating function, the following

expression for


The matrix forms of equations (17) and (18) are then

£ • £"*

# _ dF i~cT~ "%—r


It can be observed that the above interpolating function is not a radialfunction, and hence the previous theoretical considerations are not valid.

Thus, the domain integral can be recast in the form


The above optimal function suggests that a good choice of approximatingfunctions for 6 in equation 8) is given by

p6(x) = ]T \n& log R + ax + by + c (26)

m=l

Am3/m = 0m=l m=l m=l

If the thin plate splines are used to approximate the forcing terms, thenwe can obtain a particular solution


where the diffusion coefficient D was assumed equal to one, for simplicity,the velocity v^ is a linear function of x expressed as

Vx(x) = kx + C\

and the Vy component is equal to zero.The general solution of equation (28) is given by

4» = ̂ eK+Ci* (29)

Imposing the boundary conditions 0 = a at x = 0 and = b at x = 1,the constants (/> and C\ in (29) can be evaluated as

*-*• ' «

and the velocity field becomes:

The problem region is modelled as a rectangle with dimensions 1 x 0.7and the boundary discretized with 40 linear continuous elements, 17 ofwhich on each of the longer faces. The boundary conditions specify thevalues a = 300 and fo = 10 along the faces x = 0 and x = 1, respectively,with no flux in the y— direction.

A plot of the variation of the potential along the x— axis is presentedin figure 1 for k = 5. In this case, the global Peclet number is equal toP = -3.401 ± 2.5. The DRM formulations did not use any internal pointin the approximating series. It can be seen that the agreement with theanalytical solution is very good.

Figures 2 and 3 present the cases k = 20 (P = -3.401 ± 10) and k =40 (P = -3.401 ± 20). It is obvious that, as the velocity increases, thepotential distribution becomes steeper and more difficult to reproduce withnumerical models; thus, more refined discretizations are required. The DRMapproximations employed 30 internal points for k = 20 and 54 for k = 40.All solutions are still in good agreement for k = 20 but oscillations appearin both DRM formulations for k = 40, being more pronounced with the C-Dfundamental solution than Laplace's. All examples were compared with theresults obtained with a classical BEM cell formulation. Even for large valuesof the parameter t, the results with the cell formulation are very good anddisplay no oscillations whatsoever. It is important to remark that the cellformulation is convergent monotonically as the number of cells is increased,while both DRM formulations did not display monotonic convergence withincreased numbers of internal points.

The previous problem was also solved using Popov and Power domaindecomposition formulation, and using both basis approximating functions,



the usual radial function and the thin plate spline function. Two differentvalues of the parameter k were used, in the first case, k = 20, (P = -3.401 ±10) the domain was divided in three subdomains, with interfaces at x = 0.15and x = 0.3, two subdomains with interface at x = 0.25, and keeping theentire region as one domain. The density of the internal points was kept thesame in all three cases. The error for the internal potential obtained withthe formulation based on the usual radial function is given in figure 4. Wecan see that the error is higher for one and two subdomains, while for threesubdomains gives better results, although in this cases the improvement isnot very significant. The errors for the normal derivative at x = 0 andx = 1 are given in Table 1.

Table 1.

Number ofsubdomains1 domain2 subdomains3 subdomains

Error [%]x=04.78314.42404.5596

x=l7.25263.63513.8783

The error for the internal potential obtained with the thin plate splinefunction is given in figure 5, while the errors for the normal derivative inTable 2. It can be seen again that the accuracy on the evaluation of thepotential increase, as the number of subdomains increase. The maximumerror on the potential using a single domain was of 33.53%, with two domainwas of 7.13% and with three was of 3.12%.

Table 2.

Number ofsubdomains1 domain2 subdomains3 subdomains

Error [%]x=01.39180.93611.0739

x=l0.43361.75181.7525

Also, if we compare the best results for the internal potential profiles,which were obtained with three subdomains in both cases, using the usualradial function and the thin plate spline (see figure 6), we can conclude thatthe formulation with the thin plate spline function produces more accurateresults. From Tables 1 and 2 we can see that more accurate results for thenormal derivative were always obtained with the thin plate spline function.

To emphasize the increase of the accuracy of the evaluation of thepotential as the number of subdomains is increased, results for k = 30(P = —3.401 ± 15) are presented, for the formulation with the thin platespline function, with one and four subdomains. The results for the error ofthe internal potential are given in figure 7, the corresponding values of thepotential are given in figure 8, while the errors for the normal derivative inTable 3.



Table 3.

Number ofsubdomains1 domain4 subdomains

Error [%]x=01.97652.2340

x=l3.33941.4493

6 Conclusions

This paper has presented an assessment of three alternative DRM-BEMschemes for solution of convection-diffusion problems with variable velocityfields. The first two DRM algorithms produce satisfactory results whenthe velocity field is low (i.e. for diffusion-dominated problems) but developoscillations when the velocities are high.

The domain decomposition DRM approach when used in conjunctionwith a well defined basis optimal interpolation function, as the one em-ployed in this paper, always gave the best results. It is thought that theimprovement observed in the numerical results is due to the increase in theaccuracy of the interpolation functions when the subdomain technique is ap-plied. Previous works on interpolation analysis, based on radial functions,have shown that this technique can improve the estimation of the functionand its derivatives by at least three orders of magnitude [16]).

