An improved partition of unity finite element model for diffraction problems

$: An improved partition of unity finite element model for diffraction problems$
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2001; 50:2727–2740

An improved partition of unity �nite element model fordi�raction problems

P. Ortiz∗;† and E. Sanchez

CEDEX; Antonio L�opez 81; 28026 Madrid; Spain

SUMMARY

The partition of unity �nite element method (PUFEM) is explored and improved to deal with practicaldi�raction problems e�ciently. The use of plane waves as an external function space allows an e�cientimplementation of an approximate exterior non-re ective boundary condition, improving the originalproposed by Higdon for general di�raction problems. A ‘virtually’ analytical integration procedure isintroduced for multi-dimensional high-frequency problems which exhibits a dramatic decrease in thenumber of operations for a given error compared with standard integration methods. Suitable conjugategradient type solvers for the whole range of wavenumbers are used, including such cases in whichPUFEM can produce nearly singular matrices caused by ‘round-o�’ limits. Copyright ? 2001 JohnWiley & Sons, Ltd.

KEY WORDS: partition of unity; short waves; �nite elements

1. INTRODUCTION

The numerical solution of the Helmholtz equation by the ‘standard’ �nite element method(FEM) yields reliable results and a moderate number of degrees of freedom for lowand medium frequencies when the typical ‘resolution’ rule is ful�lled (see, for instance,References [1; 2]). For practical purposes this rule (basically representing the number ofnodes per wavelength) can be assumed as a constant number greater than 10 when linearshape functions are used.However, as the wave number increases, the error in the numerical solution can grow in-

�nitely even if the ‘resolution’ constraint is maintained. The cause of this is the ‘pollution er-ror’, not controlled by a constant resolution because of its dependence on the wave length [3].Thus, accurate FEM results for high wave number problems can only be reached with

meshes and approximation orders in such a way that both ‘resolution’ and ‘pollution’constraints are ful�lled. For practical implementations these conditions produce expensive

∗Correspondence to: Pablo Ortiz, CEDEX, Antonio L�opez 81, 28026 Madrid, Spain†E-mail: [email protected]

Received 4 February 2000Copyright ? 2001 John Wiley & Sons, Ltd. Revised 29 June 2000

2728 P. ORTIZ AND E. SANCHEZ

computations, and some attempts have been proposed in recent years to circumvent theserestrictions.The incorporation of some previous known (or approximately known) behaviour of the

solution is a rational way to improve the performance of standard �nite element models. Thisidea was �rst introduced by Bettess and Zienkiewicz [1], by including the harmonic variationof the solution into the shape functions (for the modelling of unbounded domains by in�niteelements). Later, improvements were also proposed for �nite elements like, for instance, inReferences [4; 5] and [6]. Bettess and Chadwick [4] introduced a method based on the Astleyprocedure for the weighted functions.The PUFEM, proposed as a new approach to the Helmholtz equation by Melenk [7], shows

an outstanding improvement of the approximation properties of the standard FEM when ‘some’previous knowledge of the behaviour of the solution is available.The method can be classi�ed in the class of ‘meshless’ methods (see a review in Reference

[8]) while a general formulation, including PUFEM, was recently introduced by Duarte andOden [9].In the PUFEM the �nite element space is constructed by multiplying the standard shape

functions (which form a partition of unity) with a space of functions having good localapproximation properties (like particular solutions of the di�erential equation). This procedureenforces the global conformity of the �nite element space with no restrictions on the localapproximation functions space, and includes ‘a priori’ information about the local behaviourof the solution.The approximation function space is constructed by plane waves travelling in prescribed di-

rections. The choice of these plane waves is critical for the e�ciency of the whole procedure,because of the fact that the PUFEM eliminates the ‘numerical pollution’ only if the chosenfunctions are ‘close enough’ to the correct solution. This situation is reached by picking outthe proper set of functions (which implies, of course, a certain knowledge of the solution)or, obviously, by the inclusion of a number of di�erent directions large enough to reach this‘asymptotic’ range. This last strategy, adopted in the original PUFEM tests for the Helmholtzequation [7] is the obvious option when no information about the solution is available‘a priori’.In this work the selected plane waves are exploited also to obtain a simple and e�cient

