Grid adaptation for modeling trapezoidal difractors by using pseudo-spectral methods for solving Maxwell equations

Transcription

1 Keldysh Institute Publication search Keldysh Institute preprints Preprint No., Zaitsev N.A., Sofronov I.L. Grid adaptation for modeling trapeoidal difractors by using pseudo-spectral methods for solving Maxwell equations ecommended form of bibliographic references: Zaitsev N.A., Sofronov I.L. Grid adaptation for modeling trapeoidal difractors by using pseudo-spectral methods for solving Maxwell equations. Keldysh Institute preprints,, No., 6 p. UL:

2 О р д е н а Л е н и н а ИНСТИТУТ ПРИКЛАДНОЙ МАТЕМАТИКИ имени М.В.Келдыша Р о с с и й с к о й а к а д е м и и н а у к N.A. Zaitsev, I.L. Sofronov Grid adaptation for modeling trapeoidal difractors by using pseudo-spectral methods for solving Maxwell equations Москва

3 N. A. Zaitsev, I. L. Sofronov, Grid adaptation for modeling trapeoidal difractors by using pseudo-spectral methods for solving Maxwell equations Abstract. We propose a series of successive one-dimensional coordinate transformations to construct an adaptive two-dimensional grid for the problem of diffraction on a grating with a trapeoidal tooth. Special attention is paid to the smoothness of the grid and grid refinement in domains of large solution gradients. Key words. Adaptive grid, Maxwell equation, difraction. Н. А. Зайцев, И. Л. Софронов, Построение адаптивной сетки для трапециевидных структур при моделировании дифракции псевдоспектральным методом решения уравнений Максвелла Аннотация. В работе предложен ряд последовательных одномерных преобразований координат для построения адаптивной двумерной сетки для задачи дифракции на решетке с трапециевидным зубом. Особое внимание уделено гладкости получаемой сетки и сгущению узлов к местам наибольших градиентов решения. Ключевые слова: адаптивная сетка, уравнение Максвелла, дифракция. Contents I. Preliminaries... II. Problem formulation... 4 III. Generation of adaptive -grid... 5 IV. Generation of adaptive x-grid... 7 V. Numerical examples... VI. eferences... 6 Работа выполнена при финансовой поддержке РФФИ, гранты 4 и -567.

4 I. Preliminaries We solve D diffraction problem described by Maxwell equations c c D H t H t E () with the dispersion law t D( r, t) E( r, t) ( ) E( r, t ) d, r ( x, y, ), P p ( t) p exp ( p ip) t exp ( p ip) t i. The scalar p-polariation case of (), normal incidence, reads: D H x c t D H c t x H E E c t x y y y x () Diffraction is considered on a grating with trapeoidal tooth. The pseudo-spectral method [] used for the solution of this problem requires high-quality grid in this periodical domain. We propose step-by-step method of generating the grid and make analysis of quality of the grid by considering various examples of values of geometric parameters.

5 4 II. Problem formulation Suppose we need to construct computational grid in D rectangular computational domain of width W and height H for a periodic with respect to x problem governed by (). The physical domain is the single spacing of a grating (W coincides with the period of the problem), see Figure. The height of the tooth of the grating is h g and its width is w g. The angle of the slope of sides of the tooth is. Let us introduce the following notations: x min, x max are the horiontal ends of the computational domain, W xmax xmin ; min, max are the vertical ends of the computational domain, H max min ; a is the base of the tooth of the grating; b is the top of the tooth of the grating; x( ), xl() are the right and left supporting lines, see Figure. For a b these lines coincide with the sides of the tooth. One has to continue the lines to the bottom and the top boundaries of the computational domain. They must have vertical tangents at the boundaries in order to use simpler transparent boundary conditions []. α wg hg x Figure. The general view of the computational domain.

6 5 III. Generation of adaptive -grid We will consider the symmetric case, i.e. when the tooth of the grating is an isosceles trapeoid. In this case xmax W /, xmin xmax ; max H /, min max ; h /, ; x ( ) x ( ) ; b g a b, x b wg hg x ( ) w cot( ) h / a g g L ( ) cot( ) /. In order to construct adaptive grid and calculate derivatives x,, such that ( ), x x(, ). use a one-to-one map / and / x we We construct the map as a superposition of simpler maps. Let us start from the ( ) transformation. max b x L() x () h g a min x min x max Figure. Supporting lines.

