A Numerical Approximation Method for a NonlocalOperator Applied to Radiation Problem

H M Nasirand T Kako

Abstract In the nite element approximation of the exterior Helmholtz problem we propose anapproximation method to implement the DtN mapping formulated as a pseudodierential operatoron a computational articial boundary The method is then combined with the ctitious domainmethod Our method directly gives an approximation matrix for the sesquilinear form for the DtNmapping The eigenvalues of the approximation matrix is simplied to a closed form and can becomputed eciently by using a continued fraction formula Solution outside the computationaldomain and the fareld solution can also be computed eciently by expressing them as operationsof pseudodierential operators An inner articial DtN boundary condition is also implemented byour method We prove the convergence of the solution of our method and compare the performancewith the standard nite element approximation based on the Fourier series expansion of the DtNoperator The eciency of our method is demonstrated through numerical examples

Introduction

We consider the following twodimensional exterior Helmholtz problem

u ku in RnO a

u

n uinc

non b

limr

pr

u

r iku

c

where is the interior of the complement of a bounded region O in R with smooth boundary on which the Neumann boundary condition b is imposed and c is the Sommerfeld radiationcondition at innity

The equation can be used to simulate the scattering phenomena of timeharmonic electromagneticor acoustic wave by an obstacle O which is sometimes called a scatterer Here uincx eikx is thetimeharmonic incident plane wave whose direction of propagation is given by the vector k and nis the outward unit normal on the scatterer see Fig

O

Ω nΓR

O

nΩ R

Figure Obstacle and articial boundary

In order to solve the exterior Helmholtz problems numerically it is a common practice to introduce an articial boundary to limit the area of computation and to prescribe an articial boundarycondition on this boundary The boundary condition is expected to absorb the outgoing wavesand to exclude any incoming waves Various articial boundary conditions have been proposed inthe literature for this purpose see Givoli Ihlenburg and the references therein The articial boundary condition that gives the solution to is given by the Dirichlet to Neumann DtN mapping

University of Peradeniya Sri Lanka Email nasirhmfeduuecacjpThe University of ElectroCommunications Chofu Japan Email kakoimuecacjp

In the nite element approximation of the problem the implementation of the DtN mapping orits approximations has been a subject of interest by many authors see for example Kako Liu Liu and Kako and the references therein As for the case of using the exact DtN mappingMacCamy and Marin used an integral representation of the DtN mapping and obtain its niteelement matrix by explicitly solving some auxiliary integral equations Keller and Givoli usedthe Fourier series representation of the DtN mapping and use the standard nite element techniqueto obtain the matrix in an innite series form see also Ernst and Heikkola et al

In this paper we propose an approximation method to implement the DtN mapping by expressingit in a form of pseudodierential operator The nite element approximation corresponding to thesesquilinear form of the pseudodierential operator is given by a matrix which we call a mixedtype approximation matrix This matrix is obtained by replacing the argument of the functionin the pseudodierential operator which in this case is the Laplacian on the unit circle by itsnite element matrix This gives a matrix in a closed form which can be eciently computed by acontinued fraction without use of the Hankel function and its derivative The computational costfor the boundary condition in this method is On where n is the number of partitions in angulardirection

When the origin of the polarcoordinate system is outside the obstacle domain one can consideran inner articial boundary that excludes the origin from the computational domain and anotherDtN boundary condition is imposed on the inner articial boundary which is also treated by ourmethod

The solution outside the computational domain and the fareld pattern are expressed in closedforms by using pseudodierential operators and our previous method can also be applied to computethe quantities

We consider the ctitious domain method to form the linear equations and use the Krylovsubspace iterative method to solve the linear system Kuznetsov et al Heikkola et al

The rest of the paper is organized as follows In Section we review the articial boundarycondition and its standard nite element approximation In Section we introduce a mixed typemethod for the articial boundary and its application in ctitious domain method In Section we consider the application of the mixed type method for the solution outside the computationaldomain and the fareld pattern In Section we prove the convergence of the solutions We presentthe results of numerical tests in Section and make some concluding remarks in Section

