TICAM REPORT 96-49October, 1996

Solution of the 3D Helmholtz Equation in ExteriorDomains of Arbitrary Shape Using

HP- Finite- Infinite Elements

K. Gerdes and L. Demkowicz

SOLUTION OF THE3D-HELMHOLTZ EQUATION IN EXTERIOR DOMAINS

OF ARBITRARY SHAPEUSING HP-FINITE-INFINITE ELEMENTS

K. Gerdes and L. Demkowicz

The Texas Institute for Computational and Applied MathematicsThe University of Texas a.t Austin

Ta.ylor Hall 2.400Austin, TX 78712, USA

Abstract

This work is devoted to a convergence and performance study of finite-infiniteelement discretizations for the Helmholtz equation in exterior domains of arbitraryshape. The proposed approximation applies to arbitrary geometries, combiningan hp FE discretization between the object and a surrounding sphere and an hpInfinite Element (IE) discretization outside the sphere with a spectral-like rep-resentation (resulting from the separation of variables) in the "radial" direction.The described approximation is an extension of our earlier work [18] restricted todomains with separable geometry only. The numerical experiments are confinedto these geometrical configurations: a sphere, a (finite) cylinder, and a cylinderwith spherical incaps, all within a truncating sphere. The sphere problem admitsan exact solution and serves as a basis for the convergence study. Solutions tothe other two problems are compared with those obtained using the BoundaryElement approach described in [11].

1 Introduction

The presented paper is motivated by our ea.rlier work on elastic sca.ttering (see [11] forthe latest results) and the new concept on various infinite elements by Burnett [6], Astleyet all [1], Cremers et all [7] and our own work [18, 10].

The eventual problem of interest deals with scattering of acoustic waves on elasticor rigid (the simplified case) objects. The mathematical formulation consists of theHelmholtz equation in the exterior domain accompanied by the Sommerfeld radiationcondi tion and Neumann boundary condi tion on the boundary of the scat terer (rigidscattering). In the elastic scattering case, the Helmholtz equation is coupled to equationsdescri bing behavior of the structure (elasto- or viscoelasto-dynamics). The presentedwork discusses t.he rigid scattering case only.

Problems of the descri bed type a.re usually solved using various versions of the Bound-a.ry Element Method (BEM). The mathematics of the BE approximation (especially int.he Galcrkin version) is well est.ablished and the method delivers reliable results in t.hewhole range of wa.ve numbers. The main drawback of the BEM is its cost - the met.hodbecomes prohibitively expensive for large wave numbers.

The approach based on the truncation of the infinite domain to a finite one andapplication of the so called open boundary conditions has always been an alternativetechnique to solve the problem. In particular, the recent versions of the Infinte Elementsbased on multi pole expansions seem to be especially attractive as they offer accuracy ofarbitrary high order and can be coupled with standard Co finite elements. The recentresults on convergence of such methods [2, 10] support reliability of such an approacha.nd add to its attractiveness.

The common property of the methods presented in [6, 1, 7] is that both approachesuse a variable number of basis functions in the radial direction. The essential differenceappears in the underlying variational formulations (though never explicitly stated in[6, 1, 7]). The scheme proposed in [1, 7] is based on a formulation in weighted Sobolevspaces and consistent with the existence and uniqueness theory by Leis [22]. This is alsothe approach which we investigated in [18] and continue to use in this work. The conceptpresented in [6] is based on a different variational formulation, also shortly discussed in[18] .

Another essential difference between the various versions of the infinite elernents hasbeen recently pointed out in [2]. The difference lies in the fact whether one does or doesnot use the complex conj ugate over the test functions (sesquilinear vs. bilinear form

1

formulation) .

1n the presented approa.ch we couple the standa.rd FEM with the IEM, ba.sed on theLeis sesquilinear variational formulation. The sca.tterer is surrounded with a truncatingsphere. In between the scatterer and the truncating sphere we use 3D hp-FE approxi-mations described in [11]. Outside the sphere, the compatible hp-discretization on thesphere is combined with a spectral like approximation in the radial direction.

Numerical results are confined to these geometries: a sphere, a (finite) cylinder anda (rinite) cylinder with spherical inca.ps. The sphere problem admits an exact solut.ionwhich is used t.o compute the convergence rates. The numerical solutions to the cylinderproblems are compared with solutions obtained by the BEM discussed in [11].

The plan of the presentation is as follows. We begin by formulating the Helmholtzequation and describing the IE in section 2. Numerical experiments are presented insection :3. Section 4 discusses some preliminary results on a-posteriori error estimationa.nd hp-ada.ptivity, and we finish the presentation with concluding remarks in section 5.

2 Coupled FE/IE Discretizations for the HelmholtzEquation in Arbitrary Exterior Domains

2.1 Notation

• n c IB? is a domain occupied by the rigid scatterer and contained in the unitsphere

• ne = JR3 - n is the domain exterior to the scatterer

• fs = {x E JR3; Ixl = I} is the surface of the unit sphere

• n~= {x E JR3; Ix I > I} is the domain exterior to the unit sphere

• f = an is the surface of the rigid scatterer

• ns = {x E JR3; Ixl ::; I} - n is the domain between the unit sphere and the rigidscatterer

The notation is illustrated in Figure 1.

2

Figure 1: Notation. Cylinder within a unit sphere.

2.2 Classical Formulation of the Problem

The goal is to find a function u = u( x) sa.tisfying:

• the Helmholtz equation in the domain exterior to the scatterer,

where k is the wa.ve number;

(2.1 )

• a Neumann boundary condition on the scatterer

Vnu=g

where 9 is a prescribed function on f;

for x E f (2.2)

• the Sommerfeld radiation condition at infinity,

(2.:3)

2.3 Variational Formulation and Existence Theory

Given a "truncated" exterior domain n~,

the Helmholtz equation is multiplied by the complex conjuga.te of a test function v,integrated over the truncated domain n~,and integrated by parts. Using the Neumannboundary condition on on~ results in the following equation, for any admissible testfunction v,

where S-y is the "truncating" sphere with radius 1· ')'. Rewriting the Sommerfeldra.dia.tion condition (2.3) in the form

au = il.:u + tp(x)01'

(2.5 )

where tp(x) = 0 (1"-2) is an unknown function, and building it into the varia.tionalformulation (2.4) by substituting for au/on = OU/01' in the corresponding boundaryterm formula (2.5), we get

