Spherical Harmonic Solution of the Robin Problem for the Helmholtz Equation in a Supershaped Shell

Spherical Harmonic Solution of the Robin Problem for the

Helmholtz Equation in a Supershaped Shell

*

Diego Caratelli1, Johan Gielis2, Ilia Tavkhelidze3, Paolo Emilio Ricci4

1_{Microwave Sensing, Signals and Systems, Delft University of Technology, Delft, The Netherlands} 2_{Department of Bioscience Engineering, University of Antwerp, Antwerp, Belgium}

3_{Faculty of Exact and Natural Sciences, Tbilisi State University, Tbilisi, Georgia} 4_{Faculty of Engineering, Campus Bio-Medico University, Rome, Italy}

Email: d.caratelli@tudelft.nl

Received August 11, 2012; revised September 11, 2012; accepted September 19, 2012

ABSTRACT

The Robin problem for the Helmholtz equation in normal-polar shells is addressed by using a suitable spherical har-monic expansion technique. Attention is in particular focused on the wide class of domains whose boundaries are de-fined by a generalized version of the so-called “superformula” introduced by Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica© _{is developed in order to validate the proposed methodology. In}

this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained. Keywords: Robin Problem; Helmholtz Equation; Spherical Harmonic Expansion; Gielis Formula; Supershaped Shell

1. Introduction

Many problems of mathematical physics and electro- magnetics are related to the Laplacian differential opera- tor. Among them, it is worth mentioning those relevant to the Laplace and Helmholtz equations. However, most of the mentioned differential problems can be solved in explicit way only in canonical domains with special symmetries, such as intervals, cylinders or spheres [1]. The solution in more general domains can be obtained by using the Riemann theorem on conformal mappings and the relevant invariance of the Laplacian [2]. However, it is clear that conformal mapping techniques can not be used in the three-dimensional case where approaches based on suitable spatial discretization procedures, such as such as finite-difference or finite-element methods, are usually adopted [3].

Different techniques have been proposed in order to solve the mentioned class of differential problems both from a theoretical and numerical point of view (e.g., rep-resenting the solution by means of boundary layer tech- niques [4], solving the corresponding boundary integral equation by iterative methods [5], approximating the relevant Green function by means of the least squares fitting technique [6], solving the linear system relevant to an elliptic partial differential equation by means of re- laxation methods [7]). However, none of the contribu-

tions already available in the scientific literature deals with the classical Fourier projection method [8] which has been extended in recent papers [9-16] in order to address boundary-value problems (BVPs) in simply con- nected starlike domains whose boundaries may be re- garded as an anisotropically stretched unit circle or sphere centered at the origin.

In this contribution, a suitable methodology, based on the theory of spherical harmonics [17], has been devel- oped in order to compute the solution of the Robin prob- lem for the Helmholtz equation in normal-polar shell-like domains. In particular, the boundaries of the considered domains are supposed to be defined by a generalized version of the so-called Gielis formula (also known as “superformula”) [18]. Regular functions are assumed to describe the boundary values, but the proposed approach can be easily generalized in case of weakened hypothe- ses.

In order to verify and validate the developed technique, a suitable numerical procedure based on the computer algebra system Mathematica© _{has been adopted. By us-}

ing such procedure, a point-wise convergence of the spherical harmonic series representation of the solution has been observed in the regular points of the boundaries, with Gibbs-like phenomena potentially occurring in the quasi-cusped points. The obtained numerical results are in good agreement with theoretical findings by Carleson [19].

*_{This research has been partly carried out under the grant PRIN/2006}

2. The Laplacian in Stretched Spherical

Coordinates

Let us introduce in the real space the usual spherical co- ordinate system: sin cos , sin sin , cos , x r y r z r              (1)

and the polar equations:



,



,

rR   (2) relevant to the boundaries of the supershaped shell which is described by the following chain of inequalities:





,







R   _{ }r R ,_{ (3)} ,

with 0  π and 0  2π. In (2), R



 ,



_are

assumed to be piece-wise functions satisfying the condition:

2

C



,





,



0, 0 π, 0 2π.

R   _R   _{  } _{ } ₍₄₎ In this way, upon introducing the stretched radius  such that:



 

,

 

 

,



, b R a R r b a              (5)

with , the considered shell-like domain can be readily obtained by assuming

0

b a  

0  , 0 π    and

2π a  .  b

Remark: Note that, in the stretched coordinate system

, ,

   the original domain is transformed into the spherical shell of radii a and b, respectively. Hence, in this system one can use classical techniques to solve the Helmholtz equation, including the eigenfunction method [1].