exterior condition. This is carried out by approximating the radiation boundary condition atin�nity by means of the Higdon boundary operator (HBC) at the external boundary. But herethe absorbing directions of the Higdon condition are selected equal to the directions of theexternal plane waves used in PUFEM (PF-HBC). This angle selection is shown to be validfor the range of wave numbers considered in the numerical test.A generalized conjugate gradient solver [10] is extended here to the PUFEM. Practical

implementations have shown that for medium and low wave numbers ‘round-o�’ limits canproduce nearly singular coe�cient matrices. A procedure is proposed here, transforming theoriginal matrix to a Hermitian positive de�nite matrix with a minimum bound for the lowereigenvalue bigger than the ‘round-o�’ orders of magnitude for the extreme cases when thisis required.In addition, a proper integration method was explored. The oscillatory behaviour of the

integrands makes the typical procedures, for instance, Gaussian quadrature, inadequate formedium and high wave numbers. For these ranges an exact integration for oscillatory inte-grands is adapted. A local coordinate rotation is introduced in such a way that the oscillatory

Copyright ? 2001 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2001; 50:2727–2740

AN IMPROVED PARTITION OF UNITY FINITE ELEMENT MODEL 2729

part of the integrand depends only on one direction, with the resultant saving on the numberof operations.This paper is organized as follows. The next section is focused on the de�nition of PUFEM

and the approximation function space proposed, as well as the procedure proposed for thearti�cial boundary. The following sections are devoted to the description of the numericalintegration procedure, the solver methodology and to the numerical experiments carried out. Arepresentative real-test case was chosen with the study of the di�raction around a circular pile,without considering the ‘a priori’ knowledge of the behaviour of the solution in order to testseverely the performance of the model for real di�raction problems. Some �nal considerationsand further works to be done conclude this paper.

2. NUMERICAL SOLUTION

The scattering and di�raction of monochromatic waves is governed by the Helmholtz equation

��+ k2�=0 on (1)

where � is the velocity potential and k the wavenumber.The boundary conditions corresponding to physical boundaries, which can be totally or

partially absorbent �a, and totally re ective �r are, respectively,

@�@n+ ik��=0 on �a (2)

and

@�@n=0 on �r (3)

with � the absorption coe�cient. At in�nity, the radiation condition should be added. Awell-known form of it was given by Sommerfeld [11].In order to deal with a numerical solution of the unbounded problem, an arti�cial boundary

�I is required where a condition must be imposed. Several options were proposed (see e.g.Givoli [12]). The multi-dimensional extensions of the exact one-dimensional form of radiationhave given good practical results, like for instance in long-wave resonance problems [2], butin general, a less re ective form is needed and is developed in Section 3.The numerical solution of the Helmholtz equation by the PUFEM is de�ned as follows.

The approximated potential �̃ is constructed as

�̃=’iqlbli (4)

where ’i are functions which form a partition of unity. Then the condition∑i’i=1

is ful�lled for all the points of the domain. (The well-known �nite element shape functionsusually form a partition of unity.) Functions ql are called ‘external’ functions and shouldhave good local approximation properties for an e�cient numerical approach of the solution.



A reasonable choice for this space is the use of particular solutions of the di�erential equation.Finally, bli are the parameters associated with points and with the functions of the externalapproximation space.By the application of the standard Galerkin method the �nal form can be established as

a(v; �̃)= a

(’jqm; ’iqlbli

)=0 (5)

where

a(v; �)=∫∇v∇� d− k2

∫v� d−

∫�v@�@nd�

and (i; j=1; N ); (l; m=1; M), where N represents the total number of points, M the totalnumber of external functions chosen and v the weighting functions.Two main spaces were proposed in the original PUFEM for the local functions ql: gener-

alized harmonic polynomials for the Helmholtz equation and plane waves. Here plane waveswere adopted because of its simplicity of implementation. Then the external functions arewritten for bidimensional problems as

ql= exp{ik(x cos �l + y sin �l

)}(6)

representing a plane wave propagating in the direction given by the angle �l. These wavescan be de�ned locally (at an element e) or globally.The original method suggests the use of a constant number of functions associated with

every point. Here a local and variable number of external plane waves is proposed.

