There are several standard and specialty numerical methods used to solve a variety of linear differential equations.
The boundary element method can be used to solve linear partial differential equations that are written in integral form.
Just like typical FDTD simulations, these solutions are based on an iterative numerical technique, but BEM occurs in the steady state.
The boundary element method can be used to calculate solutions to vibrational problems in membranes and other surfaces
Not all engineering design problems can be solved with simple formulas, and numerical simulations must be used to calculate solutions describing physical phenomena in very complex systems. Mathematicians have spent significant effort developing numerical solutions to complex problems involving partial differential equations, and these have been formalized into a number of methods. Matching the solution method to the problem being investigated is quite easy, but determining a solution can be difficult by hand.
Today’s commercial field solvers can be used to implement a set of standard numerical methods for solving complex differential equations using a series of iterative calculations. The principle methods for solving these problems are finite-difference time-domain, finite element method, and boundary element method, where the latter is sometimes called the “method of moments”. In this article, we’ll give a brief overview of the method and how it is used in commercial field solvers to investigate complex multiphysics problems.
Boundary Element Method Formulation
The boundary element method involves simplifying the solution to a differential equation by reformulating it in terms of the boundary conditions. The solution to the problem is written in terms of an integral formulation using a Green’s function for the particular system being investigated. Most textbooks do a decent job of introducing the fundamental theory used to develop the boundary element method, but there are certain points that aren’t often stated clearly. In particular, it is important to note the following:
- The boundary element method is normally discussed in terms of the solution to the Laplace equation, making it directly applicable to simple CFD problems, electrostatics (non-propagating) problems, and multiphysics problems.
- The method can be extended to Helmholtz equations and to other 1st or 2nd order linear partial differential equations in general. The Helmholtz equation is quite important, as it forms the separated spatial portion of wave equation problems.
- The method can also be extended into the time domain, a point which remains a popular research topic. It can also be used to treat certain nonlinear problems under linear approximations, similar to methods used in stability problems.
- Sources can be treated explicitly thanks to the use of Green’s theorem in the formulation of the boundary element method.
To develop the integral solution method, we need some general theory from partial differential equations.
To start, the boundary element method is based on the use of a Green’s function. This function is used to calculate the solution to a differential operator ⍙G that is excited by an impulse:
Green’s function definition for a general differential operator
A Green’s function formulation has a simple interpretation: the solution is calculated at some point r given an infinite impulse applied at r’. The solution is constructed by calculating a commutation between the solution and G, followed by integrating over all possible values of r’. To go further, we use Green’s theorem and integration by parts to define the solution in terms of an integration over the source term f(r) and the boundary conditions.
The Boundary Integral
While Green’s function is not solved directly, it is used with Green’s theorem to rewrite the solution to the desired system in terms of an integral over the boundary conditions (either as a Dirichlet, Neumann, or mixed problem). The value of the solution at any point in space r is written in terms of an integral over the function that defines the boundary conditions:
Definition of the solution in terms of Green’s function on the boundary of the solution domain
As long as the boundary conditions and sourcing term are specified and can be discretized, the problem can be solved with an integral formulation. Any of the standard integration schemes can be used to calculate the above integral and determine the solution in space.
Commercial Field Solvers and the Boundary Element Method
The role of commercial field solver applications in the boundary element method is to apply a meshing method to the boundary function, calculate the discrete solution to the associated homogeneous problem, and to calculate the discrete integral equation shown above. The boundary element method can be used in a field solver application to solve generalized problems in a variety of fields, including:
- Acoustics and vibration problems
- Linear CFD problems (incompressible fluids)
- Temperature and heat transport
If you’re a user of a commercial field solver package, it’s important that you carefully define the boundary conditions so that you can get the most accurate results. In problems involving interconnects, such as impedance calculations, commercial field solvers can easily apply a mesh and define the boundary conditions using the data in your PCB layout. For more complex problems, like airflow or heat transport in an electronic system, material constants and relevant length scales need to be defined throughout the design to ensure the results are accurate. The best field solver software can help by interfacing with your electrical design and taking data directly from your physical layout.
Cadence’s systems analysis software suite can help you build models to describe your oscillator circuits and examine the phase noise in frequency synthesizers. From powerful multiphysics boundary element method solvers to PCB layout and simulation utilities, Cadence’s software gives you access to an industry-standard toolset for systems-level electrical design and analysis. You’ll also have access to a range of simulation features you can use in power and signal integrity analysis, giving you everything you need to evaluate your system’s functionality.