Boundary Element Method for the Helmholtz Equation

Transcription

1 Journal of Mathematical Extension Vol. 9, No. 4, (205), ISSN: URL: Boundary Element Method for the Helmholtz Equation A. Mesforush Shahrood University Z. Mehraban Shahrood University Abstract. The boundary element methos is applied to the Helholtltz equation. To this end we discretized the Helmholtz equation over the boundary and conclude a system of equations. By applying the boundary condition we get a new system which gives the solution of the equation. AMS Subject Classification: 65L80; 35Axx Keywords and Phrases: Laplace equation, Helmholtz equation, boundary integral equation, discretization, boundary element method. Introduction Numerical approximation for solving a differential equation has a wide range of application in engineering and mathematics. Some of these methods are: finite difference methods, finite element methods and boundary element method. In these methods we usually discritize the equation and apply the governing equations on the suitable mesh. The most important thing in the numerical approximation is determination of the size of mesh such that the approximation solution has the suitable accuracy. Received: January 205; Accepted: May 205 Corresponding author 65

2 66 A. MESFORUSH AND Z. MEHRABAN The boundary element method is a powerful tool for numerical approximation for a differential equation. For the first time, in 872, Betti presented this method in elasticity theory []. In 903 Fredholm extended this method [2], but in 977 the boundary element method appears in some publications of Banerjee, Butterfield, Brebbia and Dominguez [3, 4]. In this paper we want to find an approximate solution for the Helmholtz equation by the boundary element method. This paper organized in three sections: in the first section, we introduce some preliminaries and also describe the structure of the boundary element method. In the second section, we applied boundary element method for Laplace equation which is a special case of helmholtz equation. Finally, in the third section we approximat the boundary element solution of Helmholtz equation. In boundary element method we use three important things. - The Gauss divergence theorem: If R d be a bounded domain and w = (w,, w d ) C () d be a vector-valued function, then for x R d we have [8] w dv = w n ds, where n = (n,, n d ) is the outward unit normal to boundary Γ. 2- The second Green identity: Let R d be a bounded domain and v, w C 2 (), then [8] (w 2 u u 2 w)dv = (w n u w n )ds. 3- The Dirac delta function: The Dirac delta function defined as [] {, X = X 0, δ(x X 0 ) = 0, X X 0, and has two following properties: δ(x X 0 ) dx =, Γ Γ f(x)δ(x X 0 ) dx = f(x 0 ).

3 BOUNDARY ELEMENT METHOD FOR The structure of the boundary element method can be decribed as follows. Consider the following boundary element value problem Au := Au xx + 2Bu xy + Cu yy + Du = f, in, u = ū, on Γ, q = n = q, on Γ 2, where the coefficients A, B, C, D are constant and Γ, Γ 2 are boundaries of domain, q is the derivative of potential function in direction of n and ū, q are given values of the flux and potential function on the boundary. In boundary element method we multiply the first equation of () by weight function w C 2 () and integrate over to get the following integral form (Au)wdv = fwdv. (2) by choosing the fundamental solution Aw = δ(x X 0 ) as the weight function and using the second Green identity, we can convert (2) to a continuous integral equation over the boundary. For applying the boundary element method we discretize the problem over the boundary of. To this end we consider = N j= j and conclude the system of equations HU = GQ. If we apply the boundary conditions, we get a new system of equations as AX = B, which gives the flux and potential u on the boundary. In this paper we want to apply the bounday element method for the Helmholtz equation. In () if we consider A =, B = 0, C =, D = k 2 and by considering 2 u = u xx + u yy, we get the Helmholtz equation as where k is the wave number. Au := 2 u + k 2 u = 0, in R 2, u = ū, on Γ, q = n = q, on Γ 2, () (3)