7 6 In -direction we have 4 supporting points: min, a, b, max. The grid in - direction should be Chebyshev s type grid and should have small grid spacing near the points a and b. First we normalie the domain in order to use sine function as a condensing map. So we need to map min on, a on, b on and max on. It is done by means of the polynomial function of fifth order: ( ) c c, c 5 c, [,] () where c5 max b/, c b c5. for max / b.7 ; and by cc ( ) c arctan( c ),, [,] ( c ) where c and c are chosen so that carctan( c) b, c arctan( c) max for max / b.7. (4) Now we concentrate grid points to the interfaces the following function: a and b by means of G G ( ) sin /, cos, [,] ; (5) where coefficient G is the approximate ratio of grid step sies near the interfaces to sies of the grid steps without this transformation. Usually one needs additional transformation in order to redistribute grid points between the inner region b and the outer region b. It can be done making use of the following function: where ( ) d d, d d, [,] (6) d, d d, here characteries a coarsening from the centre and is equal to the value of ratio ()/ () (the derivative d/ d at the middle point d ). If - is equal to

8 7 grid without transform (6) is equidistant then the ratio of grid steps at the middle point and at the outer boundary is equal to If then instead of coarsening the transform concentrates grid points to the center. However, if then map (6) is not one-to-one. In this case we can use the following map: where ( ) d sin( )/, d cos( ), [,] (7) d. Sometimes this transformation is not aggressive enough. Therefore we can use the following instead of (7): where t t( ), t, t ( t) t d sin( t)/, d cos( t), t [,], t( ) d sin( )/, t d cos( ), [,] d. (8) Now one can construct -grid using the Chebyshev s points cos ( N k)/( N ), k,..., N k and calculate derivative d/ d with the spectral accuracy, see []. IV. Generation of adaptive x-grid The essential difference between x-grid and -grid is that the problem is periodic in x- direction. Therefore all our maps must be periodic along lines const. We start the constructing of the x-grid from the eliminating of -dependence of the grid lines. For this purpose we use transform x( y, ) y asin( y/ ymax ), y[ xmax, xmax ], x a sin( y/ y ), x a cos( y/ y )/ y max y max max (9)

9 8 where x( ) y dx ( )/, d a a, sin( y / y ) sin( y / y ) max max max y x, y h /. g max () x and dx / We need also functions () a polynomial of the second order: d. The simplest way to continue x is using min a x ( ) a a a, dx / d a a, [, ], x ( ) x ( ) a, a a, a x ( ) a a, b a min a a a b a ( a min ) x( ) b b b, dx / d b b, [ b, max ], x ( ) x ( ) b, b b b a max b a ( b max ) One can use also polynomials of the third order: b b b, b x ( ) b b. min a x ( ) a a a a, dx / d a a a, [, ], x ( ) x ( ) a, a a, b a min b a ( a min ) a a min amin, a x( a) a a aa aa, b () x ( ) b b b b, dx / d b b b, [, max ], x( b) x( ) b a, b b max, ( ) and of the fourth order: b a b max max max b b b b b b b, b x ( ) b b b. 4 dx x ( ) 4, a a a a a a a a 4 a4, [ min, a], d x ( ) x ( ) a, a a ( ), a a 6 a, 4 b a 4 a min min 4 min b a ( a min ) a a min a min 4 a4min, a x( a) a a aa aa a4a 4, ()

10 9 4 dx x ( ) 4, b b b b b b b b 4 b4, [ b, max ], d x ( ) x ( ) b, b b ( ), b b 6 b, 4 b a 4 b max max 4 max b a ( b max ) b b max b max 4 b4 max, b x( b) b b bb b b b4b 4. () To provide higher smoothness one can need even the sixth order. Formulas are derived as follows. We introduce auxiliary map s min for [ min, a ]. Then x ( s) a a4s a5s a6s, dx / d 4a4s a5s 6 a6s, [ min, a] (4) where the coefficients are obtained from solution of system a a s a s a s x ( ) 4 a 5 a 6 a a 4 5 x ( ) ( ) b x a s a a a sa a6sa 4 4 a 5 a 6 a a4sa a5sa a6sa 6a s a s 5a s 5 b a (5) with and s a a min. Similarly, for b max [, ] s max x ( s) b b4s b5s b6s, dx / d 4b4 s b5 s 6 b6s, [ b, max ] (6) where the coefficients are solution of the system b 5 b 6 b b b b s b s b s x ( ) 4 5 x ( ) ( ) b x b s a b b sb b6s b 4 4 b 5 b 6 b b4sb b5s b b6sb 6b s b s 5b s 5 b a (7)