Let us consider a piece-wise function ν (x, y, z) 2 C  _{ }     cosr



sin cos , sin sin , , ,

v r   r  



u r







 

the Laplace operator, , in spherical coordinates:

2 2 2 2 2 2 2 1 1 _sin sin 1 . sin u u u r r r r r u r               _{ } _  _            (6)

In the considered stretched coordinate system,  can be represented by setting:







 

 

 



, , , , , . U b R a R u b a                   _ _    (7)

In this way, by denoting R



 ,



as R _{for the} sake of shortness, one can readily find:

, u b a U r R R   _      (8) 2 2 2 2 , u b a U r R R _2              (9) , R u U R R  U       _{ }  _     (10)













2 2 2 2 2 2 2 2 2 2 , R R R R R R u u R R R U R U R R R R           2_U                     _     _{ }            (11)













2 2 2 2 _{2 2} 2 2 2 2 , R R R R R R u U R R R U R U R R R R           2_U                     _     _{ }            (12) with R



b 



R



a



R   and   

    , and where the sub- scripts denote the partial differentiation with respect to the polar angles  and  . Substituting Equations (8)- (12) into Equation (6) finally yields Equation (13).

2 2 2 2 2 2 2 2 _{2 2} 2 2 1 sin 1 2 1 cot sin sin 1 cot sin R R b a U u R R R R R R R R R R R R R U R R R R R R R R R R R R U U U R                                     _{ }        _{  }  _{ } _{ }     _                   _  _  _ _    _              _{  }         2 2 2 2 2 _sin2 . R R R U U R R   _{   }      _    _      _{   }  _{ } _ (13)

As it can be easily noticed, upon setting R



 ,



_ 0

a and , the classical expression of the Laplacian in spherical coordinates is recovered.





,

R   _{  1}b



3. The Robin Problem for the Helmholtz

Equation

Let us consider the interior Robin problem for the Helmholtz equation in a starlike shell , whose bounda- ries are described by the polar equations

   



, rR   _{respectively:}





























2 , , , , 0, , , , , , , , , , , , , , v x y z k v x y z x y z v v x y z x y z f x y z x y z                    _        (14)

where k0 denotes the propagation constant, ˆ _



ˆ ,



   _{are the outward-pointing normal unit vectors}

to the domain boundaries __, respectively, and __,

 _{are given regular weighting coefficients.}

Under the mentioned assumptions, one can prove the following theorem. Theorem. Let:



,





_

,



_

, R R   ,            (15)



,



_



_

,



, , sin R R               (16)









2





,  ,  ,    _    _   2 , (17) and



























, ,



0 0 , sin cos ,

, sin sin , , cos ,

cos cos sin ,

n m n n m n m n m f R R R F P m              m                 

 

 (18) where:











 



, , 2π π 0 0 ! 2 1 4π ! cos , cos sin d d sin n m m n m m n n m n n m m F P m    ,                _{ }   _          

 

 (19)

m being the usual Neumann’s symbol and

 m

 

n

P  the associated Legendre function of the first kind and orders n, m. Then, the boundary-value problem (14) for the Helmholtz equation admits a classical solution v x



, ,y z



 

2 _

L

 such that the series expansion (20) holds true. In Equation (20)  

 

 1 2

 

1 π 2 n n h z  zH z denotes the spherical Hankel function of kind and order n. For each pair of indices

1, 2  

 

: n₀  0 n and 0, 1, ,

m  introduce the terms reported in Equa-tion (21), where:  

_{ }

 

_{ }

 

_{ }

 

_{ }

1 d , d n n n n n h z h z h z h z z  z          ₍₂₂₎ and

 

 

  



1

 

2 d _. d 1 m m n n m m n n nzP z n m P z P z P z z z        ₍₂₃₎

Therefore, the coefficients , , , , in (20) can

be determined by solving the infinite linear system: , n m n m A B





















 1





_

_

 2





_

_

1, , 1, , 2, , 2, , 0 0 , , , , , ,

sin cos , sin sin , cos

, ,

, , , ,

cos cos cos sin ,

n m n n n m n m n n m n m n m R R R v b a b a b a U R R P h k A m B h k A m B m b a b a                                   _{  }                             

 

 (20)









 

















 



_

_



 



_

_



























, , , , 2 , , cos , , _{, cos} sin , cos ,

cos cos sin

sin _{1 ,} sin cos _,

sin , n m m n n n m m n n m n n h kR P h kR _P k R P h kR m m m m m m _m R                              _{  }   _{  }                                                                       _ ,   (21)

