Computers & Fluids 86 (2013) 37–44

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Computers & Fluids

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Solving moving-boundary problems with the wavelet adaptive radialbasis functions method

0045-7930/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compfluid.2013.06.029

⇑ Corresponding author. Tel.: +386 1 472 11 43.E-mail address: leopold.vrankar@gmail.com (L. Vrankar).

Leopold Vrankar a,⇑, Nicolas Ali Libre b, Leevan Ling b, Goran Turk c, Franc Runovc a

a University of Ljubljana, Faculty of Natural Sciences and Engineering, Aškerceva cesta 12, Ljubljana, Sloveniab Department of Mathematics, Hong Kong Baptist University, Hong Kongc University of Ljubljana, Faculty of Civil and Geodetic Engineering, Jamova cesta 2, Ljubljana, Slovenia

a r t i c l e i n f o

Article history:Received 29 April 2012Received in revised form 18 June 2013Accepted 27 June 2013Available online 11 July 2013

Keywords:Moving-boundary problemsWavelet methodLevel set methodGlobal RBFsCompactly supported RBFsPartial differential equationsAdaptive greedy algorithm

a b s t r a c t

Moving boundaries are associated with the time-dependent problems where the momentary position ofboundaries needs to be determined as a function of time. The level set method has become an effectivetool for tracking, modeling and simulating the motion of free boundaries in fluid mechanics, computeranimation and image processing. This work extends our earlier work on solving moving boundary prob-lems with adaptive meshless methods. In particular, the objective of this paper is to investigate numericalperformance the radial basis functions (RBFs) methods, with compactly supported basis and with globalbasis, coupled with a wavelet node refinement technique and a greedy trial space selection technique.Numerical simulations are provided to verify the effectiveness and robustness of RBFs methods with dif-ferent adaptive techniques.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

One common feature of moving-boundary problems is the factthat the location of the solid–liquid interface is not known in ad-vance and has to be determined during the analysis. Movingboundary problems can be considered as time-dependent problemwhere the position of the boundary has to be determined as a func-tion of time.

Various numerical methods are known to solve the moving-boundary problems, e.g. front-tracking, front-fixing, and fixed-domain methods [1]. The Level Set Methods (LSM) have becomean effective tool for simulating the motion of free boundaries influid mechanics and other similar processes [2–7]. The governingequation of LSM are usually solved by conventional numericalmethods, e.g. the finite-difference methods and the finite-elementmethod [8,9]. More and more attention has been given to thedevelopment of meshless methods using radial basis functions(RBFs) for the numerical solution of PDEs [10]. Recently, the RBFbased meshless method has been successfully used for the numer-ical solution of moving boundary problems [11].

Adaptive techniques could be used to improve the efficiency ofnumerical methods especially in LSM moving boundary problems

in which the level set function should be accurately captured onlynear the zero contour. The computational resolution should beadaptively refined near the boundaries and could be coarsenedelsewhere. Any adaptive technique that is employed in movingboundary problems should be fast enough as the boundary evolveswith time and the location of the zero contour has to be updated atcertain time steps. In recent years, wavelet methods have beenused for signal processing and mathematical analysis. In this paper,we propose a wavelet-based adaptive scheme that utilizes thewell-known RBFs (e.g. the multiquadrics and the Wendland’s com-pactly supported basis) as the basis of the solution space.

Implementing adaptivity in RBF methods for time dependentproblems is not new. Behrens and Iske [12] used RBF (though notas a full solver) in their grid free adaptive semi-Lagrangian schemefor a transport problem. Sarra [13] utilized equidistribution tech-nique to concentrate RBF nodes in regions that needed to be re-fined. Driscoll and Heryudono [14] used an algorithm calledadaptive residual subsampling to automatically track the movingfront and refine/coarsen RBF points. The variation of their adaptivemethod based on RBF in finite difference mode is available inHeryudono’s thesis [15]. None of the methods above utilizedwavelet style refinement techniques. Hence, our work is new andis a viable alternative to available methods.

The objective of this paper is to improve the resolutions, henceaccuracy, of the meshless RBF method with some adaptive wave-let-greedy techniques to construct a more efficient approach for

38 L. Vrankar et al. / Computers & Fluids 86 (2013) 37–44

moving boundary problems.Numerical simulations are performedto verify the capability of the proposed numerical scheme in themoving boundary problems.

2. Governing equations and discretization

The level set method (LSM) is a numerical technique for track-ing different types of interfaces. The advantage of the LSM is thatnumerical computations involving curves and surfaces can be exe-cuted on fixed Cartesian grid. Therefore, the parametrization of theobject is not needed [16].

