# Using a Multiphysics Heat Finite Element Solver

April 6, 2020 Temperature map from a thermal imager or heat finite element solver

As much as I use my computers for my day job, I’ve never had my motherboard or CPU burn out due to excessive heat. Systems designers use very precise simulations and measurements as part of a thermal management strategy. In your next complex system, analyzing heat flow is critical for ensuring temperature stays within safe operating limits. If you can identify which components are creating hot spots in your board, some simple rearrangements or an added heat sink can be enough to bring down temperature and increase your system’s lifetime.

The primary tool in this type of analysis is a multiphysics heat finite element solver. There are a number of applications that can perform these simulations, but most do not take data directly from PCB design files and component libraries to perform simulations. Instead, you’ll have to manually build a mesh for your simulations, and you’ll have to manually define heat sources throughout your board. If you use an electrical-thermal simulation tool like a finite element solver, you can easily develop a thermal management strategy for your board.

## The Heat Equation with Sources and Convection

Electronic systems are not isolated from airflow and contain plenty of heat sources. As part of thermal analysis, you’ll need to account for a number of fundamental physical phenomena in your system:

• Heat dissipation in components: Current in resistive components and in switching integrated circuits causes electrical energy to be lost as heat, known as Joule heating.

• Heat conduction: Heat can transfer between hot and cold objects simply due to the objects being in contact.

• Heat convection: Airflow can carry heat between hot and cold regions of a system. In other words, cool air can take heat from a hot component, and then flow towards a cool component. The warmer air then transfers its heat to the cooler component.

• Forced laminar flow or convection: Laminar airflow or convection can be intentionally driven using a cooling fan, and this needs to be considered alongside natural convection.

Considering any of these fundamental physical processes in isolation is relatively easy and can be determined using some simple equations. You can find these equations in many engineering and physics textbooks. These equations are closed form and allow you to calculate heat generated by an electric current (Joule heating), heat transfer rates between a system, and changes in temperature due to addition or removal of heat.

Since you need to consider airflow in the system, whether it is laminar or driven by convection, you need a set of equations that links the airflow in the system to the temperature of the system. This is done by treating heat transfer and airflow as a fluid dynamics problem, as shown below.

### Heat Flow in Electronics as a Thermal and Fluid Dynamics Problem

By combining the Navier-Stokes equation, conservation of momentum, and the heat equation, we arrive at the following set of coupled nonlinear differential equations for describing heat flow and airflow in an electronic system. Navier-Stokes equation (top), heat equation (middle), and conservation of momentum (bottom).

In the first equation above, the constants before the derivative terms have their usual meaning in fluid dynamics, and the far right term accounts for any pressure gradient in the system (e.g., due to an electrical fan). T(r, t) is the temperature field of the system and u(r, t) is the airflow velocity in space and time. S is a heat source that accounts for Joule heating in your components, and P is the external pressure gradient that drives airflow.

Here, you need to define some initial airflow velocity and temperature in the system to examine how these quantities evolve in space and time. The other required inputs are the thermal conductivities of your PCB substrate, components, and any other elements, such as heat sinks and copper conductors.

### Interpreting These Equations

Interpretation of these equations is straightforward; the solution tells you how airflow in the system and heat generated by electric currents produce the temperature field found in the simulation. Here, the total heat being transferred between two points is secondary; the variable we care about in an electronic system is the temperature of the board and the components.

Once a solution is generated, the temperature at critical points can be examined as a function S and P. By calculating spatial averages at critical components, you can relate the airflow required to keep the component at a specific temperature to the amount of heat dissipated in the component. An example of this type of graph is shown below (note the normalized scales). Example results analysis results. Here, the normalized pressure gradient for driving airflow is shown as a function of normalized Joule heating in the system. Each curve represents a desired equilibrium temperature.

## Using a Forced Heat Finite Element Solver

The equations above can be solved by hand in some limited cases, and with some reasonable assumptions in limited situations. When the Reynolds number is low and we look at airflow close to the surface of a PCB, flow can be approximated as laminar, and the number of spatial variables is reduced from 3 to 1. If you treat the board as a plane with some defined initial and boundary conditions, you can easily calculate the system’s temperature as a function of space and time. This gives you a simple upper limit on the airflow required to maintain a specific temperature.

A real system is much more complicated and needs to be treated with a heat finite element solver. This type of simulation tool is ideal for multiphysics problems, where multiple physical phenomena and their interactions are considered simultaneously. The problem discussed above is one of many examples found in science and engineering.

If you’re working with a PCB, you’ll need to import your board design into the simulation tool and build a mesh for your board. This is then used to calculate finite differences in the temperature field from the initial conditions and boundary conditions. The full solution is then generated through iteration and can be visualized as a color map. You can then immediately see hot spots in your design and adjust your layout to keep temperatures low.

Extreme temperatures in your board and components are critical determinants of reliability, but you can build and execute important heat finite element solver simulations with the Celsius Thermal Solver from Cadence. 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 create an effective thermal management strategy.

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