Articial boundaries and articial boundary conditions

For the numerical treatment of the problem the unbounded domain is truncated by an articialboundary denoted by R and an articial boundary condition is introduced The articial boundaryis a circle of radius R and we denote by BR the circular domain of radius R bounded by R Theapproximate boundary value problem is then given by

u ku in R BR a

u

n uinc

non b

u

r Mu on R c

where M is the DtN mapping which we regard as a pseudodierential operator as a function of theLaplacian operator D and is given by

MD uR k

Xn

HkRn HkRn

Z

uR eind

kHkR

pD

HkRpD

uR

where we denote by Hx the Hankel function of the rst kind of order The basic denitionof pseudodierential operator can be found for example in Nirenberg and Taylor

Weak formulation and FEM

Let V HR where HsR is the Sobolev space of order s R in R and HR

HR be the trace operator Then the weak formulation of the boundary value problem isFind u V such that

au v hu viM uincn v v V

where the sesquilinear forms a h iM and respectively are

au v

ZR

u

r

v

r

ru

v

kuv

rdrd u v HR

hp qiM

Z

Mp q Rd p q HR

and f g

Z

fgd f g L

Now based on the element partitioning of the computational domain described in Subsection we form a nite dimensional subspace Vh of V The nite element approximate problem is thengiven by Find uh Vh such that

auh vh huh vhiM uincn vh vh Vh

FEM matrix of DtN mapping by the Fourier mode representation

The nite element approximation matrix corresponding to the DtN mappings given in the form of has been obtained by several authors eg Ernst

According to the nite element partitioning of R the articial boundary R is discretized bya uniform partitioning with n nodes and an equal number of intervals We use piecewise linearcontinuous functions figni as the basis for the nite element approximation The sesquilinearform corresponding to the DtN mapping is represented in terms of the Fourier modes as

huh vhiM

Xn

RMn cuhncvhn

where phn is the Fourier coecient of ph given by phn p

R

ph eind

We express ph Pn

j phjj and set ph ph ph phnT By performing theintegration we get cuhn nQhnfuh where

ph n cosnh n

h for

n Qhn einn einnn pn and h n Clearly Qhj Qhlnj for j n l Z Substituting cuhn and cvhn in we have

huh vhiM

Xn

fvhTQThnRMn nQhnfuh

nXj

fvhTQThj

Xl

RMln j lnjQhj fuh Mstd

h fuh fvh Cn

where Mstdh Qh

stdh Qh Qh is the unitary matrix formed by the rows Qhj j n and

stdh

is a diagonal matrix whose jth diagonal element is the eigenvalue ofMstdh and is given by

stdhj

RM h j

cos jh h

Xl

RMln j

ln j j

Note that stdhj stdhnj From the estimate jMn j C jnj see Masmoudi the sumtends to RMj j and cos jh h j as h Thus we get the following facts for j n

stdhjh RMj as h

jstdhj j Ch jjj

A mixed type method

We propose a method which gives an approximation matrix directly for the sesquilinear formhu viM The matrix is circulant and its eigenvalues are one term expression which can be computed eciently by means of a continued fraction see Section The standard nite elementmatrixMstd

h is then replaced by this matrix in the linear equations to be solvedWith the same partition and basis functions considered in the last section the nite element

matrices corresponding to the sesquilinear forms u v L and u v L respectively aregiven by

Ah

hCirc Bh

hCirc

where we denote by Circa b c the circulant matrix for which the main diagonal is formed by b andthe lower and upper diagonals are formed by a and c respectively

Denition A mixed type approximation matrix corresponding to the operatorMD is denedby

Mmixedh BhRMBh Ah

where the matrices Ah and Bh are given in

In the error analysis we introduce a sesquilinear form corresponding to this matrix SinceMmixed

h is circulant it can be expressed as Mmixedh Q mixed

h Q as in the standard FEM case

The jth eigenvalue ofMmixedh is given by mixed

hj RMhj Bhhj where hj

Ahhj

Bhhj

Ahhj

cos jh h and Bhhj h cos jh Clearly we have

mixedhj mixed

hnj and the similarestimates to and hold for mixed

hj as well as stdhj

Continued fraction

In this subsection we present an ecient computation of the logarithmic derivatives of the Bessel andHankel functions which appear in the DtN mappings The key idea is to use continued fraction formsfor the logarithmic derivatives These continued fractions are rapidly converging and an ecientalgorithm for computing them is readily available as the modied Lentz!s method Thompson andBarnett The continued fraction for the DtN mapping on the exterior articial boundary isgiven by