Passing formally with I ---7 00, we obta.in

(2.7)

From a general theory in [22, 27], it is known that the leading order term of uln< is ofs

the formuo(x) exp(ih)

l' '

and, consequently, both u and its gradient, Vu, are not L2-integrable over the exteriordomain, similarly as for the sphere problem. A remedy to this problem is again to employdifferent test functions in n~,of order 0 (r-3). This makes it possible to interprete theintegrals in the usual Lebesgue sense. As for the sphere problem, this particular choice

4

of t.he test functions does not allow one to build the radiation condition into the weakformulation, and the Sommerfeld condition has to be included directly in the definitionof the spaces. This leads to the definition of the following weighted Sobolev space

(2.8)

with the norm 1[1lllt" corresponding to the inner product

Two particular weights are of interest,

{

Iw(x) = ~

1,2

and a "dua1" weight

for 1· = Ix I :S 1

for 1'= Ixl > 1

{I for r = Ix I :S 1

w*(x) = 1,2 for l' = Ixl > 1

The variational formulation reads now as follows

{

Find U E Htw(ne) such that

r V 1l . Vv dne- e r U v dne = r 9 v ciS V v E Htw* (ne)Jrr Jrr Jane '

Ren1arks:

(2.10)

1. As for the sphere problem, the proposed variational formulation corresponds toan extension of the operator setting of Leis [22], where the domain of the oper-ator is restricted to a subspace of Htw(ne) consisting of all functions for whichthe (Helmholtz) operator value is in the weighted L;v*(ne) space. \i\Tith these as-sumptions, Leis proves the uniqueness and existence of solutions, showing that theresulting operator is bounded below with a constant locally independent of wavenumber k.

2. Essentially the same formulation has been proposed in [1], [7], [8] and [18].

5

The Burnett formulation is introduced in t.he same way as for t.he sphere problem. Bothsolution 'U and test function v are represented outside of t.he unit sphere in the fonn

exp(ikl') (0 A-.) + U(1.,O,¢»11.0 , 'I'l'

exp(il.:1·)vo(O,¢» + V(1·,0,¢»1"

(2.11)

where ·,.,O,¢ are spherical coordinates, where functions 1l0{fJ,rjJ) and '/Jo(O,¢» denote t.heradiation patterns, and functions U(/',O,¢», 1I(1·,0,¢» are from Hl(n.~), i.e. both U, ,/and their gradients "\7U, "\711 are square-integrable. Function II of t.his form sat.isfiesmdomatically the Sommerfeld radiation condi t.ion. Upon su b::;titu ti ng formulas (2.11)into (2.4) and cancelling out terms involving the radiation patt.erns, one can pass to t.he

limit with I - 00. Contrary to t.he weighted spaces formulation, the integral over S-yinvolving the radiation patterns will not vanish in the limit, and the integral over t.heexterior domain ~y is t.o be understood in the CPV sense described above.

2.4 Separation of Variables in O~and the Asymptotic Formof the Solution

Using the classical separation of variables approach, one arrives at the following form ofthe solution, va.lid outside the truncating unit sphere.

00 n

'/1,(1·,0, ¢» = L L hn(h)p~n(cos 0) (Anm cos(m¢» + Bnm sin(m¢»)n=O m=O

(2.12)

where P;:'( cos 0) are the Legendre functions.

Using the representation from [23], the spherical Hankel functions of the first kindmay be represented as

177.= 0

177. ~ 1

hn(kr) = I~ exp (-i~(n + 1) + ikr) t im (17. + ~, 177.) (2kl'rmitT 2 m=O 2

with

(n+~,m){ il(n+k)~(n-m+k)The terms can be reordered to obtain

I (I .. ) _ ~ exp(ih) exp(-i~(n + l)).m ( 1 )l'n 1..1 - L 1. 17. + -, 111-

1·m+l H2k)rn 2m=O

6

(2.13)

(2.14)

2.5 Definition of the hp-Infinite Element

The following shape functions in the radia.! direction were used for the sphere problemin [10]:the tria.! functions

t.he t.est functions~}i(1.) = exp(iJ.:r)

..:, <") .,

j > 1

j"21

(2.15)

(2.16)

Not.e t.he different powers of.,. in t.he denominators. The use of the sesquilinea.r fonnu-lation eliminat.ed the need of integration of the oscillatory component. exp(ih).

The infinite element shape functions were t.hen defined as tensor products of 2D finiteelement shape functions a.nd the functions introduced in (2.l5) and (2.16) respectively,I.e. a. typical infinite element trial shape function N/(1', x) was given by

'r > 1, x E I'., (2.] 7)

In order to minimize the interaction with finite elements within truncated domain Os,the IE trial shape functions are now modified as follows

(2.18)

with an identical modification for the test functions. In this way, all the shape functionscorresponding to j "2 2 will contribute to basis functions with supports outside of theunit sphere only. Inclusion of the exponential factor exp (-ik) in the new definitionforces the infinite element shape functions to coincide with the standard 2D hp shapefunctions on the surface of the sphere.

2.6 3D hp-Adaptive FE Discretization

The domain Os in between the scatterer and the truncating sphere is discretized using(triangular) prismatic hp-elements introduced in [12].

!IIasle1> element. The master prismatic element, shown in Figure 2, consists of sixvertex nodes and fifteen highe1>-o1>de.,.nodes: nine mid-edge, two mid-base, three mid-sideand one middle node. The corresponding shape functions are tensor products of the 2D

7

~I

~I

Figure 2: l'vlaster triangle and master prism

triangle shape functions Xi(~l, 6) discussed in [16] and 1D inaemental shape functionsX.i(6)

(2.19)

For j = 1,2, functions X.i(6) are the regular linear shape functions. Given a particularorder of approxima.tion q in the "vertical" (6) direction, functions X.i(6), j = 1, ... , q-l,coincide with the regular ID Lagrange shape functions of order q, vanishing at theendpoints.

Consequently, the mid-side and the middle node have two corresponding orders ofapproximations: a horizontal p and a vertical order q. For that reason, we frequentlytalk about the hpq FE approximations. For all details concerning the definition of themaster element we refer to [12].

