Pulse Scattering on Objects in the Inhomogeneous Conducting Medium

The objective of the work is to present the results of computer simulation of the problem of pulse scattering on cylinder objects buried in the inhomogeneous conducting medium. The finite difference time domain method (FD-TD) is used for solving the problem. The grid discretization in space and time...

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Дата:	2002
Автори:	Varyaniza-Roshchupkina, L.A., Kovalenko, V.O.
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Мова:	English
Опубліковано:	Радіоастрономічний інститут НАН України 2002
Назва видання:	Радиофизика и радиоастрономия
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Цитувати:	Pulse Scattering on Objects in the Inhomogeneous Conducting Medium / L.A. Varyaniza-Roshchupkina, V.O. Kovalenko // Радиофизика и радиоастрономия. — 2002. — Т. 7, № 4. — С. 435-440. — Бібліогр.: 6 назв. — англ.

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spelling	irk-123456789-1223542017-07-03T03:03:29Z Pulse Scattering on Objects in the Inhomogeneous Conducting Medium Varyaniza-Roshchupkina, L.A. Kovalenko, V.O. The objective of the work is to present the results of computer simulation of the problem of pulse scattering on cylinder objects buried in the inhomogeneous conducting medium. The finite difference time domain method (FD-TD) is used for solving the problem. The grid discretization in space and time is carried out taking into account the required stability of the realized method. The elementary absorbing boundary conditions (ABC), such as the perfect matched layer, Bayliss-Turkel annihilation operators and the Mur finite-difference scheme have been realized. Visualization of the results obtained has been carried out, and scattered field images have been constructed. Цель настоящей работы – представить результаты компьютерного моделирования рассеяния импульсов на цилиндрических объектах, погруженных в неоднородную проводящую среду. Для регуляризации задачи используется метод конечных разностей во временной области (FD-TD). Дискретизация сетки в пространстве и времени проводится с учетом обеспечения требования устойчивости реализуемого метода. Реализованы простейшие поглощающие граничные условия (ABC), такие как PML (идеально согласованный слой), операторы аннигиляции Байлисса-Туркела, конечно-разностная схема Мура. Проведена визуализация полученных результатов, построены изображения рассеянных полей. Мета цієї праці – представити результати комп'ютерного моделювання розсіювання імпульсів на циліндричних об’єктах, занурених у неоднорідне провідне середовище. Для регуляризації задачі використовується метод скінчених різниць у часовій області (FD-TD). Дискретизація сітки в просторі і часі проводиться з урахуванням забезпечення вимоги стійкості реалізованого методу. Реалізовано найпростіші поглинаючі граничні умови (ABC), такі як PML (ідеально погоджений шар), оператори анігіляції Байлісса-Туркела, скінченно-різницева схема Мура. Проведено візуалізацію отриманих результатів, побудовано зображення розсіяних полів. 2002 Article Pulse Scattering on Objects in the Inhomogeneous Conducting Medium / L.A. Varyaniza-Roshchupkina, V.O. Kovalenko // Радиофизика и радиоастрономия. — 2002. — Т. 7, № 4. — С. 435-440. — Бібліогр.: 6 назв. — англ. 1027-9636 http://dspace.nbuv.gov.ua/handle/123456789/122354 en Радиофизика и радиоастрономия Радіоастрономічний інститут НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The objective of the work is to present the results of computer simulation of the problem of pulse scattering on cylinder objects buried in the inhomogeneous conducting medium. The finite difference time domain method (FD-TD) is used for solving the problem. The grid discretization in space and time is carried out taking into account the required stability of the realized method. The elementary absorbing boundary conditions (ABC), such as the perfect matched layer, Bayliss-Turkel annihilation operators and the Mur finite-difference scheme have been realized. Visualization of the results obtained has been carried out, and scattered field images have been constructed.
format	Article
author	Varyaniza-Roshchupkina, L.A. Kovalenko, V.O.
spellingShingle	Varyaniza-Roshchupkina, L.A. Kovalenko, V.O. Pulse Scattering on Objects in the Inhomogeneous Conducting Medium Радиофизика и радиоастрономия
author_facet	Varyaniza-Roshchupkina, L.A. Kovalenko, V.O.