4 68 A. MESFORUSH AND Z. MEHRABAN 2. Boundary Element Method for the Laplace Problem In this section we apply the boundary element method to the Laplace equation which is the special case of Helmholtz equation. In equation (3) if we set k = 0, the following laplace equation is resulted: 2 u = 0, in R 2, u = ū, on Γ, q = (4) n = q, on Γ 2. Consider weight function w such that it satisfies 2 w + δ(x X 0 ) = 0, (5) where X 0 = (ξ, η) is a singular point and δ is the Dirac delta function. Considering the polar form 2 w = w (r ), where r = X r r r X 0 = (x ξ) 2 + (y η) 2. In this case for r > 0, δ(x X 0 ) = 0, one obtain w (r r r r ) = 0, where A and B are arbitrary constants. Since we look for a particular solution, we may set B = 0. Which gives after integrating twice w = A ln r + B. By considering a neighborhood of X 0 and using the integral property for dirac delta function, we obtain A =. Hence, the fundamental 2π solution for the Laplac equation becomes [] w = ( ) 2π ln. r Also note that w n = w r r n = ( r n) = 2πr 2πr 2 (r xn x + r y n y ),

5 BOUNDARY ELEMENT METHOD FOR where r x = x ξ and r y = y η. Let X 0 and consider a neighborhood of X 0 with radius ε, namely ε and consider the new domain ε with boundary ε. In this domain by letting q = n and q = w the Green formula can be n written as lim ε (u 2 w w 2 u)dv = lim (uq wq)ds + lim (uq wq)ds, ε by using (5) and the definition of the Dirac delta and X 0 / ε, one obtain lim u 2 wdv = lim uδ(x X 0 )dv = 0, ε ε using the Laplac equation (4), result in lim ε (u 2 w w 2 u)dv = 0. On the other hand, [9] lim (uq wq)ds = lim (u u(x 0 ) + u(x 0 )) q ds ε ε lim ln ε lim ln ε lim ln ε 2π ε n ds 2π ε n ds 2π ε n ds 2π 2π = lim u(x = lim u(x 0 ) 0 2π dθ + lim ε ln ε 0 ) 2π 0 2π dθ + lim ε ln ε (X 0 ) 2π ln ( ) dθ 2π n 0 = u(x 0 ). = u(x 0 ). Therefore, the boundary integral equation in is: u(x 0 ) = (wq uq )ds, (X 0 ) n which means that for given u, q on, the values of u can be determinded for the interior point of the domain.

6 70 A. MESFORUSH AND Z. MEHRABAN Let X 0 and consider a neighborhood of X 0, namely i ε and consider the domain i ε with boundary i ε ( ε ). In this domain the Green formula can be written as, lim i ε (u 2 w w 2 u)dv = lim (uq ( wq)ds ε) + lim (uq wq)ds. i ε Since the governing equation is Laplace equation and the X 0 is out of the domain i ε,we have: lim (u 2 w w 2 u)dv = 0. ε On the other hand lim (uq wq)ds = lim (u u(x 0 ) + u(x 0 )) q ds i ε ε lim ln ε lim ln ε 2π ε n ds 2π ε n ds = lim u(x 0 ) i 2π dθ + lim ε ln ε = lim u(x 0 ) 2π i 2π dθ + lim ε ln ε (X 0 ) 2π dθ 2πε n ε 0 = C(X = C(X 0 )u(x 0 ), 0 )u(x 0 ), (X 0 ) n where C(X 0 ) is a geometry-dependent parameter. It varies between zero and unity and equals 2. If the point is on a smooth part of the boundary at edges, it is related to the angle of the joining surfaces and equals internal angle C(X 0 ) =. Also 2π lim (uq wq)ds = lim (uq ε wq)ds. So the boundary integral equation in is: C(α)u(X 0 ) + uq ds = wqds. 2π 0

