The Finite Difference Method for the Helmholtz Equation with Applications to Cloaking Li Zhang Introduction • In the past few years, scientists have made great progress in the field of cloaking. Cloaking involves making an object invisible or undetectable to electromagnetic waves. • In this paper, we will use the numerical solutions to visualize the effect that the cloaking constructions have on a traveling waves. 2 The Helmholtz Equation • The wave equation 2w 2 c w 2 t (1) models the propagation of a wave travelling through a given medium at a constant speed c. 2 2 2 2 x y 3 • If we assume the solution w is separable, then we can write w( x, y, t ) u( x, y)v(t ) Substituting this into (1) gives 2v u 2 c 2 vu t which can be rewritten as 1 v u 2 2 c v t u 2 ( 2) 4 • If we assume both sides are equal to the constant s 2 , then solving (1) reduces to solving the two equations 2v 2 2 2 c s v 0 t u s 2u 0 (3) ( 4) • • Equation (3) has solutions of the form v(t ) c1 cos(t ) c2 sin(t ) • where cs 5 Approximating Derivatives • To apply the finite difference method, we will need to estimate the derivatives of a function. • Then from Taylor’s formula, h 2 '' h3 ''' f ( x h) f ( x) hf ( x) f ( x) f ( x) 2! 3! (5) h 2 '' h3 ''' f ( x h) f ( x) hf ( x) f ( x) f ( x) 2! 3! ( 6) ' ' 6 • Subtracting them and so we can approximate the derivative of f using the formula f ( x h) f ( x h) f ( x) 2h ' (7 ) • Furthermore , adding (5) and (6) gives the approximate f ( x h) 2 f ( x ) f ( x h) f ' ' ( x) h2 (8) 7 In the similarly way, we can approximate the partial derivatives of f as follows u u ( x h, y ) u ( x h, y ) ( x, y ) x 2h u u ( x, y k ) u ( x, y k ) ( x, y ) y 2k 2u u ( x h, y ) 2u ( x, y ) u ( x h, y ) ( x, y ) x 2 h2 2u u( x, y k ) 2u( x, y) u( x, y k ) ( x , y ) y 2 k2 (9 ) (10) (11) (12) 8 The Finite Difference Method • Let R a, b c, d be a rectangle in R 2 , and consider the boundary value problem 2 u s u0 • in R (13) u ( x, y ) f ( x, y ) on R • • Then apply the finite difference method to solve this problem. 9 k2 r 2 h Letting ,this we can get the following five-point formula, ru ( x h, y) ru ( x h, y) u( x, y k ) u( x, y k ) (s k 2r 2)u( x, y) 0 2 2 10 Consider the boundary value problem • in R u u 0x u( x, y) sin( ) • on R 6 • Using a 70 70 lattice resulted in the solution pictured in Figure (1) 11 • Similarly, the solution to u u 0 • in u( x, y) 1 • on x2 y 2 • is pictured in Figure (2) R R 12 Transformations • A divergence form operator acting on function u C 2 ( R2 ) • is a differential operator L of the form Lu div( Au) r x r y F ( x, y ) ((1 ) , (1 ) ) 2 r 2 r • with r x 2 y 2 13 Then for a differentiable function, define the matrix • F1 x ( x, y ) DF F2 ( x, y ) x A( x, y ) F1 ( x, y ) y F2 ( x, y ) y 1 det DF ( x, y ) where F ( x, y) ( F1 ( x, y), F2 ( x, y)) DF ( x, y)DF ( x, y) T ( x , y ) F 1 ( x , y ) 14 A Generalization of the Helmholtz Equation • In this section, we are interested in a general Helmholtz equation of the form 2 Lu s u 0 • Given a 2 2 matrix A, we wish to apply the finite difference method to solve a boundary value problem Lu s 2u 0 • in R u ( x, y ) f ( x, y ) • on R 15 • where R a, b c, d as before, and • For simplicity, let u ux and u uy • If we assume 2 1 a11 A a21 Lu div( Au) a12 a22 • we can write a11 a12 1u a111u a12 2u Au a a u a u a u 22 2 21 22 2 21 1 16 • Then we can express Lu as 2 2 Lu div( Au ) i (aij j u ) (14) i 1 j 1 • In the same manner as before from the Taylor series, we can show that 1 1 ( 2u )( x, y ) ( 2u ( x h, y ) 2u ( x h, y )) 2h 1 u ( x h, y k ) u ( x h, y k ) u ( x h, y k ) u ( x h, y k ) ( ) 2h 2k 2k u ( x h, y k ) u ( x h, y k ) u ( x h, y k ) u ( x h, y k ) 4hk (15) 17 • Utilizing (9)-(12)along with(15) in (14) allows us to approximate the equation Lu s 2u 0 • with the nine-point formula ( ( a12 a21 a b a a21 )u ( x h, y k ) ( 222 2 )u ( x, y k ) ( 12 )u ( x h, y k ) 4hk k 2k 4hk ( a11 b1 2a11 2a22 a11 b1 2 ) u ( x h , y ) ( s ) u ( x , y ) ( )u ( x h, y ) 2 2 2 2 h 2h h k h 2h a12 a21 a b a a21 )u ( x h, y k ) ( 222 2 )u ( x, y k ) ( 12 )u ( x h, y k ) 0 4hk k 2k 4hk • with 2 b j i aij i 1 18 Applications to Cloaking • We use a specific example to solve the boundary value problem. • Consider the transformation r x r y F ( x, y ) ((1 ) , (1 ) ) 2 r 2 r • where r x2 y2 19 • Using the transformation as discussed in last section, we can calculate to conclude 2 r x2 3 r DF ( x, y ) 2r xy r3 2 r y2 3 2r r r (2r 1) x 2 3 r 1 ( r 1 ) r A( x, y ) (2r 1) xy 3 ( r 1 ) r xy r3 (2r 1) xy (r 1)r 3 r (2r 1) y 3 r 1 (r 1)r 3 20 • For example, let R be the square 3,3 3,3 • Consider Lu u 0 u( x, y) sin(x ) 6 • Using the MATLAB program , the numerical solution is shown in Figure (3) 21 • For the equation Lu u 0 u( x, y) 1 2 2 x y • The numerical solutions are Figure (4). 22 • Compare Figure (1) with Figure (3) u u 0x u( x, y) sin( ) 6 Lu u 0x u( x, y) sin( ) 6 23 • Figure (2) and Figure (4) u u 0 u( x, y) 1 2 2 x y Lu u 0 u( x, y) 1 2 2 x y 24 Indistinguishable waves • Two waves are indistinguishable, if their boundary values and normal derivatives are identical. 30 30 40 40 50 50 60 60 70 70 80 80 Figure(1)and Figure(3) 0.377 0.214 0.116 0.062 0.034 0.018 Figure(2)and Figure(4) 0.161 0.044 0.018 0.009 0.005 0.003 25 • We set up the initial value 2v v 0 2 t v (0) 1, v (0) 0 t 26 27 Thank you! 28