author_sort	Varyaniza-Roshchupkina, L.A.
title	Pulse Scattering on Objects in the Inhomogeneous Conducting Medium
title_short	Pulse Scattering on Objects in the Inhomogeneous Conducting Medium
title_full	Pulse Scattering on Objects in the Inhomogeneous Conducting Medium
title_fullStr	Pulse Scattering on Objects in the Inhomogeneous Conducting Medium
title_full_unstemmed	Pulse Scattering on Objects in the Inhomogeneous Conducting Medium
title_sort	pulse scattering on objects in the inhomogeneous conducting medium
publisher	Радіоастрономічний інститут НАН України
publishDate	2002
url	http://dspace.nbuv.gov.ua/handle/123456789/122354
citation_txt	Pulse Scattering on Objects in the Inhomogeneous Conducting Medium / L.A. Varyaniza-Roshchupkina, V.O. Kovalenko // Радиофизика и радиоастрономия. — 2002. — Т. 7, № 4. — С. 435-440. — Бібліогр.: 6 назв. — англ.
series	Радиофизика и радиоастрономия
work_keys_str_mv	AT varyanizaroshchupkinala pulsescatteringonobjectsintheinhomogeneousconductingmedium AT kovalenkovo pulsescatteringonobjectsintheinhomogeneousconductingmedium
first_indexed	2025-07-08T21:33:54Z
last_indexed	2025-07-08T21:33:54Z
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fulltext	Radio Physics and Radio Astronomy, 2002, v. 7, No. 4, pp. 435-440 PULSE SCATTERING ON OBJECTS IN THE INHOMOGENEOUS CONDUCTING MEDIUM L.А. Varyanitza-Roshchupkina, V.О. Kovalenko* ) A.Ya. Usikov Institute for Radiophysics and Electronics of the NAS of Ukraine 12 Акad. Proscura St., 61085, Kharkov, Ukraine E-mail: vla@ire.kharkov.ua *) International Research Centre for Telecommunication-Transmission and Radar, Department of Informational Technology and Systems, TU Delft, Mekelweg 4, 2628 CD Delft, The Netherlands E-mail: V.Kovalenko@IRCTR.Tudelft.NL The objective of the work is to present the results of computer simulation of the problem of pulse scattering on cylinder objects buried in the inhomogeneous conducting medium. The finite difference time domain method (FD-TD) is used for solving the problem. The grid discretization in space and time is carried out taking into ac- count the required stability of the realized method. The elementary absorbing boundary conditions (ABC), such as the perfect matched layer, Bayliss-Turkel annihilation operators and the Mur finite-difference scheme have been realized. Visualization of the results obtained has been carried out, and scattered field images have been constructed. 1. Introduction Ground penetrating radars (GPR) attract more and more attention and are of great interest in recent years. The theoretical works in this area are in two directions: solution of direct problems of pulse dif- fraction on subsurface objects and inverse problems of object reshaping and relocation by the present echo-response. The direct problem solution is very important because during investigations one can obtain a set of “pictures” of fields reflected from different objects. Comparing them with the available echo-response permits to detect a required object without solving inverse problems in some cases. The direct diffrac- tion problems are reduced to solving non-stationary Maxwell’s equations with initial and boundary condi- tions. Up-to-date computers solve this sort of prob- lems by direct methods. One of such methods is the finite-difference time-domain method (FD-TD method), on the basis of which it is possible to solve vector problems of electromagnetic pulse diffraction. The given work is aimed at the FD-TD method Fig. 1. a) Position of the electric and magnetic field vector components about a cubic unit cell of the Yee space lattice; b) Space-time chart of the Yee algorithm for a one-dimensional wave propagation L.А. Varyanitza-Roshchupkina, V.О. Kovalenko 436 Radio Physics and Radio Astronomy, 2002, v. 7, No. 4 application to the problems of pulse diffraction on cylinder objects in dispersion and absorption media and investigation of scattered field on this basis. 