# Knowledge Base

In the engineering world, wave behaviors are governed by physical laws. Mathematical wave equations, typically Partial Differential Equations (PDEs) stand behind the physical laws and provide us a conceptual method for a solution. However, in the real world, waves always exist in complex environments. This may include a mix of arbitrary geometries, material properties, multiple physics and many other aspects. Pure analytical solutions of the wave equations are hard and sometimes impossible to obtain in an arbitrarily configured environment. Consequently, numerical techniques arise to address the problem of finding approximated solutions of PDEs with complex environments.

During the past decades, many numerical techniques, such as Finite Difference (FD) method, Finite Element (FE) method and Finite Volume (FV) method were developed, well studied and widely applied in simulating engineering problems. Researchers and numerical software engineers have clearly understood that each method has its own pros and cons. No single numerical technique can be suitable for all types of problems in all situations. This inspires WCT’s R&D team that a combination of the major techniques is the future trend in simulating multi-functional (smart), multi-scale, multi-physics and multi-process (parallel) engineering designs and problems.

WCT’s R&D team has spent years’ efforts in understanding the nature of the above mentioned classic numerical techniques. We’ve accumulated abundant experiences to acquire them efficiently into our software framework. Meanwhile, WCT is targeting to understand even more profound meanings of these techniques and prepare for the next generation simulators.

FDTD

Finite-difference time-domain (FDTD) is one of the major available computational electromagnetics techniques. Since it is a time-domain method, FDTD can achieve a wideband response with a single computation, the transient solutions also reflect the time evolving field behaviors in the model, which sometimes is relatively more straightforward to understand. Another significant feature is that FDTD can treat nonlinear material properties in a natural way.

FDTD relies on the stair-cased stencil to discretize the field over space. In time domain it applies central difference approximation for the time partial derivatives. Leapforg is the most popular time marching scheme that is applied in FDTD

The FDTD method requires an orthogonal grid, thus a high discretization density is always required to capture the geometric characteristics of electrically fine structures. This will lead to a large number of wasted unknowns in the electrically coarse domains. The sub-gridding technique can alleviate this issue of the FDTD method; however, it will spoil the simple data structure of the standard FDTD scheme, thus greatly increasing the computational complexity. Stability is another issue for the FDTD method. Based on the Courant-Friedrichs-Levy (CFL) condition, very small cells for electrically fine structures will lead to an extremely small time step increment, and consequently, an unaffordable number of steps in time integration.

During the past two decades, FDTD techniques have emerged as primary means to computationally model many scientific and engineering problems dealing with electromagnetic wave interactions with material structures. Current FDTD can solve EM problems range from near DC to optical frequencies. Various applications, such as geophysics, radio frequency devices and systems, bio-imaging, photonic crystals are using FDTD techniques to simulate the design and layouts.

FDTD is one of the core techniques of Wavenology software products.

FETD

The finite element method (FEM) or finite element analysis (FEA) is a numerical technique for finding approximate solutions of partial differential equations (PDEs). Finite element time domain (FETD) method is a branch of FEM, which solve problems with time-evolving variables included. The major advantage of FETD is its flexibility of modeling the geometry. By discretizing the geometry into much smaller polygons (finite elements), complex geometries can be finely represented. The basis functions defined over the element also provide flexible options to interpret the physical quantities. Such two features lead to the “hp” refinement technique of FEM, which provides much more space for error control and accuracy convergence. Moreover, FEM is one of the major methods for solving multiple types of PDEs, such as diffusion equation, Maxwell’s equation, elastic wave equation etc. Therefore multiple physics such as computational mechanics, computational fluid dynamics (CFD), computational electromagnetics, computational thermal dynamics etc. described by different PDEs can be simulated within a single FEM framework.

Although the FETD method is more flexible in geometric modeling, however, this method requires solving matrix equations, either directly or iteratively. A discretized multiscale problem usually has a great number of unknowns, viz. huge system matrices. It can be prohibitively expensive to perform operations with those huge matrices during time stepping. In addition, the robust mesher that can discretize complicated geometries is not often easy to implement, which becomes another drawback for FETD applications.

Wavenology software integrates FETD method into its engine and applies it smartly to specific problems.

