Wavelets trong Electromagnetics và mô hình thiết bị P4 potx

CHAPTER FOUR Wavelets in Boundary Integral Equations Numerical treatment of integral equations can be found in classic books [1, 2]. In this chapter the integral equations obtained from field analysis of electromagnetic wave scattering, radiating, and guiding problems are solved by the wavelet expansion method [3–7]. The integral equations are converted into a system of linear algebraic equations. The subsectional bases, namely the pulses or piecewise sinusoidal (PWS) modes, are replaced by a set of orthogonal wavelets. In the numerical example we demonstrate that while the PWS basis yields a full matrix, the wavelet expansion results in a nearly diagonal or nearly block-diagonal matrix; both approaches re- sult in very close answers. However, as the geometry of the problem becomes more complicated, and consequently the resulting matrix size increases greatly, the ad- vantages of having a nearly diagonal matrix over a full matrix will become more profound. 4.1 WAVELETS IN ELECTROMAGNETICS Galerkin’s method is a zero residual method if the basis functions are orthogonal and complete, and thus Galerkin’s method with orthogonal basis functions is generally more accurate and rapidly convergent. Two types of orthogonal basis functions are frequently utilized for electromagnetic field computation. Mode expansion method (or mode-matching method) has often been applied to solve problems due to various discontinuities in waveguides, finlines, and microstrip lines. Generally, this technique is useful when the geometry of the structure can be identified as consisting of two or more regions, which each belongings to a separable coordinate system. The basic idea in the mode expansion procedure is to expand the unknown fields in the indi- vidual regions in terms of their respective normal modes. In fact the mode expansion method is identical to Galerkin’s method which uses the normal mode functions as 100 Wavelets in Electromagnetics and Device Modeling. George W. Pan Copyright ¶ 2003 John Wiley & Sons, Inc. ISBN: 0-471-41901-X WAVELETS IN ELECTROMAGNETICS 101 the basis functions. Quite often the normal modes are made of the classical orthogo- nal series systems such as trigonometric, Legendre, Bessel, Hermite, and Chebyshev. Owing to the orthogonality of the normal modes, a sparse system of linear algebraic equations is expected to be generated by the mode expansion method. For general cases of arbitrary geometries and material distributions, however, the mode functions are often too difficult to be constructed. The second class of orthogonal basis functions consists of a group of subsectional bases, each of which is defined only in a given subsection of the solution domain. An advantage of the subsectional bases is the localization property, that is, each of the expansion coefficients affects the approximation of the unknown function only over a subdomain of the region of interest. Thus, often not only does this class of computations simplify the computation, but it also leads easily to convergent solu- tions. In the subsectional basis systems, generally, only partial orthogonality can be attained; only the pair of bases whose supporting regions do not overlap are orthog- onal. Moreover the higher the continuity order of the constructed bases is