2. General Characteristics of the Given Method The finite-difference time-domain method [1,2] is the direct solution method for Maxwell’s time-dependent curl equations. It employs no potentials. Rather, it is based upon volumetric sampling of the unknown near-field distribution (E and H ) within and sur- rounding the structure of interest, and over a period of time. The sampling in space depends on pulse du- ration and is drawn by the user. Typically, 10 to 20 samples per wavelength are needed. The sampling in time is selected to ensure numerical stability of the algorithm. FD-TD is a procedure that simulates the con- tinuous actual electromagnetic waves in a finite spa- tial region by sampled-data numerical analogs propa- gating in a computer data space. For simulations where the modeled region must extend to infinity, absorbing boundary conditions (ABC) are employed at the outer grid truncation planes which ideally per- mit all outgoing wave analogs to exit the region with negligible reflection. Phenomena such as induction of surface currents, scattering and multiple scattering, aperture penetration, and cavity excitation are mod- eled time-step by time-step by the action of the nu- merical analog to the curl equations. Time-stepping is continued until the desired late-time pulse response is observed at the field point of interest. Yee Algorithm 1. As it is shown in Fig. 1(a), the Yee algorithm [3] centers its components E and H in the tree- dimensional space in such a way that each compo- nent E is surrounded by four circulating compo- nents H , and each component H is surrounded by four circulating components E . Resulting finite-difference expressions for space derivatives used in curl operators are central by their nature and have the second order of accu- racy. The continuity of tangential components E and H remains naturally the same when passing across the boundary in case that the boundary is parallel to one of the coordinate axes of the grid. The location of components E and H in the Yee-grid and central-difference operations with these components implicitly realize two rela- tions on the Gaussian law. 2. As it is shown in Fig. 1(b), the Yee algorithm cen- ters its components E and H in time in the so- called “leapfrog” (lf) order. All calculations of E in the interesting for us three-dimensional space are made and stored in memory for a single time point, using the data of H pre-stored in the com- puter memory. Then all calculations of H in the formed space are made and stored in memory, us- ing recently-calculated data of E . This process is circular and continues till finishing the time- stepping (ts). This lf-ts process is all-explicit and therefore there are no problems related with solution of combined equations and matrix inversion. Resulting finite-difference expressions for time derivatives used in curl operators are central by their nature and have the second order of accu- racy. The resulting ts algorithm is nondissipative; i.e. oscillations of the numerical wave propagating in the grid do not falsely decay. Finite Differences By introducing the following designation ( ) , ,, , , n i j ku i x j y k z n t u∆ ∆ ∆ ∆ = for field compo- nents and using finite-difference expressions for space and time derivatives ( ) ( )[ ]1/2, , 1/2, , 2 , , , , n n i j k i j k u i x j y k z n t x u u x x + − ∂ ∆ ∆ ∆ ∆ = ∂ − +Ο ∆ ∆ ( ) ( )[ ] 1/2 1/2 , , , , 2 , , , n n i j k i j k u i x j y k z n t t u u xt + − ∂ ∆ ∆ ∆ ∆ = ∂ − + Ο ∆∆ we obtain the numerical approximation of three- dimensional numerical Maxwell's equations. 