approximation, see zerroukat [l]-[2] for details. · 2014. 5. 13. · 190 boundary elements...

On time-dependent domain integral

approximation using radial basis functions

M. Zerroukat & H. Power

7/4,4,

Abstract

Using time-dependent fundamental solution in the boundary element

for the solution of transient-diffusion type equations has a high com-

putational cost, for large number of time steps, due to the inherent

time history constraint in the integral representation. In general, the

solution for n domain and m boundary points at the fc-th time-step re-

quires an amount of computer operations of the order O(km? + knm).

This paper presents a time-marching scheme that requires a compu-

tational cost of the order of only O(m? 4- nm), where the dependence

from the past /c-steps is removed. The scheme uses the time depen-

dent fundamental solution but the time integration is performed over

one time-step only and the rest of the history integral is converted to

a domain integral and approximated using radial basis functions.

1 Introduction

The numerical solution of the time-dependent diffusion type equa-tion using the boundary integral approach have been great lv ham-

pered by the time dependence in the boundary integral formulation.

For instance, evaluating the solution require more and more compu-

tational work as time progresses. Many authors have attempted to

overcome this major drawback. These methods can be categorisedinto: (i) truncation, (ii) a time-free space transformation, i'iii) ap-

propriate weighting fundamental solution, and l i v i domain integral

Transactions on Modelling and Simulation vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-355X

190 Boundary Elements

approximation, see Zerroukat [l]-[2] for details.

This paper presents a scheme which can be considered as one

of group (iv). It employs the time-dependent fundamental solution

in the integral representation but the time integration is performed

over one time step only. The rest of the time-history integral is con-

verted to a domain integral, which is approximated using radial basis

functions. This results in huge savings in terms of cpu-time and put

the scheme on the same playing field as finite differences and finiteelements where the solution at every time step is computed from that

at the previous one only. In this paper both globally and compactly

support radial basis functions are considered and their merits are dis-

cussed.

2 Boundary integral equation for diffusion prob-

lems

Consider the general diffusion equation:

^%^ = KV2%(x,f). x€dcR\ f>0 (1)at

with certain conditions on the boundary F = <9Q, where n(x. t) de-

notes concentration at the spatial position x at time t, V the gradient

differential operator; His a bounded domain in R\ and K the dif-

fusivity. The integral equation corresponding to (1) over the entire

space-time domain can be written as [3]:

+

where u* is the free space Green's function given by:

1(3)

where d is the dimension of the problem, r — ||£ - x|| is the Euclidian

distance between the field point x and the source point £, n, is the

outward normal to the boundary F. and c(£) is a constant which


Boundary Elements 191

depends on the location of the source point £ and the local geometry

(e.g. for a smooth boundary c(£) = 1/2). Assuming that fl = t" and

t = f+i = t" + 5t, equation (2) becomes

+p"((,&) (4)

where 6t is the time step size and

(5)JO.

where

.*.«) = 77—TT exp | -%#- 1 (6)

Assuming that, given a total of TV boundary and domain colloca-

tion points, then u(x, t") can be approximated using N radial basis

functions </?, viz:

Using (7), p"(£,<5£) in (5) can be approximated by:

JV

(8)

where tu(£,Xj,5t) = J y({.x, 6t)¥?(x.x )cff2. Applying (4) to every

boundary points £ = ^j, ..... he following system of equations can

be obtained:

(9)

) -

and {p"> = [p"( i,5t) ... p"(W(f which is given

by



{A"} = ${A"} (10)

where {A"} = [AJ...Aft], $y = c;(&,Xj,<%). The notation {...} and

[...] are used for vector and matrix, respectively; [...p denotes matrix

transpose. For the coefficients matrices H and G, see Wrobel and

Brebbia [3] for details. It has to be noted that the entries of 4> depend

on only 6t and the locations of both the boundary and the internal

points. As it is a common practice to keep the same collocation

points and a constant time step throughout, the computation of <fr

is performed only once. This reduces the computation of {p"} at

every time step to simply compute the vector {A™}. The entries of

<£ consist of the integration of a known function, with a variable

parameter £ = £^, ....,£&> over a known domain.