rendered, the larger the required supporting region. Hence there exists a trade-off between the orthogonality and continuity for the subsectional basis systems. Even if complete orthogonal bases with higher-order continuity are hard to build, the subsectional bases with certain continuity order can be constructed widely (e.g. by using polynomial interpolation functions). The finite element method, which has been universally applied in engineering, is a subsectional basis method. So is the boundary element method. Because of the kind of orthogonality, or, say, localization that exists in subsectional basis systems, the differential operator equations may yield sparse systems of linear algebraic equations by using subsectional bases. However, it is also noted that the subsectional basis systems do not necessarily convert the integral operator equations into sparse systems of linear algebraic equations. Orthogonal wavelets have several properties that are fascinating for electromag- netic field computations. First, wavelets are sets of orthonormal bases of L 2 (R). They are problem-independent orthogonal bases and thus are suitable for numeri- cal computations for general cases. Second, the trade-off between orthogonality and continuity is well balanced in orthogonal wavelet systems because now the orthog- onality always holds, whether the supporting regions are overlapping or not. One can build an orthogonal wavelet system with any order of regularity, expecting larger supporting regions as higher orders of regularity are selected. Third, in addition to the advantages of the traditional orthogonal basis systems, orthogonal wavelets have a cancellation property such that they are much more certain to yield sparse systems of linear algebraic equations. Furthermore orthogonal wavelets have localization properties in both the spatial and spectral domains. Therefore the decorrelation of the expansion coefficients oc- curs both in the space and Fourier domains. Nevertheless, according to the theory of multigrid processing, one can improve convergence by operating on both fine and coarse grids to reduce both the “high-frequency” and “low-frequency” compo- nent errors between the approximate and exact solutions. In contrast, the traditional way of operating only on fine grids reduces only the “high-frequency” component. The expansion with subsectional bases actually is equivalent to the expansion on 102 WAVELETS IN BOUNDARY INTEGRAL EQUATIONS the finest scale only (in fact, the pulse function is equivalent to the scalet of Haar’s bases). On the contrary, the multiresolution analysis implemented by wavelet ex- pansion provides a multigrid method. Finally, the pyramid scheme employed in the wavelet analysis provides fast algorithms. 4.2 LINEAR OPERATORS Functional spaces and linear operators were presented systematically and rigorously in Chapter 1. In this section we will only quote the minimum prerequisite knowledge for the method of moment applications. I NNER PRODUCT  f, g. An inner product  f, g on a complex linear space is a complex-valued scalar satisfying  f, g= g, f  α f +βg, h= α f, h+βg, h  f, f =f  2  > 0iff = 0 = 0iff = 0, where the overbar denotes the complex conjugate. O PERATOR L. The linear operator L and its corresponding equation are given as Lf = g. For instance, the Poisson equation