In two dimensions, the LSM represents a closed curve in theplane as the zero value of the two-dimensional auxiliary functionU(x, t), mapping R2 � R! R, where x is a position of the interfaceand t is a moment in time. The closed curve is presented as:

C ¼ fðx; tÞjUðx; tÞ ¼ 0g: ð1Þ

The function U is termed a level set function and it is assumed totake positive values inside the region delimited by the curve Cand to take negative values elsewhere [2,3]. The level set functionis defined as a signed distance function from the interface. The mov-ing interface at given time t is determined by locating the set of C(t)for which U vanishes. The movement of the level set function is de-scribed as

@U@tþ vT

OU ¼ 0; Uðx;0Þ ¼ U0ðxÞ; ð2Þ

where U0(x) represents the initial position of the interface andvT = [m1,m2] is the continuous field depending on position x.

Suppose the time derivative is discretized by an implicit schemethen we have

Unþ1 �Un

Mtþ m1

@Unþ1

@xþ m2

@Unþ1

@y¼ 0; ð3Þ

where tk = kMt Uk = Uk(x) is the level set variable at time tk

(k = 0, 1, 2, . . .). For spatial discretization, the function Uk is approx-imated by a linear combination of radial basis functions:

UkðxÞ ¼XN

j¼1

akj ujðxÞ ¼

XN

j¼1

akj uðjjx� njjjÞ; ð4Þ

where uj(x) :¼ u(——x � nj——) is any radial basis function whosevalue depends only on the distance between x and a fixed set ofcenter points N = {n1, . . . , nN} specifying the centers of RBFs andaj(tk) is the weight of the radial basis function positioned at thejth center. Note that the exact positions of centers N are totally arbi-trary. In this paper, the centers will be adaptively placed so thattheir density is higher near the zero contour of U. On these centers,two commonly used radial basis functions, namely the globally sup-ported multiquadrics uðrÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ c2p

and a Wendland’s compactlysupported basis [17] uðrÞ ¼ ð1� rÞ8þð32r3 þ 25r2 þ 8r þ 1Þ, will beused as trial functions.

Now, the initial set of weights aj(t0) is obtained from the giveninitial conditions. Thereafter, if we impose collocation conditionalso at the set of centers N, it is straightforward to obtain the fol-lowing linear system of equations

XN

j¼1

ujðniÞMt

þ m1@ujðniÞ@x

þ m2@ujðniÞ@y

� �akþ1

j ¼UkðnjÞMt

i ¼ 1; . . . ;N:

ð5Þ

which allows us to update fakj g to fakþ1

j g (k = 0,1,2,. . .). The pre-sented material to this point is standard; readers can find out moreimplementation details in [11].

3. Adaptive placement of ‘‘centers’’

As mentioned, the zero contour of U is what we are interestedin. The first adaptive technique is wavelet-based technique thatmake our RBF centers N, which are also taken as the set of colloca-tion points, with higher density around the zero contour. Intui-tively, more collocations or data means more information thatwill allow us to better capture the details of the contour curve.

In recent years, some attempts have been made to relate theRBFs to wavelets. The concept of wavelet analysis was introducedin applied mathematics by the end of the 1980s by Daubechies[18]. Since then interest has grown rapidly in developing waveletapplications in engineering and science. Recently, the waveletanalysis has been used in RBF collocation method to adaptively re-fine mesh in regions with localized features [19–21]. The advan-tage of the adaptive wavelet technique in comparison to theconventional techniques is that the wavelet coefficients whichare used to detect regions with localized features are simply com-puted by the fast discrete wavelet transform. Therefore, the wave-let based adaptive technique is very fast which is of greatimportance in time dependent problems in which the mesh shouldbe repeatedly refined at certain time steps. In contrast, other auto-matic adaptive techniques are usually based upon the posterior er-ror indicator in which the computation of the posterior indicatoroften dramatically decreases calculation efficiency.

