# Using Finite-Difference Time-Domain Modeling to Solve Electromagnetic Problems

### Key Takeaways

• Among the numerical methods available today, the FDTD model is the most widely used numerical method to solve complex electromagnetic problems.

• The finite-difference time-domain (FDTD) method was proposed by Yee for solving Maxwell’s equations using central difference approximations of the spatial and temporal derivatives of curl equations.

• The volumetric approach makes FDTD exceptionally efficient for modeling complex structures and inhomogeneous mediums including dielectrics and conductors.

The advent of computers has simplified solving differential equations

In simple mathematical equations, the dependency of variables in the equation, time dependency, and other variations in the system are easily expressed. As systems become more complex, their mathematical representations and solution processes also get more complicated. With the advent of computers, various numerical techniques, such as the finite-difference method and the finite-difference time-domain method, have simplified solving differential equations, particularly partial derivatives. These numerical methods follow the procedure of discretization or modeling as approximations to solve an equation. Finite-difference time-domain modeling is a numerical technique used to solve problems related to interactions between biological bodies and electromagnetic fields.

## The Finite-Difference Time-Domain Method

Problems involving interactions of electromagnetic fields consist of complex mathematical equations. Wave equations and Maxwell’s equations are the expressions present in these mathematical problems. These equations are solved using analytical methods and numerical methods and by considering the initial and boundary conditions. Numerical methods have gained popularity with the emergence of fast computers. Finite-difference-based numerical methods are used to solve electromagnetic field problems regarding the geometry of the material interface and wavelengths.

Mathematician Kane Yee introduced the finite-difference time-domain (FDTD) method to solve Maxwell’s equations using central difference approximations of the spatial and temporal derivatives of curl equations. The basic Yee algorithm is suitable for regular geometry. However, the application of the Yee algorithm fails when dealing with electromagnetic problems in more complex geometry.

The FDTD method is derived from the basic Yee algorithm and is suitable for solving Maxwell’s equations in regular and irregular geometrical shapes. Among the numerical methods available today, the FDTD model is the most widely used numerical method to solve complex electromagnetic problems.

## Finite-Difference Time-Domain Modeling

In the finite-difference method (FDM), the solution region is sampled at grid points. This causes problems when the grid points lie on the interface of two mediums or at the edge of a conductor or dielectric. Volume-based, numerical FDTD modeling can overcome the problems encountered by the FDM method. The volumetric approach makes FDTD exceptionally efficient in modeling complex structures and inhomogeneous mediums, including dielectrics and conductors.

FDTD modeling can be summarised with the following steps:

1. The solution region is divided into a uniform mesh composed of cells.
2. The electric and magnetic field components are defined for each cell.
3. The electric and magnetic components are staggered, and the standard leapfrog scheme is used for time staggering. Both spatial staggering and time staggering help achieve the direct solution of electromagnetic fields in space and time.
4. The solution for electric and magnetic field components at a given instant in time is defined and stored in the computer's memory.

FDTD modeling and finite element modeling (FEM) are very similar. The concept of dividing into cells is identical to both FEM and FDTD. In FEM, matrix equations are developed from approximations, which are solved using various methods. In contrast, there are no matrix formations in FDTD modeling. The formation of the 3D Cartesian grid in FDTD utilizes space and time staggering, and solutions for the electric and magnetic fields are obtained at any given instant in time.

## The Pros and Cons of FDTD Modeling

The FDTD numerical method is used extensively in computational electromagnetic applications. The table below gives some of the pros and cons of FDTD.

 Pros Cons FDTD is a time-domain method when compared to the method of moments and FEM. FDTD is ineffective in solving unstructured non-cartesian grids. FDTD is effective for modeling complex structures and inhomogeneous mediums including dielectrics and conductors. FDTD is inaccurate when the solution region has curved boundaries. Large and complex problems can be easily solved using FDTD. FDTD is not able to represent the wave motion over long distances. The FDTD algorithm runs efficiently on parallel computers. FDTD requires explicit truncation of the unlimited physical domain through boundary conditions in open-region electromagnetic problems.

It is easy to conceptualize and execute finite-difference time-domain modeling of the solution regions in electromagnetic field problems. Cadence offers 3D FDTD electromagnetic simulation tools for tackling electromagnetic challenges in electronics, automotive, and high-performance computational systems.