The Spectral-Element Time-Domain (SETD) Method

Key Takeaways

• Many physical systems, including electronics, can have complex geometries that require high levels of discretization.

• The spectral-element time-domain method is one method for reducing the geometric complexity of a system when developing a numerical mesh for a field solver.

• This numerical method is still a subject of continuous research and can be more computationally efficient than FDTD/FDFD field solvers in certain geometries.

Spectral-element time-domain simulations let you visualize physical phenomena in complex geometries.

Modern computers and field solvers for differential equations have brought major productivity boosts to engineers in all disciplines. This is especially true in electromagnetics and computational fluid dynamics, where a set of partial differential equations needs to be solved in space and time. Research on field solvers is rich, especially regarding techniques to reduce computational burden to acceptable levels.

There are many systems that might have complex geometry which then require high discretization density when developing a numerical mesh for a field solver. The field solver you use will generally brute-force the creation of a numerical mesh to give the required accuracy. However, more elegant methods can be used to reduce mesh density, thereby decreasing the computational time required to solve the problem.

One of these methods is the spectral-element time-domain method. In this method, the geometry is a variant on a particular finite element method that was first proposed in 1984. This is one of many methods that can be implemented in commercial field solvers, including for design and evaluation of electronic devices. Some example devices include piezos and actuators as well as periodic arrays. Here’s how this method works and why you might consider using it to evaluate your next electronic device.

What Is the Spectral-Element Time-Domain Method?

The spectral element time-domain method is a numerical method for solving differential equations which uses polynomial basis functions to approximate the solution to a differential equation. In particular, the point of the spectral-element time-domain method is to expand the solution in terms of a series of trigonometric functions up to very high order. The higher the order of expansion, the more accurate the resulting solution.

This expansion is allowed as trigonometric functions and their approximations form an orthonormal basis for differential operators used in computational fluid dynamics (CFD), electromagnetics, and heat flow problems. Such trigonometric polynomials are usually orthogonal Chebyshev polynomials, Legendre polynomials, or Lagrange polynomials. This method has seen adoption outside of electromagnetics and CFD, such as in seismology.

Spectral-Element Method vs. Spectral-Element Time-Domain Method

The spectral-element time-domain method is related to another computational method known as the spectral-element method, or simply SEM. The relationship between the two is analogous to that between FDTD and FEM. FDTD is used for solutions where the amplitude and distribution of the solution in space is time varying, whereas FEM can only consider harmonic time-dependence. In the spectral-element time-domain method, the trigonometric decomposition is applied to both the space and time.

When to Use Each Type of Spectral-Element Method

The decision to use a space + time-domain discretization method vs. spatial discretization only depends on the level of numerical accuracy needed in the solution and the exact type of time dependence. For example, if the solution only requires harmonic time dependence, then the time dependence in the system’s governing differential equation solution can be eliminated with a Fourier transform, just like in FEM. Otherwise, the time domain portion of the solution is discretized in the same way as in FDTD.

Spatial Discretization and Time Discretization

The spectral-element method and spectral-element time-domain method are both built by discretizing the geometry using trigonometric functions. In other words, we care about the spatial frequency rather than the angular frequency. By applying a trigonometric polynomial expansion to the solution, the derivatives can be eliminated and rewritten in terms of a spatial frequency. This is analogous to applying the spatial Fourier transform identity:

Spatial Fourier transform identity for spectral-element method and spectral-element time-domain method.

By then applying the trigonometric series expansion as a function of k, we can develop a solution as a function of spatial frequency. This is where orthogonality between the trigonometric polynomial and the geometry becomes very useful as it eliminates many terms in the expansion that are harmonics of a lowest order k value. This is one reason why spectral-element time-domain methods can be used in periodic systems or in systems with definite symmetry; the resulting iterative equations that need to be solved can be further reduced by exploiting orthogonality, thereby reducing computation time.

Systems where the time dependence is non-harmonic can also be used. This allows treatment of a range of input signals in a complex geometry. Examples include pulses and chirped signals such as FMCW radar. In addition, spectrally and electrically nonlinear systems with feedback can produce complex time-dependent and frequency-dependent behavior that is difficult to examine in a transient simulation. The spectral-element time-domain method handles these types of solutions as well.

Are There Commercial Solvers with the Spectral-Element Time-Domain Method?

At the time of writing, there are not any spectral-element time-domain solvers that are commercially available. However, these methods and other methods for computational time reduction are active areas of research and we can expect new products to be developed around these methods. As an engineer that needs to use a field solver, your goal is to know which type of field solver is best suited for your design.

That being said, an analogous approach can be taken in any other 2nd order differential equation by applying the spatial Fourier transform identity above to reduce the spatial derivatives. This gives an equation in terms of spatial frequency that simply needs to be solved for the time dependence, i.e., using a Green’s function method. This is one approach that is useful when the system has a periodic forcing function (these are rare but are physically possible) and complex time dependence with no orthogonal basis.