3. Computer Realization of the Method For the two-dimensional region with a finite number of media having different electrical properties we can determine the ( ), ,MEDIA i j k structure for each component of the vector field with information about dielectric properties of a medium at the given point. Maxwell’s curl equations are reduced to the follow- ing finite-difference system by the described above finite differences: TM mode: ,xH i jm MEDIA= , ( ) ( )( ) 1/2 1/2 , , , 1/2 , 1/2 , n n x a xi j i j n n i j i jb z z H D m H D m E E + − − + = + − Pulse Scattering on Objects in the Inhomogeneous Conducting Medium Radio Physics and Radio Astronomy, 2002, v. 7, No. 4 437 ,yH i j m MEDIA= , ( ) ( )( ) 1/2 1/2 , , 1/2, 1/2, , n n y a yi j i j n n i j i jb z z H D m H D m E E + − + − = + − ,zE i jm MEDIA= , ( ) ( )( ) 1/21 ,, 1/2, 1/2 1/2 1/2 1/2, , 1/2 , 1/2 . nn n z a z i j b yi j i j n n n y x xi j i j i j E C m E C m H H H H ++ + + + + − − + = + − − + − TE mode: ,xE i jm MEDIA= , ( ) ( )( ) 1/21 ,, , 1/2 1/2 , 1/2 , nn n x a x i j b zi j i j n z i j E C m E C m H H ++ + + − = + − − ,yE i j m MEDIA= , ( ) ( )( ) 1/21 ,, 1/2, 1/2 1/2, , nnn i jy a y b zi j i j n z i j E C m E C m H H ++ − + + = + − − ,yE i j m MEDIA= , ( ) ( ) ( ) 1/2 1/2 , , , 1/2, 1/2 1/2, 1/2, n n z a z bi j i j nn i jx xi j n n i j i jy y H D m H D m E E E E + − −+ − + = + × − + − where updating coefficients are: , , , , , 1 1 2 2 i j i j a i j i j i j t t C σ σ ε ε    ∆ ∆  = − +         , , , , , 1 2 i j b i j i j i j ttC σ ε ε    ∆∆   = +     ∆    , , , , , , 1 1 2 2 i j i j a i j i j i j t t D ρ ρ µ µ ′ ′   ∆ ∆  = − +          , , , , , 1 2 i j b i j i j i j ttD ρ µ µ ′   ∆∆  = +     ∆    . The Source The hard source is set up simply by specifying a de- sired time function for definite components of the electric and magnetic fields in the spatial FD-TD grid. For example, in the one-dimensional TM grid the following hard source on zE should be set up at the source point si for generating a continuous sinu- soidal wave with the frequency 0f : ( )0 0sin 2 s n z iE E f n tπ= ∆ . Another common hard source is the wideband Gaussian pulse with the finite dc. The pulse is cen- tered in the time step 0n and has the 1/e character- istic decay decayn of time steps: ( )[ ]20 / 0 decay s n n nn z iE E e− −= . There is a simple method to avoid the reflexive action of the hard pulse source – to remove it from the algorithm after the pulse will decrease essentially to 0 and apply an updated field instead of the normal Yee field. In the source context we will program an equivalent of the following updating relation for the electric field at si : if ( )( )01 / 3.0decayn n n+ − ≤ ( )[ ]201 /1 0 decay s n n nn z iE E e− + −+ = else ( ) ( )( ) 1 1/2 1/2 1/2 1/2 s s s s n n z a zi i n n b y yi i E C m E C m H H + + + + − = + − Absorbing Boundary Conditions The main problem related with the FD-TD-approach to solving the problems of electromagnetic wave in- teraction is the fact that many interesting geometries are determined as open regions in which the com- puted field spatial region is unbounded in one or more coordinate directions. Clearly there is no com- puter capable to store an unbounded quantity of data and therefore the region of field calculation should be limited in size. The calculation region should be suf- ficiently large in order to surround the interesting structure, and around the external perimeter it is nec- essary to apply appropriate boundary conditions for simulating the wave propagation to infinity – so- called absorbing boundary conditions (ABC). Bayliss-Turkel operators of scattered wave annihilation For cylindrical coordinates the Bayliss-Turkel annihi- lation operator of n-order is defined as [4]: ( ) 1 4 3 2 n n k kB L r= −= +∏ , where the operator 1L c t R ∂ ∂≡ + ∂ ∂ . L.А. Varyanitza-Roshchupkina, V.