3. ARTIFICIAL BOUNDARY CONDITION

The need for arti�cial boundary conditions arises when the spatial domain of the problem athand is unbounded. In this case, a numerical treatment usually requires the introduction of anarti�cial boundary �I in order to make the computational domain �nite. The radiation conditionat in�nity for the outgoing waves is given, for example, by the well-known Sommer�eldcondition

limr→∞ r(d−1)=2

(@�s

@r− ik�s

)=0 (7)

where r is the radial coordinate, �s represents the scattered part of the total solution �=�I+�s

and d is the spatial dimension of the problem. The simplest approximation of this conditionat the arti�cial boundary is

@�s

@n+ ik�s = 0 on �I (8)

However, it is a well-known fact that this simple boundary approximation produces largespurious re ected waves on �I . To solve this problem, di�erent methods can be found in theliterature (one revision can be found in Givoli [12]).



For the particular case of PUFEM the procedure of Higdon [12; 13] can be speciallyattractive. The exterior boundary condition proposed by Higdon for the wave equation [13]is reduced for the Helmholtz equation de�ning the operator B over �s as

B�s =

[m∏p=1

(ik cos(�p)− @

@n

)]�s = 0 on �I (9)

where the angles {�p; p=1; 2; : : : ; m} represent the directions of absorption on the arti�cialboundary. The Higdon condition behaves as perfectly absorbing for a plane wave hitting theboundary �I at one of the angles �p. Generally, in traditional FEM, the incident direction ofthe scattered waves approaching the boundary is unknown a priori, so the optimal selectionof �p angles for every problem have to be determined by numerical tests or by a number ofdi�erent angles enough to reduce the re ections up to a bound.In PUFEM, due to the fact that the total problem solution is expressed explicitly as a linear

combination of plane waves of known propagation angles, it is reasonable to suppose that theincident angles of the scattered solution on the external boundary are mainly given by theangles of the plane waves used as external functions space. To show that, de�ne the localPUFEM approximation of the solution at node i as

�si ≈ �̃is =q∑l=1bli exp

[k(x cos

(�l)+ y sin

(�l))]

(10)

For this approximation the boundary operator B of the Higdon condition is

B�̃is =

q∑l=1

m∏p=1

(ik cos(�p)− ik cos (�l)) bli =0 (11)

When the absorbing directions �p are selected equal to the propagation angles �l

�p= �l or �p= �l + � (12)

the Higdon condition is exact at node i. This can be written in terms of a nodal re ectioncoe�cient Ri de�ned as

Ri=m∏p=1

cos(�p)− cos(�l)cos(�p) + cos(�l)

≡ 0

The angle selection proposed (Equation (12)) optimizes the absorption properties of Equa-tion (9). The e�ciency of this method is obviously restricted by the number of absorptionangles selected or, in other words, by the number of plane waves chosen.For practical implementations, a general m-Higdon condition is here replaced by its �rst-

order approximation:

B ∼= (ik)mm∏p=1cos(�p)− (ik)m−1

m∑p=1

(∏mj=1 cos(�

j)

cos(�p)

)@@n+O

((@@n

)2)on �I (13)

where m is again the total number of directions selected for each of the boundary nodes. Itcan be readily seen that the inclusion of the condition (13) to the �nite element formulationis very similar to the simplest one (of the type of Equation (8)).