is −  2 φ = ρ, where the linear operator L =−  2 . The adjoint L a is defined by Lf, g=f, L a g. An adjoint operator is self-adjoint if L a = L. The inverse operator of L is denoted as L −1 . For instance, the formal solution to (4.3.1) is f = L −1 g. In numerical computations we use a matrix to represent a linear operator. METHOD OF MOMENTS (MoM) 103 4.3 METHOD OF MOMENTS (MoM) Consider an operator equation Lf = g, (4.3.1) where L is a linear operator, f is the unknown function, and g is a given excitation. We first expand the unknown function f (x ) in terms of the basis functions f n (x) with unknown coefficients α n , namely f =  n α n f n . Thus L  n α n f n = g. Multiplying both sides of (4.3.1) by the weighting (testing) function w m and taking the inner product ·, ·, we obtain  n α n w m , Lf n =w m , g. In matrix form, it appears as [l mn ]|α=|g, (4.3.2) where |α=      α 1 α 2 . . . α N      N ×1 ←− unknown, [l mn ]=   w 1 , Lf 1 , w 1 , Lf 2  ··· w 2 , Lf 1 , w 2 , Lf 2  ··· ···   N ×N ←− evaluated, and |g=    w 1 , g w 2 , g . . .    N ×1 . Formally, equation (4.3.2) is solved to yield |α=[l mn ] −1 |g. 104 WAVELETS IN BOUNDARY INTEGRAL EQUATIONS There are two kinds of popular schemes in the method of moments: (i) Pulse-delta scheme. Basis functions f n = pulse functions, and testing func- tion w m = δ(x m − x), Dirac delta function (point matching). (ii) Galerkin scheme w m = f m . The pulse-delta scheme is equivalent to the rectangular rule in the numerical integra- tion, and the Galerkin scheme is a zero residual method. Example Charged Conducting Plate (zero thickness). A charged plate is de- picted in Fig. 4.1. Find the charge distribution. Solution The electrostatic potential at any point (x, y, z) in space is given by V (x , y, z) =  a −a dx   a −a dy  σ(x  , y  ) 4π|r − r  | with the unknown charge density σ(x  , y  ). An integral equation of the first kind is then formulated as V (r) =  v G(r, r  )σ (r  )d 3 r  , where the Green’s function is G(r, r  ) = 1/4π|r − r  |, and the potential on the plate surface is a constant V. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 x 10 −3 FIGURE 4.1 Charge distribution q/ on a square plate. METHOD OF MOMENTS (MoM) 105 The capacitance of the plate can be found by C = q V = 1 V  a −a dx  a −a dyσ(x, y). Therefore the main problem is to solve σ(x  , y  ) of the integral equation by the MoM, namely  a −a G(r, r  )σ (r  ) ds  = V , where G(r, r  ) is also called the integral kernel. We present this example because of its simple Green’s function, and clear physical meaning. The numerical procedures may be outlined in the following steps: (i) Define the basis functions to be the pulse functions f n (x  , y  ) =  1,(x  , y  )on S n 0, on all other S m , m = n, that is, α n applies only on S n . (ii) Approximate the charge by σ(x, y) ≈ N  n=1 α n f n . (iii) Convert the operator equation into a matrix equation. The operator equation is Lσ = V, or in the explicit form  G(r, r  )   n α n f n  ds  = V . Applying the linearity of the operator, we obtain  n α n  S n f n dx  dy  4π  (x − x  ) 2 + (y − y  ) 2 + (z − z  ) 2 = V (x, y). Take the inner product of the equation above with the testing function w m (x, y) = δ(x m − x)δ(y m − y), where (x m , y m ) is the midpoint of the patch S m . The corresponding system of equations is formed with 