7 BOUNDARY ELEMENT METHOD FOR... 7 To apply the boundary element method, the boundary is divided into small elements as = N j= j where N represents the number of elements that form the boundary. The mesh is generated only on the boundary and it is more flexible than in the case of finite element mesh. The integrals are then expressed for each element and summed up in order to evaluate the integral on the whole boundary. Considering constant element discretization and the source points situated on a smooth boundary, the boundary integral equation can be written as 2 u(x 0) + uq ds = wqds. N j= j N j= j where by definition Ĥ ij, i j, H ij = 2 + Ĥij, i = j, and we get w Ĥ ij = j n ds j, G ij = w ds j, j N H ij u j = j= N G ij q j. j= Here we know some of u j s and some of q j s, i.e.,there are N known quantities and N unknowns. Applying the boundary conditions to every midpoint of the elements, we obtain the equations in the matrix-vector form as HU = GQ. Rearranging this system with known quantities on one side and unknowns on the other, we get a linear system of equations AX = B where X is unknown values. Example 2.. Consider the Laplace equation in the rectangle plate with

8 72 A. MESFORUSH AND Z. MEHRABAN mixed boundary conditions as [7] 2 T = 0, 0 < x < 2, 0 < y <, (x, 0) = 0, T (2, y) = 0, n (x, ) = 0, T (0, y) =. n Using four constant element discretization with the collocation points at the center of the elements, H ij and G ij for i j, one obtain: H ij = q ds j = Xj+ j 2π X j r 2 (r xn x + r y n y )dx = l 4 e w k 4π rk 2 (r xk n xk + r yk n yk ), k= and G ij = wds j = j 2π X j = l 4 ( ) e w k ln. 4π r k k= Xj+ ln ( ) dx r also for i = j, we have H ij = 2, and G ij = π le 2 0 ln ( ) dx = l ( e ln r 2π ( 2 l e ) ) +, where l e = X j+ X j is the length of the element on which the integral is calculated, w k represent the weight functions from the Gauss integration scheme and r k is the distance from the i th source point to each integration points on the j th element. So H = ,

9 BOUNDARY ELEMENT METHOD FOR G = From these equations unknow values in system AX = B will be calculated as T = , T 3 = , q 2 = , q 4 = Consider these values of T in interior points, result in as where So P = (0.75, 0.5), P 2 = (, 0.5), P 3 = (.5, 0.5), T (i) = N G ik q k k= N H ik T k, k= w i G ik = w i ds k, H ik = Γ k Γ k n ds k, T P = , T P2 = , T P3 = 0.38, Therefore the exact values are T e P = 0.625, T e P 2 = 0.5, T e P 3 = Compared with the exact values, the solution shows quite a good approximation for the internal points not only for the boundary points. Figure, 2 Figure. The effect of increasing the number of elements on approximate solution of the Laplace equation

10 74 A. MESFORUSH AND Z. MEHRABAN Figure 2. The effect of increasing the number of elements on approximate solution of the Laplace equation (zoom) 3. Boundary Element Method for the Helmholtz Equation In this section we apply the boundary element method to the Helmholtz equation (3) for k = 0. Consider weight function w such that it satisfies 2 w + k 2 w + δ(x X 0 ) = 0, (6) where X 0 = (ξ, η) is a singular point. The solutions for the homogeneous case are the Hankel functions of the first and second kinds of order zero. So we write w(r) = AH (2) 0 (kr), where r = (x ξ) 2 + (y η) 2 and H (2) 0 is the Hankel function of the second kind of order zero and it is representable in terms of the 0th-order Bessel functions of the first and second kinds J 0, Y 0 as H (2) 0 (z) = J 0(z) iy 0 (z) = H () 0 (z). By using the small-argument approximation to the Hankel function as [2] 0 (kr) i 2 π log(γkr ), r 0, 2 H (2)

