Key Takeaways

• All particular solutions to differential equations rely on enforcing boundary conditions.

• The boundary conditions in a problem define how a solution to a differential equation behaves at the boundary of a system.

• Boundary conditions can be fixed values or they can be defined as some other function in space and time.

Any time you take a mathematical physics class, there is always one piece of information that clearly defines your problems: boundary conditions. But, what are boundary conditions? All solutions to differential equations only become unique in a system where boundary conditions are defined and enforced. When used alongside initial conditions, you know everything about the system in terms of its initial behavior, how the system evolves, and what happens at the boundary of a system.

Boundary conditions allow a simulation design to constrain system behavior based on what is happening in the real system being simulated. In terms of mathematics, boundary conditions can be constants, functions, or terms proportional to the value of the solution. In terms of simulation data, boundary conditions can be tabulated and defined at multiple points within the system, creating a simulation model that would be intractable to solve by hand. Simulation applications can apply standard numerical techniques to account for system boundary conditions and help you get accurate results for modeling physical phenomena.

What Are Boundary Conditions?

So, what are boundary conditions? Boundary conditions are values of the solution to a differential equation that are defined at the boundary of a system. For example, a simple system you might try to simulate is a differential equation governing a physical quantity (temperature, electric field, mechanical vibration, etc.) inside of a 3D box. In this basic example, the boundary conditions enforce the value of the physical quantity being simulated to take a specific value at the boundaries of the box. Boundary conditions are chosen and defined in order to represent the behavior of a real physical system that is being simulated.

General statement for a differential equation problem defined by a differential operator (D), a forcing function F, and some boundary conditions

Another type of condition that is enforced when solving time-dependent differential equations is initial conditions. These define the value of your physically measurable quantity at some initial point in time, which is generally defined as the beginning of your simulation. Together with the boundary conditions, these define the subsequent behavior of the system at all points in time and everywhere in space.

For simulation purposes, a designer needs to set the boundary conditions in their simulation in order to determine, at minimum, a steady state solution to the differential equation they are investigating. To determine a transient response or a driven response, such as with time-dependent sourcing, initial conditions are needed as well. 

Types of Boundary Conditions

Most problems will only involve boundary conditions up to 1st order derivatives of the solution being investigated for a given problem. In general, there are five classifications of boundary conditions for a given problem, as shown in the table below. Physically, these conditions define specific values of the solution at a boundary, continuity across the boundary (with a derivative), or both.

When the form of the problem is defined, a solution algorithm can be implemented for a given problem. For the Dirichlet, Neumann, and mixed problems, a Green’s function approach is often used with a numerical scheme used to solve an integral for the problem in terms of its forcing function. This integration approach is common in wave propagation problems (e.g., in electromagnetics) while CFD problems will generally solve the equation directly with a numerical scheme (e.g., RANS, LES, or another reduction method).

Setting Boundary Conditions

Before setting boundary conditions, an important question to remember is: how many boundary conditions are needed? The number of boundary conditions governing the solution for a given variable in a partial differential equation is equal to the order of the differential equation in that variable. For example, the heat equation is 1st order in time and 2nd order in space, so it would require 1 initial condition and 2 boundary conditions for each spatial variable. For a 3D problem, this would give a total of 7 boundary conditions to ensure the solution to the heat equation was unique. The same idea applies to other partial differential equations, including nonlinear equations, such as the governing equations in CFD simulations.

Boundary conditions have an important physical representation, which is how they are defined based on measurements or derivations in real systems. The physical interpretation for each term in the standard classifications of boundary conditions is listed below.

No matter what boundary conditions you need to accommodate in your system, you can solve the main governing equations with CFD simulation tools from Cadence. Modern numerical approaches used in aerodynamics simulations, turbulent and laminar flow simulations, reduced fluid flow models, and much more can be implemented in Cadence’s simulation tools.

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