106 WAVELETS IN BOUNDARY INTEGRAL EQUATIONS RHS =w m , V =  V (x , y)δ(x m − x)δ(y m − y) dx dy = V (x m , y m ) and LHS =  n α n  S n dx  dy   dx dy δ(x m − x)δ(y m − y) 4π  (x − x  ) 2 + (y − y  ) 2 =  n α n  S n dx  dy  1 4π  (x m − x  ) 2 + (y m − y  ) 2 . Thus a matrix equation converted from the operator equation is      l 11 l 12 ··· l 1N l 21 l 22 ··· l 2N . . . l N 1 l N 2 ··· l NN           α 1 α 2 . . . α N      =      V (x 1 , y 1 ) V (x 2 , y 2 ) . . . V (x N , y N )      . In choosing pulse basis functions, the charge is assumed to be a constant over a subarea (patch), namely l mn =  S n dx  dy  1 4π  (x m − x  ) 2 + (y m − y  ) 2 . (iv) In handling integral equations, singularity occurs when the field point (x, y), in this case (x m , y m ), lies with in the domain of integration, S n . For the di- agonal element, l mn (m = n), the integrand experiences a singularity which must be treated carefully. Analytical removal, pole extraction by an asymp- tote, and folding technique are among the popular methods for handling sin- gularities [8, 9, 10]. In the present case the method of analytic removal is applicable: l 11 =  x 1 +b x 1 −b dx  y 1 +b y 1 −b dy 1 4π  (x − x 1 ) 2 + (y − y 1 ) 2 =  b −b dx  b −b dy 1 4π  x 2 + y 2 = b 4π  1 −1 du  1 −1 dv 1 √ u 2 + v 2 , where  1 −1 du  dv 1 √ u 2 + v 2 = 8  π/4 θ=0  1/ cos θ ρ=0 ρ dρ dθ ρ FUNCTIONAL EXPANSION OF A GIVEN FUNCTION 107 = 8  π/4 0 d sin θ cos 2 θ = 8  1/ √ 2 0 dτ 1 − τ 2 = 8 · 1 2  1/ √ 2 0  1 1 − τ + 1 1 + τ  dτ = 8ln  3 + 2 √ 2 = 8ln(1 + √ 2). The numerical solution of the charge distribution on the plate is depicted in Fig. 4.1. 4.4 FUNCTIONAL EXPANSION OF A GIVEN FUNCTION In the previous section the MoM was briefly discussed. The MoM is a powerful numerical algorithm, and it has been employed to solve electromagnetic problems for a half-century. Unfortunately, the MoM matrix is full. With the help of wavelets, one can obtain sparse impedance matrices. We begin with the expansion of a given function in the wavelet bases. It is easier to expand a given function in a wavelet basis than to expand an unknown function in wavelets while solving the corresponding integral equation by the method of mo- ments. The experience we gain here will be applied in the wavelet-based MoM. From the multiresolution analysis (MRA), the nested subspaces can be decom- posed as V m+1 = W m ⊕ V m = W m ⊕ W m−1 + V m−1 = W m−1 ⊕ W m−2 ⊕ W m−3 ⊕··· and ⊕ m∈Z W m = L 2 (R). Therefore {ψ m,n } m,n∈Z is an orthonormal (o.n.) basis of L 2 (R).Forall f (x ) ∈ L 2 (R),wehave f (x) =  m,n  f (x), ψ m,n (x)ψ m,n (x). In practice, we can only approximate a given physical phenomenon with finite pre- cision. Mathematically the approximation is to project a function from the L 2 onto a subspace V m+1 = V m ⊕ W m , namely f (x)  A m+1 f (x) :=  n s m+1 n ϕ m+1,n (x), 108 WAVELETS IN BOUNDARY INTEGRAL EQUATIONS where s m+1 n =f (x), ϕ m+1,n , A m+1 f (x) is the approximation of f (x) at resolution level 2 m+1 and A m+1 is the projection operator. As m →∞, A m f (x) = f (x). Since V m+1 = V m ⊕ W m , it follows that f (x)  A m+1 f (x) = A m f (x) + B m f (x), where B m f (x) =  n d m n ψ m,n (x) and d m n =f (x), ψ m,n (x). Continuing the process, we obtain A m+1 f (x) = A m 1 f (x) + m  m  =m 1 B m  f (x), (4.4.1) where m 1 is a prescribed number, representing the