4. MODEL IMPLEMENTATION

4.1. Integration of oscillatory integrands

The integrals coming from Equation (5) are of the type

I =∫P(x; y) exp

[ik(cos �lx + sin �ly

)]exp [ik (cos �mx + sin �my)] d (14)

where P(x; y) is a polynomial. Now the goal is to obtain a one-dimensional oscillatoryintegrand in order to simplify the corresponding integration of Equation (14). If the inte-gral I is expressed in a local co-ordinate system of reference (�; �) de�ned by the unitarytransformation: (

xy

)−(x0y0

)=(cos � sin �sin � − cos �

)(��

)(15)

where the angle � is the rotation needed to get the one-dimensional oscillatory integranddesired and r0 = (x0; y0) is the position of one of the element node assumed as local originof coordinates, the integral I becomes

I = exp[ik · r0]∫H (�; �)R1(�)R2(�) d� d� (16)

and R1(�) and R2(�) are the oscillatory integrands in the � and � directions, respectively:

R1(�)= exp[ik((cos �l + cos �m

)cos �+

(sin �l + sin �m

)sin �

)�]

and

R2(�)= exp[ik((cos �l + cos �m

)sin �− (sin �l + sin �m) cos �) �]

and �nally H (�; �) is a polynomial of the same degree as P(x; y).By the use of well-known trigonometrical relations, R2(�) takes the form

R2(�)= exp[2ik cos

(�m − �l2

)(cos(�m + �l

2

)sin �− sin

(�m + �l

2

)cos �

)�]

or, �nally

R2(�)= exp[2ik cos

(�m − �l2

)sin(�m + �l

2− �)�]

(17)

It is immediately seen that R2(�) is set to one if the rotation angle is

�= 12

(�m + �l

)and then the oscillatory component in terms of � is eliminated.The integral I is now reduced to

I = exp[ik · r0]∫H (�; �) exp[ik0�] d� d� (18)



Figure 1. Global and local co-ordinate systems for the integration procedure: �1; �2; �3:integration limits in the � direction.

Figure 2. E�ciency of the numerical integration methods tested for oscillatory integrands:G.L: Gauss–Legendre integration method; AEH: Chebyshev series integration method.

Relative solution error required 10−12.

where an ‘equivalent’ wavenumber k0 is introduced as

k0 = 2k cos(�l − �m2

)cos(�l + �m

2

)

It should be remarked that now the oscillatory integrand is one dimensional. A sketch of thecomputation de�nition over an element is plotted in Figure 1.



Figure 3. Convergence behaviour of the generalized conjugate gradient solver for the wave numbersw = 0:5; 16; 32; 132 and 264. Melenk test on a square mesh of linear triangular element (16 nodes)

and 66 external functions per node.

The direction � can be seen as an e�ective incident direction (�) valid only for everypair of external waves corresponding to every integral. The oscillatory behaviour along thecoordinate � (orthogonal to the incident direction) is eliminated with the obvious saving inthe number of operations and allowing an easy application of integration methods adapted tooscillatory integrands of the type of Equation (18).To study an e�cient form to deal with the new simpli�ed form, an ‘adhoc’ method for

oscillatory integrands (Chebyshev series based [14] (AEH)) is compared with the standardGauss–Legendre numerical integration.The integration of ∫ +1

−1�r exp(ik�)

by the AEH method is given by

I =r∑l=0

r!(r − l)!

�r−l

kl+1

{sin(k�+

l�2

)− cos

(k�+

l�2

)}+1−1

(19)

This procedure can be thought as a special case of the Clenshaw–Curtis [15] formulae whenthe function in the integrand is a polynomial, and then an exact solution can be obtained.For the comparison test proposed, polynomials of degree 3 were chosen as H (�; �) (this

is the maximum degree encountered for the piecewise linear shape functions assumed in thiswork, provided the limit of the integral for the linear triangles is a linear function of �).



Figure 4. Test problem. Horizontal plane wavedi�racted by a rigid cylinder.

Figure 5. Computational domain and meshemployed (54 nodes, 52 elements).