Geometry 1'epresentation. The domain ns is viewed as a union of disjoint blocks. Foreach of the three considered configurations, the topology is the same, and the domainis covered with 8 triangular, and 8 rectangular prisms. The bottom base of each of theprisms coincides with a triangle or a rectangle on the scatterer and the top base lies onthe unit sphere. The concept is illustrated in Figures 3 to 5.

Each of the prisms is viewed as the image of a corresponding standard master trian-gular or rectangular prism. The triangles and rectangles are parametrized using explicit

8

Figure 3: Geometry model for a sphere within the unit sphere

Figure 4: Geometry model for a cylinder within the unit sphere

9

y

(-' y

y

Figure 5: Geometry model for a cylinder with spherical incaps within the unit sphere

parametrizations discussed in [15] and a linear parametrization is used in the "vertical"direction. For all the details concerning the geometric modeling, we refer to [15].

The initial mesh is generated by using the idea of an algebraic mesh generator andhp-interpolation. Given, for the reference prism numbers 11, and l of divisions in the"horizontal" and "vertical" directions respectively, (compatible for neighboring elements,the initial mesh is always regular), the reference blocks are covered with uniform, regulargrids consisting of elements k. By constructing a composition of the standard affine map1] transforming the master element k onto element k and the (restriction of) the blockparametrization Xb, a map is constructed from the master element k onto a curvilinearelement J{, identified as the image of element J{ under the particular parametrizationXb·

(2.20)

In principle, this map could be used directly to define the curvilinear element, i.e. inthe element calculations. In practice, it is approximated using the idea of isoparametricapproximation. More precisely, given a particular order of approximation for element J{

(may vary for different nodes), transformation x is replaced with its hp-interpolation.

The idea of the hp-interpolation follows from the convergence theory for hp ap-proximations [3] and has been introduced in [25] (comp. also [9]). Roughly speaking,the hp-interpolation combines the classical interpolation for vertex nodes with local

10

HJ-projections for higher-order nodes. Given a sufficiently regular function, the cor-responding hp-interpolant exhibits the same orders of convergence (in terms of both hand p) as the corresponding global HJ-projection (solution to the Laplace equation withDirichlet boundary conditions imposed using the HJ projection on the boundary).

A typical initial FE mesh for a spherical scatterer with radius 0.5, located inside ofthe unit sphere, is shown in Figures 6 and 7. In the example there are 2 layers of :3Dfinite elements with 24 elements per layer, p = 4, q = 2 and 24 elements on the surfaceof the unit sphere representing 24 infinite elements.

The same hp-interpolation technique is used during mesh refinements.

3 Numerical Experiments

3.1 Implementation Details for the Helmholtz Equation

The integration of the exponential terms in the radial direction for the infinite elementinvolves terms of the following type

(3.1)

(3.3)

JOO -c:;-- JOO exp (ih) exp (-ih) 1

1/;(1')7/},(1') dr = . c d1' = ~1 J J . 1 1'J 1'J+2 ) + ) + 1

JOO (. exp (ih) .exp (ikr))1'2 zk . - ) .1 1'J rJ+1

(·kexp (-ih) (": )exp (-ikr)) d. -z - - ) + 2 - l'

1'j+2 1'j+3

P ik(j-(]+2)) j(]+2)": + . ": +. ":)+)-1 )+) )+)+1

Having these integrals computed ahead of time the calculation of the infinite elementstiffness matrices is straightforward and reduces to standard 2D FE-like calculations.The computation of the 3D finite element stiffness matrices is also standard and theelement contributions are then assembled by a frontal solver adapted to handle complexmatrices. The element load vector is calculated similarly.

It should be noted that the final global stiffness matrix is not symmetric.

11

y

Figure G: Initial Finite Element Mesh for the rigid scattering on a sphere within theunit sphere with p=4, q=2, 2 layers of finite elements. Half of the mesh is displayed.

y

Figure 7: Initial Finite Element Mesh for the rigid scattering on a. sphere within theunit sphere with p=4, q=2, 2 layers of finite elements. Three quarters of the mesh a.redisplayed.

12

3.2 Scattering of a Plane Wave by a Rigid Sphere with Radius0.5

The derivation of the scattered wave on a rigid sphere with radius 0.5 corresponding toa.n incident plane wave can be found, e.g. in [18]. The final form of the scattered wavepS is

pS = L hn(b·)P,7.'(cosO)An.n=O

wit.h An given by

- Pinc(2n + 1)i" .")j,~(1,;1") I01·

Dhn(kT) I r 0..5

a1" 1"=0.5

3.3 Error Calculations

V n> 1

(:3.4 )

(:~.5)

Consistent with the variational formulation, the error between the numerica.l and t.heexact solution is measured in the weighted HI-norm in f[€, where

with

(3.6)

(:3.7)

a.nd(3.8)

The exact solution II for the rigid scattering of a sphere with radius 0.5 is given by (3.4)and the numerical solution in n: can be represented in the form

N M

Hh(1',x) = LLlljiN/(i,j)(1',x)j=1 i=1

(:3.9)

where N1(i.j) represent hp-basis functions of the IE and llji denote the correspondingdegrees of freedom. In ns the numerical solution is given by

nrdof

Hh - L lli<.pii=1

(:3.10)

whcre 'Pi. i = l, ... , nrdof, represents the :30 finite element shape functions. Theevaluation of Ill/, - '/lhlli.o. follows the standard FE computations. For the computationof the errol' Illl - 'Ilhlli 0-' we refer to [18].

, <

Ren1arks:

1. In the error computation the term corresponding to the Sommerfeld radiationcondit.ion is neglected; compare with (2.9).

2. The integrals in the radial direction in (3.7) are integrated exactly and a higherorder Gauss rule is used for the integration over ()n~. Similarly, a higher orderGauss rule is used for the evaluation of (3.8).

3.4 Choice of an Initial FE Mesh

Due to the oscillatory character of the solution, choosing an arbitrary initial FE/IE meshmay not lead to a.ny reasonable numerical solution. In particular, thc initial mesh hasto be chosen dependently on the wave number /.;and the prescribed Neumann boundaryconclition clata on the surface of the scatterer.

For the sphere problem it is possible to compute the best appro:rimation e1T01· in ns(:3.11)

where Vh(ns) c HI (ns) is the finite dimensional FE subspace of the Sobolev spaceHI(ns).