11 BOUNDARY ELEMENT METHOD FOR where γ =.78 and integrating (6) over a very small circle of radius ε centered at the origin, one obtain A [ + k 2 ][ i 2 π log(γkr )]dv =, 2 using the divergence theorem, the first term is converted to a line integral as i 2 2π ( log( γkr ) π 2 ) rdφ = 4i, the second term goes to zero, so 0 A = i 4, Therefore, we get the following fundamental solution for helmholtz equation as [2] Also we note that [0] q = w n = w r r n = 4i kh(2) w = 4i H(2) 0 (kr). (kr)( r n) = n xr x + n y r y 4ir kh (2) (kr), where r x = x ξ and r y = y η. To obtain the boundary integral equation, we integrate (3) over the domain with a weighting function w to get, ( 2 u + k 2 u)wdv = 0. The second Green identity, implies that w 2 udv = u 2 wdv + so (uq wq)ds, ( 2 w + k 2 w)udv + (uq wq)ds = 0. (7) By using the properties of Dirac delta gives ( 2 w + k 2 w)udv = u(x 0 ).

12 76 A. MESFORUSH AND Z. MEHRABAN So the boundary integral equation for all the interior points is u(x 0 ) + uq ds = wqds. The boundary integral equation in general case is [3] C(X 0 )u(x 0 ) + uq ds = wqds, where C(X 0 ) = α is a geometry-dependent parameter where α is internal angle. Assume that the boundary is discretized and represented by 2π N linear constant elements as = N j= j and the source points situated on a smooth boundary (α = π), so the boundary integral equation can be written as 2 u(x 0) + u w N j= n ds = w j N j= n ds. j Therefore where and N H ij u j = j= N G ij q j, j= Ĥ ij, i j, H ij = 2 + Ĥij, i = j, w Ĥ ij = j n ds j, G ij = w ds j. j By applying these boundary conditions to every midpoint of elements, we obtain the equations in the matrix-vector form HU = GQ. Rearranging this system, a linear system of equations AX = B, where X is unknow values will be resulted. Example 3.. Consider the following Helmholtz equation with k = and the domain 0 < x <, 0 < y <, the boundary conditions in [4] 2π 4

13 BOUNDARY ELEMENT METHOD FOR u + k 2 u = 0, 0 < x <, 0 < y <, u(0, y) = 0, 0 < y <, 2 u(, y) = 2 cos(πy ), 0 < y <, 4 n = 0, 0 < x <, y=0 n = π 2 y= 8 sin(πx ), 0 < x <. 4 The exact solution of this particular boundary value problem is u(x, y) = sin( πx 4 ) cos(πy 4 ). In Table, a comparison between the exact solutions and the approximate solutions for differente values of elements of equation (6) of this element is given. Also in figure 3 and 4, these results are depicted. Table. The effect of increasing the number of elements on approximate solution helmholtz equation for interior points Figure 3. The effect of increasing the number of elements on approximate solution helmholtz equation

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15 BOUNDARY ELEMENT METHOD FOR [9] Y. Khalighi and D. J. Bodony, Improved near-wall accuracy for solutions of the Helmholtz equation using the boundary element method, Center for Turbulence Research Annual Research Briefs, (2006), [0] K. Mohan Kadalbajoo and K. Arumugam, Boundary element method for elliptic differential equations with a small parameter, Applied mathematics and computation, (995), [] C. Walton Gibson, The Method of Moments in Electromagnetics, Chapman and Hall/CRC, [2] L. C. Wrobel, The Boundary Element Method: Applications in Thermo- Fluids and Acoustics, UK, [3] K. Yukio, S. Yonghao, and M. Zaheed, Regular boundary integral formulation for the analysis of open dielectric/optical waveguides, Memoirs of the Faculty of Engineering, 30 (995), [4] A. Ywhye-Teong, A beginner s course in boundary element methods, Universal Publishers, (2007), Ali Mesforush Department of Mathematics Assistant Professor of Mathematics School of Mathematics Shahrood University of Technology Shahrood, Iran ali.mesforush@shahroodut.ac.ir Zohreh Mehraban Department of Mathematics M.Sc Student of Mathematics School of Mathematics Shahrood University of Technology Shahrood, Iran zohre.mehraban@ymail.com