On the approximation of helmholtz's differential equation

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Voi,12(5),283-287, 1985. Printed in the USA. Copyright (c) 1985 Pergamon Press Ltd.

ON THE APPROXIMATION OF HELMHOLTZ'S DIFFERENTIAL EQUATION

Enayat Mahajerin Mechanical Engineering Department Saginaw Valley State College University Center, MI, 48710 USA

(Received 13 June 1985; accepted for print 5 August 1985)

Introduction

A straightforward approximation technique for Helmholtz's differential equation subjected to Robin's boundary conditions is presented. The idea is based on the collocation of the translated fundamental solution of Helmholtz's differential equation. From the known asymptotic behavior of the translated fundamental solution, a simple scheme is developed which is suitable for small computers. Accurate results at the interior as well as boundary points can be obtained without a great deal of effort. THis procedure is demonstrated by two cases and the results are Tabulated.

Analysis

The Helmholtz's partial differential equation subjected to generalized (Robin's) boundary conditions in a two-dimentional region R in the x-y plane (see FIGURE 1.) can be written as follows:

u,xx+u,yy+k2u=O

in R

(i)

au+bu,n=c

on ~ R

(2)

where D R is the boundary of R, n is outward normal to ~R, k 2 is constant (positive or negative), a and b are Robin's coefficients (a=l,b=O corresponds to Dirichlet boundary conditions; a=O,b=l corresponds to Neumann's boundary conditions), c is the prescribed boundary values, and comma denotes differentiation with respect to the argument(s) followed by the comma. The boundary value problem (]) and (2) governs a wide range of physical phenomena such as electromagnetics, fluids and solid mechanics. A variety of numerical methods have been applied to (i) and (2): finitedifferences [1], finite elements [2], and more recently boundary elements [3]. 283

284

E. MAHAJERIN

None of these methods are efficient and each has at least a major drawback. The finite difference method, although simple, is inaccurate, time consuming, and not suitable when ~ R is curved.

The finite element method is too general,

needs skill and requires considerable amounts of computational effort.

Moreover,

the finite element method has been applied to special cases only [2]. The boundary element method, although straightforward relative to the finite element method, requires complex functions such as Bessel's and Hankel's functions to express the required fundamental solution and its derivatives [3]. of of these functions the method is virtually useless.

In the absence

Also with the boundary

element method it is impossible to predict the solution on (or near) the boundary, R.

This paper represents a simple method which does not require any complex

functions and therefore suitable for small computers.

The heart of the formulation

is to select a suitable fundamental solution U, such that U,xx+U,yy+k2U= ~

(3)

subjected to no particular boundary conditions.

In equation (3) ~ is the usual

two-dimensional Delta function of fixed (field) coordinates (p,q) and of dummy (source points) variables (xs,ys); i.e, U = U(p,q; xs,ys) Introducing polar coordinates at the fixed point and seeking U as a function of distance r only, we can write rU,rr+U,r+k2rU=O The

(4)

change of variable r =kr reduces (4) to the Bessel's equation [4] of order

zero which has two linearly independent solutions Jo(r ), Bessel ~S function of the first kind and order zero, and Yo(r~), Bessel's function of the second kind and order zero. Thus, U=c where

J +c

OO

co

Y

(5)

]O

and

c I are constants.

Of these two solutions J

O

can be discarded, since it is finite at r=O, [Jo(O)=l]

and hence it can not satisfy the desired Delta function singularity. U = C l Y o ( r ~)

Therefore,

(6)

Now, we can write the solution of (i) at any interior point (pi,qi) as a superposition of translated fundamental solutions: N u(Pi,qi ) =~-]~ W(xsj,ysj).U(Pi,qi ; xsj,ysj) j=l

(pi,qi

~

R

(7)

APPROXIMATION OF HELMHOLTZ'S EQUATION

The

unknown

285

W's are the weights of point load singularities applied at

source points (xsj,ysj).