The minimization problem

is equivalent to the variational formulation

(3.12)

{Find uta E Vh(ns) such that

(uta,Vh)Hl(Os) = (U,Vh)Hl(os)

where (u, V)Hl(os) is the HI-inner product

(3.13)

14

(:3.14)

For a given wave number /..:and Neumann boundary condition data. corresponding to a.nincident plane wave restricted to the surface of the scatt.erer, the sphere with ra.dius 0.5,t.he exact solution u is given by (3.4). The best approximation error is then computedas

(:3.15)

Tables 1 and 2 show the best approximation error for an incident plane wave with"va.venumber /..:= 10 for various meshes, depending on

• t.he polynomia.l degree in the angular ("horizontal") direction p

• t.he polynomial degree in the radial ("vertical") direction q

• the number of t.he la.yers of the :3D finite elements l

• t.he number of the :3D finite elements per layer 17.

3.5 Convergence Rates

The following discussion presents convergence rates for different examples of finite-infinite element meshes. It focuses on the question of, how the number of the shapefunctions in the radial direction a.ffects the approxima.tion of the exact solution.

The first test. case deals with a problem with a finite number of terms in the exactsolution. The corresponding Neumann boundary condition for the Helmholtz equationis selected as

4

g(B, ¢;) = L p4rn (cos 0)( cos( rn¢;) + sin( rn¢;))

rn=O

The corresponding exact solution is then

4

u(r,O,¢;) = L h4(kr)P;(cosO)(A4mcos(m¢;) + B4msin(m¢;))m=O

(:3.16)

(:3.17)

with1

A4m = - ah4(kr) I 1

or r= 2"

1B4m = - ah4(kr) I 1

ar r= 2"

(:3.18)

The p-convergence rates are studied in terms of the weighted HI-error norm on ~Y,compare (:3.6). The error is analyzed as a function of the order of approximation in the

15

p q I nIlll - ur,aIIH},,(fls) lIu - llr,allL2(fl,)

IluIIH,~,(fls) IlullL2(fls)5 2 2 96 12.08 4.505 2 1 96 37.89 24.995 2 2 24 12.21 4.535 2 1 24 :37.94 25.004 2 4 96 :3.59 0.694 2 :3 96 5.93 1.54 2 2 96 l2.11 4.504 2 1 96 37.90 24.994 2 5 24 4.75 1.474 2 4 24 5.41 1.584 2 3 24 7.19 2.064 2 2 24 12.77 4.724 2 1 24 38.10 25.013 2 5 96 3.61 0.79:3 2 4 96 4.44 0.97:3 2 :3 96 6.48 1.653 2 2 96 12.39 4.563 2 1 96 37.98 24.993 2 5 24 16.13 8.463 2 4 24 16.35 8.483 2 3 24 17.02 8.593 2 2 24 20.00 9.623 2 1 24 40.81 26.37

Table 1: Best approximation error in ns in percent measured in the Hl_ and L2-norm

16

p q I n Ilu - 1traIIH,I,,(lls) Ilu - u~a 11L2(Il.,)IluIIHl,,(Il") lIullL2(lls)

5 3 2 96 2.84 0.725 :3 I 96 15.47 7.905 :3 :3 24 2.09 0.55.5 :3 2 24 :3.:33 0.885 :3 I 24 15.58 7.924 :3 4 96 1.16 0.214 :3 :3 96 1.45 0.264 :3 2 96 2.98 0.744 :3 1 96 15.51 7.904 3 .5 24 4.19 1.434 3 4 24 4.21 1.434 :3 3 24 4.30 1.444 :3 2 24 5.03 1.604 :3 1 24 16.02 8.02:3 3 5 96 2.85 0.71:3 :3 4 96 2.87 0.713 :3 3 96 3.00 0.733 3 2 96 3.97 1.013 3 1 96 15.72 7.933 3 5 24 15.97 8.453 3 4 24 15.98 8.453 3 3 24 16.01 8.453 3 2 24 16.23 8.483 3 1 24 22.17 11.54

Table 2: Best approximation error in Os in percent measured in the Hl_ and L2-norm

17

'tangential' direction p and as a function of the order of approximation q and N in theradial direction. In the examples the wave number I.: is set to 10 and up to :3 layers of:~D finite elements are used.

The following graphs, Figures 8 - 11 show the p-convergence rates for q = 2 and 2.:3, 4, .5 radial shape functions respectively. Figure 12 shows the p-convergence rates forq = 4. The initial mesh consists of 72 elements with p = 2, i.e. there are 24 infiniteelements on the surface of the unit. sphere and two layers of :3D finite elements in n,which give 48 3D finite elements. The wave number I.: = 10 is used. The elements arcthen enriched up to p = 5. The "y-axis" shows the e!Tor in percent. of the weight.edIII-norm of the exact solution and the "x-axis" shows the order of approximat.ion p. Allconvergence rates plots are on the log - log scale.

These pictures clearly indicate that at a cert.ain point. it does not make any sense toincrease the number of dof on the surface, unless t.he number of dof in radial directionis increased. Further, it is evident, t.hat only enough dof both in the angular and theradial directions, e.g. high]l and q, or more finite elements, and t.he number of dot'in radial direction of the infinite element matching the number of terms in the exact.solution does lead to a satisfactory result, i.e. the error wit.h p = 5, q = 4, N = .5is below 5%, see Figure 12. Figure 11 clearly indicates that the discretization in theangular and the radial directions of the 3D finite elements is not sufficient to obtain areliable numerical solution, although with N = 5 the number of terms in the infiniteelement representation matches exactly the number of terms in the exact solution. Thenumerical results can be improved by increasing the number of dof in the radial andthe angular directions. Figure 12 displays results for meshes with the number of dof inthe radial direction increased, which has led t.o a reasonable numerical solution, but theerror still cannot be forced below 4.7%, because of an insufficient number of dof in theangular direction. On the other hand, Figure 13 indicates that it is also not sufficient toprovide enough dof in the angular direction only. Obviously, the solution is to provide asufficient number of dof in the radial as well as in the angular direction. Figure 14 showsclearly that the discretization is sufficient in all directions. The error is below 1.3% forp = 5 and q = 3 with 192 finite elements and 96 infinite elements with N = ·5.

A second, more realistic example. is the plane wave problem, described in Section :3.2.The assumed incident plane wave generates a scattered plane wave, which has an infinitenumber of terms in the radial direction. In the computations this series is truncatedafter 10 terms.