11 with sbb max. One can also use the following infinitely smooth continuation of sides of the trapeoid to the boundaries: C ( ) xinf C t xinf, min, a ; min a dx x( b) x( ) a erf yt ( ) ; d ( ) b a dx x ( ) ( ) x a d d where the values of the integral have to be computed numerically. The difference between the variants of x() is presented in Figure. x() and xl(), current order= a (8) -4 = a x ( a ) order x = order x = order x = 4 order x = 6 order x = = min Figure. Variants of x () obtained using polynomials of various orders in the region min a [, ] (-axis is vertical) The dependence of x() on parameter C xinf is as follows: if Cxinf the curve x() (red line in Figure 4) tends to the line BED, see Figure 4. In this figure the segment BC is the linear extrapolation of the right side of the trapeoid, the segment ED is the line joining midpoints of sides BC and AC of triangle ABC. If

12 Cxinf the curve x() tends to the segment BD. Numerical experiments show that the optimal value of C xinf is about. One can see that for all the values of parameter C xinf point D is the end point of line x(). If we would like to move the end from point D we can use the following function: min a C ( ) x inf C t x inf, min, a. min a The right side of the trapeoid B =a E =min A D C Figure 4. Lay-out of x () For the interval [ b, max ] an infinitely smooth continuation of the sides of the trapeoid is achieved by the following formulas: C ( ) xinf C t xinf, b, max ; max b dx x( b) x( ) a erf yt ( ) ; d ( ) b a dx x ( ) ( ) x b d. d b (9)

13 The normaliation of the domain y [ ymin, ymax ] we do by the same way as for - axis. Therefore, we us the map ymax y ( ) max d y d sin, y cos, [,] () where d y y max / for. ymax / y 5 (see ()); and the map y( ) ymax d exp d cos t dt, y d exp d cos, [,] () where parameters d and d must meet the following conditions d exp d cos d y y, for ymax / y. or ymax / y 5. max d exp d cos d y max The following map concentrates grid points to the interfaces x x() and x xl( ) : ( ) G sin /, G cos, [,] () x Approximately G x is the ratio of grid step sie near the interfaces to grid steps sie without this transformation. Usually one needs an additional transformation in order to redistribute grid points between the inner region x x() and the outer region x x(). It can be done by using the following function: x where t( ) d sin( )/, t d cos( ), [,], ( t) t d sin( t)/, d cos( t), t [,], t t t x d ; x ()

14 characteries the value of ratio ()/ (). If x-grid without transform () is x equidistant then the ratio of grid steps at the middle point and at the outer boundary is equal to x. However, for big values, if x, map () is not one-to-one. In this case we use the following map: ( ) d exp d cos t dt, d exp d cos, [,] (4) where parameters d and d must meet the following conditions: log, d x d Now we can construct equidistant -grid exp d cos d m/ N, m,..., N m x x ( Nx is excluded because of the periodicity) and calculate derivative d/ d with the spectral accuracy, see [].. V. Numerical examples In short notation the presented transform has the following form:,,, x x x x x x x x y,, y y,,. (5) Thus for a function u( x, ) we have:

15 4 u u x x, u u d u x x, d (6) where x x x x dy d d d d d d,, x x y d d d d d d d x da da dx ( ) sin( / max ), y y. d d d sin( y / y ) The corresponding Matlab code which computes the grid and implements formulas (6) has been written. For numerical tests we take the following function max x u( x, ) exp sin x max max (7) for which u u u x, exp cos x x x. (8) max max max max In order to check the spectral accuracy of the presented method and its implementation we take the following setup: H=, W=, h, w 8, 7 ; g g (9) and the following values of parameters: Cx inf, G., Gx., _ rarefaction.5, x_ rarefaction 8. () Table shows obtained accuracy of computed values of u/ and u/ x for various numbers of grid points.

