# Using a Thermal FEA Solver for Heat Management in Your PCB

April 8, 2020 Temperature map for a 3D thermal FEA solver. Note the bond wires and pins on this SMD IC.

The world is governed by differential equations, and you can thank Newton and Leibniz for providing the foundational mathematical tools for understanding differential equations. As important as these equations are in describing natural phenomena, methods for determining analytical solutions are still a major subject of mathematics research. As difficult (or impossible) as these equations are to solve, numerical and semi-numerical techniques are often used to produce accurate solutions.

Numerical and semi-numerical solution algorithms are often derived for very specific systems of equations. However, one general class of problems that can be solved numerically with high accuracy are convective fluid flow problems. Thermal finite element analysis (FEA) simulators that consider fluid flow are ideal tools for evaluating a thermal management strategy and proposing design changes for a real PCB. Let’s look at the way thermal FEA solvers can aid your thermal management strategy and how to interpret the results.

## Why Use a Thermal FEA Solver?

Differential equations in complex geometries can be difficult to solve analytically. In most cases with real systems, only a piecewise solution exists, making an analytical solution throughout the system extremely difficult to work with. Real electronics systems can contain hundreds or thousands of parts, all with their own thermal conductivity values and unique geometry. In addition, real thermal problems with forced airflow create a coupled system of nonlinear partial differential equations that can only be solved analytically with approximations.

Because of the complexity of real systems, and the challenges to finding analytical solutions to these problems, numerical techniques are used. These techniques can produce highly accurate results if system models are constructed correctly. When a high resolution mesh is used in the system, the results are more accurate, but the computational time increases. For multiphysics problems, it is common to see FEA methods requiring hours of computation time in complex systems.

A thermal FEA simulation uses these numerical techniques to reduce ordinary and partial differential equations (possibly coupled, possibly nonlinear) to a long series of arithmetic relations. Each portion of the problem can be solved through simple iterative relationships. As part of thermal management in a real PCB, these simulations can be used to solve the fundamental equations in CFD with high accuracy and multiple sourcing terms. 3D temperature distribution in the steady state determined from a thermal FEA simulation.

## Time-Dependent vs. Time-Independent FEA

One question that often comes up in thermal FEA methods is time dependence. In other words, it may be necessary to examine how the solution to a thermal FEA problem changes in time. For example, one might like to simulate how switching on a cooling fan for a processor affects the processor’s heating rate, and how long it takes for the fan to cool the device. These systems can be analyzed in the steady state and in the transient state for given heat sources/sinks. Here are some considerations for each variation of time dependence.

### Time-Independent FEA

The heat equation, Navier-Stokes equation, and conservation of momentum are the fundamental equations used in FEA simulations. In a time-independent simulation, ignoring the time dependence in the system only yields the steady-state solution. This is equivalent to enforcing the following conditions on the fluid flow rate, temperature, system pressure field, and all heat sources in the system: Steady-state conditions for functions in the heat equation and Navier-Stokes equation.

In this case, the heat equation, Navier-Stokes equation, and conservation of momentum reduce to the following equations. Steady-state Navier-Stokes equation (top), heat equation (middle), and conservation of momentum (bottom).

These equations can be solved by enforcing boundary conditions and initial conditions. A suitable finite element meshing scheme needs to be selected, which will be discussed below.

### Time-Dependent FEA

The heat equation and Navier-Stokes equation are inherently time-dependent, and accounting for time dependence in a thermal FEA simulation shows how heat is removed from the system over time. This is where a multiphysics CFD simulation becomes critical as you need to account for airflow in the system. In this case, simply use the standard forms of the heat equation, Navier-Stokes equation, and conservation of momentum.

The challenges in time-dependent thermal FEA problems are as follows:

• Discretization in time: A finite difference scheme needs to be applied to the time variable. This must be done alongside a finite element discretization on the spatial variables.

• Computational time and memory: Once you discretize the time variable, the number of calculations in the simulation increases. If you discretize time into N steps, then the total number of calculations increases by a factor N.

• Visualization: Results from heat equation simulations are normally displayed as a heat map. Understanding time-dependent thermal FEA results effectively requires creating a video that shows how the heat map changes over the course of the simulation.

Whichever purpose you choose, though, requires a level of discretization to be used to ensure accuracy in your simulations.

### Discretization in Static and Dynamic Thermal FEA

In time-dependent (dynamic) and time-independent (static) FEA simulations all require choosing the appropriate level of discretization in space and time. This will determine the accuracy of the simulation results, as well as how the solution changes between different points in space and time. In time-domain simulations, discretization refers to the step size between successive points in time. A spatial distribution is calculated using the boundary conditions at each defined point in time and stored in the simulator’s memory.

In addition to discretization in time, discretization must also be applied in space. This is done by building specific shapes at different points in space and deriving spatial finite difference equations between each point. The image below shows the common tetragonal discretization used in a mechanical simulation. In this simulation, tetragonal discretization is preferred as it nicely approximates the curved surfaces of the gears. The same type of discretization could be used in any other finite element simulation, including thermal FEA. Mechanical FEA simulation results with tetragonal discretization.

## Open-Source vs. Commercial FEA Codes and Your PCB

If you want to program an FEA solver manually, there are many open-source codes you can use to create your simulation. However, not all open-source codes are designed to work with IPC 2581 or ODB++ file formats for PCB designs. Instead, you need a thermal FEA solver that builds simulations directly from your design data with a user-friendly interface.

When you’re designing a heat management strategy, you’ll need to qualify your system with a thermal FEA analysis tool that works directly with your PCB design data.The Celsius Thermal Solver from Cadence provides just this type of integration. You can also use the full suite of PCB design and analysis software and Cadence’s full suite of analysis tools to modify your designs and devise an effective heat management strategy.

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