Figures 15 - 17 show the p-convergence rates for wave number I.: = 2 with different q

18

100.00

211.1

Ill.?

2

Figure 8: Scattering of a wave on a rigid sphere. Experimental p-convergence rates wit.h48 ~3Dfinite elements and q = 2 and 24 infinite elements with 2 terms in the radialdirection for the Helmholtz equation with k = 10.

JO().OO

59.5

17.tJ

9.012

P-C(ll1vcrgcnc..:c nllcs

4

Figure 9: Scattering of a wave on a rigid sphere. Experimental p-convergence rates with48 :3D finite elements and q = 2 and 24 infinite elements with 3 terms in the radialdirection for the Helmholtz equation with J..~ = 10.

19

IOO.(M)

17.').

9.()l)

Figure 10: Scattering of a wave on a rigid sphere. Experimental p-convergence rateswit.h 48 :3D finite elements and q = 2 and 24 infinite elements with 4 terms in the radialdirection for the Helmholtz equation with k = 10.

100.00

"".0

17.".

9.0')

2

p-(.::()IlVcrgcncc ralCS

4

Figure 11: Scattering of a wa.ve on a rigid sphere. Experimental p-convergence rateswit.h 48 :3D fi.nite elements a.nd q = 2 and 24 infinite elements with .5 terrns in the radialdirection for the Helmholtz equation with k = 10.

20

ItXl.(Xl

4

p-Ctlllvcrgcncl': r:IIL'S

:'i7.()(

1:".6

4.772

Figure 12: Scattering of a wave on a rigid sphere. Experimental p-convergence rateswith 48 3D finite elements and q = 4 and 24 infinite elements with 5 terms in the radialdirection for the Helmholtz equation with J.~ = 10.

a.nd number of dof in radial direction. Figure 18 shows the p-convergence rates measuredin the L2-norm on [le.

For wave number k = 3 and k = 5, Figures 19 to 26 show the p-convergence ratesobtained with several layers of 3D finite elements and various numbers of dof in theradial direction. Finally, Figures 27 - 34 display the p-convergence rates for wave number1..: = 10.

One does not observe the exponential shape of the p-convergence rates plots as itmight be expected. The reason for this behavior is that only one parameter, the polyno-mial degree p in the angular direction, is varied, and that the quality of the approxima-t.ion cannot be improved, if, for example, the discretization in the radial direction is notfine enough. In summary, these graphs clearly indicate, that p-, q- and/or h-mesh refine-ments for the ;3D-FE mesh have to be performed simulta.neously, along with a. sufficientnumber of terms used in the radial direction of the infinite element.

It is evident that the investigated finite-infinite element method works well for asmaller wave number, i.e. 1..: = 2, but, at the same time, the quality of the approximationdecreases as the wave number is increased. Nevertheless, the results show that the

21

2 .'

p-cc IllvcrgclU":C nllcs

o.()

4

Figure 1:3: Scattering of a wave on a. rigid sphere. Experimental p-convergence rateswith 192 3D finite elements and q = 2 and 96 infinite elements with 5 terms in the radialdirection for the Helmholtz equation with J..: = 10.

5041

1.59

1.242

p-collvcrgclH":c rates

4

Figure 14: Scattering of a. wave on a rigid sphere. Experimental p-convergence rateswith 192 3D finite elements and q = 3 and 96 infinite elements with 5 terms in the radialdirection for the Helmholtz equation with J..: = 10.

22

~ 14

2

P-~("IV~I-gCI1CC f<lles

4

Figure 15: Scattering of a plane wave on a rigid sphere. Experimental p-convergencerates with 48 3D finite elements a.nd q = 2 and 24 infinite elements with 4 terms in theradial direction for the Helmholtz equation with I.~= 2.

6.HX

:l_14

2

p-c()nvcrgcncc rates

4

Figure 16: Scattering of a plane wave on a rigid sphere. Experimental p-convergencerates with 48 3D finite elements and q = 2 and 24 infinite elements with 6 terms in theradial direction for the Helmholtz equation with k = 2.

23

Figure 17: Scattering of a plane wave on a rigid sphere. Experimental p-convergencerates with 48 3D finite elements and q = 3 and 24 infinite elements with 6 terms in theradial direction for the Helmholtz equation with k = 2.

2.1(1

2

P-C()IlVcrgcn~c ralcs

4

Figure 18: Scattering of a plane wave on a rigid sphere. Experimental p-convergencerates in U-norm with 48 3D finite elements and q = 2 and 24 infinite elements with 6terms in the radial direction for the Helmholtz equation with k = 2.

24

7.70

2 4

Figure 19: Scattering of a plane wave on a rigid sphere. Experimenta.l p-convergencerates with 48 3D finite elements and q = 2 and 24 infinite elements with 4 terms in thera.dial direction for the Helmholtz equation with k = 3.

7.70

3.09

2

P-l:;()I1vcrgcnc.;c raiL's

4

Figure 20: Scattering of a plane wave on a rigid sphere. Experimental p-convergencerates with 48 3D finite elements and q = 2 and 24 infinite elements with 6 tenTIS in theradial direction for the Helmholtz equation with /,~= 3.

2.5

7.25

1.7:'lI .() I

2

P-L'IHIVCq;CIlCl" fates

4

Figure 21: Scattering of a plane wave on a rigid sphere. Experimental p-convergencerates with 72 3D finite elements and q = 2 and 24 infinite elements with 4 terms in theradial direction for the Helmholtz equation with k = :3.

7.24

1.75

1.612

p-convcrgcncc. rates

4

Figure 22: Scattering of a plane wave on a rigid sphere. Experimental p-convergencera.tes with 72 3D finite elements and q = 2 and 24 infinite elements with 6 terms in thera.dial direction for the Helmholtz equation with A:= 3.

26

12.24

."1..4(,T."\Ti

2

p-CtlllVClgCIICC fales

4

Figure 23: Scattering of a plane wave on a rigid sphere. Experimental p-convergencerates with 48 3D finite elements and q = 2 and 24 infinite elements with 4 terms in theradial direction for the Helmholtz equation with k: = 5.