xH x Hx

ix

i

x i

x i

where x kR and the continued fraction for the DtN mapping on the inner articial boundary isgiven by

xJ x Jx

x

x

x

where x krThese continued fractions converges for all values of and x except those in the neighborhood

of zero It converges very rapidly for x p

Fictitious domain method

In order to solve the problem with general obstacle we use the ctitious domain method to form the approximation subspace Vh For this the computational domain R is extended to actitious domain FR which is a circular annulus and includes the obstacle boundary When theobstacle is not narrow and contains a larger neighborhood of the origin R is extended inside theobstacle to form the ctitious domain When the obstacle is thin we choose the polar coordinatesystem such that the origin is outside of the obstacle and FR is obtained as the union of R andthe obstacle domain O see Fig

Now the annulus ctitious domain is partitioned by an orthogonal polar mesh The nodes of themesh next to the boundary of the obstacle O are shifted onto the boundary and the modiedquadrilateral elements in the computational domain are triangulated such that the resulting meshgives a shape regular triangulation B"orgers This leads to a locally tted mesh which istopologically equivalent to the original mesh and diers from it only in an hneighborhood of theobstacle boundary The mesh inside the obstacle domain is discarded to obtain the mesh for R

The approximation subspace Vh consists of functions uh such that the restrictions of uh in theunmodied rectangles are bilinear and the restrictions on the triangles near the obstacle boundaryare linear

Figure Fictitious domains and locally tted mesh

For more details on the ctitious domain method see Kuznetsov and Lipnikov and Heikkola

Further applications

The mixed type method can be used in other cases of radiation problems where pseudodierentialoperators appear We consider cases of an inner articial boundary computing solution outside thecomputational domain and computing the far eld pattern

Inner articial boundary

When the obstacle does not contain the origin one can introduce an inner articial boundary rwhich is a circle of radius r Then we consider the computational domain R which also excludesthe disc of radius r and we impose an inner DtN boundary condition on r given by

u

r Nu k

J krpD

JkrpD

ur on r

where Jx is the Bessel function of order Its corresponding sesquilinear form hu viN willbe added to the weak form In the nite element approximation we replace its standard FEMmatrix by the mixed type matrix Nmixed

h dened analogous to Denition

Solution outside the computational domain and far eld pattern

The solution on a circle of radius r outside the computational domain can be represented by serieswith respect to the solutions on the articial boundary For the exterior region the solution pr ur can be expressed as a pseudodierential operator form as follows

pr SD u

HkrpD

HkRpD

pR r R

and for the interior region the solution is given by

pr SD u

JkrpD

JkrpD

pr r r

The fareld pattern corresponding to the solution is obtained by using the asymptotic formula ofthe Hankel function in the solution and is given by

F D u

r

k

eipD

HkRpD

pR

In order to compute these solutions one can use the nite element method in which we applythe mixed type method The weak formulation of the generic form pr SD p is given bypr q hp qiS and hence using the uniform partition as before and using nite element methodwe get the matrix equation

BhPr Smixedh P

where the matrix Smixedh is given as in for the function S and Pr and P are column vectors

corresponding to pr and p respectively with respect to the nodal basis functions One cancancel the premultiplication of the matrix Bh on both sides of Hence computing the solutionis reduced to a matrix multiplication which can be performed eciently by using FFT Clearly thesolution at radius r is not coupled with solutions of the adjacent circles Hence in order to savecomputing time one can choose the minimum amount of circles for the solution that will providethe resolution of the waves As a rule of thumb one can choose radial intervals per wavelength

Convergence Analysis

Let au v RRrurvuv dx and bu v

RR

k uvdx and let PRh HR V R

h

be the orthogonal projection with respect to HR inner product where VRh fvh vh Vhg