A short review of the multi-resolution wavelet analysis (MRMA)is given in [19–21]: the mathematical foundation of the adaptivewavelet algorithm is multi-resolution wavelet analysis. The MRWAprojects a complicated function into a nested sequence of approx-imation subspaces {Vj+1}, Vj � Vj+1 and establishes a set of scalingfunction coefficients ajk and a set of wavelet coefficients djk, struc-tured over different levels of resolution. Each of subspaces can bedecomposed into an approximation space {Vj} and its orthogonalcomplement detail space {Wj}. The space L2 can be expanded asan approximation space plus a sum of detail spaces, i.e.L2 ¼ Vj¼j0 þ

Pj¼j0Wj. The bases of approximation and detail sub-

spaces are constructed with scaled and translated versions of func-tions /j,k and wj,k as respectively

fVjg ¼ spanf/j;k ¼ 2j=2/ð2jx� kÞg ð6ÞfWjg ¼ spanfwj;k ¼ 2j=2wð2jx� kÞg; ð7Þ

Using Vj � Vj+1 and Wj � Vj+1 we obtain

/ðxÞ ¼X

k

hk/ð2x� kÞ ð8Þ

wðxÞ ¼X

k

gk/ð2x� kÞ: ð9Þ

The sequences are respectively low-pass and high-pass filters. Thewavelet decomposition of f(x) into fj 2 Vj and gj 2Wj is

f ðxÞ ¼X

k

cj;k/j0;k þXj¼j0

Xk

wj;k: ð10Þ

An important point of the wavelet decomposition is the fast discretewavelet transform (DWT) [22] which provides a simple method oftransforming data from one level of resolution j to the coarser levelof resolution j � 1

cj�1;k ¼X

l

h2k�lcj;l ð11Þ

jj�1;k ¼X

l

g2k�ljj;l: ð12Þ

The DWT avoids the unpleasant calculation of the coefficients, c andj, which allows the wavelet coefficients to be efficiently calculatedusing fast numerical convolutions. The scaling function coefficientscj,k present a smoothed version of the function at the current scale,

L. Vrankar et al. / Computers & Fluids 86 (2013) 37–44 39

but the wavelet coefficients jj,k present the irregular behavior of thefunction (e.g. between the current scale and the next finer scale). Atthe corresponding level of resolution, the wavelet coefficients are ameasurement of the approximation accuracy. The basic idea of thewavelet scheme is to present a function with fewer degrees of free-dom while still obtaining an accurate approximation. At any level,the function under analysis is written as a sum

f ðxÞ ¼ f 1ðxÞ þ f 2ðxÞ; ð13Þ

where

f 1ðxÞ ¼X

k

cj0;k/j0;k þXj¼j0

Xk

jj;kwj;k for jj;k P g

f 2ðxÞ ¼Xj¼j0

Xk

jj;kwj;k for jj;k < g: ð14Þ

The scaling and wavelet coefficients could be obtained from the fol-lowing equation:

cj;k ¼ ðf ;/j;kÞ;jj;k ¼ ðf ;wj;kÞ: ð15Þ

For a smooth function f(x), the error is bounded by a prescribedthreshold Cg as —f(x) � f1(x)— 6 Cg, where the parameter C dependson f(x). Further details about the RBF adaptive wavelet method are

0

0

0

0

0

0

0

0

0

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t=2

Fig. 1. A schematic example of (Left) non-uniform node distribution resulted from the w

0

0

0

0

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t=2

Fig. 2. Shape of the interface calculated with

given in [19,20]. The extension of the adaptive wavelet method overirregular domain, that is not necessary for our work here, could befound in [21].

The following schematic RBF centers distribution aims to pro-vide readers some insights about the situation we are dealing with(Fig. 1).

It is well known that performance of global RBFs method is verysensitive to both the distribution of points and the selection of RBFshape parameter. In the coming section, we will overview an adap-tive technique that allows the global RBFs method to work, for anyshape parameter, by selecting a proper trial or solution sub-space.

4. Adaptive-globally versus compactly supported RBFs

As seen in Fig. 1, the RBF centers returned from adaptivetechnique we described just now are highly non-uniform. Suchnon-unform centers used to bring a lot of difficulties to the glo-bal RBF method in which users must determine an appropriateshape parameter. Our answer to this problem is to introduce an-other level of adaptivity by coupling an adaptive greedy algo-rithm and wavelet analysis techniques. The proposed methodwhich includes the greedy and wavelet adaptive algorithm is de-scribed in this section. Another solution to the non-unform cen-ters is to employ local basis instead of global. Although the

0 0.2 0.4 0.6 0.8 1

0

.1

.2

.3

.4

.5

.6

.7

.8

.9

1

Ncol cen

=177

avelet adaptive technique and (right) selected RBF centers by the greedy algorithm.

0 0.2 0.4 0.6 0.8 10

.2

.4

.6

.8

1N=849

the wavelet adaptive CSRBFs method.

0 100 200 300 400 500 600 7000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Fig. 3. Error of the area at different points in time. Time step is 0.01.