12.2~

3.46T."\Ti

2

P-C{lllvcrgcllcc rilles

4

Figure 24: Scattering of a plane wave on a rigid sphere. Experimental p-convergencerates with 48 3D finite elements and q = 2 and 24 infinite elements with 6 terms in theradial direction for the Helmholtz equation with k = 5.

27

11.1l2

2.011

I.lll

2

P-Ctlll\'l.:lgcll(;l'rales

Figure 25: Scattering of a plane wave 011 a rigid sphere. Experimenta.l p-convergencerates with 72 3D finite elements and q = 2 and 24 infinite elements with 4 terms in theradial direction for the Helmholtz equation with k = 5.

II.lll

:\.07

2.011

1.111

2

P~C(Hlvcrgcncc rates

Figure 26: Scattering of a plane wave on a rigid sphere. Experimental p-convergencerates with 72 3D finite elements and q = 2 a.nd 24 infinite elements with 6 terms in thera.dial direction for the Helmholtz equation with ,,~= 5.

28

S7.7tJ

12.')

1)-(.:(IIIVc:.I-gc:.II(.:C '-;ltc:.S

4

Figure 27: Scattering of a plane wave on a rigid sphere. Experimental p-convergencerates with 48 ;3Dfinite elements and q = 2 and 24 infinite elements with 4 terms in theradial direction for the Helmholtz equation with k: = 10.

16.5

2.342

p-convcl-gencc rates

4

Figure 28: Scattering of a plane wave on a rigid sphere. Experimental p-convergencerates with 48 3D finite elements and q = :3a.nd 24 infinite elements with 4 tenns in theradial direction for the Helmholtz equation with k = 10.

29

~7.X()

2.1 ..~

4 .'

Figure 29: Scattering of a plane wave on a. rigid sphere. Experimenta.l p-convergencerates with 48 3D finite elements and q = 2 and 24 infinite elements with 6 terms in thera.dial direction for the Helmholtz equation with 1..: = 10.

16.5.

2.342

P-(;(lIlVCI'gcncc rates

4

Figure :30: Scattering of a. plane wave on a rigid sphere. Experimental p-convergencerates with 48 :3D finite elements and q = 3 and 24 infinite elements with 6 terms in theradial direction for the Helmholtz equation with 1..: = 10.

30

2 4

Figure :31: Scattering of a plane wave on a rigid sphere. Experimental p-convergCl1cera.tes with 72 3D finite elements and q = 2 and 24 infinite elements with 2 terms in theradial direction for the Helmholtz equation with k: = ]o.

~5i.71

17.1

6.08

5.1l)

2

P-(.';tlllvcrgcncc rales

4

Figure 32: Scattering of a plane wave on a rigid sphere. Experimental p-convergencerates with 72 :3D finite elements and q = 2 and 24 infinite elements with :3 terms in theradial direction for the Helmholtz equation with k: = 10.

:31

'i'i.77

17.2

Figure :3:3: Scattering of a. plane wave on a rigid sphere. Experimental p-convergencera.tes with 72 3D finite elements a.nd q = 2 and 24 infinite elements with 4 terms in t.hera.dial direction for the Helmholtz equation with k = 10.

'i'i.711

17.2

'i.'ill

4.ll22

P-C(Hlvcrgcllt;c rules

Figure 34: Scattering of a plane wave on a rigid sphere. Experimental p-convergencerates with 72 3D finite elements a.nd q = 2 and 24 infinite elements with 6 terms in theradial direction for the Helmholtz equation with A: = 10.

32

16.4.

.lAO

1512 4

Figure :35: Scattering of a pla.ne wave on a rigid sphere. Experimental p-convergencerates with 72 3D finite elements and q = 3 and 24 infinite elements with 6 terms in theradial direction for the Helmholtz equation with k = 10.

method is reliable for higher wave numbers as well, provided sufficiently fine meshesal'(~used. The best approximation error results in Tables 1 and 2 indicate that thecost will increase with larger wave numbers. Recent results by Ihlenburg [20] on theso called polution error for the Helmholtz equation with high wave numbers imply t.hatthe number of dof for a reliable numerical solution will be larger, compared with whatfollows from the best approximation error analysis.

3.6 Numerical Results for Other Geometries

In this section, numerical results for the rigid scattering on a (finite) cylinder and a(finite) cylinder with spherical incaps are presented. Both cylinders are contained withinthe unit sphere. The Neumann boundary condition on the surface of the scatterercorresponds to an incident plane wave.

The IEM is used to compute solutions for the rigid scattering on both cylinderswith wave number A~ = 1,5,10. The finite element mesh contains three layers of finiteelements with 216 finite elem.ents per layer, p = :3 and q = 2. There are 216 infiniteelements with p = :3 and 6 radial shape functions. Altogether, 864 elements are used.

Comput.ing the numerica.l solution with the coupled FE/IE methodology t.akes fifteento twenty minutes on a DEC :3000 workstation. Numerical solutions obtained by t.heBEM a.re only presented for 1..: = 1, since the workstation was not capable of computingsolutions for k » 1, due to the memory and CPU dema.nds of the BErvi for higher wavenumbers.

4 Adaptive hp-Mesh Refinement

This section describes a strategy for performing automa.ted adaptive hp-mesh refine-ments. We intend to show only that the methodology of coupled finite/infinite elementapproximations can be embedded into the usual adaptive mesh refinement procedures./\ rigorolls errOl' estimator for the exterior Helmholt.z problelll has !lot been developedand for t.he sake of simplicity of t.he implement.ation, only a very crude error indicatorhas been used to show the possibility of using the hp-mesh refinements in cont.ext of thecoupled finite/infinite element methodology. The error indicator decribed in the nextsection will eventually have to be replaced with a more sophisticated, a.ppropriate errorestima.tor for the exterior Helmholtz problem.

4.1 An Error Indicator on Sls

The error indicator compares the numerical solution pointwise on two different meshesand provides an error indication for each finite element. More precisely, the numericalsolution Hit is computed on an initial mesh and then compared to the numerical solution'Il;~ obtained on a refined mesh. The difference between Hit and HI;" measured in theLev-norm over each finite element nl, serves as the error indicator TIL, i.e.