О. Kovalenko 438 Radio Physics and Radio Astronomy, 2002, v. 7, No. 4 The given differential operator systematically “destroys” or “annihilates” arbitrary outgoing scat- tered waves and leaves the remainder term, which is the difference process error. At any point of the grid external boundary the application of this differential operator to the local field allows us to estimate the field space derivative in the direction of outgoing wave propagation in terms of transverse space and time derivatives by using the data at points that are entirely inside the grid. The knowledge of the field space derivative in the direction of outgoing wave propagation permits to close the calculation region. The Finite-Difference Mur Scheme Let 0, n jW be the Cartesian component E or H lo- cated on the boundary 0x = of the Yee grid. Mur realized the space derivatives as central differences decomposed at the auxiliary point ( ) 1/2, j and ob- tained [5]: ( ) ( ) ( ) 1 0, 1 1 1 1, 1, 0, 0, 1, 2 0, 1 0,2 0, 1 1, 1 1, 1, 1 2 ( ) 2 2( ) ( ) 2 n j n n n j j j n n j j n n j j n n n n j j j j W c t xW W W c t x x W W c t x c t x W W y c t x W W W W + − + − + − + − = ∆ −∆− + + + ∆ +∆ ∆ + + ∆ +∆ ∆ ∆ − + ∆ ∆ +∆ + − + − the ts algorithm for W components along the boundary 0x = . Similarly it is possible to obtain analogous finite-difference expressions for Mur ab- sorbing boundary conditions on every other boundary of the grid x h= , 0y = and y h= . Berenger Perfectly Matched Layer (PML) Berenger obtained effective reflection coefficients for his absorbing boundary condition constituting 1/3000 of reflection coefficients of considered above stan- dard analytic absorbing boundary conditions of sec- ond and third order. The approach, which he called “the perfectly matched layer (PML) for electromag- netic wave absorption” [6], is based on splitting the electric and magnetic field components in the absorb- ing boundary region into single subcomponents. The TE Case Let’s consider two-dimensional Maxwell's equations for the TE polarization case. After splitting zH into two components zxH and zyH we obtain four (ear- lier we obtained three components) components for the TE case, connected by the following equations: ( ) 0 zx zyx y x H HE E t y ε σ ∂ +∂ + = ∂ ∂ , ( ) 0 y zx zy x y E H H E t x ε σ ∂ ∂ + + = − ∂ ∂ , 0 yzx x zx EH H t x µ ρ ∂∂ ′+ = − ∂ ∂ , 0 zy x y zy H E H t y µ ρ ∂ ∂′+ = ∂ ∂ , where xσ and yσ denote electric conductivity, and xρ ′ and yρ ′ denote magnetic loss. When yσ = yρ ′ =0, the PML-medium can absorb plain waves with field components ( , )y zzE H propa- gating along x , but it does not absorb waves with field components ( , )x zyE H propagating along y . We have the opposite situation when 0x xσ ρ ′= = . 4. The Program Grider1. Example The program Grider1 uses the described above method of calculation of fields scattered in dispersion and absorbing media. Let’s consider the spatial region with length 4x = m and depth 4y = m, consisting of three dielectric layers with parameters ( 1ε = , 1µ = , 0σ = , 0ρ ′ = ), ( 5ε = , 1µ = , 0.000001σ = , 0ρ ′ = ), ( 10ε = , 1µ = , 0.00001σ = , 0ρ ′ = ) and a submerged in the last layer object with dielec- tric parameters ( 1ε = , 1µ = , 100000000σ = , 0ρ ′ = ). We specify the grid 200 200× , 0.002dx = m, 0.002dy = m. To ensure solution stability the time- step should not exceed 0.04714045 ns. The following hard source on xE is set up at the point (100, 100): it is a wideband Gaussian pulse with finite dc with the central time step equal to 20, characteristic decay of 4 time-steps and the amplitude of 1 V/m. Fig. 3(a) represents distribution of the field component xE in space at the time 150t = time- steps: Fig. 2. Structure of a two-dimensional FD-TD grid having the Berenger PML ABC Pulse Scattering on Objects in the Inhomogeneous Conducting Medium Radio Physics and Radio Astronomy, 2002, v. 7, No. 4 439 The reflection (field F1) from the boundary be- tween first two media is clearly seen in Fig. 3(a). But at the present time the signal hasn’t reached yet the third medium and the object. Fig. 3(b) represents distribution of the field component xE in space at the time 150t = time- steps. In the given figure (Fig. 3(b)) one can see reflec- tions (field F1, field F2) from both boundaries be- tween dielectric media and the signal (field F3) re- flected from the conducting object. Fig. 4 represents