40 L. Vrankar et al. / Computers & Fluids 86 (2013) 37–44

shape parameter of CSRBF is still arbitrary, it is commonly ac-cepted fact that this is much less of a problem in comparisonto the problem seen in its global counterpart. The numericalmethods used in this paper were utilized for the solution ofLSM problems either with adaptive MQ RBF or with the CSRBFs.Various examples will then be provided to compare the perfor-mances of the choices of basis.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t=0.01

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t=2

Fig. 4. Shape of the bubble in

The numerical stability of the global RBF method highly de-pends on the optimal selection of its centers and the correspondingshape parameters. Intuitively, such optimal locations will dependon many factors: the PDE and its domain, the RBF basis used, thecomputational precisions, some user defined parameters, and soon. This makes finding optimal RBF centers and the correspondingshape parameters a very difficult task.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1N=698

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1N=1805

a single-vortex flow field.

Fig. 5. Shape of the bubble in an extreme situation.

L. Vrankar et al. / Computers & Fluids 86 (2013) 37–44 41

The adaptive greedy algorithm in which the system of equa-tions are partially solved by a series of sub-optimal adaptive gree-dy algorithms could be used for the selection of nearly optimalsubset of RBF centers [23,25,24]. The idea is to set the N � N matrixsystem by chosing N RBF centers and N collocation points. Next,the adaptive algorithm is employed to select an optimal subsetof Np RBF centers. The final approximation is given by the least-squares solution resulting in selection of Np RBF centers and N col-location points. Some convergence results are given in [26] and re-cent applications of the method can be found in [27–30]. Theselected row and columns are added to the previously selected ma-trix and the new approximation ak+1 is obtained by solving the sys-tem associated with k + 1 collocation points and k + 1 RBF centers.

Be reminded that the adaptive wavelet techniques produce adense mesh near the boundaries of moving boundary problemswhile keeps a relatively coarse mesh elsewhere. The very finemesh near the boundaries will result in a highly ill conditionedsystem of equations in which the stable solution is an importantissue. On the other side the greedy algorithm starts from finelydistributed set of RBF centers and selects a small and nearlyoptimal subset of it. The greedy adaptive method guaranties thatthe condition number of resulted system of equations is lowerthan a prescribed value that ensures stable solution of the sys-tem. The first step of adaptive wavelet-greedy technique is to

produce a finely distributed RBF centers near the boundariesusing the adaptive wavelet procedure described in Section 3.The next step is to utilize the adaptive greedy technique, de-scribed in Section 4, to select nearly optimal RBF centers fromthe adaptive mesh produced in previous step. The overall proce-dure of updated RBF collocation method with adaptive wavelet-greedy technique is summarized in Algorithm 1.

Algorithm 1. Wavelet node-refinement with greedy trial spaceselection

Input data:Type of RBFs (MQs or CSRBFs, Section 2)Shape parameter for MQ RBF (Section 2)Pointer to the greedy (wavelet) methodInitial and boundary conditionsType of velocity fields (see examples in Section 5)

Initialization:Generate the base collocation computational grid (e.g.25 � 25)Define the zero contour (as circular bubble, see examples inSection 5)

(continued on next page)

42 L. Vrankar et al. / Computers & Fluids 86 (2013) 37–44

if the pointer to wavelet method > 0 thenApply the adaptive wavelet method

end ifIteration:while t 6 Tfinal do

Compute the new zero contour (Section 2)if the pointer to the wavelet (greedy) method > 0 then

Apply the wavelet (greedy) procedure (Section 3 andSection 4)end ifif the initial contour area < the real contour area then

Apply the reinitialization [16]end ifSolve the problem and calculate the unknown coefficients(Section 2)Setting time t :¼ t + Dt

end while

5. Numerical results

In the simulations we used the MQ RBF basis uðrÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ c2p

with shape parameter c and compactly supported RBF spline

0

0

0

0

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t=0

0

0

0

0

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1t=0.5

Fig. 6. Shape of the bubble in

uðrÞ ¼ ð1� rÞmþpðrÞ, where p(r) is a polynomial of the Wendland[17] compactly supported (CSRBF) spline:

ð1� rÞmþ ¼pðrÞ; if 0 6 r < 1;

0; if r P 1:

�

5.1. Oriented flow

The first example is a translation of circular interface in ori-ented flow. The circular bubble of radius r = 0.15, initially cen-tered at (0.3,0.3), moves by the orientated flow in acomputational domain of size 1 � 1 with the velocity field(u,v) defined as follows:

u ¼ 0:2ðxþ 0:5Þ; ð16Þv ¼ 0:2ðyþ 0:5Þ: ð17Þ

Results obtained with the wavelet adaptive RBFs method andgreedy algorithm’s (see Fig. 1 and Fig. 2) are compared with resultsobtained in [11]. The results are comparable.