Table I. Averaged solution error obtained with HBC andPF-HBC conditions for a PUFEM mesh of 540 DOF (54nodes, N =10): HBC: arbitrary Higdon condition; PF-HBC:

pufem Higdon condition.

ka=1 ka=2 ka=3:5 ka=5

HBC 3.4% 6.3% 12.9% 21.3%PF-HBC 3.3% 6.1% 8.4% 13.1%

The relative wavelengths (wavelengths=element) varies from very long waves (0.01) to veryhigh-frequency waves (1000).Results are shown in Figure 2, where the number of operations are represented in terms

of the relative wavelength for the two methods once the relative error is settled as 10−12.The number of operations of the AEH method does not depend on the frequency, being moree�cient for relative wavelengths higher that 5. The AEH method requires additional operationsfor very low frequencies to avoid indeterminancy coming from the factor depending on thewavenumber. For this case, a Taylor expansion should replace Equation (19). However, it isclear that the ‘long-wave’ range should be covered by a Gaussian quadrature while the exactintegration is preferable for medium and high frequencies. The model developed in this worktakes this into account by selecting the optimal integration method for each calculated k0:

4.2. Conjugate gradient solver

The method of solution of the linear system obtained by the PUFEM must deal, in general,with poorly conditioned matrices. As the number of extrinsic functions increases, the systemtends to be linearly dependent (for usual ‘round-o�’ limits), in particular for the lowest rangeof wavelengths, and to lose the typical FEM ‘banded’ structure.Also, poor conditioning implies the need for higher residual norms orders for convergence

to reach the bounds prescribed in the solution error. A suitable method for this situation is



Figure 6. Real and imaginary potential around the cylinder for ka=3:5.

the generalized conjugate gradient method for non-de�nite (Hermitian) matrices [10]. Herethe Hermitian condition is obtained by solving the system premultiplied by the conjugate ofthe coe�cient matrix A: (A∗). The linear system coming from (5) is now

A∗Ab=A∗fwhere b represents the unknown parameters and f the terms coming from the boundary con-ditions. When current ‘round-o�’ limits do not permit the attainment of an adequate tolerancefor the solution error the conditioning can be improved by imposing the positive de�nitenessin the form

(A∗A+ I)b=A∗fwhere is a constant of the order of the ‘round-o�’ limit. This situation was found in thevery extreme case of a very large number of extrinsic functions (more than 64 per point forcurrent double precision computation) and very low frequencies.The performance of the method was studied for a problem of a square mesh of linear

triangular elements with 16 total nodes and 66 extrinsic functions associated to nodes (astringent condition for long and intermediate waves), with a residual norm tolerance (relative)of 3× 10−13. At the left boundary a normal incident wave is prescribed (by means of thecondition (2) adding an incident wave), while in the rest of the sides normal absorbent



Figure 7. Real and imaginary potential at r=5a for ka=3:5.

conditions (2) are imposed. In Figure 3, the residual norm in terms of the number of iterationsis shown for wavenumbers varying from 0.5 (long waves) to very short waves (k=264). Itis seen that the worst convergence conditions are for intermediate wavenumbers (k=32).When the wavenumber is higher than this order, convergence is slightly improved becausefor high wavenumbers the in uence of the mass matrix in the structure of the total systembecomes more important. However, in the medium–high-frequency range, the dependence ofthe number of iterations on the wavenumber is relatively weak.The solution error (L2) for each wavenumber (from 0.5 to 264) is, respectively: 10−7; 10−4;

10−4; 10−5; 10−6.

5. TEST CASE: DIFFRACTION AROUND A CIRCULAR PILE

The di�raction of a plane wave around a circular pile [16] has been chosen as numerical testof the method. The pile is considered to be a vertical and totally re ecting cylinder of radioa with circular exterior domain placed at rmax =5a: The incident plane wave propagates atthe x direction (Figure 4).The mesh employed is shown in Figure 5. A total number of 52 linear triangular elements

(54 nodes) have been used for all the range of wave numbers studied.