3 Domain integral approximation

Since (9) consists of computing the solution [u™+* q +i] at the

level time (n + 1) from those at level n. and the solutions {u™} and{u™~*} at the levels n and (n— 1), respectively, are already known, the

evaluation of {p™} from {p™~*}, i.e. computing {A™} from {A™~*},

is similar to solving a local Dirichlet problem, for t™~~* < t < £™,

defined by:

(n)C/ 6

Li<lX« Is) — IL \ / ) ~~~ V /

where /(x,t) and t ~*(x) are known functions. It has to be em-phasised that the boundary and initial conditions (12)-(13) are /oca/,^n-i < t < t™, and do not represent the boundary and initial con-

ditions of the global diffusion problem, for which the BEM is used.

The global boundary and initial conditions are treated in the same



manner as in standard BEM [3]. For instance, / in (12) is a simple

linear function of the boundary solutions at the time steps (n) and

(n - 1), e.g., /(x,() = (i( K(x) + ((K-Xx).

First, let us discretize (11) according to the 0- weighted schemegiving

where 0 < 6 < 1 and t" = t"~*+6t. Using the notation u" = u(x, r),(14) can be rearranged as:

^ + aV = - + / 3 V " (15)

where a = -K06t and /? = «6t(l - 9}. Applying the approximation

(7) to every collocation points Xj, the following system is obtained:

TV N

A^. , % = 1,...,N (16)

In general equation (7) or (16) can also be written with an additional

polynomial on the right-hand side, see Zerroukat [l] for details. How-

ever, in this paper and for simplicity, the polynomial is omitted due

to its negligible effect on the accuracy of the scheme. Rewriting (16)

in matrix form, viz:

{u"} = A{A"} (17)

where {u"} = [...], {A"} = M...A% &nd A is given by:

Assuming that there are N& < AT internal (domain) points and JVp =

(N — NCI) boundary points, i.e., N — (Nn 4- A/p). then the (A* x N)

matrix A can be split into: A = AQ — Ar« where

= [ . for (1 < 2 < TVn-1 < j < AT). 0 elsewhere]

= [f - for (A n < 2 < A'. 1 < ; < N). 0 elsewhere: (19)



Using the notation £A to denote the matrix of the same dimension

as A and containing the elements £5y, where y?y = Cfij, where.£ is a

linear operator, then equation (15) together with (12) can be written,in a matrix form, as:

B{A"} = Cp {A"-i} + {F"} (20)

where

B-Ca + Ar and Cp=An + pVAa, p = a, p (21)

and {F } = [0 ..... 0 J% / 2-/N] Rewriting (20) and puttingit in a simpler form, viz

} + {E*} (22)

where

D = B-*C£ and {E"} = B-i{F*} (23)

Similarly as $, B and D are computed only once, hence computing

{A™} from (A™~ } is a simple matrix-vector product operation of

order O(N). Although equation (22) is valid for any 9 € [0,1], the

value of 9 — 1/2 is used (i.e., the Crank-Nicholson scheme), hence

CQ = 2An — C/3, where /3 — K,8t/2. Knowing the initial distribution

%(x,fo), {A } can be obtained from (17), i.e. {A } = A" {u }. The

computation of the solution at any time step (n-f 1) involves solving

the system (9), with the right-hand side calculated from equation

(10), where the vector {A } is calculated from {A"~*} using equation(22).