16 5 Table. Error of derivatives versus discretiation parameters N x N Error of u/ Error of u/ x It s clear that the method has the spectral accuracy. Nevertheless, it seems that too many points needed to achieve the desired accuracy. The reason is that even a very smooth in x, -space function can has very big gradient in, -space. Let us investigate sensitivity to separate parameters of grid. Graph of function u( x, ) (7) in Cartesian coordinates is presented in Figure 5. Graph of the same function in parametric coordinates and for the following values of the parameters H, W, Gx, x, h 6, w 5, G, g g () is presented in Figure 6. Here hg H and wg W/ so the normaliation doesn t influence almost. Similar graph for Gx. is shown in Figure 7. Here u(, ) has much bigger gradient. Change of parameter G has similar effect: graph of u(, ) for parameters Gx and G. is shown in Figure 8 (other parameters are not changed). The next Figure shows the picture when we condense grid points to the interfaces in both directions: Gx., G.. Graph of u(, ) without any condensation but with coarsening x 8 is presented in Figure. The same picture with x and x /8 (concentration) is presented in Figure. The picture with x 8 and x /8 is presented in Figure. Graph of u(, ) without any condensation and coarsening but for the real positions of the interfaces ( hg, wg 8) is presented in Figure. One can see very strong deformation of the original function (see Figure 6). Graph of u(, ) when all the parameters are active is shown in Figure 4. One can see that the coarsening transform reduces the gradient near the boundaries but generates additional gradient near the center. Next set of figures shows graphs of correspondent grids.

17 6 In Figure 5 and Figure 6 -grid and its grid steps are shown when there is no coarsening in direction. Figure 7 and Figure 8 show the same picture for coarsening.5. Figure 9 and Figure show x-grid grid and its grid steps when there is no coarsening in x direction. Figure Figure show the same picture for coarsening x 8. The final grid when all the parameters are active is presented in Figure. VI. eferences [] Зайцев Н.А., Софронов И.Л. Схемы высокой точности для трёхмерных уравнений Максвелла в средах Лоренца, базирующиеся на неявных схемах переменных направлений // Препринты ИПМ им.м.в.келдыша с. UL: [] Софронов И.Л., Воскобойникова О.И., Зайцев Н.А. Прозрачные граничные условия на неравномерных сетках для двумерных уравнений Максвелла с дисперсиейсхемы высокой точности для трёхмерных уравнений Максвелла в средах Лоренца, базирующиеся на неявных схемах переменных направлений // Препринты ИПМ им.м.в.келдыша с. UL: [] Lloyd N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia,

18 x -5 Figure 5. Function u( x, ) Figure 6. Function u(, ) for parameters of grid transform H, h 6, W, w 5, G, G,,. g g x x

19 Figure 7. Function u(, ) for parameters of grid transform H, h 6, W, w 5, G., G,,. g g x x Figure 8. Function u(, ) for parameters of grid transform H, h 6, W, w 5, G, G.,,. g g x x

20 Figure 9. Function u(, ) for parameters of grid transform H, h 6, W, w 5, G., G.,,. g g x x Figure. Function u(, ) for parameters of grid transform H, h 6, W, w 5, G, G, 8,. g g x x

21 Figure. Function u(, ) for parameters of grid transform H, h 6, W, w 5, G, G,,.5 g g x x Figure. Function u(, ) for parameters of grid transform H, h 6, W, w 5, G, G, 8,.5 g g x x

22 Figure. Function u(, ) for parameters of grid transform H, h, W, w 8, G, G,,. g g x x Figure 4. Function u(, ) for parameters of grid transform H, h, W, w 8, G., G., 8,.5 g g x x

23 6 -grid, K_intervals= Figure 5. -grid when there is no coarsening in direction, 6 Grid steps: h min =.476 h max = h Figure 6. Grid steps () h when there is no coarsening in direction,

24 6 -grid, K_intervals= Figure 7. -grid for coarsening.5 6 Grid steps: h min =.594 h max = h Figure 8. Grid steps () h for coarsening.5

25 4 6 x-grid, Kx_intervals= Figure 9. x-grid when there is no coarsening in x direction, x 4 Grid steps: hx min =.854 hx max = h x x Figure. Grid steps () x h x when there is no coarsening in x direction, x

26 5 6 x-grid, Kx_intervals= Figure. x-grid for coarsening x 8 4 Grid steps: hx min =.6975 hx max = h x x Figure. Grid steps () x h x for coarsening 8 x

27 6 6 D grid, Kx_intervals=64 K_intervals= Figure. Final grid for H, h, W, w 8, G., G., 8,.5 g x x g