Figure 8. Real and imaginary components of the potential at �= � for ka=3:5.

The PUFEM approximation for the scattered wave is constructed by an external functionbasis of plane waves (Equation (6)) and �l is selected as

�l= l2�N

l=1; 2; : : : ; N (20)

The total number of external functions N has been kept constant for all the nodes of themesh.The wave numbers considered in the experiment are ka=1; ka=2:, ka=3:5 and ka=5:The test of the boundary conditions has been made applying the �rst-order approximation

of the Higdon condition on the exterior boundary (13). The two cases discussed in previoussections have been considered:

Strategy A. Arbitrary selection of absorbing directions (HBC).Strategy B. Selection of the absorbing directions equal to the plane wave external function

(PF-HBC).

The results (Table I) show a better behaviour of the PF-HBC for all the range of wavenumbers considered for a constant number of degrees of freedom.The convergence properties of PUFEM by a real di�raction problem have been also anal-

ysed. This was carried out varying the dimension of the external basis N for a �xed mesh(Figure 5) in order to reach a total average solution error �L263 per cent with respect to the



Figure 9. Real and imaginary components of the potential at �=0◦ for ka=3:5.

analytical solution. The real and imaginary components of the solution for the case ka=3:5 atr= a and r= rmax can be seen in Figures 6 and 7, respectively. The total number of externalfunctions needed in this case is N =18. For the same case, the real and imaginary parts ofthe solution along the angles �= � and �=0 are shown in Figures 8 and 9, respectively.

6. CONCLUSIONS

The use of the partition of unity method instead of standard �nite elements in di�raction prob-lems is recommended if the implementation is e�cient in terms of the integration method andthe solver adopted. The nearly analytic integration procedure proposed in this work, indepen-dent of the frequency, covers the medium- and high-frequency range with an impressivereduction in the number of operations. With respect to the solver, a robust version of the gen-eralized conjugate gradient method was developed here and future comparisons with availablefast direct solver routines will be carried out. The approximate exterior boundary conditionproposed is suitable for the de�nition of the numerical solution based on plane waves and



shows an improvement for the same basis of external functions if it is compared with theclassical Higdon condition.The better approximation properties of the exterior function space, which reduce the number

of nodes to only those needed for a proper de�nition of the domain, and the fact that theboundary condition implemented o�ers exibility in the position of the arti�cial boundary areboth attractive strategies in the search of methods less dependent on the mesh for di�ractionof waves problems.The extension of the model for di�raction–refraction of short waves is the next step in the

development and is now being carried out.

REFERENCES

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3. Ihlenburg F, Babuska I. Finite element solution of the Helmholtz equation with high wave number, Part II: theh–p version of the FEM. SIAM Journal of Numerical Analysis 1997; 34:315–358.

4. Bettess P, Chadwick E. Wave envelope examples for progressive waves. International Journal for NumericalMethods in Engineering 1995; 38:2487–2508.

5. Chadwick E, Bettess P, Laghrouche O. Di�raction of short waves modelled using new mapped wave envelope�nite and in�nite elements. International Journal for Numerical Methods in Engineering 1998; 45:335–354.

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11. Sommerfeld A. Lectures on Theoretical Physics. Academic Press: New York, 1964.12. Givoli D. Non re ecting boundary conditions. Journal of Computational Physics 1991; 94(1):1–27.13. Higdon R. Absorbing boundary conditions for di�erence approximations to the multi-dimensional wave equation.

Mathematics of Computation 1986; 47(176):437–459.14. Alaylioglu A, Evans G, Hyslop J. The use of Chebyshev series for the evaluation of oscillatory integrals. The

Computer Journal 1974; 19(3):258–267.15. Evans G. Practical Numerical Analysis Wiley: New York, 1995.16. Berkho� JCW. Mathematical models for simple harmonic linear water waves: wave di�raction and refraction.

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An improved partition of unity finite element model for diffraction problems

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