lowest resolution level. Example Expand f (x) in terms of Daubechies wavelets N = 2, where f (x) = 1 −|x | for |x |≤1. Solution The function f (x) is defined on [−1, 1], but the Daubechies are with supp {ϕ}=[0, 3] and supp {ψ}=[−1, 2]. Therefore we cannot use ϕ 0,0 nor ψ 0,0 because they are too wide. Let us choose f (x) ∼ f 4 ∈ V 4 . By the MRA V 4 = V 2 ⊕ W 2 ⊕ W 3 , where the lowest resolution level m 1 = 2. Thus f (x) ∼  n  f (x), ϕ 2,n (x)ϕ 2,n (x) +  p  f (x), ψ 2, p (x)ψ 2, p (x) +  k  f (x), ψ 3,k (x)ψ 3,k (x), where ϕ j,n (x) = 2 j/2 ϕ(2 j x −n), ψ j,k (x) = 2 j/2 ψ(2 j x −k). FUNCTIONAL EXPANSION OF A GIVEN FUNCTION 109 From the supports of supp {ϕ}, supp {ψ} and the scale, we have supp {ϕ 2,0 (x)}=  0, 3 4  supp {ψ 2,0 (x)}=  − 1 4 , 1 2  supp {ψ 3,0 (x)}=  − 1 8 , 1 4  . It can be easily verified: (1) For j = 2,ϕ 2,−6 (x) is the leftmost scalet that intercepts −1and supp {ϕ 2,−6 (x)}=  − 6 4 , − 3 4  . Note that ϕ 2,−5 (x) also intersects −1, with supp {ϕ 2,−5 (x)}=[−5/4, −2/4]. However, it is only next to the leftmost scalet. We find n =−6 by substituting n into 2 2 x −(−n) = 0 (left edge), 2 2 x −(−n) = 3 (right edge). The integer n is selected such that x solved from the right edge equation is just on the right of −1. When n =−7 is used in the left and right edge equations, the resultant interval does not intersect −1. In the same way, the rightmost scalet that intercepts 1 is found as ϕ 2,3 (x) and supp {ϕ 2,3 (x)}=[3/4, 6/4]. (2) For j = 3, the leftmost ψ 3,n (x) that intercepts −1isn =−9. In fact supp {ψ 3,−9 (x)}=[−10/8, −7/8], as solved from 2 3 x −(−9) =−1 ⇒ x =− 10 8 , 2 3 x −(−9) = 2 ⇒ x =− 7 8 . In the same manner, the rightmost ψ 3,n that intercepts 1 is n = 8. (3) For j = 2, ψ 2,−5 (x) is the leftmost basis that intercepts −1and supp {ψ 2,−5 (x)}=  − 6 4 , − 3 4  . The rightmost basis that intercepts 1 is ψ 2,4 and supp {ψ 2,4 (x)}=[3/4, 6/4]. In conclusion, f (x) ≈ 3  n=−6  f,ϕ 2,n ϕ 2,n (x) + 4  m=−5  f,ψ 2,m ψ 2,m (x) + 8  k=−9  f,ψ 3,k ψ 3,k (x). [...]... the Battle–Lemarie) wavelets, although the technique is applicable to other wavelets The Franklin wavelets have computational simplicity, symmetry, and approximately closed form As a result the computational cost of a matrix filled by the Franklin wavelets is almost the same as that of MoM by the triangle basis functions The 134 WAVELETS IN BOUNDARY INTEGRAL EQUATIONS Battle–Lemarie wavelets can be expressed... solve the integral equations on bounded intervals using wavelets as basis functions, the treatment of edges must be carried out with caution There are several techniques, and we list the most commonly used ones below: 118 • • • • WAVELETS IN BOUNDARY INTEGRAL EQUATIONS Coordinate transformation Periodic wavelets Intervallic wavelets Weighted wavelets Here we apply coordinate transformation to b I =... Periodic wavelets MoM method 5.0 4.0 3.0 2.0 1.0 0.0 0 30 60 90 120 150 180 Azimuth Angle,Φ FIGURE 4.11 Scattering coefficient for conducting elliptic cylinder 133 FAST WAVELET TRANSFORM (FWT) It should be mentioned that this problem can be worked out using standard wavelets As long as the boundary curve has a closed contour, there is no need to employ the intervallic wavelets, nor the periodic wavelets. .. p−2k p √ = 2− j/2 2 = 2−( j−1)/2 (4.6.3) PERIODIC WAVELETS 123 In summary (1) For large j, the wavelets