3.1 Global radial basis functions

Using global radial functions, the construction of elements y^, 1 <

i>j < TV, of A are computed using either multiquadrics or thin platesplines. Due to their high accuracy, these radial functions became

widely used in interpolation literature. These are:

(i) Multiquadric (for generality, it is written with a variable shape

parameter Sj):



(ii) m^-order thin plate splines:

y(xt, Xj) = <p-(rij) = r\™ log(r ) m — 1, 2, 3, ... (25)

where r - = ||x — Xj|| is the Euclidian norm. Since y? given by (24) is

C°° continuous while (25) is only C^™~~* continuous, the multiquadric

can be used directly, while higher order thin plate splines must be

used, for higher order partial differential operators. For diffusion type

equations, a choice of m = 2 for thin plate splines, i.e., second-order

thin plate splines is used. Given the optimum shape parameter, the

multiquadric is usually slightly more accurate. However, computingthe shape parameter is still an intensive research subject. In the

absence of a simple procedure by which the optimum shape parameter

can be computed, the relationship given Hardy [4] can be used, i.e.

Sj — s — 4 x rmin, where r^m = mm (||x; - Xj||, 1 < i,j < N).

When using global radial functions, the resulting interpolation

matrix A is fully populated. The system may also become ill con-

ditioned if very smooth radial functions are used on dense and large

number of points. Although pre-conditioning methods [5] can be sued

to deal with this problem, both the cpu-time and ill-conditioning in-

crease with increasing the number of collocation points. This limits

the use of global radial basis functions to a maximum number of col-

location points, which depends on the power of the computational

platform. Domain decomposition [6] can also be used to reduce the

scale of ill-conditioning and introduces some sparsity in the A matrix.

3.2 Compactly supported radial basis functions

The procedure to compute {A } from {A ~*} is the same as with

global radial basis functions, except the construction of A and 3>

When compactly supported radial basis functions are used, , 1 <2, j < N, are given by [7]:

(2 o)J tor r,j > 1

where T{J — ||xi—Xj|| /Q, and Cj — c(x%) is the compact support of

at x^. As for <J>^-, 1 < z, j < N. in equation (10 . they are reducedto an integral over a simple support domain (%,-. instead of the global

domain fi, viz:



*« = (£;> xj, 6t) = / y(£, x, <5t)0(x, Xj)oKl; with x €fy (27)./a,-

where £2j is the support domain at Xj, which is a distance of length

2c(xj), a circle of radius c(x ) or a sphere of radius c(xj) with Xj at

the centre, for one-, two- or three-dimensional problems, respectively;

ty = {x, Hx-XjH < c(xj)}. (j)(rij] in (26) and (27) is a positive

definite radial function, which according to Wendland [7], can be

computed for any dimension d and a required smoothness 2k using(subscripts 2, j are dropped from r for neatness) :

,H' (28).7=0

with I = [d/2\ + k + 1 ([y\ denotes the largest integer less than or

equal to y) and the coefficients a'- can be computed recursively for

anv 0 < 77i < k — 1 using:

= 0, m > 0 (29);=o •' *-

For convenience, the radial functions given by Wendland [7]. for the

obvious cases of d = 1, ...5. are reproduced in Table 1. It can be seen

from Table 1 that only odd d's are tabulated. This due to the fact the

construction of functions for an even space d = 2ra using equation (28)

leads to the same functions for an odd space d = 2n + 1. Therefore,

for d — 2n any function positive definite for 5<^2n+i ^ri be used.

The entries for the space dimension d in Table 1 give the maximum

possible space dimension where the basis function is positive definite.The functions given for a space d are also optimal for the space k <

d. Since the diffusion equation involves second-order derivatives, a

minimum of C^ is required for d>.

When using compactly support radial basis functions, the matrices

involved are sparse. This bypass the previously mentioned problems



related to global radial basis functions. Therefore, the systems can

be efficiently inverted using an iterative method instead of using a

direct solver. In this case, an iterative solver based on the Biconjugate

Gradient Method (BGM) is used to solve the sparse systems [8]. This

results in further reduction in the overall cpu-time of the scheme.