distribution of the field com- ponent xE at the point (100, 110) during 300 time- steps. The echo-response of the transmitted signal is also easily observable in the given figure (Fig. 4). For comparison let’s consider the same spatial region with only parameters of layers ( 1ε = , 1µ = , 0σ = , 0ρ ′ = ), ( 8ε = , 1µ = , 0.000005σ = , 0ρ ′ = ), ( 15ε = , 1µ = , 0.00005σ = , 0ρ ′ = ) that correspond to moisture saturation. For this case we obtain following distribu- tion of the field component xE at the point (100, 110) during 300 time-steps: We can see a typical modification of pulse shape for this case on Fig. 5. Fig. 3. Distribution of the field component xE in space at the time a) 150t = time-steps; b) 250t = time-steps Fig. 4. Distribution of the field component xE at the point (100, 110) during 300 time-steps L.А. Varyanitza-Roshchupkina, V.О. Kovalenko 440 Radio Physics and Radio Astronomy, 2002, v. 7, No. 4 5. Conclusion As it is seen from the above-said material the finite difference time domain method is very useful for realization on computer and gives totally valid results at sufficiently small spatial and time domain discreti- zation. Thus, using the given method and Grider1 program realizing it one can investigate in future scattered fields on various objects in different media. References 1. A. Taflove. Computational electrodynamics. The Fi- nite-Difference Time-Domain Method. Artech House. Boston-London. (1995). 2. A.F. Peterson, S.L. Ray, R. Mittra. Computational Methods for Electromagnetics. IEEE PRESS. New York. Oxford University Press. Oxford, Tokyo, Mel- bourne. (1997). 3. K.S. Yee. Numerical solution of initial boundary value problems involving Maxwell’s equations in iso- tropic media. IEEE Trans. Antennas Propagat, AP-14. (1966). 4. A. Bayliss, E. Turkel. Radiation boundary conditions for wave-like equations. Comm. Pure Appl. Math. vol. 23. (1980). 5. G. Mur. Absorbing boundary conditions for the finite- difference approximation of the time-domain electro- magnetic field equations. IEEE Trans. Electromag- netic Compatibility, EMC-23. (1981). 6. J.P. Berenger. A perfectly matched layer for the ab- sorption of electromagnetics waves. J. Computat. Phys., vol. 114. (1994). РАССЕЯНИЕ ИМПУЛЬСОВ НА ОБЪЕКТАХ В НЕОДНОРОДНОЙ ПРОВОДЯЩЕЙ СРЕДЕ Л.А. Варяница-Рощупкина, В.О. Коваленко Цель настоящей работы – представить результаты компьютерного моделирования рассеяния импульсов на цилиндрических объектах, погруженных в неодно- родную проводящую среду. Для регуляризации задачи используется метод конечных разностей во временной области (FD-TD). Дискретизация сетки в пространстве и времени проводится с учетом обеспечения требова- ния устойчивости реализуемого метода. Реализованы простейшие поглощающие граничные условия (ABC), такие как PML (идеально согласованный слой), опера- торы аннигиляции Байлисса-Туркела, конечно- разностная схема Мура. Проведена визуализация полу- ченных результатов, построены изображения рассеян- ных полей. РОЗСІЯННЯ ІМПУЛЬСІВ НА ОБ’ЄКТАХ В НЕОДНОРІДНОМУ ПРОВІДНОМУ СЕРЕДОВИЩІ Л.А. Варяниця-Рощупкіна, В.О. Коваленко Мета цієї праці – представити результати комп'ю- терного моделювання розсіювання імпульсів на цилін- дричних об’єктах, занурених у неоднорідне провідне середовище. Для регуляризації задачі використовується метод скінчених різниць у часовій області (FD-TD). Дискретизація сітки в просторі і часі проводиться з урахуванням забезпечення вимоги стійкості реалізова- ного методу. Реалізовано найпростіші поглинаючі гра- ничні умови (ABC), такі як PML (ідеально погоджений шар), оператори анігіляції Байлісса-Туркела, скінчен- но-різницева схема Мура. Проведено візуалізацію отриманих результатів, побудовано зображення розсія- них полів. Fig. 5. Distribution of the field component xE at the point (100, 110) during 300 time-steps

Pulse Scattering on Objects in the Inhomogeneous Conducting Medium

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