1,1, , , , 1,1, , , , 1,2, , , , 1,2, , , , 2,1, , , , 2,1, , , , 2,2, , , , 2,2, , , , 1,1, , , , 1,1, , , , 1,2, , , , 1,2, , , , 2,1, , , , 2,1, , , , 2,2, , , , 2,2, p q n m p q n m p q n m p q n m p q n m p q n m p q n m p q n m p q n m p q n m p q n m p q n m p q n m p q n m p q n m X Y X Y X Y X Y X Y X Y X Y X Y                1, , , 1, , , 0 0 2, , , 2, , , , , , , n m p q n n m p q n m n m p q n m p q n m p q A B A B                       _ _                

 

              (24) where:

 











 



2π π , , 1 , , , , ,₂ 0 0 ! cos 2 1 , cos sin d d sin 4π ! q q n m p p q n m p q q p X P q p q  ,          _        

 

 _{ }   (25)

 











 



2π π , , 1 , , , , ,₂ 0 0 ! cos 2 1 , cos sin d d sin 4π ! q q n m p p q n m p q q p Y P q p q  ,          _        

 

 _{ }   (26) with 1, 2 and ₀ .

Proof: Upon noting that in the stretched coordinate system

, 0, 1, ,

p q  p

, ,

   introduced in the x y z, , space, the considered domain turns into the spherical shell of radii a and b, one can readily adopt the usual eigenfunc- tion method [1] in combination with the separation of variables (with respect to



,

r  and  ). As a conse- quence, elementary solutions of the problem can be searched in the form:





















     

, , , , , , , , . u r b r R a r R U R R P                     _  _ _  _             (27)

Substituting into the Helmholtz equation, one easily finds that the functions P

   

  , , and  

 

must satisfy the ordinary differential equations:

 

_{   }

_{ }

2 2 2 2 2 d d 2 0, d d P r P r r r kr P r r r      (28)

 

2

_{ }

2 2 d 1 d sin 0, sin d d sin               _  _        (29)

 

_{ }

2 2 2 d 0, d         (30) respectively. The parameters  and  are separation constants, whose choice is governed by the physical re- quirement that at any fixed point in space the scalar field



, ,

u r  must be single-valued. So, by setting:



, m    (31)



 

2 0 1 , n n n     



(32) one can easily find:

 

 amcosm bmsinm ,     (33)

 

,



cos



, m n m n c P     (34)

where a b c_m, ,_{m n m,}   denote arbitrary constants. In order to identify the radial function P

 

 introduced in (27), it is convenient to set:

   

12

 

.

P r  kr   r (35) In this way, it is readily shown that 

 

r satisfies:

 

_{   }

2

_{ }

2 2 2 2 d d 1 0, d 2 d r r r r kr n r r r     _{ }    _  _         (36)

and, hence, is a cylinder function of half order that, without loss of generality, can be expressed as a linear combination of ordinary Hankel functions of first and second kind must, so that:

 

 1

_{ }

 2

_{ }

_,

n n n n

P r d h kr e h kr (37) with d en, n. Therefore, the general solution of the

Robin problem (14) can be searched in the form:







  





 





1 1, , 1, , 0 0 2 2, , 2, , , ,

cos cos sin

cos sin . n m n n n m n m n m n n m n m u r P h kr A m B h kr A m B m   m           _     _

 

(38) Enforcing the Robin boundary condition yields:









































, , , , , , , ˆ , , , , , , , F u u R R u R u R              ,                                (39) where:

















, , _ˆ1 , , ˆ , , , , 1 ˆ , sin u r u r u r r r r u r r                        (40)

and:

















2 ˆ _ˆ ˆ , , ˆ , . 1 , r                         (41)

Hence, combining equations above and using a classi-cal harmonic projection method, the Equations (21)-(26) follow after some algebraic manipulations.

It is worth noting that the derived expressions still hold under the assumption that R



 ,



_{are piecewise}

con-tinuous functions, and the boundary values are described by square integrable, not necessarily continuous, func-tions, so that the relevant spherical harmonic coefficients

, , ,

n m n m

  _{in Equation (19) are finite quantities.}

4. Numerical Procedure

In the following numerical examples, let us assume, for the boundaries of the considered annulus, general polar equations of the type:

  





0 1 , 1 _{sin cos} 2 4 1 1

sin sin cos ,

2 4 2 x y z x x x y y z y z R a b d a b a d d                                     (42)

which provides an extension, to the three-dimensional case, of the “superformula” introduced by Gielis in [18] . Very different characteristic geometries, including ellip-soids, ovaloids, and Lamé-type domains (also called “superellipsoids”) can be obtained by assuming suitable values of the parameters a a a b b d d dx, y, z, , ,x y x, y, z,x,

0

, ,

y z

     _{in (42). It is to emphasized that almost all}

three-dimensional normal- polar shell-like domains can be described, or closely approximated, by the considered formula.