In addition to reducing the magnitude of the maximum error, one ofthe major advantages of the domain decomposition approach is the uniformdistribution of the errors obtained over the entire domain of the problem.

Further work in progress with the domain decomposition approach, ad-dresses the questions of the optimal manner of domain splitting, size ofsubdomain and collocation of internal points. Preliminary results obtainedfor case of Peclet number in the order of thousands appears to validate theproposed formulation.

References

[1] Mikhlin S.G., Mathematical physics, an advanced course, North-Holland, Amsterdam, 1970 (This book was original published in Rus-sian (1968)).

[2] Courant R. and Hilbert D., Methods of mathematical physics, Vol //,John Wiley and Sons, New York, 1962.

[3] Enokizono, M. and Nagata, S., Convection-diffusion analysis at highPeclet number by the boundary element method, IEEE Transactionson Magnetics, Vol. 28(2), 1651-1654, 1992.



[4] Nardini, D. and Brebbia, C.A., The solution of parabolic and hyper-bolic problems using an alternative boundary element formulation, inBoundary Elements VII, Springer-Verlag, Berlin, 1985.

[5] Brebbia, C.A. and Skerget, P., Diffusion-convection problems usingboundary elements, Advances in Water Resources, Vol. 7, 50-57, 1984.

[6] Wrobel, L.C. and DeFigueiredo, D.B., Numerical analysis of convec-tion diffusion problems using the boundary element method, Int. J.Num. Meth. Heat and Fluid Flow, Vol. 1, 3-18, 1991.

[7] Partridge, P.W. and Brebbia, C.A., The dual reciprocity boundarymethod for the diffusion convection equation, Proc. IX Int. Conf. onComputational Methods in Water Resources, Computational Mechan-ics Publications, Southampton, and Elsevier, Amsterdam, 1992.

[8] Wrobel, L.C. and DeFigueiredo, D.B., A dual reciprocity boundaryelement formulation for convection-diffusion problems with variablevelocity fields. Engineering Analysis with Boundary Elements, Vol.8(6), 312-319, 1991.

[9] Popov V. and Power H., A domain decomposition on the dual reci-procity approach, Boundary Element Communications, Vol. 7, No. 1,1996.

[10] Partridge, P.W., Brebbia, C.A. and Wrobel, L. The Dual ReciprocityBoundary Element Method. Computational Mechanics Publications,Southampton Boston, 1992.

[11] Powell, M.J.D. Radial Basic Functions for Multivariable Interpolation:a Review, in Algorithms for Approximation, eds. J.C. Mason andM.G. Cox, Oxford University Press, Oxford 1987.

[12] Powell, M.J.D. The Theory of Radial Function Approximation in1990, presented at the Numerical Analysis Summer School, Lancaster1990.

[13] Micchelli, C.A. Interpolation of Scattered Data: Distance Matriceand Conditionally Positive Definite Functions, Constr. Approx., Vol.2,1986.

[14] Golberg, M.A. and Chen, C.S., The Theory of Radial Basis Functionsapplied to the BEM for Inhomogeneous Partial Differential Equations,Boundary Elements Communications, Vol.5, No. 2, 1994, 57-61.

[15] Golberg M.A., The numerical evaluation of particular solutions in theBEM, A review, Boundary Element Communications, Vol.6, No.3,1995, 99-106.



[16] Kausa, E.J. and Carlson, R.E., Radial Basis Functions: A class ofgrid-free, seatered data approximations, Computational Fluid Dy-namics, Vol. 3, No. 4, 1995, 479-496.



350

0300

250

200

150

100

50

AnalyticalLaplace DRM

ooooo C-D DRMooooo Internal cells

0.0 0.2 0.4 0.6 0.8 1.0

X

Fig- 1: Potential distribution for k = 5

AnalyticalLaplace DRM


0.0 0.2 0.4 0.6 0.8 1.0

Fig. 2: Potential distribution for k = 20


0 -

-50

AnalyticalLaplace DRW



0.0 0.2 0.4 0.6

X

0.8 1.0

Fig. 3: Potential distribution for Jb = 40

;%) ii10987

«- 6

€§- 5— 4

321

T3-,-.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0X

Fig. 4: The error for the internal potential for the usual radial functionusing one, two or three subdomains



20

15

10

0 *0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

X

Fig. 5: The error for the internal potential for the thin plate spline functionusing one, two or three subdomains

9-0

O thin plate spline

"" ""•"0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

X

Fig. 6: Error for the internal potential profiles with the usual radial andwith the thin plate spline function with three subdomains



w

«•

•̂

LUU

908070

6050

"4030

2010n

OO 1 domains ',A 4 domains / •

i \i \ '

A. / \

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0X

Fig. 7: Comparison of the accuracy for the internal concentration profilefor one and four subdomains, using the thin plate spline function

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

100

Fig. 8: Comparison of the accuracy for the internal concentration profilefor one and four subdomains, using the thin plate spline function


The solution of transport problems using dual the ... · terms expressed as domain integrals,...

Documents

Transcript of The solution of transport problems using dual the ... · terms expressed as domain integrals,...