What You Can Takeaway
The boundary element method is a numerical solution technique for linear partial differential equations with integral solutions.
Integral solutions are formulated in space as a Green’s function solution for the spatial part.
There is an associated differential equation for the temporal solution, although the steady state solution or harmonic solution is normally considered to account for time-independent sources.
CFD simulation results showing laminar flow along a board surface
Partial differential equations can be analyzed as simple systems using analytical methods, but the real world is seldom as simple as we would like to believe. Real systems with complex geometries require numerical methods to generate solutions, often with idealized boundary conditions and sources. This may sound vague, but numerical methods can be used to generate realistic physical results in the face of approximated inputs. In electronics, approximated sources include heat generated by components, external airflow from fans, and current density throughout an IC or PCB.
An important input into any system of coupled partial differential equations is the boundary conditions. In the presence of sources and known boundary conditions, linear partial differential equations can be reformulated as integral equations. This is where the boundary element method can be used to determine an appropriate solution for the system. Here’s what you need to know if you plan to use the boundary element method in acoustics, CFD, and electromagnetics problems.
What is the Boundary Element Method in Acoustics?
Studying acoustics is a great way to introduce the boundary element method in other fields as the methods are analogous. If you’re not a hardcore mathematician, it is still rather easy to understand the boundary element method. The boundary element method is defined for linear differential equations where a Green’s function can be calculated.
Consider a physical quantity A governed by a partial differential equation in space and time. The Green’s function G tells you the value of A at some spatiotemporal coordinate pair (r, t) due to a source located at (r’, t’). This is a standard formulation in electromagnetic textbooks, but it is also used for the boundary element method in acoustics, CFD, the heat equation, the Laplace equation, and other multidimensional physical quantities.
It’s probably easiest to understand the boundary element method in terms of a simple example from acoustics. Consider the acoustic wave equation in 3 dimensions with a time-dependent velocity potential f(r, t) as an acoustic source. The wave equation for the velocity potential A(r, t) and the associated Green’s function G(r, r’, t, t’) is:
Wave equation for acoustic velocity potential and the associated Green’s function.
The goal in solving this type of problem is to use the boundary conditions in Green’s theorem to define a surface integral over the Green’s function. The problem is normally rewritten in the Fourier domain and solved as a function of the spatial variables and frequency, although the integral below considers the time domain directly and can treat instances where a Fourier transform of f(r,t) is undefined. The solution A(r, t) can then be converted back to the time domain with a Fourier transform and comparing the result to the initial conditions.
Note that, in deriving a Green’s integral shown below, that G satisfies homogeneous boundary conditions. Because the value and derivatives of A(r, t) are defined at the boundary of the system, they are used in Green’s identities to derive the integral shown below, giving a full solution for A(r, t):
Solution to the wave equation shown above using a Green’s function.
Rather than solving the differential equation directly with FDTD, FVM, or FEM, the problem requires discretizing the boundary conditions and the remaining regions in the system. This allows the Green’s function to be calculated along the boundary using a standard numerical integration technique when invoking Green’s identities.
Boundary Element Method in Other Problems
The boundary element method can be used in many other problems. One example from optics is in Kirchoff’s integral, which is used to solve diffraction problems; this involves calculating the electromagnetic field in a volume of space in terms of a surface integral at the boundary of that space, thus it is natural to use the boundary element method for this problem. In electronics, the principle domains where the boundary element method can be used are in electromagnetics, heat conduction, and CFD problems.
EM Wave and Heat Conduction Problems
Heat conduction in the absence of airflow is just a diffusion problem, where the physical quantity of interest is the temperature field. For electromagnetics problems, the relevant physical quantities are the electric and magnetic potential functions, or the electric and magnetic fields, which obey their own wave equations. Time-dependent sources can be considered in the standard boundary element problem, just like the boundary element method in acoustics.
These types of problems are elementary to solve in the steady state as they are essentially a Laplace equation problem. For the EM wave equation with oscillating sources, the electromagnetic field and radiated power can be calculated easily in the frequency domain, which reduces the number of variables from 4 to 3. The initial and boundary conditions still need to be specified and included in the Green’s function integral.
Heat conduction results in the steady state near an IC on a PCB.
The Navier-Stokes equation and the other equations in CFD problems form a set of nonlinear coupled differential equations. In order to use the boundary element method for this system, the equations need to be linearized. Either the nonlinear terms need to be ignored as they are small, or an approximation needs to be applied in order to make the set of equations linear. This is equivalent to examining small changes in the flow field, temperature field, and heat flow around the sources in the system.
Even in the case where the equations are coupled, there is a method to solve this set of equations using a convolution involving a Green’s function for the system. Take a look at this research article for an example method for finding a Green’s function for a system of coupled diffusion equations or convection-dispersion equations. The result can then be used in the standard boundary element method for acoustics.
When you need to use the boundary element method in acoustics, electromagnetics, CFD, or heat conduction problems, you first need to create your system in a set of PCB design and analysis features that integrate with a 3D field solver. The Celsius Thermal Solver and SI/PI Analysis Point Tools from Cadence integrate into a complete system for analyzing CFD equations, which can then be used for thermal and acoustics problems in complex geometries.
If you’re looking to learn more about how Cadence has the solution for you, talk to us and our team of experts.