Table 1: Compactly supported radial basis functions for d = 1, ..., 5

3)

4 Numerical example

For simplicity, the boundary element scheme using (9), with {p™}computed in the normal convolution- type integration, which con-tains the original time-history, will be referred to as DBEM (Direct

Boundary Element Method) while the scheme where the computa-

tion of {p™} is performed using (10) will be referred to as MQBEM

(Multiquadrics Boundary Element Method), TPSBEM (Thin Plate

Splines Boundary Element Method) and CSBEM (Compactly Sup-

ported RBFs Boundary Element Method! when multiquardics, thin

plate splines and compactly supported radial basis functions are used

for yp, respectively. For validation, a simple one-dimensional diffusion

problem is considered. However, the conclusions reached for this sim-ple case is expected to remain valid for other more complicated prob-

lems in higher dimensions. Let us consider the following boundaryvalue problem governed by the following equations:

(30)



tz(0,i) = u(l,t) = 0 , 4>o (31)

** , 0 < x < 1/2, (32)

The analytical solution is given by [9]:

. OO s / '

«(*> *) = S E I £ »* ( IT»=i ^ V ^

Table 2 shows a comparison between the different schemes using

AT = 21 points. For CSBEM, the compact support 0.45 < c(x) <

0.65, where c(x) = 0.2z + 0.45 for x < 1/2. and c(x) = -0.2o; + 0.65

for x > 1/2. It can be seen that all schemes have a good agree-

ment with the analytical solution. It can also be seen that overall the

DBEM is more accurate. However, the results obtained by MQBEM,

TPSBEM and CSBEM are also accurate by any standard of engineer-

ing applications (up to three digits), but with a substantial gain in

cpu-time. For instance, the speedup factor (ratio of cpu-time DBEM

over cpu-time of present methods) at any time step k is in the order

of O(k)j while it is in the order of O(k/2) when taking into account

the accumulated gains from the initial time [lj.As TV increases, the gain in cpu-time using iterative, instead of di-

rect, solvers became substantial and CSBEM outperforms TPSBEM

and MQBEM from the cpu-time point of view, yet still producing

relatively comparable results. The cpu-time gain using BGM instead

of Gaussian Elimination is in the order of O(N) — O(N ), depend-

ing on the degree of sparsity, which increases or decreases as the

size of the compact support decreases or increases, respectively [2].

This allow the possibility of using a large number of points without

any substantial penalty on the cpu-time for CSBEM. The size of the

compact support on the overall accuracy of the scheme was also in-

vestigated, where, as expected, the accuracy increases as c(x) — > 1[2]. However, as c(x) — * 1 the matrices become fully populated again

and therefore a compromise between a relatively fair sparsity and anacceptable accuracy has to be adopted. Furthermore, tests were car-

ried out with variable, adaptive compact support cix), and the results

suggest that the accuracy can be tuned accordingly, by varying the

compact support with sharp field variations or increased geometrical

complexity [2].



5 Conclusions

The numerical results clearly demonstrate that despite the fact that

the direct boundary element method is in general more accurate, the

proposed schemes achieve a relatively comparable accuracy but with

huge savings in the computational cost. Unlike the direct method,

the present schemes removes the time-history dependence and allows

the solution at any time step to be computed with a relatively con-

stant cpu-time. This put the schemes on the same level playing field

with finite-differences and finite elements methods when it comes to

computational cost. The local nature of compactly supported radial

basis functions, gives the scheme some flexibility for heterogeneous

mediums and results in sparse matrices, which allow the problem

to be solved using a sufficiently large number without a substantial

penalty on cpu-time. Adaptive points density and compact support

can also be easily incorporated to deal with regional variations in

the smoothness of the solution and geometrical complexities. Fur-

thermore, unlike element based integration, the radial-basis function

integration scheme is remarkably simple, especially in higher dimen-sions.

Acknowledgements: This research project is supported by the De-

partment of Trade and Industry of the United Kingdom, and forms

o/ ZAe acfzcm COST- 2 o/ ZAe European,

References

[l] M. Zerroukat, A fast boundary element algorithm for time-

dependent potential problems, Appl. Math. Modelling, (in press).

[2] M. Zerroukat, A boundary element scheme for diffusion problems

using compactly supported radial basis functions, Submitted to:

Engineering Analysis with Boundary Elements.