(4.1 )

In practice, TIL is obtained by comparing the values of H~ and ul;, at the correspondingGauss points in each finite element nl, that is

V Gauss point x,gp E nl (4.2)

4.2 An Adaptive Mesh Refinement Strategy

The following adaptive mesh refinement strategy, based on the error indicator above,intends to illustrate only the possibility of changing both p and 11, to obtain a non

34

h-p

I'lI"cs.Stln'

k = J.()

I-cal t,;OlllpOIH'1l1

Figure :36: FEMjIEM. Sca.ttering of a plane wave on a rigid cylinder with sphericalincaps, k = 1. Real part of the scattered pressure obtained with a mesh of 648 3D finiteelements (I = 3, n = 216) of order p = 3, q = 2, and 216 infinite elements with 6 termsin the radial direction

h-p DEM

Pl·cssurc

k = 1.0

real t,;olllponcni

Figure 37: BEM. Scattering of a plane wave on a rigid cylinder with spherical incaps,k = 1. Real part of the sca.ttered pressure obtained with a mesh of 216 2D finite elementsof order p = 3

35

h-p

prcsSlln'

k = 1.0

illlag CIHllp~lIK'1l1

Figure 38: FEMjIEM. Scattering of a plane wave on a rigid cylinder with sphericalincaps, I.: = 1. Imaginary part of the scattered pressure obtained with a mesh of 648 3Dfinite elements (I = 3, n = 216) of order p = 3, q = 2, and 216 infinite elements with 6terms in the radial direction

h-p OEM

prCSSUI·C

k = 1.0

inlug C()InpOnCnl

Figure 39: BEN!. Scattering of a plane wave on a rigid cylinder with spherical incaps,k = 1. Imaginary part of the scattered pressure obtained with a mesh of 216 2D finiteelements of order p = 3

36

h-p

II:'M/I:I:'M

pn.:ssun:

k = 1.0

ahsolute VahH"

Figure 40: FEM/IEM. Scattering of a plane wave on a rigid cylinder with sphericalincaps, h: = 1. Absolute value of the scattered pressure obtained with a mesh of 648 3Dfinite elements (I = 3, 11, = 216) of order p = 3, q = 2, and 216 infinite elements with 6terms in the radial direction

h-p DEM

pressure

k = 1.0

absolutc valuc

Figure 41: BEM. Scattering of a plane wave on a rigid cylinder with spherical incaps,k = 1. Absolute value of the scattered pressure obtained with a mesh of 216 2D finiteelements of order p = 3

37

h-p

IEM/FEM

k. = .'\.0

leal <':t)lllp()II~'lll

Figure 42: FEM/IEM. Scattering of a plane wave on a rigid cylinder with sphericalincaps, !.: = 5. Real part of the scattered pressure obtained with a mesh of 648 3D finiteelements (I = 3, 11, = 216) of order p = 3, q = 2, and 216 infinite elements with 6 termsin the radial direction

h-p

IEM/FEM

pressure

k - 5.0

inlag C0l11pOncnl

Figure 43: FEM/IEM. Scattering of a plane wave on a rigid cylinder with sphericalincaps, k = 5. Imaginary part of the scattered pressure obtained with a mesh of 648 3Dfinite elements (I = 3, 11, = 216) of order p = 3, q = 2, and 216 infinite elements with 6terms in the radial direction

38

h-p

II::M/I:I::M

k = ."'l,O

"hslI!tllc vaillc

Figure 44: FEM/IEM. Scattering of a plane wave on a rigid cylinder with sphericalincaps, A: = 5. Absolute value of the scattered pressure obtained with a mesh of 648 3Dfinite elements (l = 3, n = 216) of order p = 3, q = 2, and 216 infinite elements with 6terms in the radial direction

h-p

IEM/FEM

p,'cssurc

k = 10,0

real conlponenl

Figure 45: FEM/IEM. Scattering of a plane wave on a rigid cylinder with sphericalincaps, k = 10. Real part of the scattered pressure obtained with a mesh of 648 3Dfinite elements (I = 3, n = 216) of order p = 3, q = 2, and 216 infinite elements with 6terms in the radial direction

39

h-p

II~MII:I~M

pressurc

k = 10.0

ilnag C(lI11PtHICIlI

Figure 46: FEMjIEM. Scattering of a plane wave on a rigid cylinder with sphericalincaps, I.: = 10. Imaginary part of the scattered pressure obta.ined with a mesh of 648:3]) finite elements (l = 3, 17, = 216) of order p = 3, q = 2, and 216 infinite elements with6 terms in the radial direction

h-p

IEM/FEM

prcssUI'C

k = 10.0

absolute value

Figure 47: FEMjIEM. Scattering of a plane wave on a rigid cylinder with sphericalincaps, I.: = 10. Absolute value of the scattered pressure obtained with a mesh of 648:3D finite elements (l = 3, 17, = 216) of order p = 3, q = 2, and 216 infinite elements with6 terms in the radial direction

40

Il-p

IEM/I:I~M

pressun."

k ~ 1.0

H."a I c.:OlllpOnL'1l1

Figure 48: FEM/IEM. Scattering of a plane wave on a. rigid cylinder, /.;= 1. Real partof the scattered pressure obtained with a mesh of 648 3D finite elements (l = 3, 11, = 216)of order p = 3, q = 2, and 216 infinite elements with 6 terms in the radial direction

h-p IlEM

k ~ 1.0

real conlponenl

Figure 49: BEM. Scattering of a plane wave on a rigid cylinder, k = 1. Real part of thescattered pressure obtained with a mesh of 216 2D finite elements of order p = :3

41

Il-p

II:M/F'-':M

Figure 50: FEMjIEM. Scattering of a plane wave on a rigid cylinder, k = 1. Imagina.rypart of the sca.ttered pressure obtained with a mesh of 648 3D finite elements (I = 3,n = 216) of order p = 3, q = 2, and 216 infinite elements with 6 terms in the radialdirection

h-pllEM

pressure

k = 1.0

imag componenl

Figure 51: BEM. Scattering of a plane wave on a rigid cylinder, I.: = 1. Imaginary partof the scattered pressure obtained with a mesh of 216 2D finite elements of order p = 3

42

h-p

k = 1.0

ahsolule vallI{'

Figure 52: FEMjIEM. Scattering of a plane wave on a rigid cylinder, I.: = 1. Absolutevalue of the scattered pressure obtained with a mesh of 648 3D finite elements (l = 3,11, = 216) of order p = 3, q = 2, and 216 infinite elements with 6 terms in the radialdirection