We dene a sesquilinear form on HR corresponding to the mixed type method as

hp qimixedMh Mmixed

h PRh p PR

h q Cn p q HR

With astdM u v au v bu v hu viM and amixedMh uh vh auh vh buh vh

huh vhimixedMh we have the following problems

E astdM u v hf vi v V

E mixedh amixed

Mh uh vh hf vhi vh Vh

where hf vi uincr v In the following we denote by kks s R the norm on the Sobolevspace Hs R or R Ciarlet and Lion

Theorem Let u V be the solution of E Then there exists h such that for all h h there exist unique solutions uh Vh of E mixed

h such that

limh

ku uhkR

To prove the theorem we need some lemmas Let u and uh be the solutions of E and E mixedh

respectively and put eh u uh Since is smooth u HR Equating E and E mixedh

and adding and subtracting hu vhimixedMh we have

aeh vh beh vh heh vhimixedMh rhu vh

where rhu v hu viM hu vimixedMh Now we have the following lemmas

Lemma There exists a constant Ch with limh Ch and h such that for all h h

jrhu vh j Ch kukRkvhkR for all vh Vh

Lemma For every there exists a constant C h with limh C h such that

jamixedMh eh eh j kehkR C h kukR

Lemma There exist two constants Ch and Ch with limh Ch limh Ch such

that

jbeh eh j Ch kehkR Ch kukR

Proof Proof of Theorem Since ReM for all R Koyama we have Re heh ehimixed

Mh

Considering the real part of kehkR aeh eh amixedMh eh eh beh eh heh ehimixed

Mh

and lemmas and we have

kehkR Re amixedMh eh eh Re beh eh

kehkR C h kukR Ch kehkR Ch kukR

Hence we have Ch kehkR C h Ch kukR Choosing small enoughsuch that Ch we get kehkR C h Ch kukR as h

For the uniqueness if f then u by the solvability of E Then by the last inequalityeh uh Proof Proof of Lemma First we establish an estimate for kphksR s R Analogous to and with Mj j s and R we have

kphksR

nXj

Xl

ln j slnj jQhj phj

nXj

j sj jQhj phj C

nXj

h j sjQhj phj

due to the fact that j hsinjh jh Ch for j n

We write rhu vh huPRh u vhiMhPR

h u vhiMhu vhimixedMh I II From

standard estimates kIPRh u kmR Chm kukR m we get kIPR

h u k

R

Ch

kukR by interpolationSince the DtN operator is a bounded operator from HR into H

R Masmoudi we have

jI j CkI PRh u k

Rkvhk

R Ch

kukRkvhk

R

For the treatment of II we adjust the index range as n j n for simplicity Wehave from the estimates and for stdhj and mixed

hj that for an arbitrarily xed j there

exists Cj h with limh Cj h such that jstdhj mixedhj j Cj h h for all jjj j

and jstdhj mixedhj j Ch jjj for all j Now denoting Qu jQhj

PRh uj and Qvh

jQhj fvhj for brevity and using we getjII j jMstd

h Mmixedh PR

h u fvh j

Xjjjj

jstdhj mixedhj jQuQvh

Xjjjj

jstdhj mixedhj jQuQvh

Cj h Xjjjj

hQuQvh CXjjjj

h jjj QuQvh

Cj h kPRh ukRkvhkR

Cjjj

Xjjjj

h jjj Qu jjj Qvh

Cj h Cjjj kukRkvhk

R

Hence adding I and II we have jrhu vh j Ch kukRkvhk

R Ch kukRkvhkR

with Ch Ch Cj h Cjjj For an arbitrary we rst choose j such that

jjj and then we can see that there exists h such that Ch Cj h for

all h h

Proof Proof of Lemma By we have amixedMh eh eh amixed

Mh eh uuh amixedMh eh uvh

rhu u vh eh and hence

jamixedMh eh eh j CkehkRku vhkR Ch ku vh ehkRkukR

Ch Ch kukRkehkR Ch hkukR kehkR C h kukR

where C h Ch Ch

Ch h as h

Proof Proof of Lemma There exists a unique w HR such that astdM v w v k eh

for all v V and

kwkR CkehkR

where C is a constant independent of eh and w Using we have for all vh Vh

beh eh astdM eh w amixedMh eh w rheh w

amixedMh eh w vh rhu vh rheh w

amixedMh eh w vh rhuw vh rhuw rheh w

Now from the boundedness of rh in HR lemma and with the use of orthogonal projectionPh V Vh with respect to H