To satisfy equation (2) we need to select N boundary

points (xbi,Ybi); i.e, let (pi,qi)

~ (xbi,Ybi)~ ~R

and substitute (7)

into (2) to obtain: N

N

Wj U + b i j~--i

ai

Wj U,n

of N linear

= ci

i=l,2,...N

equations

(8)

This

is a sys t e m

which may be solved for the

W's.

Once the W's have been determined, the solution at any given field

point can be obtained from (7). Selecting the translated U's; i.e, locating source points away from the boundary ~R,

has two important advantages:

(i) When solution at the boundary points are desired, since 2 2+ 2 r = (xbi-xsj) (Ybi-ysj) =~ O, no difficulty will be encountered. (ii) Since (xs,ys)'s are located away from the boundary, r can be chosen such that the following asymptotic series can be employed [5]: V

0

= (2/n~r)3

sin(r -~/4)

r > 25

(9)

Therefore; we select U=

(r)-3

U,n=U,r

sin(r

- ~r/4)

(i0)

.r ~ , n=k 2 ( r~) -3/2 [cos(r

- ~/4)-.Ssin(r

- T[/4)/r

.(xcose Where

~ is

the

angle

that

the

normal

].

+ysine

n makes with x-axis.

)

The

l

constant ci(2/~ )2 appearing in both U and U,n is discarded, its effect being absorbed by the W's.

Equations (lO) and (ll) are the only require-

ments in the system of equations (8) which is now completely constructed. Gaussian elimination or iteration techniques [6] may be applied to (8) to obtain the W's. Examples

The forgoing procedure was applied to the following cases: A square plate, ITx~, u,xx+u,yy+2u=O and

(see FIGURE i.) subjected to in R

(ll)

286

E. M A H A J E R I N

Y xs, ys

I 1

2

C

I

L N v

R

1 I

X

o

A

OR

t___~ 8R FIG I. Geometry Case i.

on

OA

u,n=O

on

AB

u=-cosx

on

BC

on

OA

u=

u=-cosy

on

AB

u=-cosx

on

BC

u=cosy

on

CO

u=

The

cosx

Case 2.

exact

solution

was

U=COSX.

Both cases were solved using on]y

cosx

OR.

system of equations

16 boundary

points.

boundary

~R

(8) was solved using Guassian

The author

used additional

calculated

the corresponding

points

results was not significant.

equidistant.

elimination.

16 source S=25 The

The results

for both cases are shown in TABLE 1

terms in the asymptotic solutions

According]y

, at a distance

These points were not necessarily

for 7 interior as well as boundary

CO

cosy

points were selected along an extended units away from

and

expansion

at the same points,

for Yo and

the change in the

APPROXIMATION OF HELMHOLTZ'S EQUATION

287

TABLE 1.

x,y

u Case 1

u Exact

u Case 2

1.57080,1•57080

0

2•35619,2.35619

•528197

.5

•490041

2•74889,2•74889

.862865

.853553

.847902

3•14159,3•14159

•997311

1.O

•998628

3.14159,1.57080

0

0

0

3•14159,2.35619

.706525

.707106

•693451

2.61799,2•61799

.761051

.75

•738649

0

Conclusions An approximate technique for Helmhotz's differential equation subjected to generalized boundary conditions has been presented.

It employ~

collocation

of translated fundamental solutions for which an asymptotic series is successfully substituted•

Unlike boundary element or finite element methods, it does

not require complex functions or a great deal of computational effort. Tabulated results for two cases indicate that this method may be adopted to solve a variety of physical boundary value problems•

References 1. 2. 3. 4. 5. 6.

L. Lapidus and G. F. Pinder, Numerical Solution of Partial Differential Equations in Science & Engineering, John Wiley, (1982) O. C. Zienkiewics, The Finite Element Method, 3rd ed., McGraw-Hill, (1979) C. A. Brebbia, The Boundary Element Method for Engineers, John Wiley,(1978) C. N. Watson, A Treatise of the Theory of Bessel Functions, 2nd. ed., Cambridge University Press, London, (1952) J. J. Tuma, Engineering Mathematics Handbook, 2nd ed. McGraw-Hill, (1979) J. R. Westlake, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations, John Wiley, (1968)