FVTD

The finite volume method is the third major method for solving partial differential equations. Similar to the finite difference method or finite element method, the problem domain is discretized and physical quantities are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh which can also be interpreted as a finite element. In the finite volume method, a divergence volume integral is always transformed into a surface integral which introduces the concept of “flux” that is flowing through the finite volume. Finite volume method is widely used in computational fluid dynamics packages, since it is convenient to be formulated over unstructured meshes.

ECT

A conformal finite-difference time-domain (CFDTD) method has been put forward to model curved perfectly conducting objects. The method deforms the grid locally to accommodate the curvature of the PEC surface, and therefore, avoids the staircase error of the conventional FDTD method in approximating boundaries. However, sometimes the deformed grid may be too small to obtain a stable solution. This instability can be removed by using a smaller time step, at the expense of more times steps being required. An efficient, yet accurate, technique, the enlarged cell technique (ECT) can be used to obtain a stable solution without the need to reduce the time step. This technique has a clear geometric and physical explanation and is easy to implement. Unlike CFDTD methods, the ECT sacrifices neither accuracy nor efficiency in treating small distorted cells.

SETD

The spectral element time domain method can be considered as a special class of the general finite element method with a different choice of nodal points and quadrature integration points which are based on Gauss–Lobatto–Legendre (GLL) polynomials. It has the advantages of spectral accuracy and block-diagonal mass matrix due to the orthogonality of the basis function and GLL quadrature. With the inexpensive inversion of the block-diagonal mass matrices, this method requires only a trivial sparse matrix-vector product at each time step, thus significantly reducing CPU time. In addition, the stiffness matrices are independent of the element shape. Thus the huge memory requirement in traditional FETD method can be dramatically reduced.

SPICE

SPICE (Simulation Program with Integrated Circuit Emphasis) is a general-purpose circuit simulation program. It implements the Modified Nodal Analysis (MNA) for nonlinear dc, nonlinear transient, and linear ac circuit problems. Circuits may contain resistors, capacitors, inductors, mutual inductors, independent voltage and current sources, four types of dependent sources, lossless and lossy transmission lines, switches, uniform distributed RC lines, and the five most common semiconductor devices: diodes, BJTs, JFETs, MESFETs, and MOSFETs.

Over the past several decades, SPICE has had a significant impact on circuit design methodology. Such tools have been widely applied in both industry and academic communities and nearly every electronic design automation (EDA) software package will integrate SPICE functionalities for circuit system simulations. Efforts have been made to improve SPICE performance. Equivalent circuit modeling has also received great interest and a significant work has been devoted to formulate compact, yet complete, models of both passive and active devices of various kinds. With these efforts, SPICE has become the most popular tools in the field of circuit system design.

Wavenology EM has completely integrated SPICE package for circuit system simulations. It can also perform co-simulation of both circuit systems and EM field systems simultaneously.

DGTD

Discontinuous Galerkin (DG) methods are also popular for solving PDEs. They combine features of the FE and the FV framework by employing piecewise continuous basis and testing functions locally in decomposed sub-domains and apply the boundary flux to communicate between sub-domains. More recently, it is also realized that DG methods also reveal the natural connection between FETD and FDTD methods by applying domain decomposition and special basis functions to the FETD method.

DG methods are characterized as being high-order accurate, able to model complex geometries, efficient, stable, and highly parallel. Discontinuous Galerkin Time-Domain (DGTD) methods have more recently been employed for the solution of Maxwell’s equations. Unlike commercially available electromagnetic software, the DGTD method is a high-order, finite-element based solution of coupled curl Maxwell’s equations. Since it is based on a finite-element discretization, it can be accurately and efficiently applied to model very complex structures with arbitrary geometries and complex materials (e.g., anisotropic and/or dispersive materials). Also, since it is based on the solution of the time-dependent Maxwell’s equations, either linear or non-linear materials can be treated. Because the method is a high-order method (this includes high-order spatial discretization and time discretization), the overall dimensionality of the problem can be significantly reduced. This also allows for controllable accuracy. Furthermore, the DGTD method is based on a domain decomposition of the finite-element mesh. Consequently, it is a highly-parallel algorithm and is easily adaptable to take full advantage of modern computing systems. As a result, the DGTD method is a very robust method that can be hybridized with many other solvers to simulate complex problems such as the full model of modern radio frequency systems. In the mean time, DGTD method has revealed us a much more profound meaning of the connection between classic numerical methods for solving PDEs. Therefore, DG methods have in particular received considerable interest for problems with a dominant first-order part, e.g. in electrodynamics, and plasma physics.