In order to assess the performance of the proposed

methodology in terms of numerical accuracy and con- vergence rate, the relative boundary error has been evaluated according to Equation (43), where  is the usual _L2 norm, and



_{, ,}

N N

U U   



denotes the partial sum of order N relevant to the spherical harmonic expansion representing the solution of the bound-ary-value problem for the Helmholtz equation (see Equa-tion (44)).

Remark: It is to be noted that, where the boundary values exhibit a rapidly oscillating behavior, the number N of terms in the spherical harmonic expansion approxi-mating the solution of the problem should be increased accordingly in order to achieve the desired numerical accuracy. Example Upon assuming in (42) ax ay az 2, 4,bx by _  _ _  _  _ 4 5, 1, 3, d_y d_z _x _y _z x y z x d_d _d _ d nd      

a 0 2, the shell  turns to feature a cuboidal shap _{f x y z}



_{, ,}



_{n 2x cos3y e}2y23z21

    and

e. Let si



, ,



f x y z x y z3 3 3ex y z x 2y3z be the f

bing the bo ary values. Pr

unc-ovided that the propagation constant is k 2π

tions descri und

 , and  2 5, 1,  1 10, 1 2

_ _   are the weighting coefficients in the Robin condition, the relative boundary error eN as

function of the number N of terms in the truncated ries expansion (44) exhibits the behavior shown in Figure 1. As it appears from Figure 2, the selection of the expan-sion order N 11

se

 leads to a very accurate representa-tion of the solurepresenta-tion vN



x y z, ,



, whose spherical

har-monic coefficients A, , , ,n m 1, 2







are

plot-ted in Figure 3. Remark: It has be

n m and B

en observed that _L2 norm of the dif-ference between the exact solution and the relevant ap-proximation is generally negligible. Point-wise conver-gence seems to be verified in the considered domains, with the only exception of a set of measure zero consist-ing of quasi-cusped points. In the neighbourhood of these points, oscillations of the truncated order solution, recall- ing the classical Gibbs phenomenon, usually take place.

2 2 2 2 , N N N N N U U U F U F e F F                                      (43)









 





_

_

 





_

_

1 1, , 1, , 0 0 2 2, , 2, , , ,

, , cos cos sin

, , cos sin . N n m N n n n m n m n m n n m n m R U P h k A m B b a R h k A m B m b a                       _            _       

 

m (44)

Figure 1. Relative boundary error eN as function of the order N of the truncated spherical harmonic expansion representing

the solution of the considered Robin problem for the Helmholtz equation in the supershaped shell described by the generalized Gielis formula with parameters

 , , , , , _ _ _ _ _ _ _ _ _ _ _ _ _ _ 2 4 4 5 1 x y z x y x y z x y x y z 3 a a a b b d d d d d dz ν ν ν . _ 0 2 ν (a) (b) Figure 2. Boundary behavior along  (a) and  (b) of the partial sum UN



  , ,



pershaped sh

    of order N 11 representing 

the solution of the considered Robin lem for th lmholtz equation in the su ell

5. Conclusion

A harmonic projection meth

ordinate , has obin for xpressions of the solution of the con- Ps can be derived by using classical

adrature rules, so overcoming the need for cum er- h as finite-difference or oposed approach has been lidated by dica

puter-aided algebra tool Mathematica . A point-wise convergence of the expan-

prob e He  .

od, in combination with the some numerical techniques suc finite-element methods. The pr adoption of a suitable stretched co system

been developed for solving the R problem the Helmholtz equation in supershaped shell. In this way,

successfully va means of a de ted numerical procedure based on the com

©

analytically based e sidered class of BV

qu b

(a) (b)

Figure 3. Magnitude of the coefficients A, ,n m and B, ,n m



 1 2,



relevant to the spherical harmonic expansion vN



x y z, ,





N 11



representing the solution dere the Helmholtz equation in the supershape

with the only exception of a set of measure zero consist- ing of the quasi-cusped points along the boundary of the problem domain. In these p

und d

series.

[3] A. Bondeson, T. Rylander and P. Ingelstrom, “Computa- tional Electro ience, New Y 2005.

thematics, Vol. 44

of the consi d Robin problem for d shell  .

oints, Gibbs-like oscillations may occur. The computed results are fo to be in goo agreement with the theoretical findings on Fourier

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