[3] L.C. Wrobel and C.A. Brebbia, Boundary Element Methods in

Heat Transfer, Computational Mechanics Publications & ElsevierApplied Science. Southampton, 1992.

[4] R.L. Hardy, Multiquadric equations of topography and other ir-

regular surfaces, J. Geophysical Research 176. 1905-1915 1971).



[5] N. Dyn, D. Levin and S. Rippa, Numerical procedures for global

surface fitting of scattered data by radial functions, SIAM J. Sci.

Statist. Comput. 7, 639-659 (1986).

[6] M.R. Dubai, Domain decomposition and local refinement for mul-

tiquadric approximations, J. Appl. Sci. Comp. 1, 146-171 (1994).

[7] H. Wendland, Error estimates for interpolation by compactly

supported radial basis functions of minimal degree, to ap-

pear in: Journal of Approximation Theory, Reprints in:

http://www.num.math.uni-goettingen.de/wendland/papers.html

[8] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery,

Numerical Recipes in Fortran (2nd ed.), Cambridge University

Press, Cambridge, 1992.

[9] H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, 2nd

edn, Clarendon Press, Oxford, 1959.



Table 2: Comparison of DEEM, MQBEM, TPSBEM, CSBEiM and

X

t =

0

0

0

0

0

t =

0

0

0

0

0

t-

0

0

0

0

0

t =

0.

0.

0.

0.

0.

t =

0.0.

0.

0.

0.

corre

solut"(t.\ —

= 0.10

= 0.2

.09673

.09746

.09791

.09580

.09676

= 0.4

08343

08432

08500

08197

08335

= 0.6

06928

06974

07046

06722

06910

= 0.8

05705

05726

05798

05476

05683

= 1.0

04690

04697

04766

04458

04667

or every !

•spend to

ions, res;I 1 'i V

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0.

0.

0.

0.0.

0.

0.

0.

0.

0.

0.

', 1D

>ec

0.25

.22886

.23265

.23269

.22935

.22977

19137

19374

19435

18968

19219

15780

15904

15991

15475

15836

12974

13034

13132

12598

13009

10661

10684

10787

10253

10681

the first

BEM, N

tively. 'lit .

0

0

0

0

0

0

0

0

0

0

0.

0.

0.

0.

0.

0.

0.0.

0.

0.

0.0.

0.

0.

0.

se

IQ]Fhe(rr.

0.40

.31889

32719

32681

32031

32088

25901

26300

26324

25782

26048

21224

21428

21495

20941

21333

17427

17527

17619

17038

17503

14316

14357

14463

13867

14366

cond, thi

BEM, TT

average-f- \ /* , _ _

0

0

0

0

0

0

0

0

0

0

0.

0.

0.

0.

0.

0.

0.

0.0.

0.

0.

0.

0.

0.

0.

rd

>s

ati

0.55

33317

34145

34110

33426

33546

26922

27290

27313

26799

27088

22039

22215

22280

21760

22161

18094

18168

18260

17707

18178

14864

14883

14989

14413

14920

, fourth

BEM, C

nsolute (~. +M .

0

0

0

0

0

0

0

0

0

0

0

0.

0.

0.

0.

0.

0.

0.0.

0.

0.

0.

0.

0.

0.

an

SB

;rrc,.u.

0.7

.26500

26850

26841

26546

26668

21923

22092

22134

21760

22054

18040

18104

18172

17742

18129

14829

14540

1492214449

14586

12135

12173

12262

11763

12220

1

3

2

0

0

1

1

2

0.

0.

0.

3.

0.

0.

0.3.

0.0.

0.

3.

d the fifth re

EM and An,

>r £• t) is defi

) x 10^

24252

03109

96790

83388

92126

33258

56387

21612

65801

60852

99974

11268

45511

308367470658451

31084

25876

67341

77181

>w of dat;

alytical

ined asT__ C

Analytically and Numerically respectively; the boundary solutions are not

included in ~e(t) since they axe given.


approximation, see zerroukat [l]-[2] for details. · 2014. 5. 13. · 190 boundary elements...

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