1 EEE 431 Computational Methods in Electrodynamics Lecture 7 By Dr. Rasime Uyguroglu

2 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

3 FDTD References [1]M.N.O. Sadiku, Numerical Techniques in Electromagnetics, CRC Press, 2001, pp [2] A. Taflove, Computational Elertodynamics: the Finite-Difference Time-Domain Method, Artech, 1995 [3] A. Taflove, S.C. Hagness, same as above, 2nd ed., Artech, 2000 [4] K. Kunz and R. Luebbers, Finite-Difference Time-Domain Method for Electromagnetics, CRC Press, 1993 [5]Kane S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat., vol. AP-14, No. 3, pp , May,1966.

4 References Introduction to the Finite Time Domain (FDTD) Technique, S.Connor

5 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) The FDTD method, proposed by Yee, 1966, is another numerical method, used widely for the solution of EM problems. It is used to solve open-region scattering, radiation, diffusion, microwave circuit modeling, biomedical etc. problems.

6 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) One of the most important concerns of the FDTD method is the requirement of the artificial mesh truncation (boundary) conditions. These conditions are used to truncate the solution domain and they are known as absorbing boundary conditions (ABCs), as they theoretically absorb fields. Imperfect ABCs create reflections and the accuracy of the FDTD method depends on the accuracy of the ABCs.

7 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) The following advantages make the FDTD method popular: Its a direct solution of Maxwell’s equations, no integral equations are required and no matrix inversions are necessary. Its implementation is easy and it is conceptually simple. It can be applied to the three-dimensional, arbitrary geometries. It can be applied to materials with any conductivity.

8 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) The FDTD method has also the following disadvantages: When the FDTD method is applied, the object and its surroundings must be defined. Since computational meshes are rectangular in shape it is difficult to apply the method to the curved scaterers. It has low order of accuracy and stability unless fine mesh is used

9 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) In this method the coupled Maxwell’s curl equations in the differential form are discretized, approximating the derivatives with centered difference approximations in both time and space domains.

10 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) The six scalar components of electric and magnetic fields are obtained in a time-stepped manner. The space domain includes the object and it is terminated by Absorbing Boundary Conditions (ABCs).

11 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Basic Finite-Difference Time-Domain Algorithm Differential Forms of Maxwell’s Equations

12 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) In linear, isotropic and homogeneous materials and are related to and with the following constitutive relations:

13 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Also is related to as, in a conducting medium.

14 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Substituting the constitutive relations into the Maxwell’s equation’s, we can write six scalar equations in the Cartesian coordinate system.

15 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Electric Field Intensity:

16 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Magnetic field intensity:

17 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Yee Algorithm: Yee algorithm solves for both electric and magnetic fields in time and space using the coupled Maxwell’s curl equations.

18 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) the Yee algorithm centers its and field components in three-dimensional space so that every component is surrounded by four circulating components, and every component is surrounded by four circulating components.

19 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Unit cell of the Yee space lattice.

20 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) The computational domain is divided into a number of rectangular unit cells According to Yee algorithm and field components are separated by in time.

21 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

22 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Finite Differences ( Discretization ) A space point in a uniform rectangular lattice is denoted as

23 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Here, and are the lattice space increments in the x, y, and z coordinate directions respectively and i, j, and k are integers. If any scalar function of space and time evaluated at a discrete point in the grid and at a discrete point in time is denoted by u, then

24 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Using central, finite difference approximation in space, i.e w.r.t.x:

25 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Using central, finite difference approximation in time:

26 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Ex

27 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Ey

28 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Ez

29 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Hx

30 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Hy

31 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Hz