R inner product we have

jrhuw j jrhuw Phw j jrhu Phw j CkukRkw PhwkR Ch kukRkPhwkR Ch Ch kukRkwkR

jrheh w j jrhI Ph eh w j jrhPheh w j CkI Ph ukRkwkR Ch kehkRkwkR ChkukR Ch kehkR kwkR

Using jbeh ehj k kehk

R the boundedness of amixed

M and rh the fact thatinfvhVh kw vhkR ChkwkR and we have from

k kehk

R CkehkR kukR inf

vhVhkw vhk

fChkehkR Ch ChkukRgkwkR

kehkRfChkehkR ChkukRg

kehk

R CfC

hkehk

R C

hkuk

Rg

where Ch Ch Ch Ch Ch Ch Rearranging the inequality completes theproof

Numerical tests and results

We present in this section some of the results of numerical testings of our method for variousexamples We compare the eciency of our mixed type method with the standard FEM

All computations were carried out on VTAlpha Mhz MB RAM with Linux operatingsystem environment with double precision arithmetic using object oriented C codes The iterationscheme in solving the system of linear equations using ctitious domain method we use the transposefree quasi minimal residual TFQMR by Freund The residual tolerance was set to

Convergence testing

To test the convergence of the computed solutions and compare with the standard FEM solutions asthe mesh size decreases we consider an example of a circular obstacle of radius r with articialboundary radius R We choose the wave numbers k and and the incident waveas a plane wave in the xaxis direction For the nite element mesh we choose orthogonalpartition with size nr n ranging between For the standard nite elementapproach the innite series in eigenvalues are computed until machine precision is achieved Theresulting separable linear system is solved by using fast direct method with FFT

The maximum errors ku ustdh kR ku umixed

h kR against the angular partition size areshown in Fig a The maximum error between the two computed solutions kustdh umixed

h kR isshown in Fig b in logarithmic scale Both solutions converge linearly as well as their dierence

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

10 100 1000 10000 100000

mixed k = pi std k = pi

Mixed k = 2 pi std k = 2 pi

Mixed k = 10 pi std k = 10 pi

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

10 100 1000 10000 100000

k = pi k = 2 pi k = 10 pi

max

|u

- u

|st

d

mix

edh

h

Angular partition size (nt) Angular partition size (nt)

std

hm

ax|u

- u

|

,m

ax|u

- u

|m

ixed

h

κ = 2πκ = 10π

κ = π

κ=2π

κ=10π

κ = π

(a) (b)

Figure Convergence

Eciency testing

To test the computing time dierence between the methods we consider the rst test example aboveand an example with an elliptic shape obstacle with axes a and b The wave number

k We choose the articial boundary radius R and The radial and angular partitionsizes nr and n respectively The results are shown in Table

Table Comparison between MTM and standard FEMR Obstacle Std FEM time sec MTM time sec Iterations Circle both Ellipse both Circle both Ellipse both

We also considered an arc shaped obstacle and the Helmholtz resonator with the domain truncated by inner and outer articial boundaries The scattering waves and the fareld pattern are computed by using the formula for solution outside the computational domain From the fareld patternthe radar cross section RCS is computed by using the formula RCS log

jF j which

is in decibel units The total waves real part for circular arc with waves number k andscattering waves real part for the Helmholtz resonator with wave number k and their radarcross sections are shown in Fig

Figure Wave pattern for antenna and Helmholtz resonator

Figure Radar cross sections for antenna and Helmholtz resonator

Conclusions

In this paper we proposed a mixed type method for the nite element approximation of nonlocalradiation boundary condition written in the form of pseudodierential operator We dened amixed type approximation matrix to approximate the sesquilinear form corresponding to the DtNoperator The method is also eciently applied to compute the solution of the radiation problemoutside the computational domain and to compute the fareld pattern

Numerical tests show that the mixed type method is computationally ecient The convergenceis conrmed for the mixed type method and is observed to be of the same order as the standardnite element approximation

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