In Fig. 3, the normalized errors [11] of the area as a function oftime are presented. The error of the area is calculated for the wave-let adaptive CSRBFs method and the MQ RBFs method and greedyalgorithm combination.

0 0.2 0.4 0.6 0.8 10

.2

.4

.6

.8

1N=655

0 0.2 0.4 0.6 0.8 10

.2

.4

.6

.8

1N=1419

an advection flow field.

L. Vrankar et al. / Computers & Fluids 86 (2013) 37–44 43

5.2. Bubble in a single-vortex flow field

The following examples were also presented in article [31].First, we consider a circular bubble of radius r = 0.15, initially cen-tered at (0.5,0.75), advecting in a steady, non-uniform vorticityfield in a computational domain of size 1 � 1 with the velocity field(u,v) defined as follows:

u ¼ � sin2ðpxÞ sinð2pyÞ; ð18Þv ¼ sin2ðpyÞ sinð2pxÞ: ð19Þ

This velocity field possesses vortical flows which deform andpossibly tear any interface carried with the flow. In Fig. 4, we cansee that the resulting vortex field spins fluid elements, stretchingthe bubble into long and thin body which spirals around the centerof the domain.

Second, we consider a circular bubble of radius r = 0.25, initiallycentered at (0.5,0.75) in a computational domain of size 1 � 1 withthe velocity field (u,v) defined as follows:

u ¼ sinð4pðxþ 1=2ÞÞ sinð4pðyþ 1=2ÞÞ; ð20Þv ¼ cosð4pðxþ 1=2ÞÞ cosð4pðyþ 1=2ÞÞ: ð21Þ

The velocity field has the ability to force the bubble to undergo ex-treme topological changes. The results are presented in Fig. 5.

5.3. Bubble in an advection flow field

Many engineering processes include phase transitions. Com-puter simulations provide a useful tool for studying such transi-tions. We consider a circular bubble of radius r = 0.15, initiallycentered at (0.4,0.56), moved by a shear flow in a computationaldomain of size 1 � 1 with the velocity field (u,v) defined as follows:

u ¼ sinð2pxÞ sinð2pyÞ; ð22Þv ¼ � cosð2pyÞ sinð2pxÞ: ð23Þ

The function U is also evaluated on a 49 � 49 grid in order to findthe zero contours. The results are presented in Fig. 6.

6. Conclusions

This paper outlines the continuation of work done on an alter-native approach to the conventional level set methods for solvingtwo-dimensional moving-boundary problems. In this case, thewavelet adaptive RBFs method was added to the earlier approach,which was a combination of MQ RBFs and the adaptive greedyalgorithm.

More examples are presented, in particular oriented flow, bub-ble in a single-vortex flow field and bubble in a advection flowfield.

In the conventional level set methods, the level set equation issolved to evolve the interface using a capturing Eulerian approach.The solution procedure requires the appropriate choice of the up-wind schemes, reinitialization algorithms and extension velocitymethods, which may require excessive amount of computationalefforts. In our case, we did some reinitialization in some time stepsbut the number of required reinitialization has decreased afterapplying the proposed adaptive techniques. Actually, our examplesshow that the solutions are much more stable (less artifacts appearin the corners) if the new method (LSM + CSRBF + adaptive wave-let) is employed. The time step was 0.01. We can also conclude thatthere is instability as the time becomes larger, but the simulationshave shown that the instability is less pronounced if the adaptivewavelet CSRBF method was used. This has very beneficial influenceon the numerical effectiveness of the method.

Figs. 2 and 4 show that we can expect more accurate results ifthe wavelet adaptive RBF method is used. We can clearly empha-size that the greedy-adaptive wavelet method is much more suit-able for complicated problems (see Figs. 5 and 6). In the futurework, in the case of greedy algorithms, we are going to check thepossible influence of row-weighting for non-uniform grids, but inthe case of the CSRBF adaptive method, it is necessary to check ifit is possible to implement the method without the adaptive sup-port. We can also conclude that the greedy is developed to workon well-spread but scattered data. Our non-uniform centers re-quires weighting of some sort.

Acknowledgements

This work has been financed by the Slovenian Research Agency(ARRS) through the research program Geotechnology (P0-0268),and Structural mechanics (P2-0260). The authors wish to thankthe anonymous reviewers for their valuable comments that helpedto improve the manuscript.

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