are greatly compressed within [0, 1] Hence p ϕ j,k (x) = ϕ j,k (x) (2) In contrast, for small enough j, ϕ j,k (x) is chopped into pieces of length 1, which are shifted onto [0, 1] and added up, yielding the periodic wavelets The constructed periodic wavelets of Coifman, Daubechies, and Franklin are... algebraic equations 4.6 PERIODIC WAVELETS 4.6.1 Construction of Periodic Wavelets Consider a periodic function with period 1, namely f (x + 1) = f (x) ⇔ f (x − 1) = f (x) Then, the wavelet coefficients on a given scale j f, ψ j,k = f, ψ j,k+2 j Show RHS = f (x)2 j/2 ψ(2 j x − k − 2 j ) d x = f (x)ψ[2 j (x − 1) − k]2 j/2 d x = f (u + 1)ψ(2 j u − k)2 j/2 du PERIODIC WAVELETS 121 f (u)ψ(2 j u − k)2 j/2... constructed periodic wavelets of Coifman, Daubechies, and Franklin are depicted in Figs 4.6, 4.7, and 4.8 4.6.2 Properties of Periodic Wavelets p p It can be verified that ψ j,k , ϕ j,k form an orthonormal basis system possessing the same MRA properties as the regular wavelets do, for example, p 1 p ψ j,k , ϕ j,k = p 0 p p ψ j,k (x)ϕ j,k (x) d x = 0 p ψ j,k , ψ j,k = δk,k p p p p V j = V j+1 V0 ⊂ V1... p 2,0 0.00 1.00 2.00 −2.00 1.00 2.00 − 2.00 0.80 p 1,1 (c) 4.00 0.60 (b) Magnitude Magnitude 4.00 x 0.60 0.80 1.00 p 2,3 2.00 0.00 −2.00 0.00 0.20 0.40 (g) FIGURE 4.6 Periodic Coifman wavelets x (h) 0.60 125 PERIODIC WAVELETS Daubechies φ p 0 0, 2 Daubechies ψ p 0,0 2 1 1 0 0 −1 −1 −2 0 0.2 0.4 0.6 0.8 1 −2 0 0.2 Daubechies ψ 1,0 0.4 0.6 0.8 1 0.8 1 Daubechies ψ 1,1 p p 3 3 2 2 1 1 0 0 −1 −1 −2 −2... −2 −2 −3 0 0.2 0.4 0.6 0.8 1 −3 0 0.2 0.4 0.6 0.8 1 p Daubechies ψ 2,3 p Daubechies ψ 2,2 4 4 3 3 2 2 1 1 0 0 −1 −1 −2 −2 −3 0 0.2 0.4 0.6 0.8 1 −3 0 0.2 0.4 FIGURE 4.7 Periodic Daubechies wavelets 0.6 0.8 1 126 WAVELETS IN BOUNDARY INTEGRAL EQUATIONS p Franklin φ 0, 0 p Franklin ψ 0,0 2 2 1 1 0 0 −1 −1 −2 0 0.2 0.4 0.6 0.8 1 −2 0 0.2 p Franklin ψ 1,0 0.4 0.6 0.8 1 0.8 1 p Franklin ψ 1,1 3 3 2 2 1... 2 2 1 1 0 0 −1 −1 −2 −2 0 0.2 0.4 0.6 Franklin 0.8 1 0 0.2 ψp 2,2 0.4 0.6 Franklin 4 3 2 2 1 1 0 1 4 3 0.8 ψp 2,3 0 −1 −1 −2 −2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 FIGURE 4.8 Periodic Franklin wavelets 0.6 0.8 1 PERIODIC WAVELETS = = 1 2j 2j = 0 ∞ r ∈Z −∞ = r ∈Z = r ∈Z d xψ(2 j x + 2 j − k)ψ(2 j x + 2 j +1 127 −k ) d yψ[2 j y + 2 j ( − ) − k]ψ(2 j y − k ) 2 j ψ(2 j y − 2 j r − k)ψ(2 j y − k ) d y ψ j,2 j... of the equation above has only one term of r = 0, because k = 0, 1, , 2 j − 1 4.6.3 Expansion of a Function in Periodic Wavelets Example Expand a periodic function f (x) where f (x) = −x, 2x − x 2 , −1 ≤ x < 0 0 ≤ x < 1 Solution Let us expand f (x) in V3 in terms of periodic wavelets Note that f (x) here has a period of 2, instead of 1 Thus we first map x ∈ [−1, 1] onto t = [0, 1] by the coordinate . normal mode functions as 100 Wavelets in Electromagnetics and Device Modeling. George W. Pan Copyright ¶ 2003 John Wiley & Sons, Inc. ISBN: 0-471-41901-X WAVELETS IN ELECTROMAGNETICS 101 the. WAVELETS IN BOUNDARY INTEGRAL EQUATIONS • Coordinate transformation. • Periodic wavelets. • Intervallic wavelets. • Weighted wavelets. Here we apply coordinate transformation to I =  b a f (x  )G(x,. systems of linear algebraic equations. Orthogonal wavelets have several properties that are fascinating for electromag- netic field computations. First, wavelets are sets of orthonormal bases of L 2 (R). They