-

II \

/ \

-

h-pDEM

pressure

k = 1.0

ahsolule value

Figure 53: BEM. Scattering of a plane wave on a rigid cylinder, k = 1. Absolute valueof the scattered pressure obtained with a mesh of 216 2D finite elements of order p = 3

43

11-P

p'·CSSIII·C

k = .:".0

I-cal COlllponClll

Figure 54: FEMjIEM. Scattering of a plane wave on a rigid cylinder, I.: = 1. Real partof the scattered pressure obtained with a mesh of 648 3D finite elements (l = 3, n = 216)of order p = 3, q = 2, and 216 infinite elements with 6 terms in the radial direction

h-p

[EM/FEM

prcssurc

k = 5.0

ilTIag COlllponcnt

Figure 55: FEMjIEM. Scattering of a plane wave on a rigid cylinder, k = 5. Imaginarypart of the scattered pressure obtained with a mesh of 648 3D finite elements (I = 3,n = 216) of order p = 3, q = 2, and 216 infinite elements with 6 terms in the radialdirection

44

h-p

k = ."\,0

ahsolulc vallie

Figure 56: FEM/IElVl. Scattering of a plane wave on a rigid cylinder, k = 5. Absolutevalue of the scattered pressure obtained with a mesh of 648 3D finite elements (I = :3,n = 216) of order p = 3, q = 2, and 216 infinite elements with 6 terms in the radialdirection

h-p

IEM/FEM

pressure

k = 10.0

real <":Olllponcnl

Figure 57: FEM/IEM. Scattering of a plane wave on a rigid cylinder, /.;= 10. Real partof the scattered pressure obtained with a mesh of 648 3D finite elements (I = 3, n = 216)of order p = 3, q = 2, and 216 infinite elements with 6 terms in the radial direction

4.5

h-p

II ~MIr:1 ':M

k = to.O

illlag c:otnponCl11

Figure 58: FEM/IEM. Scattering of a plane wave on a rigid cylinder, I.: = 10. lmaginar.ypart of the scattered pressure obtained with a. mesh of 648 3D finite elements (I = :3,n = 216) of order p = 3, q = 2, and 216 infinite elements with 6 terms in the radialdirection

h-p

[EM/FEM

pressure

k = 10.0

ahsolUlC v,alue

Figure 59: FEM/IEM. Scattering of a plane wa.ve on a rigid cylinder, k = 10. Absolutevalue of the scattered pressure obtained with a mesh of 648 3D finite elements (I = :3,n = 216) of order p = 3, q = 2, and 216 infinite elements with 6 terms in the radialdirection

46

uniform hp-finite-infinit.e element mesh. The error indicator, given in the previous sec-tion, is computed for every :3D FE, after solving the scattering problem using the FE-IEmethodology. Each element nt, for which the relative error indicator 17/" is greater t.han(1. treshhold value T, will be refined, i.e. if

T ryl ( .3)171 = - > T 4.,17t

where ryt denotes the maximum element error indicator over the whole mesh. The com-pa.tibility between the FE and IE approximations is enforced by refining infinite element.sadjacent to the refined finite elements. This means that an enrichement. of t.he polyno-mial degree in the angular direction of t.he :3D FE or/and the h4-refincment. of the :3DFE will also be carried out for the adjacent IE.

The scattering problem is solved on the refined mesh, and the error indicators areca.lculated again, with the whole procedure repeated. More precisely, the follO\vinga.lgorithm has been used.

1. Compute the numerical solution on an initial mesh.

2. Perform a uniform p-refinement.

3. Compute the numerical solution on the refined mesh.

4. Compute the error indicator using the numerical solutions obtained in steps 1 and3.

5. Perform a non-uniform p-refinement for elements, where ry[ > T.

6. Compute the numerical solution on the non-uniform mesh.

7. Compute the error indicator using the numerical solutions obtained in steps 3 and6.

8. Perform a non-uniform h-refinement for elements, where 17[ > T.

9. Compute the numerical solution on the non-uniform hp-mesh.

Figure 60 shows a sample hp-finite element mesh obtained with the above procedureby assuming an incident plane wave with wave number k = 5 and using for the initialmesh: N = 6, i.e. six shape functions in the radial direction, q = 2 for the polynomialdegree in the radial direction of the finite elements, p = 2 for the polynomial degree inthe angular direction, and 2 layers of 3D finite elements with 24 elements per layer, i.e.there are also 24 infinite elements present. The value for T is selected as T = :3.5%.

47

,2

i

Figure 60: Adapted hp-finite-infinite element. mesh.

5 Conclusions

The work reports on numerical experiments on hp-adaptive, coupled IE/FE discretiza-tions for the 3D Helmholtz equation in exterior domains of arbitrary shape. The IE/FEapproximation presents a challenging alternative not only for the traditional boundaryelements (BE) but also for the formulations based on various implementations of the socalled Dirichlet-to-Neumann (DtN) operator, see ego [19].

Contrary to the BE and DtN methods, the IE approximation is local and can beeasily built into any FE code using standard Co elements, including the hp-adaptive FEdiscretizations.

The presented numerical results were confined to three cases:

• a sphere within the unit sphere,

• a (finite) cylinder within the unit sphere,

• a (finite) cylinder with spherical incaps within the unit sphere.

48

The sphere problem admits an exact solution and was the basis for a numericalconvergence study of the coupled FEIVI/IEM. In the remaining two cases the IE/FE so-lutions were compared with numerical solutions obtained using the BE method presentedin [11]. The presented numerical evidence indicates that the method converges.

In conclusion, several remarks suggest themselves:

1. The numerical results indica.te that it is impossible, in general, to resolve the exte-rior boundary-value problem with a finite number of approximating basis functionsin the radial direction.

2. The method is stable for arbitrary lip FE discretizations in between the scattererand the truncating surface, making the lip mesh adaptat.ion possi ble.

:3. In a more sophisticated implementation, the number of basis functions in the radialdirection may vary from a node to a node and can be adapted during the solution.

4. Derivation of an a-posteriori error est£mato'J' indicating the error correspondingboth to the 3D FE and the IE (in t.he radial direction) discretizations is necessaryfor a successful large scale implementation of the method.

Acknowledgement: The work has been supported by the Office of Naval Researchunder Contract N00014-89-.J-14.51.

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